scholarly articles on implementation research

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

• Advanced Search
• Journal List
• Front Public Health

Enhancing the Impact of Implementation Strategies in Healthcare: A Research Agenda

Byron j. powell.

1 Department of Health Policy and Management, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States

2 Cecil G. Sheps Center for Health Services Research, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States

3 Frank Porter Graham Child Development Institute, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States

Maria E. Fernandez

4 Center for Health Promotion and Prevention Research, School of Public Health, University of Texas Health Science Center at Houston, Houston, TX, United States

Nathaniel J. Williams

5 School of Social Work, Boise State University, Boise, ID, United States

Gregory A. Aarons

6 Department of Psychiatry, University of California, San Diego, La Jolla, CA, United States

Rinad S. Beidas

7 Department of Psychiatry, Center for Mental Health, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, United States

8 Department of Medical Ethics and Health Policy, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, United States

9 Leonard Davis Institute of Health Economics, University of Pennsylvania, Philadelphia, PA, United States

Cara C. Lewis

10 MacColl Center for Healthcare Innovation, Kaiser Permanente Washington Health Research Institute, Seattle, WA, United States

Sheena M. McHugh

11 School of Public Health, University College Cork, Cork, Ireland

Bryan J. Weiner

12 Department of Global Health, Department of Health Services, University of Washington, Seattle, WA, United States

The field of implementation science was developed to better understand the factors that facilitate or impede implementation and generate evidence for implementation strategies. In this article, we briefly review progress in implementation science, and suggest five priorities for enhancing the impact of implementation strategies. Specifically, we suggest the need to: (1) enhance methods for designing and tailoring implementation strategies; (2) specify and test mechanisms of change; (3) conduct more effectiveness research on discrete, multi-faceted, and tailored implementation strategies; (4) increase economic evaluations of implementation strategies; and (5) improve the tracking and reporting of implementation strategies. We believe that pursuing these priorities will advance implementation science by helping us to understand when, where, why, and how implementation strategies improve implementation effectiveness and subsequent health outcomes.

Introduction

Nearly 20 years ago, Grol and Grimshaw ( 1 ) asserted that evidence-based practice must be complemented by evidence-based implementation. The past two decades have been marked by significant progress, as the field of implementation science has worked to develop a better understanding of implementation barriers and facilitators (i.e., determinants) and generate evidence for implementation strategies ( 2 ). In this article, we briefly review progress in implementation science and suggest five priorities for enhancing the impact of implementation strategies. We draw primarily upon the healthcare, behavioral health, and social services literature. While we hope the proposed priorities are applicable to studies conducted in a wide range of contexts, we welcome discussion regarding potential applications and enhancements for contexts outside of healthcare, such as community and public health settings ( 3 ) that often involve different types of stakeholders, interventions, and implementation strategies.

Implementation strategies are methods or techniques used to improve adoption, implementation, sustainment, and scale-up of interventions ( 4 , 5 ). These strategies vary in complexity, from discrete or single component strategies ( 6 , 7 ) such as computerized reminders ( 8 ) or audit and feedback ( 9 ) to multifaceted implementation strategies that combine two or more discrete strategies, some of which have been branded and tested using rigorous designs [e.g., ( 10 , 11 )]. Implementation strategies can target a range of stakeholders ( 12 ) and multilevel contextual factors across different phases of implementation ( 13 – 16 ). For example, strategies can address patient ( 17 ), provider ( 18 ), organizational ( 19 ), community ( 20 , 21 ), policy and financing ( 22 ), or multilevel ( 23 ) factors.

Several taxonomies describe and organize the types of strategies available ( 6 , 7 , 24 – 26 ). Similarly, taxonomies of behavior change techniques ( 27 ) and methods ( 28 ) describe components of strategies at a more granular level. Both types of taxonomies promote a common language, inform implementation strategy development and evaluation by facilitating consideration of various “building blocks” or components of multifaceted and multilevel strategies, and improve the quality of reporting in research and practice.

The evidence base for implementation strategies is steadily developing. Initially, single-component, narrowly focused strategies that were effective in earlier studies were selected in subsequent studies despite differences between the clinical problems and contexts in which they were deployed ( 29 ). That approach was based on the assumption that strategies would be effective independent of the implementation problems being addressed ( 29 ). This “magic bullet” approach has led to limited success ( 30 ), prompting recognition that strategies should be selected or developed based upon a thorough understanding of context, including the causes of quality and implementation gaps, an assessment of implementation determinants, and an understanding of the mechanisms and processes needed to address them ( 29 ).

Evidence syntheses for discrete, multifaceted, and tailored implementation strategies have been conducted. The Cochrane Collaboration's Effective Practice and Organization of Care (EPOC) group has been a leader in this regard, with 132 systematic reviews of strategies such as educational meetings ( 31 ), audit and feedback ( 9 ), printed educational materials ( 32 ), and local opinion leaders ( 33 ). Grimshaw et al. ( 34 ) note that while median absolute effect sizes across implementation strategies are similar (see Table ​ Table1), 1 ), the variation in observed effects within each strategy category suggests that effects may vary based upon whether or not they address determinants (barriers and facilitators). Indeed, determinants at multiple levels and phases may signal the need for multifaceted and tailored strategies that address key determinants ( 13 ).

Evidence for common implementation strategies targeting professional behavior change.

Table updated from Grimshaw et al. ( 34 ), and draws upon Cochrane Reviews from the Effective Practice and Organization of Care (EPOC) group ( 38 ) .

While the use of multifaceted and tailored implementation strategies is intuitive and has considerable face validity ( 29 ), the evidence regarding their superiority to single-component strategies has been mixed ( 37 , 39 , 40 ). A review of 25 systematic reviews ( 39 ) found “no compelling evidence that multifaceted interventions are more effective than single-component interventions” (p. 20). Grimshaw et al. ( 34 ) provide one possible explanation, emphasizing that the general lack of an a priori rationale for the selection of components (i.e., discrete strategies) in multifaceted implementation strategies makes it difficult to determine how these decisions were made. They may have been selected thoughtfully to address prospectively identified determinants through theoretically- or empirically-derived change mechanisms, or they may simply be the manifestation of a “kitchen sink” approach. Wensing et al. ( 41 ) offer a complementary perspective, noting that definitions of discrete and multifaceted strategies are problematic. A discrete strategy such as outreach visits may include instruction, motivation, planning of improvement, and technical assistance; thus, it may not be accurate to characterize it as a single-component strategy. Conversely, a multifaceted strategy including educational workshops, educational materials, and webinars may only address provider knowledge and fail to address other important implementation barriers. They propose that multifaceted strategies that truly target multiple relevant implementation determinants could be more effective than single-component strategies ( 41 ).

A systematic review of 32 studies testing strategies tailored to address determinants concluded that tailored approaches to implementation were more effective than no strategy or a strategy not tailored to determinants; however, the methods used to identify and prioritize determinants and select implementation strategies were not often well-described and no specific method has been proven superior ( 37 ). The lack of systematic methods to guide this process is problematic, as evidenced by a review of 20 studies that found that implementation strategies were often poorly conceived, with mismatches between strategies and determinants (e.g., barriers were identified at the team or organizational level, but strategies were not focused on structures and processes at those levels) ( 42 ). A multi-national program of research was undertaken to improve the methods of tailoring implementation strategies ( 43 ), but tailored strategies had little impact on primary and secondary outcomes ( 40 ). Questions remain about the best methods to develop tailored implementation strategies.

Five priorities need to be addressed to increase the public health impact of implementation strategies: (1) enhance methods for designing and tailoring; (2) specify and test mechanisms of change; (3) conduct more effectiveness research on discrete, multifaceted, and tailored strategies; (4) increase economic evaluations; and (5) improve tracking and reporting. Table ​ Table2 2 provides examples of studies that have pursued each priority with rigor.

Five priorities for research on implementation strategies.

Enhance Methods for Designing and Tailoring Implementation Strategies

Implementation strategies are too often designed in an unsystematic manner and fail to address key contextual determinants ( 13 – 16 ). Stakeholders may rely upon inertia (i.e., “we've always done things this way”), one size fits all approaches, or utilize what Martin Eccles has called the ISLAGIATT principle (i.e., “it seemed like a good idea at the time”) ( 53 ). Consequently, strategies are not always well-matched to the contexts in which they are deployed, including the interventions to be implemented, settings, stakeholder preferences, and implementation determinants ( 37 , 42 , 54 ). More rational, systematic approaches to identify and prioritize barriers and link strategies to overcome them are needed ( 37 , 42 , 55 – 57 ). A number of methods have been suggested. Colquhoun and colleagues ( 56 ) found 15 articles with replicable methods for designing strategies to change healthcare professionals' behavior, and Powell et al. ( 55 ) proposed Intervention Mapping ( 58 ), concept mapping ( 59 ), conjoint analysis ( 60 ), and system dynamics modeling ( 61 ) as methods to aid the design, selection, and tailoring of strategies. These methods share common steps (identification of barriers, linking barriers to strategy component selection, use of theory, and user engagement), and have potential to make the process of designing and tailoring implementation strategies more rigorous ( 55 , 56 ). For example, Intervention Mapping is step-by-step approach to developing implementation strategies using a detailed and participatory needs assessment and the identification of implementers, implementation behaviors, determinants, and ultimately, behavior change methods and implementation strategies that influence determinants of implementation behaviors. Some work has been done to compare different methods for assessing determinants ( 62 ); however, several questions remain. How can determinants be accurately and efficiently assessed (ideally leveraging implementation frameworks)? Can perceived and actual determinants be differentiated? What are the best methods for prioritizing determinants that need to be proactively addressed? When should determinant assessment take place given that new challenges are likely to emerge during the course of implementation? Who should be involved in this process? Each of those questions has resource implications. Similarly, questions remain about efficiently linking prioritized determinants to effective and pragmatic implementation strategies. How can causal theory be leveraged or developed to guide the selection of implementation strategies? Can pragmatic tools be developed to systematically link strategies to determinants? Approaches to designing and tailoring implementation strategies should be tested to determine whether they improve implementation and clinical outcomes ( 55 , 56 ). Given that clinical problems, clinical and public health interventions, settings, individuals, and contextual factors are highly heterogeneous, there is much to gain from developing generalizable processes for designing and tailoring strategies.

Specify and Test Mechanisms of Change

Studies of implementation strategies should increasingly focus on establishing the processes and mechanisms by which strategies exert their effects rather than simply establishing whether or not they were effective ( 29 , 63 , 64 ). The National Institutes of Health ( 64 ) provides this guidance:

Wherever possible, studies of dissemination or implementation strategies should build knowledge both on the overall effectiveness of the strategies, as well as “how and why” they work. Data on mechanisms of action, moderators, and mediators of dissemination and implementation strategies will greatly aid decision-making on which strategies work for which interventions, in which settings, and for which populations.

Unfortunately, it is not common that mechanisms are even mentioned, much less tested ( 63 , 65 , 66 ). Williams ( 63 ) emphasizes the need for trials that test a wider range of multilevel mediators of implementation strategies, stronger theoretical links between strategies and hypothesized mediators, improved design and analysis of multilevel mediation models in randomized trials, and an increasing focus on identifying implementation strategies and behavior change techniques that contribute most to improvement. Developing a more nuanced understanding of mechanisms will require researchers to thoroughly assess the context of implementation and describe causal pathways by which strategies exert their effects, moving beyond a broad identification of determinants and articulating mediators, moderators, preconditions, and proximal and distal outcomes ( 67 ). Examples of this type of approach and guidance for their development can be found in Lewis et al. ( 67 ), Weiner et al. ( 23 ), Bartholomew et al. ( 58 ), and Highfield et al. ( 44 ). Additionally, drawing more heavily upon theory ( 66 , 68 , 69 ), using research designs that maximize ability to make causal inferences ( 70 , 71 ), leveraging methods that capture and reflect the complexity of implementation such as systems science ( 61 , 72 , 73 ) and mixed methods ( 74 – 76 ) approaches, and adhering to methods standards for studies of complex interventions ( 77 ) will help to sharpen our understanding of how implementation strategies engage hypothesized mechanisms. Work to link implementation strategies and behavior change techniques to hypothesized mechanisms is underway ( 67 , 78 ), which promises to improve our understanding of how, when, where, and why implementation strategies are effective.

Conduct More Effectiveness Research on Discrete, Multi-faceted, and Tailored Implementation Strategies

There is a need for more and better effectiveness research on discrete, multifaceted, and tailored implementation strategies using a wider range of innovative designs ( 70 , 79 – 82 ). First, while a number of discrete implementation strategies have been described ( 6 , 7 , 24 , 25 ) and tested ( 38 ), there are gaps in our understanding about how to optimize these strategies. There are over 140 randomized trials of audit and feedback, but Ivers et al. ( 83 ) conclude that there is much to learn about when it will work best and why, and how to design reliable and effective audit and feedback strategies across different settings and providers. Audit and feedback is an example of how complex implementation strategies can be. The ICeBERG group ( 69 ) pointed to the fact that even varying five modifiable elements of audit and feedback (content, intensity, method of delivery, duration, and context) produces 288 potential combinations. These variations matter ( 84 ), and there is a need for tests of audit and feedback and other discrete implementation strategies that include clearly described components that are theoretically and empirically derived, and well-operationalized. The results of these studies could inform the use of discrete strategies and their inclusion in multifaceted strategies.

Second, there is a need for trials that give insight into the sequencing of multifaceted strategies and what to do if the first strategy fails ( 39 ). These strategies could be compared to discrete/single-component implementation strategies or multifaceted strategies of varying complexity and intensity with well-defined components that are theoretically aligned with implementation determinants. These strategies could be tested using MOST, SMART, or other variants of factorial designs that can evaluate the relative impact of various components of multifaceted strategies and inform their sequencing ( 70 , 85 ).

Finally, tests of strategies that are prospectively tailored to different implementation contexts to address specific implementers, implementation behaviors, or determinants are needed ( 37 ). This work could involve comparisons between tailored and non-tailored multifaceted implementation strategies ( 86 ), as well as tests of established and innovative methods that could inform the identification, selection, and tailoring of implementation strategies ( 55 , 56 ).

Increase Economic Evaluations of Implementation Strategies

Few studies include economic evaluations of implementation strategies ( 87 , 88 ). For example, in a systematic review of 235 implementation studies, only 10% provided information about implementation costs ( 87 ). The dearth of economic evaluations severely limits our ability to understand which strategies might be feasible for different contexts, as some decision makers might underestimate the resources required to implement and sustain EBPs, while others might over-estimate them and preemptively limit themselves from implementing EBPs that could benefit their communities ( 89 ). Incorporating economic analyses into studies of implementation strategies would provide decision makers more complete information to guide strategy selection, and would encourage researchers to be more judicious and pragmatic in their design and selection of implementation strategies, narrowing attention to strategies and mechanisms hypothesized to be most essential. If methods for designing and tailoring strategies can be improved such that complex multifaceted strategies are proven superior to single-component or less complex multifaceted strategies ( 39 ) and tailored strategies are proven superior to more standard multifaceted strategies ( 37 , 40 , 43 , 55 ), economic evaluations will be instrumental in demonstrating whether improvements in implementation are worth added costs. Practical tools for integrating economic evaluations within implementation studies have been developed, such as the Costs of Implementing New Strategies (COINS) method ( 89 ) which was developed to address the need for standardized methods for analyzing cost data in implementation research that extend beyond the cost of the clinical intervention itself ( 90 ). For example, the original COINS study presented a head-to-head trial of two implementation approaches; although one approach was significantly more costly, the implementation outcomes achieved were superior enough to warrant the additional resources ( 91 ). Increasing the number and relevance of economic evaluations will require the development of a common framework that promotes comparability across studies ( 88 ).

Improve Tracking and Reporting of Implementation Strategies

Developing a robust evidence base for implementation strategies will require that their use be contemporaneously tracked and that they be reported in the literature with sufficient detail ( 92 ). It is often difficult to ascertain which implementation strategies were used and how they might be replicated. Part of the challenge is the iterative nature of implementation. Even if strategies are meticulously described in a study protocol or trial registry, it is often unrealistic to expect that they will not need to be altered as determinants emerge across implementation phases ( 13 , 93 , 94 ). These changes are likely to occur within and between implementing sites in research studies and applied efforts ( 50 , 51 ), and without rigorous methods for tracking implementation strategy use, efforts to understand what strategies were used and whether or not they were effective are stymied. Even when strategies are reported in study protocols or empirical articles, there are numerous problems with their description, including inconsistent labeling; lack of operational definitions; poor description and absence of manuals to guide their use; and lack of a clear theoretical, empirical, or pragmatic justification for how the strategies were developed and applied ( 4 ). Poor reporting clouds the interpretation of results, precludes replication in research and practice, and limits our ability to synthesize findings across studies ( 4 , 92 ). Findings from systematic reviews illustrate this problem. For example, Nadeem et al. ( 95 ) review of learning collaboratives concluded that, “reporting on specific components of the collaborative was imprecise across articles, rendering it impossible to identify active quality improvement collaborative ingredients linked to improved care.”

A number of reporting guidelines could be leveraged to improve descriptions of strategies ( 4 , 96 – 100 ). Proctor et al. ( 4 ) recommend that researchers name and define strategies in ways that are consistent with the published literature, and carefully operationalize the strategy by specifying: (1) actor(s) , (2) action(s) , (3) action target(s) , (4) temporality , (5) dose , (6) implementation outcomes affected , and (7) theoretical, empirical, or pragmatic justification . Specifying strategies in this way has the potential to increase our understanding of not only which strategies are most effective, but more importantly, the processes and mechanisms by which they exert their effects ( 29 , 67 ). Additional options that provide structured reporting recommendations include the Workgroup for Intervention Development and Evaluation Research (WIDER) recommendations ( 99 , 100 ), the Simplified Framework ( 96 ) and its extension [AIMD; ( 97 )], and the Template for Intervention Description and Replication (TIDieR) checklist ( 98 ). Though not specific to the reporting of implementation strategies, the Standards for Reporting Implementation Studies ( 101 ) and Neta et al. ( 102 ) reporting framework emphasizes how critical it is to report on the multilevel context of implementation. The use of any of the existing guidelines would enhance the clarity of strategy description. We believe that developing approaches to tracking implementation strategies ( 50 , 51 ), and assessing the extent to which they are pragmatic (e.g., acceptable, compatible, easy, and useful) for both research and applied efforts is a high priority. Further, efficient ways of linking empirical studies with study protocols to gauge the degree to which strategies have been adapted or tailored over the course of an implementation effort would be helpful. Failing to improve the quality of reporting will negate other advances in this area by hindering replication.

Implementation science has advanced considerably, yielding a more robust understanding of implementation strategies. Several resources can inform the use of implementation strategies, including established taxonomies of implementation strategies ( 6 , 7 , 24 , 25 ) and behavior change techniques ( 27 , 28 ), repositories of systematic reviews ( 38 , 103 , 104 ), methods for selecting and tailoring implementation strategies ( 40 , 55 , 56 ), and reporting guidelines that promote replicability ( 4 , 98 – 100 ). Nevertheless, questions remain and further effectiveness research and methodological development are needed to ensure that evidence is effectively translated into public health impact. Advancing these priorities will lead to a better understanding of when, where, why, and how implementation strategies exert their effects ( 29 , 63 ).

Author Contributions

BP conceptualized the paper and wrote the first draft of the manuscript. All other authors contributed to the writing and approved the final manuscript.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding. BP was supported by grants and contracts from the NIH, including K01MH113806, R25MH104660, UL1TR002489, R01MH106510, R01MH103310, P30A1050410, and R25MH080916. NW was supported by P50MH113840 from the NIMH. RB was supported by grants from the NIMH through R21MH109878 and P50MH113840. CL was supported by R01MH106510 and R01MH103310 from the NIMH. SM was supported by a Fulbright-Health Research Board Impact Award.

• Research article
• Open access
• Published: 14 May 2020

Identifying and selecting implementation theories, models and frameworks: a qualitative study to inform the development of a decision support tool

• Lisa Strifler 1 , 2 ,
• Jan M. Barnsley 1 ,
• Michael Hillmer 1 , 3 &
• Sharon E. Straus   ORCID: orcid.org/0000-0002-6106-832X 1 , 2 , 4  

BMC Medical Informatics and Decision Making volume  20 , Article number:  91 ( 2020 ) Cite this article

8072 Accesses

11 Citations

Metrics details

Implementation theories, models and frameworks offer guidance when implementing and sustaining healthcare evidence-based interventions. However, selection can be challenging given the myriad of potential options. We propose to inform a decision support tool to facilitate the appropriate selection of an implementation theory, model or framework in practice. To inform tool development, this study aimed to explore barriers and facilitators to identifying and selecting implementation theories, models and frameworks in research and practice, as well as end-user preferences for features and functions of the proposed tool.

We used an interpretive descriptive approach to conduct semi-structured interviews with implementation researchers and practitioners in Canada, the United States and Australia. Audio recordings were transcribed verbatim. Data were inductively coded by a single investigator with a subset of 20% coded independently by a second investigator and analyzed using thematic analysis.

Twenty-four individuals participated in the study. Categories of barriers/facilitators, to inform tool development, included characteristics of the individual or team conducting implementation and characteristics of the implementation theory, model or framework. Major barriers to selection included inconsistent terminology, poor fit with the implementation context and limited knowledge about and training in existing theories, models and frameworks. Major facilitators to selection included the importance of clear and concise language and evidence that the theory, model or framework was applied in a relevant health setting or context. Participants were enthusiastic about the development of a decision support tool that is user-friendly, accessible and practical. Preferences for tool features included key questions about the implementation intervention or project (e.g., purpose, stage of implementation, intended target for change) and a comprehensive list of relevant theories, models and frameworks to choose from along with a glossary of terms and the contexts in which they were applied.

Conclusions

An easy to use decision support tool that addresses key barriers to selecting an implementation theory, model or framework in practice may be beneficial to individuals who facilitate implementation practice activities. Findings on end-user preferences for tool features and functions will inform tool development and design through a user-centered approach.

Peer Review reports

Over 100 different theories, models and frameworks exist to guide effective implementation and sustainability of evidence-based interventions or programs [ 1 , 2 ]. The myriad of implementation theories, models and frameworks differ in complexity, such as their aim, scope and intended target for change. For example, they may describe the different stages of implementation (e.g., process models); identify barriers and facilitators that influence implementation (e.g., determinant frameworks); or predict or explain implementation success by offering an underlying mechanism or theory of change (e.g., implementation theories) [ 3 ]. Further, some theories, models and frameworks are broad and address the entire implementation process, while others focus on a particular implementation aspect such as intervention sustainability. Implementation theories, models and frameworks also operate at one or more levels of change, from a health system to an individual. In many cases, using multiple theories, models and frameworks is useful to inform or address the scope and aims of an implementation project and to guide intervention development and testing at multiple levels [ 4 , 5 , 6 ].

Despite a growing interest in the appropriate selection and use of implementation theories, models and frameworks [ 7 , 8 , 9 , 10 , 11 ], it can be difficult to sift through and make sense of the various options available – especially when most are used in practice only once or with limited justification [ 2 , 12 ]. For instance, participants in an implementation practice training course [ 13 ] reported that they struggled to identify and select suitable theories, models or frameworks to guide their work. Studies also suggest that implementation theories, models and frameworks may not be used appropriately [ 8 , 14 ].

Implementation researchers and practitioners looking to identify a theory, model or framework to inform their work can access existing tools and publicly available resources such as guidance documents (e.g., [ 15 , 16 , 17 ]). For example, drawing on their personal experience working with novice implementation practitioners, Lynch and colleagues [ 10 ] suggested five questions to consider when selecting a theory, model or framework: who are you working with, when in the process are you going to use theory, why are you applying theory, how will you collect data and what resources are available. Birken and colleagues [ 9 ] developed a checklist of 16 criteria (organized within four categories: usability, validity, applicability, acceptability) for implementation researchers or practitioners to consult when selecting a theory, model or framework. A major limitation identified by the tool developers is the prerequisite of a candidate list of suitable theories, models or frameworks to draw from and compare [ 9 ]. Rabin and colleagues developed a database of models and frameworks, www.dissemination-implementation.org , however the content is based on the findings of a narrative review of theories, models and frameworks [ 18 ] and is not comprehensive.

To address this problem, we propose to use the findings from a rigorous scoping review of over 300 implementation theories, models and frameworks [ 2 ] to develop a decision support tool, with input from implementation researchers and practitioners using qualitative research methods. A decision support tool provides structured guidance to help users make an explicit decision [ 19 ]. In this case, a decision support tool may facilitate appropriate selection of one or more implementation theories, models or frameworks by engaging the user to answer key questions, resulting in relevant options to consider. The decision support tool will be developed using rigorous methods guided by theory and evidence on user-centered design and implementation science. The overarching approach will be informed using the Knowledge-to-Action Cycle [ 20 ] and the United Kingdom Medical Research Council Framework for Development and Evaluation of Complex Interventions [ 21 ]. These methods have been used for creation of other decision support tools [ 22 ]. As tool development is not the focus for this paper, details on the methods will be described in a subsequent development and evaluation paper.

To inform tool development, we sought the perspectives of implementation researchers and practitioners working in healthcare. Specifically, this study aimed to identify 1) barriers and facilitators to identifying and selecting implementation theories, models and frameworks in research and practice, and 2) preferences for features (i.e., content items) and functions of the proposed decision support tool.

Thorne’s interpretive descriptive approach [ 23 ] guided all aspects of this research, including the design and analysis. Interpretive description is grounded in traditional qualitative methodologies (e.g., phenomenology) that are derived from the social sciences; yet, it is oriented toward applied health disciplines such as implementation practice and designed to address real-world knowledge gaps [ 23 ].

Study design

We used Thorne’s interpretive descriptive approach to elicit the perspectives of implementation researchers and practitioners through individual interviews. We chose to conduct individual, semi-structured interviews to understand individual perspectives, including challenges and successes related to identifying and selecting implementation theories, models and frameworks in research and practice. While focus groups would have allowed for group interactions and may have helped participants generate and share their ideas [ 24 ], we were most interested in individual opinions and decision processes [ 23 ]. Therefore, we felt that interviews would be more informative for tool development. Feasibility was also a factor, as our participants were from a wide geographic area. We followed the Consolidated Criteria for Reporting Qualitative Research checklist [ 25 ] (Additional file  1 ). We obtained research ethics board approval from Unity Health Toronto (REB #16–335) and the University of Toronto (REB #33907). Ethics approval covered recruitment at the conferences and workshops, which covered the study participants in the United States (USA) and Australia. Verbal informed consent was approved by the ethics boards and obtained (and audio-recorded) from all participants using a predetermined script prior to the phone interview.

Participant selection

Eligible study participants included implementation researchers and practitioners (e.g., administrators, clinicians, knowledge brokers) working in healthcare environments such as hospitals, academic research centers or universities, or broader community settings (e.g., public health or regulatory organizations). We defined implementation researchers as individuals who conducted implementation science, and implementation practitioners as individuals who facilitated implementation practice activities (including those who provided support through training and capacity building or knowledge brokering activities).

Study recruitment followed three approaches. First, we recruited in person at two international implementation conferences, one held in the USA in 2016 and one in Canada in 2017. At both conferences, we presented a poster on our scoping review of implementation theories, models and frameworks [ 2 ], distributed study information sheets to attendees who stopped to read the poster, and collected contact information from individuals who were interested in participating in our study. We then sent a personalized email to each individual to verify their interest and eligibility and schedule a phone interview. Second, we sent a personalized email to past participants of an implementation practice training course developed by the Knowledge Translation Program (St. Michael’s Hospital, Unity Health Toronto, Canada) [ 13 ] and delivered in Canada and Australia between 2015 and 2017. Third, we asked study participants to share the study information sheet with colleagues who might be interested in participating. We sent a personalized email to individuals referred to us by study participants. Up to two more emails were sent to non-responders.

These different recruitment approaches were selected because they targeted diverse implementation researchers and practitioners who were interested in, and had experience with, implementation. The sample was expected to reflect the perspectives of our target end-users of the proposed decision support tool. A sample size of 20–30 participants was expected to provide sufficient information to answer the research question through semi-structured interviews and was considered a feasible range given the available resources [ 23 , 26 ].

Data collection

Interviews were conducted over the phone by one investigator (LS) between September 2017 and January 2018. A semi-structured interview guide (Additional file  1 ) was prepared and revised as needed throughout data collection. Part 1 of the interview explored the barriers and facilitators to identifying, selecting and using implementation theories, models or frameworks in research and practice. It included participants’ views and understanding of theories, models and frameworks and the processes used for considering one or more to inform their implementation activities. The interview guide questions were informed loosely by the Theoretical Domains Framework [ 27 ] as a starting point, to allow for inductive analysis. Direct questions inquiring about perceived barriers and facilitators were also included to allow for free-flowing discussion. The Theoretical Domains Framework is a validated determinant framework [ 28 ] that has been applied in numerous implementation studies to uncover the underlying barriers to and facilitators of behaviour change. Further, the framework includes a comprehensive set of barriers at the individual or person level, along with the organizational-level (e.g., groups of individuals), which we felt were most important to understand when developing a decision support tool to meet the needs of our targeted end-user. Part 2 of the interview explored the features and functions of a hypothetical decision support tool that would be important to participants as target end-users of the tool. The interview guide was reviewed by and pilot tested with three individuals, all experienced in qualitative research and implementation science and practice, and one of whom was also a clinician. Each interview lasted 30–60 min and was audio-recorded and transcribed verbatim.

Data analysis

Following an interpretive descriptive approach, we conducted a thematic analysis of the data to synthesize meanings across codes and generate a narrative of the key themes to inform subsequent tool development [ 23 , 29 ]. Data analysis occurred concurrently with data collection. We used NVivo 12 qualitative data analysis software (QSR International, Cambridge, MA) to organize and code the transcripts. Once the audio-recorded interviews were transcribed and verified for accuracy, they were de-identified using a master linking log, prior to being imported into NVivo. After reading through the first few transcripts to become familiar with the data, we used open coding to create codes from the text and drafted a coding framework. This coding framework was revised iteratively throughout data collection and analysis. All data were coded inductively by a single investigator (LS), with a subset of 20% (i.e., 5 transcripts in total) coded by a second investigator (JB) with high concordance achieved. This duplicate coding process was done at the start and end of data collection to ensure consistency of themes. Representative quotes from participants were selected to support the themes and study findings. The final manuscript was shared with participants for feedback on the research findings.

Participant characteristics

Twenty-four individuals consented to participate: 16 were from Canada, seven from the USA and one from Australia (Table  1 ). One eligible participant declined consent due to a confidentiality agreement with their current employer. Of the eligible workshop participants contacted, 2 were not reached due to undeliverable email addresses and 33 did not respond to our email invitation. Participants were recruited until no new themes were identified; therefore, not all workshop participants were sent a study invitation. Participants worked in a variety of healthcare environments including hospitals, academic research centers, universities, government organizations, and regulatory organizations. Participants had a range of experience supporting implementation activities in healthcare environments and reported working in implementation for 1.5 to over 20 years. Of the 24 participants, 11 (46%) had completed a “Practicing Knowledge Translation” course developed by the Knowledge Translation Program at St. Michael’s Hospital, Unity Health Toronto, Canada [ 13 ]. In terms of knowledge, 14 (58%) participants rated themselves as very or extremely knowledgeable or familiar with implementation theories, models and frameworks, and 13 (54%) as very or extremely confident in selecting and applying them to their work. Sixteen (67%) participants reported frequently or always selecting an implementation theory, model or framework and applying it to their work.

Barriers and facilitators to identifying and selecting implementation theories, models or frameworks

Four broad categories and 10 factors, generated from the data, influenced identification and selection of implementation theories, models and frameworks and were relevant to tool development (Fig.  1 ). Illustrative interview excerpts are presented in Tables  2 and 3 .

Categories and factors influencing the identification and selection of an implementation theory, model or framework

Category 1: characteristics of the individual or team conducting implementation

Factor 1: attitudes about the importance of selecting theories, models and frameworks.

Participants reported having a general understanding of theories, models and frameworks and described several uses in implementation research and practice. For example, many participants found Nilsen’s 2015 taxonomy [ 3 ] was useful for defining a theory versus a model versus a framework and referred to the taxonomy when describing their similarities and differences. Some participants said their understanding was grounded in their learnings from the “Practicing Knowledge Translation” course. Others described their understanding of implementation theories, models and frameworks in terms of their clinical or health discipline, such as the Iowa Model for Evidence-based Practice to Promote Quality Care [ 30 ] which originates in the nursing field. In general, frameworks and models were described as being descriptive and useful for clarifying aspects of a complicated process. Theories were viewed as being more explicit about how certain phenomena are operating and how change might be occurring.

Participants mentioned using 28 different implementation theories, models and frameworks to inform their work (Table  4 ). Participants described the important role that theories, models and frameworks play in advancing implementation understanding, especially regarding planning, developing and sustaining effective interventions and implementation strategies. Some of the described uses of theories, models and frameworks included: informing the research question; justifying and organizing an implementation project; guiding the selection and tailoring of implementation strategies; helping to achieve intended outcomes; and analyzing, interpreting, generalizing, or applying the findings of an implementation project. Other benefits to their use included providing a good starting point for implementation, providing a systematic or pragmatic approach for implementation, avoiding overlooking key categories or processes of implementation, and increasing methodological rigor. Participants commented on the importance of engaging in practices that are informed by theories, models and frameworks and evidence.

While all participants agreed on the utility of frameworks and models, such as the Knowledge-to-Action Cycle [ 20 ], a few were skeptical of the value of using theory to enhance knowledge of the complexity of implementation; they preferred to avoid selecting a formal theoretical approach. Others lacked experience with theory-driven implementation. A few believed that implementation practitioners may not feel the same level of “pressure” to use a theory, model or framework in their role compared to an implementation researcher.

Factor 2: knowledge of existing implementation theories, models and frameworks

Knowledge of existing implementation theories, models and frameworks and where to find them were perceived to be important. Some participants struggled to identify new theories, models or frameworks to inform their work, and identified their lack of knowledge of the breadth of options as an important barrier. Most participants favoured one or more implementation theories, models or frameworks and used them repeatedly, stating that it was easy to use what was familiar. Many did not follow an explicit process for identifying a new theory, model or framework. Access to a comprehensive repository or database of existing implementation theories, models and frameworks was perceived as helpful. Participants also suggested having at least one implementation team member with up-to-date knowledge of what implementation theories, models and frameworks exist, where to find them and their uses.

Factor 3: training related to implementation theories, models and frameworks

Participants talked about the relationship between selecting implementation theories, models and frameworks in research or practice and their training experience. For example, most participants selected theories, models and frameworks for which they received specific training. Major barriers to selection included inadequate background or research training in implementation theories, models and frameworks, and lack of training or expertise in implementation research methods or practice. Some participants spoke about the challenge of getting others (e.g., senior administrators, healthcare providers) to buy into the use of a certain theory, model or framework, especially if they were not familiar with the application of theory. Facilitators to selection included gaining appropriate training through participation in capacity building activities, such as accessing implementation workshops, conferences, coaching, mentoring, train-the-trainer approaches or communities of practice. Examples included working with someone who was formally trained on the theory, model or framework, or receiving feedback from implementation experts who used it to inform their work.

Category 2: characteristics of the implementation theory, model or framework

Factor 4: language and terminology used to describe the theory, model or framework.

Language and terminology were key factors for identification and selection. Participants described the language used in implementation theories, models and frameworks as “complex”, “abstract”, “complicated” and “confusing”. In particular, the use of jargon and lack of clear construct definitions were identified as major barriers. Further, several participants struggled with overlapping constructs, and the inconsistent terms used to describe them across theories, models and frameworks. For example, the same term or definition may be used for different constructs, or different terms or definitions may be used for the same constructs. A few participants commented on the inaccurate and inconsistent use of the term theory versus model versus framework, both in research and in practice settings. This appeared to be common with theories versus frameworks (e.g., calling something a theory but referring to a framework). Facilitators included the importance of clear and concise language, and clearly-defined constructs to help differentiate among the various theories, models and frameworks.

Factor 5: fit of the theory, model or framework to the implementation project

Another key factor for identification and selection was the level of fit or appropriateness of the theory, model or framework to the implementation project. Specifically, a poor fit between the context in which the theory, model or framework was developed or had been applied, and the context of the implementation project was identified as a major barrier. For example, many theories, models and frameworks were developed for a specific condition or health behaviour and had not yet been applied in different contexts. Important aspects of the context included the research question, purpose or goal; health problem; setting; population; and level of behaviour change. Evidence that the theory, model or framework had been applied in practice in a similar context (such as relevant examples of applications in the literature) facilitated appropriate selection. Participants stated that seeing a description of the contexts in which the theory, model or framework was previously used was helpful when determining fit. Being aware of a theory, model or framework’s underlying assumptions and its limitations also informed appropriateness and applicability. Other related challenges included the interchangeability, compatibility and adaptability of implementation theories, models and frameworks. For example, some participants struggled with the trade-offs of selecting one theory, model or framework over another. Participants perceived that guidance on comparing different options would facilitate appropriate selection. Some noted that theories, models and frameworks often overlap or are highly derivative of each other, which adds to the complexity of combining more than one within an implementation project. It was deemed helpful to highlight theories, models or frameworks that fit well together, such as the research by Michie and colleagues linking Capabilities Opportunities Motivation Behaviour with the Theoretical Domains Framework [ 31 ]. For others, implementation theories, models or frameworks that allowed for some modification were appealing, but participants struggled with how to modify or change aspects to improve fit while maintaining fidelity to key elements.

Factor 6: ease of use of the implementation theory, model or framework

Ease of use in practice was perceived to influence selection of a theory, model or framework. Some participants described implementation theories, models and frameworks as “not intuitive to use” and difficult to operationalize in the context of their own implementation project, even when the theory, model or framework was viewed as a relevant option. Facilitators to selection and use included existing online tools and publicly available resources, such as websites dedicated to specific theories, models or frameworks (e.g., the Consolidated Framework for Implementation Research). In terms of measurement challenges, a few participants cited a lack of relevant measures for key variables across theories, models and frameworks, as well as variability in the extent to which measures were developed to assess constructs. Participants preferred theories, models or frameworks that were “highly actionable”, “pragmatic” and “easy to operationalize” in practice, with detailed processes for the measures themselves that were compatible with their setting.

Factor 7: evidence supporting the implementation theory, model or framework

Empirical evidence of effectiveness, including strength of evidence supporting the theory, model or framework, influenced selection. Implementation theories were described as “fairly loose” and “without solid evidence” compared to theories in other scientific fields (e.g., physical sciences). Further, within a theory, model or framework, the level of evidence was perceived to be uneven across domains or specific processes. A summary of the evidence supporting a theory, model or framework, including the evidence used to create it and evidence of its effectiveness, was deemed to be an important facilitator. Participants also felt it was important that the theory, model or framework constructs and concepts had face validity and made sense in terms of the implementation research question or goal.

Categories 3 and 4

Other important barriers and facilitators mentioned by participants were related to characteristics of the healthcare environment (Category 4) and, to a lesser extent, characteristics of the implementation intervention or project (Category 3).

Availability of resources (Factor 10) within complex healthcare environments (Category 4), such as time, staffing and capacity, funding and access to data were identified as both barriers and facilitators to selection. Many participants also described a “tension” between time and robustness of implementation. For example, a lack of time to invest in the understanding and use of a theory, model or framework (e.g., competing demands or pressure to fix the problem right away) was a major barrier, while taking the time to create an implementation plan that included consideration of theories, models or frameworks at implementation onset was a facilitator. Theory, model or framework selection was also influenced by staff and stakeholder support, such as having an inadequate number of project staff available or being the sole implementation practitioner within an organization. It was deemed important to “assemble the right people at the right table” to avoid siloed practice and redundancy.

Finally, a few participants mentioned factors related to the implementation project (Category 3), such as consideration of the purpose, problem or goal and intended outcome (Factor 8). For instance, it may be inappropriate to select a theory when part of the research question or outcome of an implementation project was to further develop theory. Another relevant factor that presented a challenge to selection was the level of intervention complexity (Factor 9), including the type of intended behaviour change (e.g., individual, program, practice, policy), and the implementation stage (e.g., planning, evaluation, sustainability) for the project.

Features and functions of a decision support tool

Participants were enthusiastic and receptive to the idea of a decision support tool targeted to implementation practitioners. The following key features and functions were suggested to inform tool development. Illustrative interview excerpts are presented in Table  5 .

Features or content items

Most importantly, the tool should include a comprehensive list of existing implementation theories, models and frameworks to choose from. Suggested content items included characteristics of the theories, models and frameworks matched with characteristics of the end-user’s implementation project (e.g., aim, scope and level of change). Participants suggested organizing the theories, models and frameworks according to their purpose (including their intended aim, scope and level of change) to align them with end-users’ needs. Alternatively, one participant (ID1) suggested starting with the project end goal or outcome, and reviewing theories, models and frameworks that include that outcome as a relevant construct. Many participants also suggested including the context in which the theories, models and frameworks have been applied, along with links to seminal articles and examples of real-world use. Linking the tool with seminal articles would allow end-users to see examples of what has been done, and perhaps gauge ease of use, as well as where the literature may or may not be saturated. Some participants suggested summarizing the evidence supporting each theory, model and framework to highlight those that have been validated. A few participants suggested content items related to the availability of implementation resources, such as the project timelines, number of stakeholders, guidance and team expertise, and financial support.

Participants suggested that the tool be simple and easy to use by the target end-user (i.e., implementation practitioners). They identified that it should provide the user with a modest set of key questions or prompts that start off broad and become more specific. For example, the tool could respond to the user’s input by guiding them toward more specific theories, models and/or frameworks. The tool should also be practical in that the level of content detail fits the intended tool audience and purpose. Being highly accessible through an open access web-based platform was also important. Further, accommodating a team-based approach (e.g., permitting access and use of the tool by an entire multi-disciplinary implementation team) would foster collaboration. Other suggested features included: interactive viewing or search capabilities (e.g., clicking on an interactive theory, model or framework diagram or figure for more information, or searching by key word or construct name); webinars or instructional videos led by experts on when (and how) to use the theory, model or framework; the use of “storytelling” (e.g., case studies) to increase personal connection; and built-in chat room capabilities to connect or collaborate with and receive feedback from others in the field who have experience selecting and using the implementation theory, model or framework. Finally, a few participants suggested an embedded evaluation component whereby users may consent to complete a survey to provide feedback on the tool.

Our findings revealed that factors related to the theory, model or framework, the individual or team conducting implementation and the implementation project are critical to consider when developing a decision support tool. Key barriers to selection related to characteristics of the theory, model or framework included: inaccurate and inconsistent language, poor fit with the implementation context, lack of appropriate measures and limited empirical evidence of effectiveness. These findings are supported by a recent, international survey of over 200 implementation researchers and practitioners who rated ‘empirical support’ and ‘application to a specific population or setting’ as the most important criteria for selection; nevertheless, survey respondents also reported selecting a theory, model or framework based on convenience or familiarity [ 1 ]. Similarly, we found that a lack of knowledge of and familiarity with existing implementation theories, models and frameworks, along with a lack of proper training on their use, were key individual/team-level barriers to selection. These knowledge and skills barriers were not surprising given the abundance of implementation theories, models and frameworks coupled with low citation rates in the literature, indicating they are not commonly used [ 2 , 12 ]. Our study reaffirmed this finding by demonstrating that a group of implementation researchers and practitioners with high self-rated knowledge and experience generated a list of 28 theories, models, and frameworks, which represent less than 20% of those identified in a scoping review. While there may be benefits to selecting a highly-cited theory, model or framework (such as comparability of results across populations or health behaviours [ 32 ] or greater availability of resources for operationalization and measurement [ 12 ]), a systematic and comprehensive approach to theory, model and framework identification and selection is necessary to advance implementation science and practice.

There are numerous determinant frameworks that we could have chosen to inform our interview guide. For example, our team recently mapped over 300 implementation theories, models and frameworks to Nilsen’s taxonomy [ 3 ] and identified over 50 determinant frameworks targeting at least individual-level change; however, many did not include a comprehensive set of barriers and facilitators ( unpublished data ). Our findings on the barriers and facilitators to selection of a theory, model or framework, in the context of informing a decision support tool, are supported by the Theoretical Domains Framework. For example, the domain ‘knowledge’ considers having the knowledge to locate and understand existing theories, models and frameworks. The ‘skills’ and ‘beliefs about capabilities’ domains focus on having the skills required to know how to select a theory, model or framework in practice and considers how easy or difficult this task is for an individual or team. The ‘social/professional role and identity’ and ‘optimism’ domains consider attitudes about the importance of using theories, models and frameworks, specifically whether an individual believes that selecting and using them is part of their role as an implementation researcher or practitioner and that doing so will benefit their implementation work. The ‘goals’ and ‘intentions’ domains focus on wanting to select and use theories, models and frameworks and then making a conscious decision to include them in implementation work, for example, by using a decision support tool. Finally, the ‘environmental resources’ domain considers having the time and funds to invest in the selection process.

A decision support tool addressing our findings on barriers and facilitators to selection might include a comprehensive list of theories, models and frameworks, a glossary of key terms, the contexts in which the theories, models and frameworks have been developed and applied (including examples of application), and any available evidence to support their validity. Other suggested features for consideration during tool development included the purpose, goal or intended outcome of the implementation project as well as the target population and the intended target for change. It would be quite challenging as tool developers, to systematically categorize existing theories, models and frameworks according to factors such as the amount of time or funding required for use; it may be more beneficial for end-users to reflect on these environment-level factors as key considerations associated with the selection of a particular theory, model or framework from the options provided by the tool. Findings on end-user preferences for tool features and functions will inform tool development and design through a user-centered approach [ 33 ].

Limitations

The following study limitations should be considered. First, we used a convenience sample of implementation conference and course attendees. As a result, close to half of our participant sample completed a “Practicing Knowledge Translation” course. As such, we were mindful during recruitment to ensure representatives from different types of healthcare environments, roles, and level of experience. Although we did not intend to saturate these fields given our sample size, we did obtain saturation of themes and had a good sample size for qualitative interviews [ 23 ]. Second, we chose to interview implementation researchers and practitioners with some implementation practice experience (i.e., as the target end-users of our tool) because we felt that this experience would be necessary to identify the underlying barriers and facilitators. As such, all study participants described having a baseline understanding of at least a few implementation theories, models and frameworks. While many participants rated their knowledge and confidence with identifying, selecting and using implementation theories, models and frameworks as fairly high, for many this rating reflected their knowledge and confidence regarding the theories, models or frameworks that they were most familiar with and used repeatedly to guide their work.

Individuals who are doing implementation work face many challenges, including how to identify and select appropriate implementation theories, models and frameworks to inform their projects. Key barriers to selection identified in this study included inconsistent language, poor fit and limited knowledge about and training in theories, models and frameworks. These barriers, together with the findings of our scoping review on existing theories, models and frameworks, will inform and tailor the features and functions of a proposed decision support tool for use by implementation practitioners. Our findings from this interview-based study suggest the tool should be easy to use, accessible and feature questions about the implementation project’s purpose, scope and intended target for change, in addition to presenting a comprehensive list of relevant theories, models and frameworks and the contexts in which they were applied.

Availability of data and materials

Not applicable.

Birken SA, Powell BJ, Shea CM, Haines ER, Alexis Kirk M, Leeman J, et al. Criteria for selecting implementation science theories and frameworks: results from an international survey. Implement Sci. 2017;12(1):124.

Article   Google Scholar  

Strifler L, Cardoso R, McGowan J, Cogo E, Nincic V, Khan PA, et al. Scoping review identifies significant number of knowledge translation theories, models, and frameworks with limited use. J Clin Epidemiol. 2018;100(Complete):92–102.

Nilsen P. Making sense of implementation theories, models and frameworks. Implement Sci. 2015;10(1):53.

Grol R, Grimshaw J. From best evidence to best practice: effective implementation of change in patients' care. Lancet. 2003;362(9391):1225–30.

Estabrooks CA, Thompson DS, Lovely JJE, Hofmeyer A. A guide to knowledge translation theory. J Contin Educ Heal Prof. 2006;26(1):25–36.

Glanz K, Rimer BK, Viswanath K. Health behavior and health education. Theory, research, and practice. 4th ed. San Francisco: Wiley; 2008.

Google Scholar  

The Improved Clinical Effectiveness through Behavioural Research Group (ICEBeRG). Designing theoretically-informed implementation interventions. Implement Sci. 2006;1(1):4.

Davidoff F, Dixon-Woods M, Leviton L, Michie S. Demystifying theory and its use in improvement. BMJ Qual Saf. 2015;24(3):228.

Birken SA, Rohweder CL, Powell BJ, Shea CM, Scott J, Leeman J, et al. T-CaST: an implementation theory comparison and selection tool. Implement Sci. 2018;13(1):143.

Lynch EA, Mudge A, Knowles S, Kitson AL, Hunter SC, Harvey G. “There is nothing so practical as a good theory”: a pragmatic guide for selecting theoretical approaches for implementation projects. BMC Health Serv Res. 2018;18:857.

May CR, Cummings A, Girling M, Bracher M, Mair FS, May CM, Murray E, Myall M, Rapley T, Finch T. Using normalization process theory in feasibility studies and process evaluations of complex healthcare interventions: a systematic review. Implement Sci. 2018;13(1):80.

Skolarus TA, Lehmann T, Tabak RG, Harris J, Lecy J, Sales AE. Assessing citation networks for dissemination and implementation research frameworks. Implement Sci. 2017;12(1):97.

Moore JE, Rashid S, Park JS, Khan S, Straus SE. Longitudinal evaluation of a course to build core competencies in implementation practice. Implement Sci. 2018;13(1):106.

Field B, Booth A, Ilott I, Gerrish K. Using the knowledge to action framework in practice: a citation analysis and systematic review. Implement Sci. 2014;9:172.

Department of Veterans Health Administration, Health Services Research & Development, Quality Enhancement Research Initiative. Implementation Guide. 2013; Available at: https://www.queri.research.va.gov/implementation/ . Accessed 8 Jan 2020.

Center for Research in Implementation Science and Prevention, University of Colorado Anschutz Medical Campus. Dissemination and implementation in health training guide and workbook. 2013; Available at: http://www.crispebooks.org/ . Accessed 8 Jan 2020.

Straus SE, Tetroe J, Graham ID. Knowledge translation in health care. Moving from evidence to practice. 2nd ed. Oxford: Wiley Blackwell; 2013.

Book   Google Scholar  

Tabak RG, Khoong EC, Chambers DA, Brownson RC. Bridging research and practice: models for dissemination and implementation research. Am J Prev Med. 2012;43(3):337–50.

Timmings C, Khan S, Moore JE, Marquez C, Pykal K, Straus SE. Ready, set, change! Development and usability testing of an online readiness for change decision support tool for healthcare organizations. BMC Med Inform and Decis Mak. 2016;16:24.

Graham ID, Logan J, Harrison MB, Straus SE, Tetroe J, Caswell W, et al. Lost in knowledge translation: time for a map? J Contin Educ Heal Prof. 2006;26(1):13–24.

Craig P, Dieppe P, Macintyre S, Michie S, Nazareth I, Petticrew M. Developing and evaluating complex interventions: the new Medical Research Council guidance. BMJ. 2008;337:a1655.

Kastner M, Straus SE. Application of the knowledge-to-action and Medical Research Council frameworks in the development of an osteoporosis clinical decision support tool. J Clin Epidemiol. 2012;65(11):1163–70.

Thorne S. Interpretive description: qualitative research for applied practice. 2nd ed. New York: Routledge; 2016.

Parsons MGJ. A guide to the use of focus groups in health care research: part 1. Contemp Nurse. 2000;9(2):169–80.

Article   CAS   Google Scholar  

Tong A, Sainsbury P, Craig J. Consolidated criteria for reporting qualitative research (COREQ): a 32-item checklist for interviews and focus groups. Int J Qual Health C. 2007;19(6):349–57.

Nastasi B. Qualitative research: sampling & sample size considerations. 2010; Available at: https://my.laureate.net/Faculty/docs/Faculty%20Documents/Forms/AllItems.aspx . Accessed 1 Oct 2016.

Michie S, Johnston M, Abraham C, Lawton R, Parker D, Walker A, et al. Making psychological theory useful for implementing evidence based practice: a consensus approach. Qual Saf Health Care. 2005;14(1):26–33.

Cane J, O’Connor D, Michie S. Validation of the theoretical domains framework for use in behaviour change and implementation research. Implement Sci. 2012;7(1):37.

Braun VCV. Using thematic analysis in psychology. Qual Res Psychol. 2006;3:77.

Iowa MC, Buckwalter KC, Cullen L, Hanrahan K, Kleiber C, McCarthy AM, et al. Iowa model of evidence-based practice: revisions and validation. Worldviews Evid-Based Nurs. 2017;14(3):175–82.

Michie S, van Stralen MM, West R. The behaviour change wheel: a new method for characterising and designing behaviour change interventions. Implement Sci. 2011;6:42.

Bastani R, Glenn BA, Taylor VM, Chen MS, Nguyen TT, Stewart SL, et al. Integrating theory into community interventions to reduce liver cancer disparities: the health behavior framework. Prev Med. 2010;50(1–2):63–7.

Mao J, Vredenburg K, Smith PW, Carey T. The state of user-centered design practice. Comm ACM. 2005;48(3):105–9.

Download references

Acknowledgements

We would like to thank Melissa Courvoisier, Dr. Julia Moore and Dr. Rae Thomas for their assistance with recruitment from the Practicing Knowledge Translation course; Christine Marquez for her qualitative research expertise; and all the individuals who participated in the interviews, for their support and contribution to this work.

Lisa Strifler is funded by a Canadian Institutes of Health Research Banting Doctoral Research Award (#146261). Sharon E. Straus is funded by a Tier 1 Canada Research Chair in Knowledge Translation and the Mary Trimmer Chair in Geriatric Medicine. The funders had no role in the design of the study, the collection, analysis or interpretation of data, or the writing of the manuscript.

Author information

Authors and affiliations.

Institute of Health Policy Management & Evaluation, University of Toronto, 155 College Street, Toronto, Ontario, M5T 3M6, Canada

Lisa Strifler, Jan M. Barnsley, Michael Hillmer & Sharon E. Straus

Knowledge Translation Program, Li Ka Shing Knowledge Institute, St. Michael’s Hospital, Unity Health Toronto, 209 Victoria Street, Toronto, Ontario, M5B 1W8, Canada

Lisa Strifler & Sharon E. Straus

Ontario Ministry of Health and Long-Term Care, 900 Bay Street, Toronto, Ontario, M7A 1R3, Canada

Michael Hillmer

Department of Geriatric Medicine, University of Toronto, 27 King’s College Circle, Toronto, Ontario, M5S 1A1, Canada

Sharon E. Straus

You can also search for this author in PubMed   Google Scholar

Contributions

LS and SES conceptualized and designed the study. LS, JMB and SES collected, analysed and/or interpreted the data. LS drafted the manuscript and JMB, MH and SES provided input and revised the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Sharon E. Straus .

Ethics declarations

Ethics approval and consent to participate.

Research ethics board approval was obtained from Unity Health Toronto (REB #16–335) and the University of Toronto (REB #33907). Ethics approval covered recruitment at the conferences and workshops, which covered the study participants in the USA and Australia. Verbal informed consent was approved by the ethics boards and obtained and recorded at the start of the phone interview using a predetermined script.

Consent for publication

Competing interests.

All authors declare no potential (or perceived) conflicts of interest.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Additional file 1..

Supplemental file with COREQ checklist (Appendix 1), interview guide (Appendix 2), and citations for theories, models and frameworks used by participants (Appendix 3).

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . The Creative Commons Public Domain Dedication waiver ( http://creativecommons.org/publicdomain/zero/1.0/ ) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

Reprints and permissions

About this article

Cite this article.

Strifler, L., Barnsley, J.M., Hillmer, M. et al. Identifying and selecting implementation theories, models and frameworks: a qualitative study to inform the development of a decision support tool. BMC Med Inform Decis Mak 20 , 91 (2020). https://doi.org/10.1186/s12911-020-01128-8

Download citation

Received : 06 November 2019

Accepted : 07 May 2020

Published : 14 May 2020

DOI : https://doi.org/10.1186/s12911-020-01128-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Implementation
• Decision support

BMC Medical Informatics and Decision Making

ISSN: 1472-6947

• General enquiries: [email protected]

• Open access
• Published: 11 April 2024

Organizing the dissemination and implementation field: who are we, what are we doing, and how should we do it?

• Gretchen J. R. Buchanan   ORCID: orcid.org/0000-0002-5186-0145 1   na1 ,
• Lindsey M. Filiatreau 2   na1 &
• Julia E. Moore 3   na1  

Implementation Science Communications volume  5 , Article number:  38 ( 2024 ) Cite this article

128 Accesses

4 Altmetric

Metrics details

Two decades into its tenure as a field, dissemination and implementation (D&I) scientists have begun a process of self-reflection, illuminating a missed opportunity to bridge the gap between research and practice—one of the field’s foundational objectives. In this paper, we, the authors, assert the research-to-practice gap has persisted, in part due to an inadequate characterization of roles, functions, and processes within D&I. We aim to address this issue, and the rising tension between D&I researchers and practitioners, by proposing a community-centered path forward that is grounded in equity.

We identify key players within the field and characterize their unique roles using the translational science spectrum, a model originally developed in the biomedical sciences to help streamline the research-to-practice process, as a guide. We argue that the full translational science spectrum, from basic science research, or “T0,” to translation to community, or “T4,” readily applies within D&I and that in using this framework to clarify roles, functions, and processes within the field, we can facilitate greater collaboration and respect across the entire D&I research-to-practice continuum. We also highlight distinct opportunities (e.g., changes to D&I scientific conference structures) to increase regular communication and engagement between individuals whose work sits at different points along the D&I translational science spectrum that can accelerate our efforts to close the research-to-practice gap and achieve the field’s foundational objectives.

Peer Review reports

Contributions to the literature

Providing clarity regarding the distinct groups of individuals involved in D&I science and practice from researchers to the communities impacted by the change and outline key roles of these unique sets of actors.

Specifying the range of activities, from theoretical research to applied implementation, involved in D&I science and practice using a translational structure.

Identifying existing gaps (e.g., poor integration of research into existing implementation efforts) that impede attainment of the shared vision of D&I science and practice and propose solutions to these gaps.

Introduction

Though still in its infancy, the field of dissemination and implementation science (D&I) [ 1 , 2 ] is facing challenges related to the growing gap between the science and practice of implementation [ 1 , 3 , 4 ]. D&I is the scientific study of translating research findings and evidence-based interventions into everyday practice; in the current state of the D&I literature, this often means that a practice developed by one group of actors is being implemented into the everyday practice of others [ 5 ]. A premortem by Beidas and colleagues [ 4 ] highlighted several factors stagnating the field, including closure of the evidence-to-practice gap [ 6 , 7 , 8 , 9 ], insufficient impact, and inability to align timelines and priorities with partners [ 1 ]. This commentary aims to establish further clarity regarding who “we” are as a field, what we are doing, and how we can collectively work to achieve shared goals of improved population health in D&I. This refers to the collective “we” of those engaged in D&I work.

In clarifying key components of D&I, important lessons can be drawn from more established fields. For example, when reflecting on disciplines such as mathematics and physics, one notes the emergence of two broad areas of scholarship—theoretical and applied—within these fields. These scholarship areas fill distinct, but important roles within their fields. Here, the authors posit that D&I science could be similarly broken down into theoretical and applied scholarships. In this paper, we, the authors, elaborate on the functions of these differential scholarships, and the functions of professionals working in the large and ever-growing field of implementation practice.

While many have noted D&I aims “to promote the adoption and integration of evidence-based practices, interventions, and policies into routine health care and public health settings to improve the impact on population health,” [ 10 ] specificity in how to achieve this outcome has been elusive. In this article, we propose that the field must first define the actors and audiences across the implementation spectrum and how each group connects with others. Subsequently, the field can strengthen the infrastructures that facilitate these connections. In this article, we aim to address the rising tension between implementation scientists, implementation support practitioners, delivery systems [ 11 ], and communities by proposing a path forward that is community-oriented and grounded in equity, thereby upholding every actor’s place at the D&I table. We draw on principles well-established in the field of translational science to better align D&I towards both improved ideas and real-world impact. We note that our mental model as authors is that success for D&I would be defined as impact at the community or population levels. We recognize this is not the mental model held by all people working in D&I, but believe even for those whose focus is not on population impact, we can collectively work together to achieve these outcomes and impact practice [ 12 ].

Who are we?

To date, much of the discussion around the direction of D&I has been researcher-centric [ 13 ]. To promote greater equity within the discipline (i.e., to reduce disparities in whose voices are heard within the field of D&I), we would like to expand the existing discourse to include the entire spectrum of professionals who work in implementation, including communities, delivery systems, implementation support practitioners, intermediaries, non-implementation science researchers (e.g., interventionists), and applied and theoretical D&I researchers. Including the entire implementation workforce in a description of the field provides opportunities to see where practitioners have not been empowered to exert influence and to change these inequities. While D&I professionals are likely to fill more than one role at a time or during their careers and may hold perspectives that are therefore representative of a number of these D&I actors, we would like to re-center the current conversation within D&I around implementation support practitioners and delivery systems specifically to uphold our commitment to those most directly affected by D&I efforts.

Communities and individuals impacted by the change

Communities and the individuals who comprise them play a critical role in the success or failure of efforts to implement evidence-based or informed programs and practices (EBPs) within a particular setting [ 14 , 15 , 16 , 17 ]. Aligned with this principle, there has been a shifting focus from using community-based to community-led research methods across academic disciplines [ 18 , 19 ]. Funding agencies have also begun to recognize the need for greater community involvement in research, with current directives to engage community partners across the research spectrum [ 20 ]. As suggested by others, strengthening relationships between communities and individuals working at all levels of implementation should remain a priority in closing the evidence-to-practice gap and upholding equity in D;I; indeed, it is essential [ 21 ].

Practitioners—implementation support practitioners and delivery systems

Implementation has been happening for the entirety of human history. While several scientific fields (e.g., political science, medicine) began formally investigating processes of D&I in the mid-to-late twentieth century—thereby laying the foundation for current research in this area— the distinct field of D&I only emerged in the past few decades, prompted by repeatedly observed barriers to the successful implementation of EBPs [ 5 , 22 ].

“Implementation practitioners” are professionals comprised of two distinct groups: implementation support practitioners [ 23 , 24 ] (e.g., administrators, policy-makers) are involved in planning, engagement, co-creation, strategy selection, capacity building, monitoring, and evaluation; delivery systems (e.g., front-line managers at organizations implementing an EBP) are responsible for implementing the actual practices with professionals, organizations, and the public [ 11 ]. Identifying professionals engaged in implementation practice can be difficult as there is inconsistency and terminology; for example, there are over 30 job titles associated with implementation support practitioner roles (see Fig.  1 ). “Delivery systems” are often unaware of the D&I field or their role as end-users. Implementation researchers appropriately identifying and connecting with delivery systems and implementation support practitioners is key to closing the evidence-to-practice gap and improving impact [ 4 ].

Professional job titles of individuals working directly in implementation or implementation support as identified through the Center for Implementation (In preparation for an event about the roles of implementation support practitioners, an open call was sent out to members of an online community of professionals supporting implementation. People were asked for their current or previous job titles that included an implementation component.)

Intermediaries

Globally, there are several intermediary organizations serving to translate findings from D&I to support the implementation of EBPs by delivery systems and implementation support practitioners (e.g., the Collaborative for Implementation Practice; Center for Evidence and Implementation in Australia; Impact Center at the University of North Carolina; Center for Effective Services in Ireland; the Nigerian Implementation Science Alliance). These organizations employ implementation support practitioners and bridge the implementation research-to-practice divide by providing training in implementation-related skills and creating tools to support the selection of appropriate implementation strategies. For example, one intermediary has a mini-course providing an introduction to implementation that has enrolled over 10,000 individuals. Millions of research, government, and philanthropic dollars are being invested in these organizations [ 25 , 26 , 27 , 28 ]. As implementation researchers and intermediaries, the authors regularly hear from organizations, communities, and individuals that they struggle to access supports in implementation science to address their needs in implementing evidence The demand for this type of work often outpaces the supply, and researchers and funders alike state a clear need for additional resources linking implementation science and practice [ 29 , 30 , 31 , 32 , 33 ].

Researchers

To better clarify the full spectrum of implementation researchers, researchers whose work is primarily centered on the advancement of implementation ideas (e.g., theory, methods, or framework (TMF) development) are referred to as theoretical implementation scientists and those whose work is primarily centered on the direct use of implementation concepts as a method to achieve better clinical or programmatic outcomes as applied implementation scientists . Scientists may work on both theoretical and applied projects but tend to focus their programs of research in one or the other and may even identify as one or the other.

Non-D&I researchers are also becoming increasingly interested in D&I, as evidenced by the growing number of D&I training institutes globally (e.g., HIV, Infectious Disease and Global Health Implementation Research Institute (HIGH IRI); University College Cork Implementation Science Training Institute; University of Nairobi Implementation Science Fellowship; Training Institute for Dissemination and Implementation Research in Health (TIDIRH)) [ 34 ]. Non-D&I researchers are individuals from distinct substantive areas (e.g., HIV, cancer prevention) who are interested in applying D&I to their work but have limited training in this area. These researchers often aim to draw from the TMFs and evidence from D&I to design, implement, and scale EBPs. They may benefit from increased collaboration with individuals who have worked more squarely in D&I.

What are we doing?

We, the paper’s authors, entered the field of D&I with the goal of bridging the research-to-practice gap to better improve the lives of people in our areas of scholarship (HIV, mental health). Yet, we have found that our substantively distinct bodies of applied D&I research have unfolded in such a way that we are all currently involved in a range of theoretical implementation research. This journey has not been without difficulty—the further we moved from our applied work and what grounded our science, the less impact we felt we were having. While we found theoretical research important, we felt as though our roles and functions within D&I were less clear. This lack of clarity in our professional self-concept ultimately helped us identify that D&I is not monolithic. Through conversation, we found that articulating the spectrum of theoretical to applied D&I helped us regain the clarity we needed to continue advancing our science. We believe these realizations could also be beneficial to other D&I professionals.

Leveraging translational science to find clarity

There is extensive literature on moving research findings into practice [ 35 ], but the translation of D&I knowledge into practice has received much less attention [ 1 ]. Moreover, there is insufficient understanding of which actors are involved at which stages along this spectrum, how each stage contributes to the field, and how these stages, and actors at each of these stages, can connect and achieve shared goals. In Fig.  2 , the authors draw on the translational spectrum to address these limitations. The traditional translational spectrum aims to streamline the “bench to bedside” approach and defines the continuum of basic science (stage T0) to public health science (stage T4) [ 36 ]. D&I science has long been placed in the T3–T4 segments of the traditional translational spectrum [ 36 ]. However, we argue that the full translational spectrum, from T0 through T4, is applicable to D&I. This distinction is often at the core of the tension observed within the field and where our personal struggles in our shifting identities and relationship with D&I research emerged.

The translational spectrum applied to implementation science

In the traditional translational spectrum, T0, “pre-clinical research,” includes bench science and aims to define mechanisms, targets, and strategies for intervention on a general level. In D&I, theoretical implementation scientists work on the development of TMFs, and elicitation, description, and modeling of mechanisms. Many of the foundational papers that guide implementation research to date stem from work at this stage [ 37 , 38 , 39 , 40 , 41 , 42 , 43 ]. T1, “translation to humans,” includes Stage 1 clinical trials and proof of concept science and aims to develop new methods of diagnosis, treatment, and prevention in highly controlled settings. In D&I, theoretical and applied implementation researchers focus on translating theoretical constructs (i.e., TMFs) to actual people and developing methods to test these constructs. Examples of this type of research include measurement of implementation domains such as context (e.g., the Organizational Readiness for Change measure) [ 44 ] and implementation outcomes (e.g., the NoMAD measure from Normalization Process Theory) [ 45 ]. T2, “translation to patients,” includes Stages 2–3 clinical trials and aims to develop clinical applications and evidence-based guidelines for a given disease. In D&I, applied implementation researchers focus on identifying implementation constructs relevant to a specific situation, intervention, context, or population where the researchers aim to understand how best to implement. Traditional randomized controlled trial designs are often used in this stage. Individuals working at this stage may test bundled strategies, interrogate the “active ingredients” in strategies [ 46 ], or test strategies in varied contexts.

An interesting phenomenon occurs in the T3–4 range. Acknowledging the contributions of researchers and practitioners, we see a split whereby researchers continue to serve as the primary actors in one branch of the translational spectrum, while practitioners become the primary actors in another branch of the spectrum. T3, “translation to practice,” includes comparative effectiveness trials and clinical outcome studies and aims to evaluate real-world effectiveness. In D&I, implementation support practitioners come into a principal role. Individuals working in this capacity use the results of T0–2 to plan implementation projects, sometimes in the form of quality improvement-type projects. In parallel, T3 applied implementation researchers are primarily monitoring or evaluating implementation projects’ real-world effectiveness; this could involve research using pragmatic or naturalistic methods whereby researchers partner with healthcare delivery systems or organizations to better understand real-world implementation or effectiveness outcomes. T4 involves population-level outcomes research and monitoring improvements in morbidity and mortality to impact policy or system change. In D&I, implementation support practitioners and delivery systems scale EBPs up and out. Implementation researchers working at stage T4 define the implementation workforce, develop surveillance systems, and evaluate the effects of evidence-informed implementation on project successes. Intermediaries are prime partners in this work. Additional work is needed to establish clear evidence about what is and is not working on a broad scale and in what contexts [ 42 , 47 ].

Defining the translational spectrum for D&I facilitates the process of identifying a “home base” for individuals involved in D&I science, thereby improving self-concept clarity and making clear how individuals can foray into upstream and downstream segments to better link their research with that of others. In keeping with findings from workplace self-concept clarity literature [ 48 , 49 ], when we claim our places in the spectrum, we can improve our effectiveness and avoid burnout [ 50 ]. Specifically, we can improve our capacity to clearly generate research questions, identify colleagues, and expand the impact of our work.

How should we do it?

As has been noted by others [ 21 , 51 , 52 ], there is a significant disconnect between individuals working in distinct roles within the field of D&I, particularly between those operating at the two ends of the D&I translational spectrum. By interacting more often and intentionally across the entirety of the D&I process, we as a field could develop significant synergy and produce actionable solutions more quickly to achieve shared goals.

Asking and answering the right question

Fundamental respect for the work of actors at every level of the implementation spectrum, fostered by regular communication, is essential in resolving our identity crises, achieving our shared goals, and upholding equity within the field [ 21 ]. One fundamental way for theoretical implementation scientists to demonstrate respect for implementation practitioners is to ask research questions that implementation practitioners want answered [ 52 ]. Implementation practitioners have critical theoretical questions that arise while implementing programs and policies in their specific contexts. For example, implementation practitioners regularly assess organizational readiness for change before altering or implementing a new program or policy (as recommended in the implementation science literature). Yet when the assessments suggest that sites are not ready to implement the intended change, there is little guidance from implementation science about how to best address this issue. A common suggestion is to prioritize “ready” sites [ 53 ]. This approach is likely to perpetuate existing inequities or disparities, as “ready” sites are often the sites that are least in need of additional resources and supports, and leaves “non-ready” sites with no plan for reaching a sufficient level of readiness. What strategies can increase readiness? Another example involves the need for a more concrete understanding of the effects of adaptation. While the field might agree adaptation is often important to the scale-up and scale-out of EBPs, many adaptation tools [ 54 , 55 ] are designed for researchers as opposed to practitioners looking for guidance in understanding if the adaptations they propose will influence the effectiveness of the original EBP. How can D&I measures be made more accessible for implementation practitioners? These are just two examples of many.

Working with existing implementation efforts

Evaluating existing processes and successes of implementation practitioners can also galvanize efforts, improve impact of D&I, and uphold equity in D&I. Delivery systems are continually implementing “the thing” and have been for years. Connecting with existing implementation efforts and studying the effectiveness of implementation strategies being actively used by delivery systems is critical to supporting the ongoing work of these individuals [ 2 , 21 , 56 ]. In many ways, this can shortcut science more quickly to a clearer understanding of what works when and for whom, and improve the likelihood of establishing sustainable practices and policies that are feasible, acceptable, and appropriate [ 23 , 24 ]. This approach is also consistent with the principles of community-based participatory research, including respect for lived experience and tailoring interventions to the needs of the community [ 57 , 58 ].

Fostering increased communication

Increased communication among actors across the D&I translational spectrum is critical, as previously noted [ 3 , 52 , 59 ]. To again draw from the successes of other fields, the International AIDS Society is a group of over 13,000 members worldwide that “unite(s) scientists, policymakers and activists to galvanize the scientific response, build global solidarity and enhance human dignity for all people living with and affected by HIV” [ 60 ]. The International AIDS Society hosts two conferences that rotate annually with a shifting focus between research and practice. Using this model, which has been repeatedly shown to be highly impactful, individuals working at all stages of the HIV implementation science spectrum can engage in, learn from, and contribute to dialogue with others with distinct perspectives and roles in the discipline, thereby improving equity concerning whose voices are centered and uplifted in global agenda-setting efforts. As such, the field of D&I could benefit from an organization akin to the International AIDS Society and agenda-setting practices and conference structures employed by this Society [ 61 , 62 , 63 ].

Developing tools to directly support real-world D&I

Tools that facilitate the translation of D&I into practice are also critical to achieving shared goals [ 1 ]. Again, the field of D&I can look to adjacent fields to learn how they have successfully scaled. For example, the Institute for Healthcare Improvement (IHI), whose mission is to improve health and healthcare worldwide, has scaled the use of quality improvement methods. Over 30 years, they have worked in 42 countries and have had over 7 million online course enrollments [ 64 ]. Part of IHI’s model has been to develop practical and easy-to-use improvement tools. A critique of implementation science is that existing frameworks are complicated and difficult to use [ 3 , 4 ]. If the field of D&I learned from the success of IHI and developed tools that help professionals operationalize implementation science in practice, it would support the broader use of D&I to improve outcomes.

Aligning funding mechanisms and priorities

Funding agencies should increase requirements and supports for community inclusion and implementation throughout the research process. Researchers currently prioritize funding agency policies and expectations, which may not allow enough time for building sustainable community relationships and co-creation of work. A shift in funding agencies’ research calls and approach to awarding research dollars is necessary to build capacity for long-term academic-community partnerships [ 65 , 66 , 67 ]. Implementation science-related funding calls from the National Institutes of Health, UK Research and Innovation, the Global Alliance for Chronic Diseases, the South African Medical Research Council, and other funding agencies could more intentionally include requirements for this type of work.

Key actions are needed for the field of D&I to self-actualize: (1) Uphold everyone’s place at the implementation table while centering the wants and needs of those most directly affected by implementation efforts; (2) Clarify where on the translational spectrum work is being done by whom and where the gaps in both sufficient volume of work and translation of that work lie; and (3) Facilitate regular communication across the spectrum, from theoretical implementation scientists to implementation practitioners and vice versa. Ideally, this work should be done with researchers and practitioners around the globe. If these three tasks are accomplished, we as a field will be able to reverse the tides and bridge the implementation research-to-practice gap, instead of letting it continue to grow.

Availability of data and materials

Not applicable.

Abbreviations

• Dissemination and implementation

Human immunodeficiency virus

Acquired immunodeficiency syndrome

Westerlund A, Sundberg L, Nilsen P. Implementation of implementation science knowledge: the research-practice gap paradox. Worldviews Evidence-Based Nurs. 2019;16(5):332–4. https://doi.org/10.1111/wvn.12403 .

Article   Google Scholar  

Metz A, Jensen T, Farley A, Boaz A. Is implementation research out of step with implementation practice? Pathways to effective implementation support over the last decade. Implement Res Pract. 2022;3:263348952211055. https://doi.org/10.1177/26334895221105585 .

Rapport F, Smith J, Hutchinson K, et al. Too much theory and not enough practice? The challenge of implementation science application in healthcare practice. J Eval Clin Pract. 2022;28(6):991–1002. https://doi.org/10.1111/jep.13600 .

Article   PubMed   Google Scholar  

Beidas RS, Dorsey S, Lewis CC, et al. Promises and pitfalls in implementation science from the perspective of US-based researchers: learning from a pre-mortem. Implement Sci. 2022;17(1):1–15. https://doi.org/10.1186/s13012-022-01226-3 .

Eccles MP, Mittman BS. Welcome to implementation science. Implement Sci. 2006;1(1):1–3. https://doi.org/10.1186/1748-5908-1-1 .

Article   PubMed Central   Google Scholar  

Taylor SP, Kowalkowski MA, Beidas RS. Where is the implementation science? An opportunity to apply principles during the COVID-19 pandemic. Clin Infect Dis. 2020:6–8. https://doi.org/10.1093/cid/ciaa622 .

Lyon AR, Comtois KA, Kerns SEU, Landes SJ, Lewis CC. Closing the science–practice gap in implementation before it widens. In: Albers B, Shlonsky A, Mildon R, eds. Implementation Science 3.0. Springer International Publishing; 2020:295–313. https://doi.org/10.1007/978-3-030-03874-8_12 .

Ploeg J, Davies B, Edwards N, Gifford W, Miller PE. Factors influencing best-practice guideline implementation: Lessons learned from administrators, nursing staff, and project leaders. Worldviews Evidence-Based Nurs. 2007;4(4):210–9. https://doi.org/10.1111/j.1741-6787.2007.00106.x .

Bernhardt JM, Mays D, Kreuter MW. Dissemination 2.0: closing the gap between knowledge and practice with new media and marketing. J Health Commun. 2011;16(SUPPL. 1):32–44. https://doi.org/10.1080/10810730.2011.593608 .

National Cancer Institute. Implementation Science. 2020. Accessed 12 Sept 2023. https://cancercontrol.cancer.gov/is/about .

Wandersman A, Duffy J, Flaspohler P, et al. Bridging the gap between prevention research and practice: The interactive systems framework for dissemination and implementation. Am J Community Psychol. 2008;41(3–4):171–81. https://doi.org/10.1007/s10464-008-9174-z .

Boulton R, Sandall J, Sevdalis N. The Cultural Politics of ‘Implementation Science.’ J Med Humanit. 2020;41(3):379–94. https://doi.org/10.1007/s10912-020-09607-9 .

Article   PubMed   PubMed Central   Google Scholar  

Jensen TM, Metz AJ, Farley AB, Disbennett ME. Developing a practice-driven research agenda in implementation science : Perspectives from experienced implementation support practitioners. Published online. 2023. https://doi.org/10.1177/26334895231199063 .

Iwelunmor J, Blackstone S, Veira D, et al. Toward the sustainability of health interventions implemented in sub-Saharan Africa: a systematic review and conceptual framework. Implement Sci. 2016;11(1). https://doi.org/10.1186/s13012-016-0392-8 .

Baptiste S, Manouan A, Garcia P, Etya’ale H, Swan T, Jallow W. Community-led monitoring: when community data drives implementation strategies. Curr HIV/AIDS Rep. 2020;17(5):415–21. https://doi.org/10.1007/s11904-020-00521-2 .

Anderson KA, Dabelko-Schoeny H, Koeuth S, Marx K, Gitlin LN, Gaugler JE. The use of community advisory boards in pragmatic clinical trials: The case of the adult day services plus project. Home Health Care Serv Q. 2021;40(1):16–26. https://doi.org/10.1080/01621424.2020.1816522 .

Ramanadhan S, Davis M, Donaldson ST, Miller E, Minkler M. Participatory Approaches in Dissemination and Implementation Science. In: Brownson RC, Colditz GA, Proctor EK, editors. Dissemination and Implementation Research in Health: Translating Science to Practice. New York: Oxford University Press; 2023. p. 212.

Chapter   Google Scholar  

Ricalde MCA, Annoni J, Bonney R, et al. Understanding the Impact of Equitable Collaborations between Science Institutions and Community-Based Organizations: Improving Science through Community-Led Research. Bioscience. 2022;72(6):585–600. https://doi.org/10.1093/biosci/biac001 .

Fernandez ME, Ten Hoor GA, van Lieshout S, et al. Implementation mapping: using intervention mapping to develop implementation strategies. Front Public Heal. 2019;7(JUN):1–15. https://doi.org/10.3389/fpubh.2019.00158 .

National Institutes of Health. All of Us: Local Community and/or Participant Advisory Boards (C/PABs). 2023. Accessed 13 Sept 2023. https://allofus.nih.gov/about/who-we-are/all-us-community-and-participant-advisory-boards .

Brownson RC, Kumanyika SK, Kreuter MW, Haire-Joshu D. Implementation science should give higher priority to health equity. Implement Sci. 2021;16(1):28. https://doi.org/10.1186/s13012-021-01097-0 .

Morris ZS, Wooding S, Grant J. The answer is 17 years, what is the question: understanding time lags in translational research. J R Soc Med. 2011;104(12):510–20. https://doi.org/10.1258/jrsm.2011.110180 .

Bührmann L, Driessen P, Metz A, et al. Knowledge and attitudes of implementation support practitioners—findings from a systematic integrative review. PLoS One. 2022;17(5 May):1–25. https://doi.org/10.1371/journal.pone.0267533 .

Article   CAS   Google Scholar  

Albers B, Metz A, Burke K. Implementation support practitioners- A proposal for consolidating a diverse evidence base. BMC Health Serv Res. 2020;20(1):1–10. https://doi.org/10.1186/s12913-020-05145-1 .

PCORI dissemination and implementation funding initiatives. Accessed 29 Feb 2024. https://www.pcori.org/impact/putting-evidence-work/pcori-dissemination-and-implementation-funding-initiatives .

United States Agency for International Development. USAID’s Implementation Science Investment. Accessed 29 Feb 2024. https://www.usaid.gov/fact-sheet/usaids-implementation-science-investment .

National Heart, Lung and BI. Implementation Science Branch. Accessed 29 Feb 2024. https://www.nhlbi.nih.gov/about/translation-research-and-implementation-science/implementation-science .

Zurynski Y, Smith CL, Knaggs G, Meulenbroeks I. Funding research translation: how we got here and what to do next. Aust N Z J Public Health. 2021;45(5):420–3. https://doi.org/10.1111/1753-6405.13131 .

Holmes B, Hamilton AB. Three opportunities to boost implementation science at a critical time of need. Heal Published online. 2021. https://doi.org/10.18865/ed.29.S1.77 .

Planning team for the Pathways to Prevention (P2P) Workshop on Achieving Health Equity in Preventive Services and the Office for Disease Prevention portfolio analysis team. We Need More Implementation Science To Improve Health Equity in Clinical Preventive Services. Director’s Messages. https://prevention.nih.gov/about-odp/directors-messages/2022/we-need-more-implementation-science-improve-health-equity-clinical-preventive-services . Published October 14, 2022. Accessed 29 Feb 2024.

Implementation Science Takes Off at Brown. 2023. Accessed 29 Feb 2024. https://psych.med.brown.edu/news/2023-10-02/bridge-growth .

Davis R, D’Lima D. Building capacity in dissemination and implementation science: a systematic review of the academic literature on teaching and training initiatives. Implement Sci. 2020;15(1):97. https://doi.org/10.1186/s13012-020-01051-6 .

Boyce CA, Barfield W, Curry J, et al. Building the next generation of implementation science careers to advance health equity. 2019;29:77–82. https://doi.org/10.18865/ed.29.S1.77 .

Osanjo GO, Oyugi JO, Kibwage IO, et al. Building capacity in implementation science research training at the University of Nairobi. Implement Sci. 2016;11(1):1–9. https://doi.org/10.1186/S13012-016-0395-5 .

Straus SE, Ma JT, Graham I. Defining knowledge translation. Review. 2009;181:165–8. https://doi.org/10.1503/cmaj.081229 .

National Center for Advancing Translational Science. Transforming Translational Science. https://ncats.nih.gov/files/NCATS-factsheet.pdf . Accessed 24 Mar 2020. 2017;Fall.

Proctor E, Silmere H, Raghavan R, et al. Outcomes for implementation research: Conceptual distinctions, measurement challenges, and research agenda. Adm Policy Ment Heal Ment Heal Serv Res. 2011;38(2):65–76. https://doi.org/10.1007/s10488-010-0319-7 .

Damschroder LJ, Aron DC, Keith RE, Kirsh SR, Alexander JA, Lowery JC. Fostering implementation of health services research findings into practice: A consolidated framework for advancing implementation science. Implement Sci. 2009;4(1):1–15. https://doi.org/10.1186/1748-5908-4-50 .

Aarons GA, Hurlburt M, Horwitz SMC. Advancing a conceptual model of evidence-based practice implementation in public service sectors. Adm Policy Ment Heal Ment Heal Serv Res. 2011;38(1):4–23. https://doi.org/10.1007/s10488-010-0327-7 .

Tabak RG, Khoong EC, Chambers DA, Brownson RC. Bridging research and practice: Models for dissemination and implementation research. Am J Prev Med. 2012;43(3):337–50. https://doi.org/10.1016/j.amepre.2012.05.024 .

Walsh-Bailey C, Tsai E, Tabak RG, et al. A scoping review of de-implementation frameworks and models. Implement Sci. 2021;16(1):1–18. https://doi.org/10.1186/s13012-021-01173-5 .

Nilsen P, Bernhardsson S. Context matters in implementation science: A scoping review of determinant frameworks that describe contextual determinants for implementation outcomes. BMC Health Serv Res. 2019;19(1):1–21. https://doi.org/10.1186/s12913-019-4015-3 .

Powell BJ, Waltz TJ, Chinman MJ, et al. A refined compilation of implementation strategies: Results from the Expert Recommendations for Implementing Change (ERIC) project. Implement Sci. 2015;10(1):1–14. https://doi.org/10.1186/s13012-015-0209-1 .

Shea CM, Jacobs SR, Esserman DA, Bruce K, Weiner BJ. Organizational readiness for implementing change: A psychometric assessment of a new measure. Implement Sci. 2014;9(1):1–15. https://doi.org/10.1186/1748-5908-9-7 .

Finch TL, Girling M, May CR, et al. Improving the normalization of complex interventions : part 2 - validation of the NoMAD instrument for assessing implementation work based on normalization process theory ( NPT ). BMC Med Res Methodol. 2018;18(135):1–13.

Google Scholar  

Desveaux L, Nguyen MD, Ivers NM, et al. Snakes and ladders: A qualitative study understanding the active ingredients of social interaction around the use of audit and feedback. Transl Behav Med. 2023;13(5):316–26. https://doi.org/10.1093/tbm/ibac114 .

Edwards N, Barker PM. The importance of context in implementation research. J Acquir Immune Defic Syndr. 2014;67:S157–62. https://doi.org/10.1097/QAI.0000000000000322 .

Gray CE, Mcintyre KP, Mattingly BA, Jr GWL. Interpersonal Relationships and the Self-Concept. Springer International Publishing; 2020.  https://doi.org/10.1007/978-3-030-43747-3 .

Wu P, Liu T, Li Q, Yu X, Liu Z, Tian S. Maintaining the working state of firefighters by utilizing self-concept clarity as a resource. BMC Public Health. 2024;24(1):1–11. https://doi.org/10.1186/s12889-024-17896-1 .

Balundė A, Paradnikė K. Resources linked to work engagement: the role of high performance work practices, employees’ mindfulness, and self-concept clarity. Soc Inq into Well-Being. 2016;2(2):55–62. https://doi.org/10.13165/SIIW-16-2-2-06 .

Harvey G, Rycroft-Malone J, Seers K, et al. Connecting the science and practice of implementation – applying the lens of context to inform study design in implementation research. Front Heal Serv. 2023;3(July):1–15. https://doi.org/10.3389/frhs.2023.1162762 .

Tabak RG, Padek MM, Kerner JF, et al. Dissemination and Implementation Science Training Needs: Insights From Practitioners and Researchers. Am J Prev Med. 2017;52(3):S322–9. https://doi.org/10.1016/j.amepre.2016.10.005 .

Atkins BR, Allred S, Hart D. Philanthropy’s Rural Blind Spot. Stanford Soc Innov Rev. Published online 2021. https://tableau.dsc.umich.edu/t/UM-Public/views/IndexofDeepDisadvantage/CountiesCitiesMap?:isGuestRedirectFromVizportal=y&:embed=y .

Stirman SW, Baumann AA, Miller CJ. The FRAME: An expanded framework for reporting adaptations and modifications to evidence-based interventions. Implement Sci. 2019;14(1):1–10. https://doi.org/10.1186/s13012-019-0898-y .

Miller CJ, Barnett ML, Baumann AA, Gutner CA, Wiltsey-Stirman S. The FRAME-IS: a framework for documenting modifications to implementation strategies in healthcare. Implement Sci. 2021;16(1):36. https://doi.org/10.1186/s13012-021-01105-3 .

Boaz A. Lost in co-production: To enable true collaboration we need to nurture different academic identities . LSE; 2021. p. 1–4. https://blogs.lse.ac.uk/impactofsocialsciences/2021/06/25/lost-in-co-production-to-enable-truecollaboration-we-need-to-%0Anurture-different-academic-identities/ .

Seifer S. Walking the Talk: Achieving the Promise of Authentic Partnerships. Partnersh Perspect. 2007;IV(I):1–12.

Ramanadhan S, Davis MM, Armstrong R, et al. Participatory implementation science to increase the impact of evidence-based cancer prevention and control. Cancer Causes Control. 2018;29(1):363–9. https://doi.org/10.1007/s10552-018-1008-1 .

Shelton RC, Brownson RC. Enhancing Impact: A Call to Action for Equitable Implementation Science. Prev Sci Published online. 2023. https://doi.org/10.1007/s11121-023-01589-z .

International AIDS Society. Home Page. Accessed 2 Oct 2023. https://www.iasociety.org .

Cahn P, McClure C. Beyond the first 25 years: The International AIDS Society and its role in the global response to AIDS. Retrovirology. 2006;3(1):2004–6. https://doi.org/10.1186/1742-4690-3-85 .

Kort R. 5th International AIDS Society Conference on HIV Pathogenesis, treatment and prevention: summary of key research and implications for policy and practice - operations research. J Int AIDS Soc. 2010;13(SUPPL. 1):1–6. https://doi.org/10.1186/1758-2652-13-S1-S5 .

Gayle H, Wainberg MA. Impact of the 16th International Conference on AIDS: can these conferences lead to policy change? Retrovirology. 2007;4:2–3. https://doi.org/10.1186/1742-4690-4-13 .

IHI Marks 30 Years of Quality Improvement in Health Care Worldwide. BusinessWire. https://www.businesswire.com/news/home/20211028005209/en/IHI-Marks-30-Years-of-Quality-Improvement-in-Health-Care-Worldwide . Published 28 Oct 2021.

Elwood WN, Corrigan JG, Morris KA. NIH-Funded CBPR: self-reported community partner and investigator perspectives. J Community Health. 2019;44(4):740–8. https://doi.org/10.1007/s10900-019-00661-6 .

Teufel-Shone NI, Schwartz AL, Hardy LJ, et al. Supporting new community-based participatory research partnerships. Int J Environ Res Public Health. 2019;16(1):1–12. https://doi.org/10.3390/ijerph16010044 .

Minkler M, Blackwell AG, Thompson M, Tamir HB. Community-based participatory research: implications for public health funding. Am J Public Health. 2003;93(8):1210–3. https://doi.org/10.2105/AJPH.93.8.1210 .

Download references

Acknowledgements

We would like to thank Drs. Cory Bradley and Donny Gerke for their contributions in early conceptualization of this paper and to colleagues who took the time to review and provide feedback prior to submission.

GB was supported by the National Institute of Mental Health grant T32MH019960 at Washington University (PI: Leopoldo J. Cabassa) during a portion of manuscript development. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Mental Health.

Author information

Gretchen J. R. Buchanan, Lindsey M. Filiatreau and Julia Moore are co-first authors.

Authors and Affiliations

Department of Family Medicine and Community Health, Hennepin Healthcare Research Institute, Minneapolis, MN and University of Minnesota Medical School, MN, Minneapolis, USA

Gretchen J. R. Buchanan

Division of Infectious Diseases, School of Medicine, Washington University in St. Louis, MO, St. Louis, USA

Lindsey M. Filiatreau

The Center for Implementation, ON, Toronto, Canada

Julia E. Moore

You can also search for this author in PubMed   Google Scholar

Contributions

GB, LF, and JM equally participated in the conception, drafting, and revising of the manuscript, and they have approved the manuscript as submitted. GB, LF, and JM agree to be personally accountable for their own contributions and to ensure that questions related to the accuracy or integrity of any part of the work, even ones in which the author was not personally involved, are appropriately investigated, resolved, and the resolution documented in the literature.

Corresponding author

Correspondence to Gretchen J. R. Buchanan .

Ethics declarations

Ethics approval and consent to participate, consent for publication, competing interests.

Author JM is the Director of The Center for Implementation and previously led the implementation team at the Knowledge Translation Program, St. Michael’s Hospital. Several examples are drawn from direct experience in these roles. LF and GB declare that they have no competing interests.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . The Creative Commons Public Domain Dedication waiver ( http://creativecommons.org/publicdomain/zero/1.0/ ) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

Reprints and permissions

About this article

Cite this article.

Buchanan, G.J.R., Filiatreau, L.M. & Moore, J.E. Organizing the dissemination and implementation field: who are we, what are we doing, and how should we do it?. Implement Sci Commun 5 , 38 (2024). https://doi.org/10.1186/s43058-024-00572-1

Download citation

Received : 10 November 2023

Accepted : 20 March 2024

Published : 11 April 2024

DOI : https://doi.org/10.1186/s43058-024-00572-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Implementation science
• Translational science
• Translational science spectrum
• Translational research continuum
• Evidence to practice gap
• Implementation support practitioners

Implementation Science Communications

ISSN: 2662-2211

• Submission enquiries: Access here and click Contact Us
• General enquiries: [email protected]

Cluster 60W: Technological Innovation through Art, Culture, and Society (Shababi) Spring 24

• Academic Texts/Secondary Sources
• Primary Sources
• Citing Your Sources

Sciences and Technology Studies

Humanities and arts, interdisciplinary databases.

• UC Library Search This link opens in a new window This is a way to search across almost all our databases at once! You can use subject filters on the left to include or exclude different subjects, and the resource type filter to choose the type of source (newspaper article vs. book, etc.)

Article Search Tips

• Finding articles
• Citing sources and avoiding plagiarism
• Breaking down academic articles
• Finding review articles
• Off-campus access

This quick introduction to searching for academic journal articles in databases will help you make your searches more efficient and more effective:

Avoiding plagiarism can be more tricky than it seems at first glance. Complete this activity to learn strategies for avoiding plagiarism.

Few of us can recall every detail and argument from academic articles - they’re packed with so much information! It’s best to create reading strategies that help you focus on comprehending the most essential elements of an academic article.  Breaking Down Academic Articles  is a webcomic tutorial that walks you through the most important aspects of an academic article, to help you prepare for class discussion on the article(s) or take away essential information from the article to support future research.

Note:  a transcript for each slide can be located at the bottom of each slide. Click the button to preview transcript of slide for description of art in each panel and the script of tutorial guide.

Looking for an efficient way to get a research overview on a topic? A review article is a great place to start.

A review article provides an analysis of the state of research on a set of related research questions. Review articles often: summarize key research findings, reference must-read articles, describe current areas of agreement as well as controversies and debates, point out gaps in knowledge and unanswered questions, suggest directions for future research.

Check out this quick overview of finding review articles in Web of Science, PubMed, Google Scholar, and more.

The Library's online subscription resources can always be accessed from computers and wireless networks on campus. However, off-campus access is restricted to current UCLA, students, faculty, and staff who have set up their computer using one of the following methods.

VPN on Mac with Natalie

"I chose VPN for my mac because I need to be able to access the full text of articles on different browsers." - Natalie, Environmental Science

VPN on Windows with Michael

"I chose VPN because I like the security it provides and the control it gives me as a user to manually enable or disable it when I'm browsing online." - Michael, Public Affairs

Additional Platforms

• Proxy for Safari on a Mac with Kate

For more information:

• UCLA OnLine Proxy Server A simple browser setting which will automatically divert you to a UCLA logon page when you first access a restricted site.
• Virtual Private Networking (VPN) A program you can download and install, or use the built-in version on many computers or mobile devices. VPN software must be logged on manually before you access a restricted site, but works for all programs on your computer.
• << Previous: Home
• Next: Primary Sources >>
• Last Updated: Apr 15, 2024 8:12 AM
• URL: https://guides.library.ucla.edu/shababi24

The Scholarly Kitchen

What’s Hot and Cooking In Scholarly Publishing

Honest Signaling and Research Integrity

• Data Publishing
• Metrics and Analytics
• Research Integrity

Editor’s Note: Today’s post is coauthored by Tim Vines and Ben Kaube. Ben is a cofounder of Cassyni, a platform for helping publishers and institutions create and engage communities of researchers using seminars.

Assessing research integrity issues for articles submitted to an academic journal is a daunting prospect: the journal must somehow use the submitted text, figures, and (if available) accompanying data to uncover unethical behavior by the authors. This behavior (such as fabricating or altering data, etc.) may have taken place long before the article itself was written. 

However, these methods primarily focus on detecting bad behavior. To complement this approach, there’s a compelling need for indicators of “honest signaling” in research — a concept that, instead of aiming to catch bad behavior, captures commitment to ethical research practices.

It is instructive to draw analogy to the evolutionary biology concept known as “honest signaling.” This principle illustrates how certain traits in the animal kingdom, such as the peacock’s extravagant tail or the bright colors of some jumping spiders, serve as ‘honest’ markers of the individual’s fitness . These traits are energetically costly to produce and maintain, signifying that only individuals in prime condition can afford these displays. Thus, these signals are a reliable indicator of an individual’s fitness and hence their quality as a mate.

Translating this concept to the realm of research integrity, we seek analogous signals within researcher behavior — practices that are sufficiently demanding of time, effort, or resources, such that they are unlikely to be undertaken by those not genuinely committed to ethical research. Honest signaling in this context would encompass activities that are both visible and verifiable, shedding light on the researcher’s dedication to transparency and integrity:

• Open Science Practices: This includes the sharing of raw data, pre-registration of studies, publishing of preprints, and sharing of the code and scripts to operate command line software. These practices not only require a significant investment of time and resources but also open the researcher’s work to scrutiny and verification by the wider community.
• Community Engagement: Active participation in the scientific community through presentations at conferences, seminars, and poster sessions acts as a form of honest signaling. Such engagements are not only resource-intensive, but also expose one’s research to critique and validation by peers. Moreover, maintaining a visible online presence (e.g., personal webpages and ORCID profiles) and contributions to journal publishing outside of authorship roles (e.g., writing peer reviews and membership on editorial boards) further exemplifies a researcher’s commitment to open discourse and collaboration.

These signals, akin to the peacock’s tail, are not just markers of the quality and credibility of the individual’s work but also serve to signal their commitment to the principles of research integrity.

Any indicators of honest signaling should be considered contextually, acknowledging that open research practices differ between fields and engagement with scholarly communities varies regionally. This is particularly true when considering researchers facing systemic barriers such as those in the Global South. This suggests that simple scores of honest signaling will be overly reductive and that honest signaling indicators should be evaluated qualitatively as part of the editorial process.

The continued growth of digital platforms and the increase in online data post-COVID offer new opportunities to operationalize honest signaling indicators. By aggregating data from society meetings, departmental web pages, and conference programs, it becomes possible to capture engagement of researchers within their respective fields at scale and produce a suite of indicators for research integrity purposes (e.g., what is the publication record of author X? have they presented similar work at relevant conferences or seminar series? Does the manuscript have accompanying data or code?).

One important consideration when aggregating data from many venues is the difficulty of author disambiguation. ORCID has an important role to play here , but the usual challenge of preventing false positives is compounded by the fact that many research outputs associated with honest signaling have traditionally lacked persistent identifiers and high quality metadata. For example posters and seminar recordings have only recently started receiving persistent identifiers. 

In biology, honest signals are an indicator of an individual’s fitness, and while some of those signals – like the peacock’s tail – might be eye-catching to us, they are fine tuned to be evaluated by individuals in the same species. Similarly, in the world of research, it will always be the experts in one’s own field that will be best positioned to pick up on and assess honest signaling.

The concept of honest signaling offers a promising framework for verifying research integrity in a manner that complements existing tools aimed at detecting misconduct. By focusing on behaviors that are costly in terms of time and effort and indicate a researcher’s commitment to ethical practices, this approach not only helps identify instances of integrity but also encourages a culture of openness and engagement within the scientific community. Operationalizing indicators of “honest signaling”, while challenging, presents an opportunity to foster transparency and accountability in academic research, ensuring that it remains a pursuit marked by ethical conduct and genuine discovery.

Tim Vines is the Founder and Project Lead on DataSeer, an AI-based tool that helps authors, journals and other stakeholders with sharing research data. He's also a consultant with Origin Editorial, where he advises journals and publishers on peer review. Prior to that he founded Axios Review, an independent peer review company that helped authors find journals that wanted their paper. He was the Managing Editor for the journal Molecular Ecology for eight years, where he led their adoption of data sharing and numerous other initiatives. He has also published research papers on peer review, data sharing, and reproducibility (including one that was covered by Vanity Fair). He has a PhD in evolutionary ecology from the University of Edinburgh and now lives in Vancouver, Canada.

Ben Kaube is a cofounder of Cassyni, a platform for helping publishers and institutions create and engage communities of researchers using seminars.

3 Thoughts on "Honest Signaling and Research Integrity"

Tim, you provide an interesting thought piece, but I think that you’ve pushed “honest signaling” way past its original meaning and conflated several practical and operational issues of journal publishing into the amorphous concept of “honesty.”

First, the concept of “honest signaling” in biology is contestable, because one needs to ascribe intent behind the action. Is the peacock being “honest” when displaying his tail, or just showing his sexual fitness? And what about all of the examples of mimicry and false signals in the biological world? Is a Viceroy butterfly being “dishonest” by mimicking the pattern of the toxic monarch butterfly so it won’t be a bird’s lunch or is it just being strategic?

Second, you list a number of things that authors can do to increase “honest signals” in publishing, such as depositing datasets, preprints, posters, etc. “Transparency” would be a better, more accurate term here than the amorphous construct of “honesty.” Similarly, we could use the term “longevity” to describe how many papers have been published previously by an author; “peer validation” to describe how colleagues view this author; “merit” to describe the institutions (labs, universities, funders) that vouch for that individual, etc. Honesty is not a yes/no checkbox–I’m not accusing you of writing this–but the logic of your data collection leads an evaluator to make an honesty evaluation.

Like citations or alt-metrics, I fear that your intent on collating these “honesty signals” and using them in the manuscript decision-making process will be counter-productive. Eugene Garfield kept arguing until his death that citations and Impact Factor scores should not be used alone and only in context. Look how far honesty got him.

• By Phil Davis
• Apr 16, 2024, 9:23 AM
• Reply to Comment

Nice post. Could adequate methods detail such as presence of RRIDs and a reasonable explanation of the limitations of the study not be a quality signal?

As you know these are nontrivial additions to the paper, that signal quality and integrity. Cell line RRID inclusion is associated with a reduction in the use of problematic cell lines as we showed in 2019 (Babic paper).

• By Anita Bandrowski
• Apr 16, 2024, 9:35 AM

Excellent essay. I must admit I liked it in part because it fits my view that research integrity and reliability is enhanced through greater transparency, not by hiding honest signaling or the lack thereof behind double-blind reviewing. Your essay seems to target editorial offices to look for honest signaling, but I think a large part also falls to the peer reviewers. The argument for blinding reviewers to signaling, honest or otherwise, is that blindness prevents status bias wherein manuscripts from Prof. Big Name from Elite University are welcomed in the Journal of High Impact Factors where author No Name from Southern Latitude University is not. I’m not sure what the best antidote is for the status bias issue, but I don’t think hiding honest signaling from reviewers is it.

• By Chris Mebane
• Apr 16, 2024, 9:41 AM

Leave a Comment Cancel reply

Notify me of follow-up comments by email.

Related Articles:

Next Article:

• Search Menu
• Advance articles
• Author Guidelines
• Submission Site
• Open Access
• About International Mathematics Research Notices
• Editorial Board
• Advertising and Corporate Services
• Journals Career Network
• Self-Archiving Policy
• Dispatch Dates
• Journals on Oxford Academic
• Books on Oxford Academic

Article Contents

1 introduction, 2 preliminaries, 3 proof of the main theorem for deformed hermitian yang–mills and the z-critical equations, 4 example of the wall-chamber decomposition: blowup of |$\mathbb{p}^{2}$| in two points, 5 the special case of the j-equation: finite set of optimal destabilizers in the small volume limit and applications, 6 flow aspects, acknowledgments.

• < Previous

The Set of Destabilizing Curves for Deformed Hermitian Yang–Mills and Z-Critical Equations on Surfaces

• Article contents
• Figures & tables
• Supplementary Data

Sohaib Khalid, Zakarias Sjöström Dyrefelt, The Set of Destabilizing Curves for Deformed Hermitian Yang–Mills and Z-Critical Equations on Surfaces, International Mathematics Research Notices , Volume 2024, Issue 7, April 2024, Pages 5773–5814, https://doi.org/10.1093/imrn/rnad256

• Permissions Icon Permissions

We show that on any compact Kähler surface existence of solutions to the Z-critical equation can be characterized using a finite number of effective conditions, where the number of conditions is bounded above by the Picard number of the surface. This leads to a first PDE analogue of the locally finite wall-chamber decomposition in Bridgeland stability. As an application we characterize optimally destabilizing curves for Donaldson’s J-equation and the deformed Hermitian Yang–Mills equation, prove a non-existence result for optimally destabilizing test configurations for uniform J-stability, and remark on improvements to convergence results for certain geometric flows.

The question of characterizing optimally destabilizing test configurations is central in the study of K-stability and the circle of ideas surrounding the Yau–Tian–Donaldson conjecture in Kähler geometry. It is natural to expect that analogous concepts can be of interest also for other stability notions and other geometric PDE.

In this work we focus on the study of obstructions and existence criteria for the Z-critical equation introduced by Dervan–McCarthy–Sektnan [ 22 , 24 , 37 ] as a general framework extending and encompassing several geometric PDE of interest in Kähler geometry, including the important special case of the deformed Hermitian Yang–Mills (dHYM) equation (see [ 15 ] for an introduction and [ 13 ] for a survey in the context of algebro-geometric stability and references therein). The latter is moreover closely related to the J-equation introduced by Donaldson [ 26 ], which is realized as a “small volume limit” in the dHYM setting, while at the same time being closely related to the constant scalar curvature equation and associated stability notions (e.g., K-stability). Existence criteria for the J-equation have been studied in a large body of work (see, for example, [ 9 , 14 , 35 , 46 ]) culminating in [ 7 ] and [ 16 , 47 ] with a characterization of existence in terms of uniform J-stability, mirroring the Yau–Tian–Donaldson conjecture, and relating stability to analytic conditions tested on subvarieties, as conjectured by Lejmi-Székelyhidi [ 35 ]. Likewise, a number of similar results are known for the dHYM equation [ 7 , 11 ]. A further and key aspect of the Z-critical equation is that it aims to highlight connections to Bridgeland stability [ 5 ], which has generated a lot of interest due to its conjecturally far-reaching connections with birational geometry and mirror symmetry. Indeed, their approach closely mirrors Bayer’s notion of a polynomial central charge [ 1 ].

In what follows, let # denote any of the J-equation, deformed Hermitian Yang–Mills (dHYM) equation, or Z-critical equation (note that we single out the dHYM equation to emphasize certain aspects specific to this much studied special case).

Verify the existence of a solution to # via a finite number of effective/numerical conditions.

The present work deals with Question 1 in the case of Kähler surfaces, providing first results in the direction of the J-equation, dHYM equation and Z-critical equation, which have the advantage that they all can be studied using obstructions arising from subvarieties, as opposed to those coming from general test configurations. A secondary intention is moreover for this paper to serve as a “base case” in an induction on dimension argument in forthcoming work, illustrating wall-chamber decompositions for Z-critical equations in higher dimension.

A main goal of this work is to improve on these existence criteria. We will give precise statements in each of the respective cases of the J-equation, dHYM equation and Z-critical equation. However, they all fall under the structure of the following “meta theorem”:

  Theorem 1.1 (Meta theorem). Let |$X$| be a compact Kähler surface and fix a compact subset |$K$| (of a suitable subset of) of |$V\times H^{1,1}(X,{\mathbb{R}})$|⁠ . Then there exists a set of finitely many curves |$\mathcal V_{K} = \left \{E_{1},\cdots , E_{\ell }\right \}$| on |$X$| depending only on |$K$| and # such that the equation # with underlying initial data and topological constraint |$(d_{\#},\alpha ) \in K$| is solvable if and only if the following finitely many conditions hold: $$\begin{align*} & \int_{E_i} \tau_{\#}(d_{\#},\alpha)> 0, \ \ \textrm{ for all } i = 1,2,\dots,\ell \end{align*}$$ All the curves |$E_{i} \in \mathcal{V}_{K}$| have negative self-intersection, and if moreover the image of |$K$| under |$\tau _\#$| is contained in the convex hull of |$k$| pseudoeffective classes |$\alpha _{j} \in H^{1,1}(X,\mathbb{R})$|⁠ , then the cardinality |$|\mathcal{V}_{K}|$| of the set of “test curves” is bounded above by |$k\rho (X)$|⁠ . If |$X$| is moreover projective, then the cardinality of |$\mathcal V_{K}$| is bounded above by |$k\rho (X)-k$|⁠ .

(1) In other words, we prove the existence of a finite set of “destablizing curves” to the numerical criteria [ 7 , 11 , 14 , 16 , 47 ], such that the set of curves can moreover be taken uniform across compact sets of initial data. Note that this is a novelty already in the much studied special case of the dHYM equation.

(2) The hypotheses imposed are precisely the natural ones studied in the general literature for both the dHYM and Z-critical equations, and the “meta theorem” covers these cases of interest. More precisely, the above hypotheses hold automatically when specializing to the dHYM equation, and arise naturally from the point of view of “subsolutions” in the more general setting of the Z-critical equation (see [ 24 ]).

(3) A main novelty in the strategy of proof is the replacement of Siu decomposition (e.g., as used by Song-Weinkove [ 46 ]) by Zariski decomposition, which turns out to offer several advantages. In particular, it makes certain aspects of previous results for geometric flows more precise (in Section 6 we discuss the J-flow, line bundle mean curvature flow, and dHYM flow from this point of view).

(4) In the case of the Z-critical equation Theorem 1.1 can be interpreted as a first PDE analogue of locally finite wall-chamber decompositions in Bridgeland stability. This is a main motivation behind the above result, as elaborated upon in Section 1.2 .

(5) The main technical simplifications that allow the strategy of proof to work on surfaces, is first the reduction to a complex Monge–Ampére equation, and second, that intersection theory takes a particularly simple form in dimension |$2$|⁠ . It would, however, be natural to conjecture a similar statement in higher dimension |$n \geq 3$|⁠ , based on generalizations of Yau’s solution of the Calabi conjecture [ 52 ], the characterisation of the Kähler cone due to Demailly-Paun [ 20 ], and the Zariski decomposition of arbitrary pseudoeffective classes on compact complex manifolds due to Boucksom [ 3 ].

If |$X$| is a compact Kähler surface with no negative curves, then the equation # admits a solution for any admissible initial data |$(d_{\#},\alpha )$| (see Corollary 1.6 and Corollary 1.12 for the precise statements) .

1.1 Main results for the deformed Hermitian Yang–Mills equation

(1) The deformed Hermitian Yang–Mills equation ( 1 ) admits a solution with respect to |$(\beta ,\alpha ) \in K$|⁠ .

• (2) For all irreducible curves |$E \subseteq X$| we have $$\begin{align*} & \int_E \alpha + \cot(\hat\Theta(\beta,\alpha))\beta>0. \end{align*}$$
• (3) For |$i= 1, \cdots , \ell $| we have $$\begin{align*} & \int_{E_i} \alpha + \cot(\hat\Theta(\beta,\alpha))\beta>0. \end{align*}$$

The new result here is |$(2) \Leftrightarrow (3)$| stating that it is enough to test the numerical condition |$(2)$| on a finite set of “test curves”, where moreover this set of curves is completely uniform of the choice of |$(\alpha ,\beta ) \in K$| across any given compact subset |$K$|⁠ . This is a consequence of our proofs using Zariski decomposition, rather than Siu decomposition of currents. In fact, in the case of the dHYM equation the set of curves can moreover be taken independent of the scaling |$\alpha \mapsto k\alpha $| as |$k \rightarrow +\infty $|⁠ , so that the same “test curves” can be used for all rescalings of the dHYM equation including its small volume limit, the J-equation, as explained in Section 5.2 . The equivalence |$(1) \Leftrightarrow (2)$| is due to [ 33 ].

Suppose that |$X$| is a compact Kähler surface with no curves of negative self-intersection. Then the deformed Hermitian Yang–Mills equation can always be solved for all pairs |$(\beta ,\alpha )$| of Kähler classes and every Kähler form |$\theta \in \beta $|⁠ .

  Corollary 1.7. In the setup of Theorem 1.4 , let $$\begin{align*} & \widetilde{c_{\beta,\alpha}}:= \frac{\int_X(\alpha^2 - \beta^2)}{2\int_X\alpha\cdot\beta} \in \mathbb{R} \end{align*}$$ and $$\begin{align*} & \tau_{\textrm{dHYM}}(\beta,\alpha):= \alpha - \widetilde{c_{\beta,\alpha}}\beta \in H^{1,1}(X,\mathbb{R}). \end{align*}$$ Consider moreover the set $$\begin{align*} & \mathcal{U}:= \left\{(\beta,\alpha) \in \mathcal{C}_X \times \mathcal{C}_X: \textrm{{The dHYM equation (1) is solvable with respect to} } (\beta,\alpha). \right\} \end{align*}$$ of |$(\beta ,\alpha ) \in \mathcal{C}_{X} \times \mathcal{C}_{X}$| such that equation ( 1 ) admits a solution |$\omega \in \alpha $|⁠ . Fixing any compact set |$K \subseteq \mathcal{C}_{X} \times \mathcal{C}_{X}$|⁠ , the restriction |$\mathcal{U}_{\vert K}$| is cut out by a finite number of open conditions depending only on |$K$|⁠ , namely $$\begin{align*} & \mathcal{U}_{\vert K} = \left\{(\beta,\alpha) \in K: \int_{E_i} \tau_{\textrm{dHYM}}(\beta,\alpha)> 0, i = 1, \cdots, \ell\right\}, \end{align*}$$ where the curves |$E_{i}$| are those appearing in Theorem 1.4 . In particular, the boundary |$\partial \mathcal{U}_{\vert K}$| is defined by $$\begin{align*} & \partial\mathcal{U}_{\vert K} = \left\{(\beta,\alpha) \in K: \prod_{i = 1}^\ell \int_{E_i} \tau_{\textrm{dHYM}}(\beta,\alpha) = 0 \right\}, \end{align*}$$ which forms a real algebraic set in |$K$| of real codimension one, giving rise to a locally finite wall-chamber decomposition of the space of dHYM equations.

Note that this immediately implies that dHYM-positivity (and thus solvability of the dHYM-equation) is an open condition under perturbation of the underlying data |$(\beta ,\alpha ) \in \mathcal{C}_{X} \times \mathcal{C}_{X}$|⁠ . Such openness would already follow from general theory, since the dHYM equation reduces to a complex Monge–Ampère equation. Nonetheless, the above Corollary 1.7 makes this openness rather explicit.

1.2 Main results for the Z-critical equation and a wall-chamber decomposition in the spirit of Bridgeland

The following main result is stated under the full generality of the positive volume condition:

(1) There exists a choice of lifts of |$\Omega $| such that the line bundle |$L \to X$| admits a solution to the Z-critical equation ( 2 ).

• (2) The class $$\begin{align*} & \tau(\Omega,L):= s\left(c_1(L) + \frac{1}{2}\eta(\Omega,L)\right)\end{align*}$$ satisfies $$\begin{align*} & \int_{E_i} \tau(\Omega,L)> 0 \quad \forall i = 1,\cdots, \ell,\end{align*}$$ where |$s$| is the sign of the non-zero real number $$\begin{align*} &\int_X \left(c_1(L)+\frac{1}{2}\eta(\Omega,L)\right)\cdot \beta \in{\mathbb{R}}.\end{align*}$$

Let |$X$| be a compact Kähler surface and |$S$| any finite set of holomorphic line bundles on |$X$|⁠ . Let |$V_{S}$| denote the set of stability data |$\Omega $| that have positive volume at each |$L \in S$|⁠ . Then, |$V_{S}$| admits a locally finite wall-chamber decomposition.

The analogous statement holds more generally when |$L$| is replaced by an arbitrary |$(1,1)$| -cohomology class |$\alpha \in H^{1,1}(X,{\mathbb{R}})$|⁠ . In that case, one replaces |$\textrm{ch}(L,h)$| in equation ( 2 ) with |$\exp (\chi ):= 1 + \chi + \frac{1}{2}\chi ^{2}$|⁠ , where |$\chi \in \alpha $| is the sought-after (1,1) form which solves the equation.

Let |$X$| and |$L$| be as in the Theorem 3.13 . Suppose |$X$| does not admit any curves of negative self-intersection. Let |$\Omega $| be a choice of stability data defining a polynomial central charge |$Z_{\Omega }$| and such that |$V(\Omega ,L)> 0$|⁠ . Then, for any lift |$\tilde \Omega $| satisfying the volume form hypothesis, the |$Z_{\Omega }$| -critical equation admits a solution on |$L$|⁠ .

Finally, let us remark that we hope to apply similar techniques in higher dimension in the future (especially in dimension |$3$|⁠ , where Pingali [ 38 ] has remarked on the close relationship between the dHYM equation and reductions to Monge-Ampére type equations). Such generalizations involve several additional hurdles related to the big cone and numerical properties of the Zariski decomposition and requires the development of further new techniques.

1.3 Optimal destabilizers for the J-equation and dHYM equation

Motivated by well-known (often conjectural) connections between existence of solutions to geometric PDEs and suitable algebraic or analytic stability notions, we also ask about optimal destabilizers to the respective equations above.

  Theorem 1.13. Suppose that |$([\theta ],[\omega ]) \in H^{1,1}(X,\mathbb{R})$| such that |$[\theta ]^{2} < [\omega ]^{2}$|⁠ . Then the infima $$\begin{align*} & \inf_{E \subseteq X} \left( \int_E \tau_{\textrm{dHYM}}([\theta],[\omega]) \right) \end{align*}$$ and $$\begin{align*} & \inf_{E \subseteq X} \left( \int_E \tau_{\textrm{J}}([\theta],[\omega]) \right) \end{align*}$$ are realized by the same finite set of curves $$\begin{align}& \left\{E \subseteq X \;: \int_{E} \textbf{a}([\theta],[\omega]) = 0 \right\},\end{align}$$ (3) where |$\mathbf{a}([\theta ],[\omega ]) \in H^{1,1}(X,\mathbb{R})$| is the |$(1,1)$| -class defined in Proposition 5.8 . In particular, if the sets of optimally J/dHYM-destabilizing curves are non-empty, then they coincide, and are explicitly given by ( 3 ). In other words, the numerical existence criteria for J and dHYM equations can be tested for the same finite set of curves. As an alternative point of view, this shows that the set of optimally dHYM-destabilizing curves can be chosen uniformly across any rescaling |$\alpha \mapsto k\alpha $|⁠ , |$k> 0$|⁠ , to the small volume limit, and these are precisely the curves that witness the “non-positivity” of the class |$\mathbf{a}([\theta ],[\omega ])$| in the Nakai–Moishezon criterion. We stress that while the sets of optimally J/dHYM-destabilizing curves coincide when non-empty, this does of course not imply the existence of any destabilizing curves, nor does it imply that the dHYM and J-equations are solvable at the same time (as in the existence criteria one integrates different |$(1,1)$| -forms |$\tau _{\textrm{J}}$| and |$\tau _{\textrm{dHYM}}$| over these same curves).

In the case of optimal destabilizers on the side of algebro-geometric stability, we show that the situation is quite different to that of curves. More precisely, building crucially on work of Hattori [ 31 ] we observe that optimally destabilizing test configurations (defined with respect to the minimum norm, see ( 4 )) never exist across rational classes:

Let |$X$| be a smooth projective surface and |$L,H$| |$\mathbb{Q}$| -line bundles on |$X$|⁠ . Then no optimally destabilizing normal and relatively Kähler test configuration exists for |$(X,L,H)$|⁠ .

(1) X admits at least one negative curve.

(2) There exist |$L$| and |$H$| ample line bundles on |$X$| such that |$(X,L)$| is |$J^{H}$| -stable but not uniformly |$J^{H}$| -stable.

The above sheds additional light on a recent example in [ 31 ], where they provided a polarized surface |$(X,L),$| which is J-stable but not uniformly. In fact, by Theorem 1.15 such examples exist in abundance (namely, precisely on any projective surface |$X$| carrying a curve of negative self-intersection). As shown in [ 31 , Corollary 7.5] such examples can be useful in constructing examples of K-stable but not uniformly K-stable polarized normal pairs, touching upon fundamental questions relating to the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics. Note, however, that Hattori’s construction applies to very singular surfaces so that the entropy term in the non-Archimedean Mabuchi functional vanishes. Producing examples of K-stable but not uniformly K-stable smooth surfaces appears to be very challenging and would likely require new ideas.

1.4 Flow aspects

As a final application of the main Theorem 1.1 we remark on improvements to certain results on singularities of the J-flow, the line bundle mean curvature flow, and a dHYM flow, due to Song-Weinkove [ 46 ], Takahashi [ 49 ], and Fu-Yau-Zhang [ 29 ], respectively. We refer the reader to Section 6 for the statements and proofs.

1.5 Outline of the paper

Section 2 deals with preliminaries.

Section 3 includes proofs of the main theorems for the dHYM equation (Theorem 3.7 ) and the Z-critical equation (Theorem 3.13 ). Note that the former is a consequence of the latter, but we write out both proofs for clarity of exposition. We also state Corollary 3.16 .

In Section 5 we first state and prove the main theorem (Theorem 5.3 ) in the special case of the J-equation, characterize the set of optimally destabilizing curves for the J-equation and the dHYM equation (Theorem 1.13 ), and prove non-existence (in general) of optimally destabilizing test configurations for the J-equation, which also involves comparing J-stability and uniform J-stabiity (Theorem 1.15 ).

In Section 4 we exemplify aspects of Corollary 3.16 for |$\mathbb{P}^{2}$| blown up in two points.

In Section 6 we finally apply the main theorem 1.1 to state and discuss improvements to certain results on singularity formation and convergence of geometric flows.

In this section we recall certain well-known and classical facts about the theory of positivity on Kähler surfaces. We will nevertheless state results in more generality than we need them, so that we can emphasise that the methods can be brought to bear upon problems in higher dimensions, though with numerous additional difficulties that would be interesting to address in future work.

2.1 Positive cones in cohomology

Let |$X$| be a compact Kähler manifold of dimension |$n$|⁠ . Recall that the Kähler cone |$\mathcal C_{X}$| is the convex open cone in |$H^{1,1}(X,{\mathbb{R}})$| comprising precisely those cohomology classes, called Kähler classes , which can be represented by a Kähler metric. The closure |$\mathcal N_{X}:= \overline{\mathcal C_{X}}$| of |$\mathcal C_{X}$| is called the nef cone , comprising the nef classes .

We also recall the following notion, due to Boucksom (see [ 3 , Definition 2.2]), of classes that are “nef in codimension one”. A class |$\alpha \in H^{1,1}(X,{\mathbb{R}})$| is called a modified nef class if, for every |$\varepsilon> 0$| and some (hence any) choice of Kähler metric |$\omega $| on |$X$|⁠ , there exists a closed (1,1) current |$T$| representing |$\alpha $| such that |$T + \varepsilon \omega \geq 0$| and the Lelong number |$\nu (T,D) = 0$| along any analytic subvariety |$D \subseteq X$| of codimension one. The modified nef classes comprise |$\mathcal{M}\mathcal{N}_{X}$|⁠ , the modified nef cone .

The Neron-Severi group |$NS(X)_{\mathbb{R}}$| of |$X$| is the subpace of |$H^{1,1}(X,{\mathbb{R}})$| generated by the first Chern classes of holomorphic line bundles on |$X$|⁠ . Throughout this paper, we shall denote its dimension by |$\rho (X): = \dim _{\mathbb{R}} NS(X)_{\mathbb{R}}.$| Clearly, |$\rho (X) \leq h^{1,1}(X)$|⁠ .

(1) |$\int _{X} \alpha ^{2}> 0$|⁠ ,

(2) |$\int _{X} \alpha \cdot \beta> 0$|⁠ ,

• (3) For every irreducible curve |$E \subseteq X$| of negative self-intersection, we have $$\begin{align*} & \int_E \alpha> 0. \end{align*}$$

On a compact Kähler surface |$X$|⁠ , we have |$\mathcal{P}_{X}^{+} \subseteq \mathcal{B}_{X}$|⁠ .

We have |$\mathcal N_{X} \subseteq \overline{\mathcal P^{+}_{X}}$|⁠ . By duality, we conclude |$\overline{\mathcal P^{+}_{X}} = \overline{\mathcal P^{+}_{X}}^{*} \subseteq \mathcal N_{X}^{*} = \mathcal E_{X}$|⁠ , since |$\overline{\mathcal P^{+}_{X}}$| is self-dual. This shows that |$\overline{\mathcal P^{+}_{X}} \subseteq \overline{\mathcal B_{X}}$|⁠ , and the result about the interior |$\mathcal P^{+}_{X}$| of |$\overline{\mathcal P^{+}_{X}}$| follows.

2.2 The Zariski decomposition

A classical theorem of Zariski [ 53 ] states that any effective divisor |$D$| on a surface can be uniquely decomposed as a sum of |$\mathbb Q$| -divisors |$D = Z + N$| where Z is nef, |$N$| has negative-definite intersection matrix and |$Z \cdot N = 0$|⁠ . In [ 3 ] Boucksom generalised this decomposition to arbitrary pseudoeffective classes on compact complex manifolds. His result, in the context of Kähler manifolds, is the following. (See [ 3 , Sections 3.2, 3.3].)

(1) For every |$\tau \in \mathcal E_{X}$|⁠ , |$Z(\tau ) \in \mathcal{M}\mathcal{N}_{X}$| is a modified nef class.

• (2) For every |$\tau \in \mathcal E_{X}$|⁠ , |$N(\tau )$| is the class of a unique effective |${\mathbb{R}}$| -divisor, that is, $$\begin{align*} & N(\tau) = \sum_{i = 1}^\ell a_i [E_i] \end{align*}$$ for some |$a_{i}> 0$| and |$[E_{i}]$| the classes of analytic subvarieties |$E_{i} \subseteq X$| of codimension one forming an exceptional family . The current of integration along the divisor |$\sum a_{i} E_{i}$| is the unique closed positive current in the class |$N(\alpha )$|⁠ . Moreover, the classes |$[E_{i}]$| are linearly independent in |$H^{1,1}(X,{\mathbb{R}})$|⁠ . In particular, |$\ell \leq \rho (X)$|⁠ .
• (3) For every |$\tau \in \mathcal E_{X}$|⁠ , we have $$\begin{align*} & \tau = Z(\tau) + N(\tau). \end{align*}$$

(4) The map |$N$| is convex.

  Example 2.4. Let |$X$| be the Fermat quartic hypersurface in |$\mathbb P^{3}$| given by the locus |$x_{0}^{4} + x_{1}^{4} + x_{2}^{4} + x_{3}^{4} = 0$|⁠ . Let |$\xi = \exp (2\pi \sqrt{-1}/8)$|⁠ , which is a primitive eighth root of unity. Then, writing $$\begin{align*} & x_0^4 + x_1^4 + x_2^4 + x_3^4 = \left(\prod_{j\in\left\{1,3,5,7\right\}}(x_0 - \xi^j x_1)\right)+ \left(\prod_{j\in\left\{1,3,5,7\right\}}(x_2 - \xi^j x_3)\right) \end{align*}$$ we see immediately that the sixteen lines $$\begin{align*} & \ell(j_1,j_2):= \left\{ [\xi^{j_1}x_1:x_1: \xi^{j_2}x_3:x_3] \in \mathbb P^3 \; | \;[x_1:x_3] \in \mathbb P^1\right\} \quad j_1,j_2 \in \left\{1,3,5,7\right\} \end{align*}$$ lie on |$X$|⁠ . Since |$X$| is a smooth degree |$4$| hypersurface, by adjunction we get that |$X$| is a K3 surface, so the genus formula for an irreducible curve |$C \subseteq X$| reduces to $$\begin{align*} & 2g(C) - 2 = C^2 \end{align*}$$ from which it follows that the only curves of negative self-intersection on |$X$| are smooth rational curves, all of which have self-intersection equal to |$-2$|⁠ . Now note that |$\ell (j,k)$| and |$\ell (r,s)$| meet transversely in a point precisely when |$j = r$| or |$k=s$| but not both. So, for fixed and distinct |$j,k \in \left \{1,3,5,7\right \}$|⁠ , any three of the four lines |$\ell (j,j), \ell (j,k),\ell (k,k),\ell (k,j)$| have an intersection matrix (with respect to an appropriate basis) given by $$\begin{align*} & \left(\begin{matrix} -2 & 1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -2 \end{matrix}\right), \end{align*}$$ which is negative-definite, and hence form an exceptional family. On the other hand, it is easy to verify that the divisor $$\begin{align*} & P(j,k):= \ell(j,j)+\ell(j,k)+\ell(k,k)+\ell(k,j) \end{align*}$$ is nef. So, it follows that any convex combination $$\begin{align*} & \tau = a[\ell(j,j)]+b[\ell(j,k)]+c[\ell(k,k)]+d[\ell(k,j)] \in H^{1,1}(X,{\mathbb{R}}), \end{align*}$$ where |$a, b, c, d \geq 0$| are non-negative real numbers has its Zariski decomposition given by $$\begin{align*} & Z(\tau) = \min(a,b,c,d)[P(j,k)] \end{align*}$$ and $$\begin{align*} & N(\tau) = \alpha-\min(a,b,c,d)[P(j,k)], \end{align*}$$ since the latter, by the above, has irreducible components that form an exceptional family.

2.3 Zariski decomposition and birational transformations

The Zariski decomposition is moreover well-behaved under birational transformations. More precisely, we have the following.

  Lemma 2.5. Suppose |$X$| is a compact Kähler manifold and |$\pi : Y \to X$| a birational morphism. Then, for any pseudoeffective class |$\tau $| on |$X$| we have $$\begin{align*} & N(\tau) = \pi_*N(\pi^*\tau). \end{align*}$$ In particular, if |$\pi : Y \rightarrow X$| is the blowup of a compact Kähler surface |$X$| at a point |$p \in X$|⁠ , with |$E \subseteq Y$| the exceptional divisor, then for any pseudoeffecitve class |$\tau $| on |$X$|⁠ , the support of |$N(\pi ^{*}\tau )$| is composed of the strict transform of the curves making up the support of |$N(\tau )$|⁠ , plus possibly the exceptional divisor |$E$|⁠ .

The first part is [ 3 , Lemma 5.8], and the second part follows immediately from the formula for the pullback of a divisor under a blowup map.

2.4 Variational setup and stability

  Lemma 2.6. ([ 35 , Proposition 13]) Suppose |$X$| is a smooth projective surface. For slope test configurations |$\Phi _{C,\kappa } \in \mathcal{H}^{\textrm{NA}}(X,[\omega ])$| we have $$\begin{align*} & \mathbf{E}_{\theta,\omega}^{\textrm{NA}}(\Phi_{C,\kappa}) = A_1\kappa^2 + B_1\kappa^3, \end{align*}$$ $$\begin{align*} & ||\Phi_{C,\kappa}|| = A_2\kappa^2 + B_2\kappa^3, \end{align*}$$ where |$A_{1},A_{2},B_{1},B_{2}$| are real numbers satisfying $$\begin{align*} & A_1 = c_{\theta,\omega}\int_C \omega - \int_C \theta, \; \; B_1 = - \left(\frac{2[\theta].[\omega]}{3[\omega]^2} \right) C^2, \; \; A_2 = \int_C \omega> 0, \; \; B_2 = -\frac{2}{3}C^2.\end{align*}$$

The proof of this result follows from standard intersection theory when |$[\theta ]$| and |$[\omega ]$| are rational classes. It also extends to |$[\theta ],[\omega ] \in H^{1,1}(X,\mathbb{R})$| by continuity. In the purely transcendental setting this is related to the study of Kähler slope stability in [ 48 ].

For background on test configurations we refer to [ 4 , 25 , 27 , 42 , 45 , 51 ] and references therein. For variational methods and stability for the J-equation we refer to [ 14 ]. In the case of the dHYM equation similar techniques have been developed and studied in [ 10 , 12 , 33 ] and followup work. For the Z-critical equation see [ 22 , 24 , 37 ].

In this section we prove the main theorems for the deformed Hermitian Yang–Mills and Z-critical equations, and map out a number of consequences of these results.

3.1 A meta proposition

As a key preparation for the proof of our main result, we make the following rather general observation:

  Proposition 3.1 (Meta proposition). Let |$X$| be a compact Kähler surface with |$\omega $| a Kähler form. Let |$\tau \in \mathcal B_{X} \subseteq H^{1,1}(X,\mathbb R)$| be any big cohomology class that is not Kähler. Then, there exists a (non-empty) finite set of irreducible curves |$\left \{E_{1}, \cdots , E_{\ell }\right \}$| such that $$\begin{align}& \int_{E} \tau \leq 0 \iff E = E_{i}\ \textrm{for some}\ i=1,\cdots, \ell.\end{align}$$ (5) Moreover, the intersection matrix |$(E_{i}\cdot E_{j})$| of the curves |$E_{i}$| is negative-definite and their classes |$[E_{i}] \in H^{1,1}(X,\mathbb R)$| are linearly independent. In particular, |$\ell \leq \rho (X)\leq h^{1,1}(X)$| where |$\rho (X) = \dim _{\mathbb R} NS(X)_{\mathbb R}$|⁠ . Moreover, if |$X$| is projective, then |$\ell \leq \rho (X) -1.$|

Now suppose, in full generality of the proposition, that |$\tau $| is not Kähler. Then, |$\tau - \varepsilon [\omega ]$| is not nef but still contained in the big cone |$\mathcal B_{X}$| for any |$\varepsilon> 0$| small enough. Now if |$\int _{E} \tau \leq 0$| for some curve |$E \subseteq X$|⁠ , we have |$\int _{E} (\tau - \varepsilon [\omega ]) < 0$|⁠ , so the set of such curves is contained among the prime components of |$N(\tau - \varepsilon [\omega ])$|⁠ , say |$E_{1}, \cdots , E_{s}$|⁠ . After relabelling if necessary, we may suppose |$\int _{E_{j}} \tau \leq 0$| precisely for |$j = 1, \cdots , \ell $|⁠ . Then clearly we have |$\ell \leq s \leq \rho (X)$| (and if |$X$| is projective, |$s \leq \ell \leq \rho (X)-1$|⁠ ). Finally, the claim about the intersection matrix follows because any submatrix of a negative-definite matrix is itself negative-definite.

3.2 Global finiteness of |$\mathcal{S}_{\tau }$| in compact sets

For the sequel we introduce the following terminology:

• (1) The set of destabilizing curves with respect to |$\tau $| is the finite set $$\begin{align*} &\mathcal D_\tau:= \left\{ E \subseteq X\ \textrm{irreducible curve}\ | \ \int_E \tau \leq 0 \right\}. \end{align*}$$

(2) A set |$\mathcal S$| of irreducible curves is a set of candidate destabilizers for |$\tau $| if |$\mathcal D_{\tau } \subseteq \mathcal S$|⁠ .

• (3) The set of Zariski negative curves for |$\tau $| is the finite set $$\begin{align*} & \textrm{Neg}(\tau):= \bigcap_{\varepsilon> 0} \left\{E \subseteq X\ \textrm{irreducible curve}\ | \ E\ \textrm{is a prime component of}\ N(\tau - \varepsilon[\omega])\right\}.\end{align*}$$

(1) Note that |$\operatorname{Neg}(\tau )$| does not depend on the Kähler class |$[\omega ]$| we choose.

(2) It is clear from the proof of the Proposition 3.1 that |$\mathcal D_{\tau } \subseteq \textrm{Neg}(\tau )$|⁠ , and hence |$\mathcal S_{\tau }:= \textrm{Neg}(\tau )$| always provides a natural set of candidate destabilizers for |$\tau $|⁠ .

(3) For any big class |$\tau $|⁠ , the closed positive current |$N(\tau )$| appearing in the Zariski decomposition is the divisor part in the Siu decomposition of a current of minimal singularities (see [ 3 ]).

(4) In previous literature, a slightly weaker analogue of Proposition 3.1 appeared in [ 46 , Proposition 4.5], which uses the Siu decomposition of closed positive (1,1) currents and a regularisation result due to Demailly (see the proof of [ 46 , Proposition 4.5] as well as [ 34 ] and references therein). The Zariski decomposition has the added advantage that the negative part is precisely given by the Siu decomposition of a current of minimal singularities (see [ 3 , Section 3.2] for reference).

  Lemma 3.4. Let |$X$| and |$\omega $| be as in Proposition 3.1 , and let |$K \subseteq \mathcal B_{X}$| be any subset contained in the positive cone over the strict convex hull of finitely many pseudoeffective classes |$\tau _{1},\dots ,\tau _{k}$| on |$X$|⁠ . Then $$\begin{align*} & \mathcal S_K:= \bigcup_{\tau \in K} \mathcal D_{\tau}\end{align*}$$ is a finite set of curves of negative self-intersection, of cardinality |$|\mathcal{S}_{K}| \leq k\rho (X)$|⁠ , and is a set of candidate destabilizers for every |$\tau \in K$|⁠ . If |$X$| is projective, then the cardinality of |$\mathcal S_{K}$| does not exceed |$k\rho (X) - k$|⁠ .

  Proof. Suppose that |$ \tau = \sum _{i = 1}^{k} a_{i}\tau _{i}, \; \; a_{i}> 0. $| Then |$\int _{C} \tau \leq 0$| implies |$\int _{C} \tau _{i}\leq 0$| for some |$i$|⁠ . Hence |$C$| is in the set of destabilizers $$\begin{align*} & \mathcal{D}_{\tau_i} \subseteq \bigcup_i \mathcal{D}_{\tau_i} \end{align*}$$ which is of cardinality at most |$k \max _{i}|\mathcal{D}_{\tau _{i}}| \leq k\rho (X)$| (or |$\leq k\rho (X) - k$| if |$X$| is projective), by Proposition 3.1 . For future use we record the elementary fact that the above in particular applies if |$K$| is a compact subset of |$\mathcal{P}_{X}^{+}$|⁠ :

Any compact subset |$K \subseteq \mathcal{P}_{X}^{+}$| is contained in the convex hull of finitely many points in |$\mathcal{P}_{X}^{+}$|⁠ .

  Proof. Without loss of generality, assume that |$K$| is connected. We may then choose an open covering of |$K$| by means of open cubes |$C_{\nu }$| whose closure is in |$\mathcal{P}_{X}^{+}$|⁠ . By compactness there is a finite open subcover $$\begin{align*} & \bigcup_{\nu \in I, |I| < +\infty} C_{\nu}. \end{align*}$$ Taking the convex hull of enough of the (finitely many) vertices of |$C_{\nu }$|⁠ , |$\nu \in I$|⁠ , this set clearly contains |$K$|⁠ .

(1) The class |$\alpha $| is Kähler.

• (2) We have $$\begin{align*} & \int_{E_i} \alpha> 0 \end{align*}$$ for all |$i \in \left \{1,\dots ,\ell \right \}$|⁠ .

3.3 Proof of the Main Theorem for the deformed Hermitian Yang–Mills equation

• (1) For any choice of Kähler metric |$\theta $| in |$\beta $|⁠ , there exists a smooth (1,1) form |$\omega $| which is a solution to the deformed Hermitian Yang–Mills equation $$\begin{align*} & \operatorname{Im}\left(e^{-\sqrt{-1}\hat\Theta(\beta,\alpha)}(\theta+\sqrt{-1}\omega)^2\right)=0, \end{align*}$$ and the solution |$\omega $| is a Kähler form in |$\alpha $| if |$\int _{X}(\alpha ^{2} -\beta ^{2})>0.$|
• (2) For every curve |$E\subseteq X$|⁠ , we have $$\begin{align*} & \int_{E}\tau(\beta,\alpha)>0. \end{align*}$$
• (3) For |$i = 1,\cdots , \ell $|⁠ , we have $$\begin{align*} & \int_{E_i}\tau(\beta,\alpha)>0. \end{align*}$$

  Proof. From the discussion preceding the statement of the Theorem, we see that $$\begin{align*} & \int_X\tau(\beta,\alpha)^2 = (1+\cot^2(\hat\Theta(\beta,\alpha)))\int_X \beta^2> 0. \end{align*}$$ Moreover, observe that $$\begin{align*} & \int_X \tau(\beta,\alpha)\cdot\alpha = \frac{1}{2}\int_X(\beta^2 + \alpha^2)> 0. \end{align*}$$ and so $$\begin{align*} & \tau(\beta,\alpha) \in \mathcal P^+_X. \end{align*}$$ Thus, the continuous map $$\begin{align*} &\mathcal C_X \times \mathcal C_X \to H^{1,1}(X,{\mathbb{R}}), \; \; (\beta,\alpha)\mapsto \tau(\beta,\alpha) \end{align*}$$ takes |$K$| onto a compact subset |$\tilde K$| of |$\mathcal P^{+}_{X}$|⁠ . Now, by Lemma 3.5 , |$\tilde K$| is contained in the convex hull of finitely many points of |$\mathcal P^{+}_{X}$| and by Lemma 3.4 , there exists a non-negative integer |$\ell \geq 0$| and finitely many curves of negative self-intersection |$E_{1},\cdots ,E_{\ell }$| such that a class |$\tau \in \tilde K$| is Kähler if and only $$\begin{align*} & \int_{E_i} \tau> 0 \end{align*}$$ for |$i= 1, \cdots , \ell .$| The proof that these curves satisfy the conclusion of the Theorem now proceeds via a standard argument whereby we reduce the equation to a complex Monge-Ampère equation. For the sake of completeness, we recall this simple argument. Suppose |$(\beta ,\alpha )\in K$| is such that the deformed Hermitian Yang–Mills equation admits a smooth solution |$\omega $| for a choice of Kähler form |$\theta \in \beta $|⁠ . Then, from the above discussion, we see that $$\begin{align*} & (\omega+\cot(\hat\Theta(\beta,\alpha)\theta)^2 = (1+\cot^2(\hat\Theta(\beta,\alpha)))\theta^2. \end{align*}$$ From this equality of forms, it follows that the (1,1) form |$(\omega + \cot (\hat \Theta (\beta ,\alpha )))$| is Kähler. Indeed, the equality obviously implies that |$\omega + \cot (\hat \Theta (\beta ,\alpha ))\theta $| is non-degenerate and defines the same orientation as the Kähler form |$\theta $|⁠ . On a surface, this condition is equivalent to definiteness. But the topological fact that the class |$\tau (\beta ,\alpha )$| is in |$\mathcal P^{+}_{X}$| ensures that the form |$\omega + \cot (\hat \Theta (\beta ,\alpha )$| is positive-definite. This shows that the class |$\tau (\beta ,\alpha )$| is a Kähler class, and proves that (1) implies (2). It is obvious that (2) implies (3). Finally, let there be given any choice of Kähler metric |$\theta \in \beta $|⁠ . Then (3) implies, by our choice of the curves |$E_{i}$|⁠ , that the class |$\tau (\beta ,\alpha )$| is a Kähler class. By Yau’s solution of the Calabi conjecture [ 52 ], this implies that we can find a Kähler metric |$\chi \in \tau (\beta ,\alpha )=\alpha + \cot (\hat \Theta (\beta ,\alpha ))\beta $| such that $$\begin{align*} & \chi^2 = (1 + \cot^2(\hat\Theta(\beta,\alpha)))\theta^2. \end{align*}$$ But then |$\omega = \chi - \cot (\hat \Theta (\beta ,\alpha ))\theta $| satisfies the deformed Hermitian Yang–Mills equation. This shows that (3) implies (1). Finally, note that if |$\int _{X}(\alpha ^{2}-\beta ^{2})>0$| then clearly |$\cot (\hat \Theta (\beta ,\alpha ))<0$| and so the form |$-\cot (\hat \Theta (\beta ,\alpha ))\theta $| is a Kähler class, and therefore, so is |$\omega = \chi -\cot (\hat \Theta (\beta ,\alpha ))\theta $|⁠ .

The hypothesis |$\int _{X} (\alpha ^{2}-\beta ^{2})> 0$| is usually formulated as the phase hypothesis |$\hat \Theta (\beta ,\alpha )\in (\frac{\pi }{2},\pi )$|⁠ , and is called the supercritical regime .

Suppose |$X$| is a compact Kähler surface that admits no curves of negative self-intersection. Then, the deformed Hermitian Yang–Mills equation always admits a solution for any pair of Kähler classes |$\beta ,\alpha $| and any choice of Kähler metric |$\theta \in \beta $|⁠ .

Take |$K = \left \{(\beta ,\alpha )\right \}$| in 3.7 and observe that we must have |$\ell = 0$|⁠ , so the third condition is vacuously satisfied.

A version of the above Corollary can be derived (with slightly different hypotheses) from [ 29 , Theorem 1.4], although in their work the emphasis is more on the study of a dHYM flow. We also remark that the above Theorem 3.7 is a special case of Theorem 3.13 below.

  Proof of Corollary 1.7. The only claim that requires justification is the claim about the boundary |$\partial \mathcal U|_{K}$| being of real codimension one, the rest of the claims following easily from the proof of Theorem 3.7 . To justify this last claim, we must show that any given “wall” defining |$\partial \mathcal U$| is a real algebraic submanifold of |$\mathcal C_{X} \times \mathcal C_{X}$| of codimension one. More precisely, given any curve |$E$| appearing in Theorem 1.4 , the locus $$\begin{align*} & W_E:= \left\{ (\alpha,\beta)\in \mathcal C_X \times \mathcal C_X \ | \ \int_E \tau_{\textrm{dHYM}}(\alpha,\beta) = 0\right\} \end{align*}$$ is a real algebraic submanifold of codimension one. The fact that it is real algebraic is straightforward, since it is the zero locus of the composition of the real algebraic maps $$\begin{align*} & (\alpha,\beta)\mapsto \tau_{\textrm{dHYM}}(\alpha,\beta)=\alpha + \frac{\beta^2-\alpha^2}{2\beta\cdot \alpha}\beta, \quad \quad \tau \mapsto \int_E \tau. \end{align*}$$ (The first map is algebraic on the open set |$ \left \{(\alpha ,\beta )\in H^{1,1}(X,\mathbb R) \ | \ 2\beta \cdot \alpha \neq 0\right \}$| which certainly contains |$\mathcal C_{X} \times \mathcal C_{X}$|⁠ .) Therefore, it suffices to prove that 0 is a regular value of the map $$\begin{align*} & (\alpha,\beta) \mapsto \int_E \tau_{\textrm{dHYM}}(\alpha,\beta).\end{align*}$$ To this end, assume |$(\alpha ,\beta )$| is a zero of the above map and define the function $$\begin{align*} & f(\varepsilon):= \int_E \left(\alpha + \varepsilon \beta + \frac{\beta^2 - (\alpha + \varepsilon\beta)^2}{2\beta\cdot (\alpha + \varepsilon \beta)}\beta\right). \end{align*}$$ Clearly, |$f(0) = 0$| and a straightforward calculation shows that $$\begin{align*} & f^\prime(\varepsilon) = \int_E\left( \beta + \frac{(-2\alpha_\varepsilon\cdot\beta)(2\beta\cdot\alpha_\varepsilon)-(\beta^2 - \alpha_\varepsilon^2)(2\beta^2)}{(2\beta\cdot\alpha_\varepsilon)^2}\beta\right)=\frac{-\beta^2}{\beta\cdot\alpha_\varepsilon} \int_E \left(\frac{\beta^2 - \alpha_\varepsilon^2}{2\beta\cdot\alpha_\varepsilon}\beta\right) \end{align*}$$ where we have written |$\alpha _{\varepsilon } = \alpha + \varepsilon \beta $|⁠ . This means that $$\begin{align*} & f^\prime(0) = \frac{-\beta^2}{\beta\cdot\alpha}\int_E \frac{\beta^2 - \alpha^2}{2\beta\cdot\alpha}\beta = \frac{\beta^2}{\beta\cdot \alpha}\int_E \alpha \neq 0, \end{align*}$$ where the last equality follows from the fact that |$f(0) = 0$| by assumption.

3.4 Proof of the Main Theorem for the Z-critical equation

  Lemma 3.11. Let |$\Omega = (\beta ,\rho ,U)$| be a choice of stability data defining a polynomial central charge on a projective surface |$X$|⁠ , with a fixed lift $$\begin{align*} & \tilde\Omega = \left(\theta,\rho_0+\rho_1 t+\rho_2 t^2, 1 + \tilde U_1 + \tilde U_2\right), \end{align*}$$ and let |$L\to X$| be a holomorphic line bundle on |$X$| such that |$\varphi (L) \neq \arg (\pm \rho _{0})$|⁠ . Then the forms |$\tilde \eta (\tilde \Omega ,L)$| and |$\tilde \gamma (\tilde \Omega ,L)$| are given by $$\begin{align} \tilde\eta(\tilde\Omega,L) &= \frac{2}{c_{0}}\left(c_{0}\tilde U_{1} + c_{1}\theta\right), \end{align}$$ (9) $$\begin{align} \tilde\gamma(\tilde\Omega,L) &= \frac{2}{c_{0}}\left(c_{0}\tilde U_{2} + c_{1}\theta\wedge\tilde U_{1} +c_{2}\theta^{2}\right),\end{align}$$ (10) where |$c_{k} = \textrm{Im}(\rho _{k})\cot \varphi (L)-\textrm{Re}(\rho _{k})$|⁠ .

  Corollary 3.12. Let |$X, L$| and |$\Omega $| be as in the Lemma. If $$\begin{align*} &V(\Omega,L):= \int_X \frac{1}{4}\eta(\Omega,L)^2 - \gamma(\Omega,L)> 0\end{align*}$$ then there exists a choice of lift |$\tilde \Omega $| that satisfies the volume form hypothesis for |$L$|⁠ .

  Proof. Clearly, if the numerical inequality |$V(\Omega ,L)>0$| is satisfied, then the class $$\begin{align*} &\frac{1}{4}\eta(\Omega,L)^2 - \gamma(\Omega,L)\in H^4(X,{\mathbb{R}})\end{align*}$$ contains a volume form |$v$|⁠ . Fix any lift |$\tilde \Omega _{0} = (\theta ,\rho ,1+\tilde U_{1} + \tilde U_{2})$| of |$\Omega $|⁠ . Then, by the |$\partial \bar \partial{}$| -lemma, there exists a real valued (1,1) form |$\zeta $| on |$X$| such that $$\begin{align*} &v = \frac{1}{4}\tilde\eta(\tilde\Omega_0,L)^2 - \tilde\gamma(\tilde\Omega_0,L) + \sqrt{-1} \partial\bar\partial{\zeta}.\end{align*}$$ Setting |$\tilde U_{1}^{\prime } = \tilde U_{1}$| and $$\begin{align*} &\tilde U^\prime_2 = \tilde U_{2} - \frac{\sqrt{-1}}{2}\partial\bar\partial{\zeta},\end{align*}$$ we see immediately from ( 9 ) that if |$\tilde \Omega = (\omega ,\rho ,1+\tilde U^{\prime }_{1}+\tilde U^{\prime }_{2})$| then $$\begin{align*} &v = \frac{1}{4}\tilde\eta(\tilde\Omega,L)^2 - \tilde\gamma(\tilde\Omega,L)\end{align*}$$ is a volume form.

In this notation, Theorem 1.4 can be stated as follows.

(1) For every |$\Omega \in K$| and every lift |$\tilde \Omega $| satisfying the volume form hypothesis at |$L$|⁠ , the |$Z_{\Omega }$| -critical equation admits a solution.

• (2) For |$i = 1, \cdots , \ell $|⁠ , we have $$\begin{align*} & s(\Omega,L)\left(\int_{E_i} c_1(L) + \frac{1}{2}\eta(\Omega,L) \right)> 0\end{align*}$$ where |$s(\Omega ,L)$| is the sign of the nonzero real number $$\begin{align*} & \int_{X}\left(c_1(L)+\frac{1}{2}\eta(\Omega,L)\right)\cdot\beta \in{\mathbb{R}}. \end{align*}$$

Note that for any line bundle |$L$| the set of |$\Omega $| satisfying |$V(\Omega ,L)>0$| and |$\varphi _{\Omega }(L) \neq \arg (\rho _{0})$| (the natural “phase” hypotheses in the context of the Z-critical equation) is non-empty and open. This can easily be seen by perturbing the stability vector |$\rho $| (if required) and then scaling |$\beta \mapsto k\beta $| for |$k>0$| large.

  Proof of Theorem 3.13. Let a compact subset |$K \subseteq \mathcal C_{X} \times (\mathbb C^{*})^{3} \times \bigoplus _{i} H^{i,i}(X,\mathbb R)$| be given such that each element |$\Omega = (\beta ,\rho ,U) \in K$| defines a valid polynomial central charge with |$\varphi _{\Omega }(L)\neq \arg (\pm \rho _{0})$| and such that |$V(\Omega ,L)> 0$|⁠ . Recall that the topological constant |$\varphi (L)= \varphi _{\Omega }(L)$| is chosen precisely so that $$\begin{align*} & \int_X \operatorname{Im}\left(e^{-\sqrt{-1}\varphi_\Omega (L)}Z_\Omega(L)\right) = 0.\end{align*}$$ In our notation, this is equivalent to $$\begin{align*} &0 = \int_X\left(c_1(L)^2 + c_1(L)\cdot \beta(\Omega,L) + \gamma(\Omega,L)\right) = \int_X\left(c_1(L) + \frac{1}{2}\eta(\Omega,L)\right)^2 - V(\Omega,L).\end{align*}$$ Therefore, the inequality |$V(\Omega ,L)>0$| implies that the class |$\sigma (\Omega ,L):= c_{1}(L) + \frac{1}{2}\eta (\Omega ,L)$| has positive self-intersection. So, by the Hodge-Index Theorem, either |$\sigma (\Omega ,L)$| or its negative is in the cone |$\mathcal P^{+}_{X}.$| Let |$s(\Omega ,L)\in \left \{\pm 1\right \}$| be defined by the condition that $$\begin{align*} &\tau_Z(\Omega,L):= s(\Omega,L)\tau(\Omega,L) \in \mathcal P^{+}_X.\end{align*}$$ Clearly, we have $$\begin{align*} &s(\Omega, L) = \operatorname{sign}\left(\int_X \sigma(\Omega,L)\cdot\beta\right).\end{align*}$$ Thus, the map $$\begin{align*} &\mathcal C_X \times (\mathbb C^*)^3\times \bigoplus_i H^{i,i}(X,{\mathbb{R}})\to H^{1,1}(X,{\mathbb{R}}), \; \; \Omega \mapsto \tau_Z(\Omega,L)\end{align*}$$ is continuous and takes |$K$| onto a compact subset |$\tilde K = \tau _{Z}(K)$| of |$\mathcal P^{+}_{X}$|⁠ . By Lemma 3.5 , |$\tilde K$| is contained in the convex hull of finitely many points of |$\mathcal P^{+}_{X}$|⁠ . Now, by Lemma 3.4 , there exist finitely many curves |$E_{1}, \cdots , E_{\ell }$| of negative self-intersection such that for each |$\tau \in \tilde K$|⁠ , |$\tau $| is Kähler if and only if $$\begin{align*} & \int_{E_i} \tau>0 \end{align*}$$ for |$i = 1, \cdots , \ell .$| Finally, according to [ 24 , Section 2.3.3] this is equivalent to the existence of a subsolution for the |$Z_{\Omega }$| -critical equation, and by [ 24 , Theorem 2.45], equivalent to the existence of a solution. This completes the proof.

Let |$X$| and |$L$| be as in the Theorem 3.13 . Suppose |$X$| does not admit any curves of negative self-intersection. Let |$\Omega $| be a choice of stability data defining a polynomial central charge |$Z_{\Omega }$| and such that |$\varphi (L)\neq \arg (\pm \rho _{0})$| and |$V(\Omega ,L)> 0$|⁠ . Then, for any lift |$\tilde \Omega $| satisfying the volume form hypothesis, the |$Z_{\Omega }$| -critical equation admits a solution on |$L$|⁠ .

In Theorem 3.13 , take |$K = \left \{\Omega \right \}$| and note that necessarily |$\ell = 0$|⁠ . Thus, the second of the two equivalent conditions is vacuously true.

  Corollary 3.16. Fix a finite set |$S$| of holomorphic line bundles |$L_{i}\to X$| (for |$i = 1, \cdots , k$|⁠ ) on a compact Kähler surface |$X$|⁠ . Let |$V_{S}$| denote the set of triples |$\Omega = (\beta ,\rho ,U)$| in |$\mathcal C_{X} \times ({\mathbb{C}}^{*})^{3} \times \bigoplus H^{i,i}(X,{\mathbb{R}})$| such that |$Z_{\Omega }(L_{i})$| lies in the upper half-plane, |$\varphi _{\Omega }(L_{i})\neq \arg (\pm \rho _{0})$| and |$V(\Omega ,L_{i})> 0$| for each |$i = 1, \cdots , k$|⁠ . Let |$\mathcal U_{i}$| denote the subset of |$V_{S}$| comprising those |$\Omega $| such that for any choice of lift |$\tilde{\Omega }$| of |$\Omega $| satisfying the volume form hypothesis for |$L_{i}$|⁠ , the Z-critical equation ( 7 ) admits a solution. Then, |$\mathcal U_{i}$| is an open subset of |$V_{S}$| and for any compact subset |$K$| of |$V_{S}$|⁠ , the set |$\mathcal U_{i} \cap K$| is cut out by the finitely many real algebraic inequalities $$\begin{align*} & W_{ij}(\Omega):= \int_{E_{ij}} \tau_Z(\Omega,L_i)> 0, \; \; j = 1, \cdots, \ell_i \end{align*}$$ where |$E_{i1}, \cdots , E_{i\ell _{i}}$| are the curves appearing in Theorem 3.13 . In particular, if |$C$| is any connected component of $$\begin{align*} & K \setminus \bigcup_{i,j} \left\{W_{ij}(\Omega)=0\right\} \end{align*}$$ then for any |$i_{0} \in \left \{1,\cdots , k\right \}$|⁠ , we have |$\Omega \in \mathcal U_{i_{0}}$| for some |$\Omega \in C$| if and only if |$\Omega \in \mathcal U_{i_{0}}$| for every |$\Omega \in C$|⁠ .

  Proposition 3.17. Let |$L$| be a holomorphic line bundle and let |$V$| denote the set comprising stability data |$\Omega = (\beta ,\rho ,U) \in \mathcal C_{X} \times (\mathbb C^{*})^{3} \times \bigoplus H^{i,i}(X,\mathbb R)$| for which |$Z_{\Omega } (L)$| lies in the upper half-plane, |$\varphi _{\Omega } (L) \neq \arg (\pm \rho _{0})$| and |$V(\Omega ,L)> 0$|⁠ . Let |$E$| be any curve on |$X$| such that $$\begin{align*} &\int_E \tau_Z(\Omega,L) = 0\end{align*}$$ for some |$\Omega \in V$|⁠ . Then the locus $$\begin{align*} &W_E = \left\{ \Omega \in V \ | \ \int_E \tau_Z (\Omega, L) = 0\right\}\end{align*}$$ is a real codimension one submanifold of |$V$|⁠ .

  Proof. Recall that the class |$\tau _{Z}(\Omega ,L)$| is given, up to a sign, by $$\begin{align*} &\tau_Z = \pm\left(c_1(L) + \frac{1}{2}\eta(\Omega,L)\right),\end{align*}$$ where $$\begin{align*} &\eta(\Omega,L) = \frac{2}{c_0}\left(c_0U_1+c_1\beta\right)=2\left(U_1+\frac{c_1}{c_0}\beta\right)\end{align*}$$ and |$c_{k} = c_{k}(\Omega , L) = \operatorname{Im}(\rho _{k})\cot \varphi _{\Omega }(L) - \operatorname{Re}(\rho _{k}).$| Recall that |$c_{0} \neq 0$|⁠ ; this is a consequence of the hypothesis that |$\varphi _{\Omega }(L)\neq \arg (\pm \rho _{0})$|⁠ . Fix $$\begin{align*} &\Omega = (\beta, \rho_0 + \rho_1 t + \rho_2 t^2,U)\end{align*}$$ such that $$\begin{align*} &\int_E \tau_Z(\Omega,L) = 0\end{align*}$$ for some curve |$E$| and consider the family of stability data given by $$\begin{align*} &\Omega_\varepsilon:= \left(\beta,\rho_0+\rho_1 t + \rho_2 t^2,U + \frac{\varepsilon}{\int_X \beta^2}\beta^2\right)\end{align*}$$ for |$\varepsilon \in \mathbb R$| small. Then, we have $$\begin{align*} &Z_{\Omega_\varepsilon}(L) = Z_\Omega(L) + \frac{\varepsilon}{\int_X \beta^2}\int_X \beta^2 \cdot(\rho_0 + \rho_1 \beta + \rho_2 \beta^2)\cdot ch(L) = Z_\Omega(L) + \varepsilon \rho_0.\end{align*}$$ Writing |$a:= \operatorname{Re}(\rho _{0}), b:= \operatorname{Im}(\rho _{0})$|⁠ , we get that $$\begin{align*} &\frac{d}{d\varepsilon}\Bigr|_{\varepsilon=0}\cot\varphi_{\Omega_\varepsilon}(L)= \frac{a\operatorname{Im}Z_\Omega(L) - b\operatorname{Re}Z_\Omega(L)}{(\operatorname{Im}Z_\Omega(L))^2}=-\frac{c_0}{\operatorname{Im}Z_\Omega(L)}\end{align*}$$ recalling that |$c_{0} = b \cot \varphi _{\Omega }(L) - a \neq 0$|⁠ . (Throughout, we write |$c_{k} = c_{k}(\Omega ,L)$| for brevity, reserving the more elaborate notation |$c_{k}(\Omega _{\varepsilon },L)$| for the perturbed constants.) A simple calculation now shows that $$\begin{align*} &\frac{d}{d\varepsilon}\Bigr|_{\varepsilon=0}\frac{c_1(\Omega_\varepsilon,L)}{c_0(\Omega_\varepsilon,L)} = -\frac{1}{c_0 \operatorname{Im}Z_\Omega(L)}\left(c_0 \operatorname{Im}\rho_1 - c_1 \operatorname{Im}\rho_0\right) = -\frac{1}{c_0 \operatorname{Im}Z_\Omega(L)}\operatorname{Im}\left(\frac{\rho_0}{\rho_1}\right)\neq 0\end{align*}$$ the last inequality following from our hypotheses and the very definition of a stability datum (see ( 6 )). This shows that the function $$\begin{align*} &f(\varepsilon):= \int_E \tau_Z(\Omega_\varepsilon,L)\end{align*}$$ satisfies $$\begin{align*} &f^\prime(0) =\frac{d}{d\varepsilon}\Bigr|_{\varepsilon=0}\left( \int_E c_1(L) + U_1 + \frac{c_1(\Omega_\varepsilon,L)}{c_0(\Omega_\varepsilon,L)}\beta \right) = -\frac{1}{c_0 \operatorname{Im}Z_\Omega(L)}\operatorname{Im}\left(\frac{\rho_0}{\rho_1}\right)\int_E \beta \neq 0\end{align*}$$ and hence zero is a regular value of the map $$\begin{align*} &\Omega \mapsto \int_E \tau_Z(\Omega,L).\end{align*}$$

  Remark 3.18 (Counter-example to globally finite wall-chamber decomposition). It is worth remarking that this wall-chamber decomposition cannot in general be globally finite. Indeed, there exist Kähler surfaces which admit infinitely many distinct curve classes with negative self-intersection. (For example, the blowup of |$\mathbb P^{2}$| in nine points in general position has infinitely many smooth rational curves of self-intersection |$-1$|⁠ .) If |$X$| is any such surface, and |$E$| is any curve on |$X$| with negative self-intersection, then consider the family of stability data given by $$\begin{align*} &\Omega_r:= \left(\beta, \frac{r}{E^2} \sqrt{-1} - \frac{1}{\int_E \beta}t + \sqrt{-1} t^2, 1 + [E] \right)\end{align*}$$ where |$\beta $| is any Kähler class on |$X$| with |$\int _{X} \beta ^{2} = 1$| and |$r> 0$| is a positive constant whose value we shall vary in the range |$r \in (0, 2]$|⁠ . (In particular, we have set |$U_{2} = 0$|⁠ .) We wish to consider the Z-critical equations associated to this family of stability data on the trivial line bundle |$L = \mathcal O_{X}$|⁠ . One verifies quite easily that $$\begin{align*} &Z_{\Omega_r}(L) = -1 + \sqrt{-1}\end{align*}$$ independently of |$r$| and therefore |$\varphi _{\Omega _{r}}(L) = \frac{3\pi }{4} \neq \arg (\pm \frac{2}{3E^{2}}\sqrt{-1}) = \pm \frac{\pi }{2}$|⁠ . A straightforward calculation then shows that $$\begin{align*} &V(\Omega_r,L) = \frac{E^2(r-2)}{r}+\left(\frac{E^2}{r\int_E \beta}\right)^2> 0\end{align*}$$ as |$r\in (0,2].$| In other words, all the hypotheses of Theorem 3.13 are satisfied. However, another straightforward calculation shows that $$\begin{align*} &\int_E \tau_Z(\Omega_r,L) = \frac{E^2(1-r)}{r},\end{align*}$$ so the assignment |$\Omega _{r} \mapsto \int _{E} \tau _{Z}(\Omega _{r},L)$| changes sign as |$r$| crosses the value |$r=1$|⁠ . In other words, the stability data |$\Omega _{r}$| cross the wall $$\begin{align*} &W_E = \left\{ \Omega \ | \ \int_E \tau_Z(\Omega,L) = 0 \right\}\end{align*}$$ defined by the curve |$E$|⁠ . Thus, every curve of negative self-intersection gives rise to a wall which has non-empty intersection with the space of admissible stability data for the trivial bundle, and there are therefore infinitely many distinct walls whenever there are infinitely many distinct curve classes with negative self-intersection.

Wall-chamber decomposition of a slice of the “space of Z-critical equations” on |$\textrm{Bl}_{p_{1},p_{2}}\mathbb{P}^{2}$|⁠ , with chambers such that (I): only |$L_{1}$| is |$Z_{\Omega _{s}}$| -stable. (II): only |$L_{2}$| is |$Z_{\Omega _{s}}$| -stable. (III): both |$L_{1}$| and |$L_{2}$| are |$Z_{\Omega _{s}}$| -stable. (IV): neither |$L_{1}$| nor |$L_{2}$| is |$Z_{\Omega _{s}}$| -stable.

Clearly one can produce many more examples and carry out a more general analysis of the above decomposition without much additional difficulty, in particular by making different choices for varying the stability datum |$\Omega $|⁠ . Other examples can be treated similarly as long as the boundary of the nef cone of |$X$| is sufficiently well understood. We leave a more systematic study of similar and other examples to future work.

The J-equation is realized as the so-called small volume limit of the deformed Hermitian Yang–Mills equation, and can thereby be seen as a simpler special case of the former. It is, however, interesting in its own right, as it is also closely connected to other geometric PDE of interest in Kähler geometry, such as the constant scalar curvature equation. Below we comment on this “special case” in some further detail, including optimal destabilizing curves for the J-equation, and test configurations, with relation to J-stability.

For the sequel we moreover use the following terminology, which is a variant of Definition 3.2 for the J-equation:

  Definition 5.1. The set of J-destabilizing curves for the pair |$(\theta ,\omega )$| is defined as $$\begin{align*} & \mathcal D_{\theta,\omega}:= \mathcal D_{\tau_{\theta,\omega}}:= \left\{E \subseteq X \; \textrm{irreducible} \; \textrm{curve}: \int_E \tau_{\theta,\omega} \leq 0 \right\}, \end{align*}$$ where |$\tau _{\theta ,\omega }$| is the class $$\begin{align*} & \tau_{\theta,\omega}:= 2\frac{\int_X \omega\wedge\theta}{\int_X \omega^2}[\omega] - [\theta]. \end{align*}$$

(1) By Theorem 5.3 below the set |$\mathcal D_{\theta ,\omega }$| is finite.

(2) The form |$\tau _{\theta ,\omega }$| of course only depends on |$[\theta ]$| and |$[\omega ]$| in |$H^{1,1}(X,\mathbb{R})$|⁠ . Brackets are often removed in indices to alleviate notation.

5.1 Proof of the Main Theorem for the J-equation

We then have the following main theorem for the J-equation. While the idea is the same as for Theorems 1.4 and 1.9 , it requires a separate proof.

• (1) For any Kähler metric |$\theta \in \beta $|⁠ , there exists a Kähler metric |$\omega \in \alpha $| solving the J-equation $$\begin{align}& \textrm{Tr}_{\omega}\theta = c.\end{align}$$ (14)
• (2) For all irreducible curves |$E \subseteq X$| we have $$\begin{align*} & c \int_{E} \omega - \int_{E} \theta> 0 \end{align*}$$ for any smooth forms |$\omega \in \alpha $| and |$\theta \in \beta .$|
• (3) For all |$E \in \mathcal{V}_{K}$| we have $$\begin{align*} & c \int_{E} \omega - \int_{E} \theta> 0 \end{align*}$$ for any smooth forms |$\omega \in \alpha $| and |$\theta \in \beta .$|

(1) |$(1) \Leftrightarrow (2)$| is the solution of a conjecture of Lejmi-Székelyhidi [ 35 ], recently proven in [ 16 , 47 ].

• (2) The equivalence |$(2) \Leftrightarrow (3)$| is more generally valid for smooth |$(1,1)$| -forms |$\theta $| for which |$\beta := [\theta ]$| is in the open convex cone $$\begin{align*} & \mathcal P_X^+ = \left\{\gamma \in H^{1,1}(X,\mathbb R) \ | \ \gamma \cdot \alpha> 0, \gamma^2 > 0\right\}, \end{align*}$$ as explained in the proof below.

  Proof of Theorem 5.3. That |$(2) \Rightarrow (3)$| is obvious. It remains to prove that |$(3) \Rightarrow (2)$|⁠ . To this end, let |$\omega $| be a Kähler form on a compact Kähler surface |$X$| with cohomology class |$\alpha = [\omega ] \in \mathcal C_{X} \subseteq H^{1,1}(X,\mathbb R)$|⁠ , and let |$\theta $| be any smooth real (1,1)-form on |$X$| such that its class |$\beta = [\theta ]$| lies in |$\mathcal P_{X}^{+}$|⁠ . We then observe that if $$\begin{align*} & \tau:= \frac{2 \int_X \theta\wedge\omega}{\int_X \omega^2} [\omega] - [\theta] \end{align*}$$ then we have |$\tau \cdot [\omega ] = [\theta ]\cdot [\omega ]> 0$| and |$\tau ^{2} = [\theta ]^{2}> 0$|⁠ , so |$\tau \in \mathcal P_{X}^{+}$|⁠ . Suppose moreover that equation ( 14 ) does not admit a solution with respect to |$(\theta ,\omega )$|⁠ . By [ 9 , Theorem 2] this is equivalent to |$\tau $| not being a Kähler class. By Lemma 2.2 we, however, have |$\mathcal B_{X} \supseteq \mathcal P_{X}^{+}$|⁠ , and hence |$\tau \in \mathcal{B}_{X} \setminus \mathcal{C}_{X}$|⁠ . Then, by Proposition 3.1 , there exist at most finitely many curves |$E \subseteq X$| such that $$\begin{align*} & \int_E \tau \leq 0.\end{align*}$$ Finally, the image of the compact set |$K$| under the map $$\begin{align*} & \mathcal{P}_X^+ \times \mathcal{C}_X \ni ([\theta],[\omega]) \mapsto \tau_{\theta,\omega}:= c_{\theta,\omega}[\omega] - [\theta] \in \mathcal{P}_X^+ \end{align*}$$ is clearly compact, so by Lemma 3.5 it is contained in the convex hull of a finite number of classes |$\alpha _{1}, \dots , \alpha _{k}$|⁠ . By Lemma 3.4 this implies that |$|\mathcal{V}_{K}| \leq k\rho (X)$|⁠ . Finally, the fact that any J-destabilizing curve must already appear in the list |$\left \{E_{1}, \dots , E_{\ell }\right \}$| is due to Proposition 3.1 . Since clearly |$\mathcal{C}_{X} \subseteq \mathcal{P}_{X}^{+}$| this finishes the proof. As a direct consequence of the above proof, we highlight the following more precise cardinality bound:

  Corollary 5.5. In the notation of the proof of Theorem 5.3 , suppose that the image of |$K$| under the map $$\begin{align*} & \mathcal{P}_X^+ \times \mathcal{C}_X \ni ([\theta],[\omega]) \mapsto \tau_{\theta,\omega}:= c_{\theta,\omega}[\omega] - [\theta] \in \mathcal{P}_X^+ \end{align*}$$ is contained in the convex hull of |$k$| pseudoeffective classes. Then $$\begin{align*} & \ell = |\mathcal{V}_K| \leq k\rho(X) \end{align*}$$

In particular, this gives a good control of the set of J-destabilizing curves as we vary the underlying classes |$([\theta ],[\omega ]) \in K \subseteq \mathcal{C}_{X} \times \mathcal{C}_{X}$|⁠ , but also as we modify our surface |$X$| via certain birational transformations. Applications of this latter point would be interesting to explore further in the future.

5.2 Optimally destabilizing curves for the J-equation and deformed Hermitian Yang–Mills equation

Along linear perturbations |$\theta _{t}:= (1-t)\omega + t\theta $|⁠ , |$t \geq 0$|⁠ , the set |$\mathcal{D}^{\textrm{J}}_{\theta _{t},\omega }$| is independent of |$t \in [0,+\infty )$|⁠ . It is non-empty precisely when |$\Delta _{\textrm{NM}}(\theta ,\omega ) \leq 0$|⁠ .

The independence in |$t$| is an immediate consequence of the linearity of the function |$[0,+\infty ) \ni t \mapsto \Delta ^{\textrm{pp}}_{\theta _{t}}(\omega )$|⁠ , see [ 43 , Lemma 15], and the straightforward observation that every curve optimally destabilizes at |$t = 0$|⁠ . The non-emptyness is an application of Theorem 5.3 , as the infimum over a finite set is always realized.

(1) The classes |$[\omega _{t}]:= (1-t)\mathbf{a} + t[\theta ]$| are Kähler for all |$t \in [0,1)$|⁠ ,

(2) |$\mathbf{a} = [\omega _{1}]$| is not Kähler, and

(3) |$[\omega ] = [\omega _{t}]$| for some |$t \in (0,1)$|⁠ .

Geometrically, |$\mathbf{a}$| is thus the first non-Kähler, but big, |$(1,1)$| -class following a given direction in the Kähler cone (in case |$[\theta ] = K_{X}$| this is often referred to as the nef threshold ). As this class |$\mathbf{a}$| is big but not Kähler, the Nakai–Moishezon criterion implies that there must exist curves |$C \subseteq X$| such that |$\mathbf{a}.[C] = 0$|⁠ . There is a finite number of such curves, and it turns out that they play a central role for both the dHYM equation and the J-equation, as described by the following result:

  Proposition 5.8. (Optimally J-destabilizing curves) Let |$([\theta ],[\omega ])$| be Kähler classes on |$X$|⁠ . In the above notation, if |$\mathbf{a}:= \mathbf{a}([\theta ],[\omega ]) \in H^{1,1}(X,\mathbb{R})$| is in |$\mathcal{P}_{X}^{+}$| and |$\Delta _{\textrm{NM}}(\theta ,\omega ) \leq 0$|⁠ , then the set of optimally J-destabilizing curves is given by $$\begin{align*} & \mathcal{D}^{\textrm{J}}_{\theta,\omega} = \left\{E \subseteq X \; \vert \; \int_E \mathbf{a} = 0 \right\}, \end{align*}$$ and the cardinality |$|\mathcal{D}^{\textrm{J}}_{\theta ,\omega }|$| does not exceed |$\rho (X)$|⁠ .

  Proof. The proof is purely cohomological. First note that along |$(\theta ,\omega _{t})$| we have $$\begin{align*} & \Delta_{\textrm{NM}}(\theta,\omega_t) = c_{\theta,\omega_t} - t^{-1}, \end{align*}$$ as follows from the computation in [ 43 , Lemma 16]. On the other hand, it is a direct consequence of Theorem 5.3 that the above infimum can be taken over a finite set, and is thus realized by a curve |$E \subseteq X$|⁠ , that is, $$\begin{align*} & \Delta_{\textrm{NM}}(\theta,\omega_t) = c_{\theta,\omega_t} - \frac{\int_E \theta}{\int_E \omega_t}. \end{align*}$$ A necessary and sufficient condition for being optimally destabilizing is therefore that $$\begin{align*} & \int_E [\omega_t] = \int_E (1-t)\mathbf{a} + t[\theta] = t\int_E [\theta], \end{align*}$$ which happens precisely if |$\int _{E} \mathbf{a} = 0$|⁠ . The cardinality bound follows from Proposition 3.1 , since $$\begin{align*} & \mathcal{D}^{\textrm{J}}_{\theta,\omega} = \left\{E \subseteq X \; \vert \; \mathbf{a}.[E] = 0 \right\} \subseteq \mathcal{S}(\theta,\omega). \end{align*}$$ In other words the curves that ‘most destabilize’ the numerical condition relevant to the J-equation are precisely the curves that obstruct the boundary class |$\mathbf{a} \in \partial \mathcal{C}_{X}$| from being Kähler, according to the Nakai–Moishezon criterion. This turns out to be a general truth coming from the dHYM equation (see below), and the above result will then illustrate that the set of optimally dHYM-destabiizing curves is preserved in the small volume limit, as |$[\omega ] \mapsto [k\omega ]$|⁠ , with |$k \rightarrow +\infty $|⁠ , in a suitable sense.

  Proposition 5.9. (Optimally dHYM-destabilizing curves) In the setup of Proposition 5.8 , suppose that |$\mathbf{a}([\theta ],[\omega ]) \in \mathcal{P}_{X}^{+}$|⁠ , |$[\theta ]^{2} < [\omega ]^{2}$|⁠ , and |$\Delta _{\textrm{dHYM}}(\theta ,\omega ) \leq 0$|⁠ . Then the set of all optimally dHYM-destabilizing curves is given by $$\begin{align*} & \mathcal{D}^{\textrm{dHYM}}_{\theta,\omega} = \left\{E \subseteq X \; \vert \; \int_E \mathbf{a} = 0 \right\}. \end{align*}$$

The proof is a repetition of that for Proposition 5.8 , with |$c_{\theta ,\omega }$| replaced by |$(\widetilde{c_{\theta ,\omega }})^{-1}$|⁠ .

The statement of the theorem is immediately deduced by combining Propositions 5.8 and 5.9 above.

5.3 Remarks on optimally destabilizing test configurations for J-stability

  Corollary 5.10. Suppose that |$\theta $| and |$\omega $| are Kähler forms on |$X$|⁠ . Assume moreover that |$\tau :=\tau _{\theta ,\omega }$| is not a Kähler class. Then there is a non-empty finite set of curves |$E_{1}, \dots , E_{\ell }$| on |$X$|⁠ , of cardinality not exceeding |$\rho (X)$|⁠ , such that $$\begin{align*} & \Delta^{\textrm{alg}}(\theta,\omega) = \Delta(\theta,\omega) = \Delta_{\textrm{NM}}(\theta,\omega) = \inf_{i = 1,\dots, \ell} \frac{\int_{E_i} \tau}{\int_{E_i} \alpha} \end{align*}$$

  Lemma 5.11. Suppose |$X$| is a smooth projective surface and that |$([\theta ],[\omega ]) \in \mathcal{C}_{X} \times \mathcal{C}_{X}$| such that |$\Delta ^{\textrm{alg}}(\theta ,\omega ) \leq 0$|⁠ . Let moreover |$C \subseteq X$| be any irreducible curve on |$X$|⁠ . Then we always have a strict inequality $$\begin{align}& \Delta^{\textrm{alg}}(\theta,\omega) < \frac{\textrm{E}_{\theta}^{\textrm{NA}}(\Phi_{C,\kappa})}{||\Phi_{C,\kappa}||}\end{align}$$ (16) for every |$\kappa \in (0,\bar{\kappa }_{C})$|⁠ .

  Proof. Introduce the shorthand notation $$\begin{align*} & A_C:= c_{\theta,\omega}\int_C \omega - \int_C \theta, \; \; B_C:= -\frac{2[\omega].[\theta]}{3[\omega]^2}C^2. \end{align*}$$ As a consequence of Lemma 2.6 , the condition for |$\textrm{J}^{\theta }$| -semistability is that |$ A_{C} + B_{C}\kappa \geq 0 $| is satisfied for all curves |$C \subseteq X$| and for all |$\kappa \in (0,\bar{\kappa }_{C})$| smaller than the Seshadri constant $$\begin{align*} & \bar{\kappa}_C:= \sup \left\{ \epsilon> 0: \pi^*c_1(L) - \epsilon[E] > 0\right\}. \end{align*}$$ Likewise, the condition for |$J^{\theta }$| -stability is the same except that the inequality above is required to be strict, that is, |$A_{C} + B_{C}\kappa> 0$| for all curves |$C \subseteq X$| and for all |$\kappa \in (0,\bar{\kappa }_{C})$|⁠ . Now assume that |$\Delta ^{\textrm{alg}}(\theta ,\omega ) = 0$|⁠ , that is, |$(X,[\omega ])$| is |$\textrm{J}^{\theta }$| -semistable but not uniformly |$\textrm{J}^{\theta }$| -stable. We then claim that the infimum in ( 16 ) is not realized by any relatively Kähler test configuration |$\Phi _{C,\kappa }$|⁠ . To see this, first of all note that semistability implies |$A_{C} \geq 0$|⁠ . We then treat the cases |$C^{2} < 0$| and |$C^{2} \geq 0$| separately: in the case |$C$| is a negative curve, the leading term |$B_{C}> 0$|⁠ , so that |$A_{C} + B_{C}\kappa> 0$| for any positive |$\kappa> 0$|⁠ . If we instead assume that |$C^{2} \geq 0$|⁠ , then |$B_{C} \leq 0$| and Theorem 5.3 implies that |$A_{C}> 0$| with a strict inequality (since |$A_{C} \leq 0$| would force |$C$| to be one of the finitely many curves of negative self-intersection produced by the theorem, contradicting |$C^{2} \geq 0$|⁠ ). The case |$B_{C} = 0$| thus clearly does not lead to a destabilizing relatively Kähler test configuration |$\Phi _{C,\kappa }$|⁠ , and it remains to treat the case when |$A_{C}, B_{C}> 0$|⁠ . If we then suppose for contradiction that |$A_{C} + B_{C}\kappa = 0$| for some |$\kappa \in (0,\bar{\kappa }_{C})$|⁠ , then the openness of the interval of admissible |$\kappa $| implies that |$\kappa + \epsilon \in (0,\bar{\kappa }_{C})$| for |$\epsilon> 0$| small enough, and at the same time |$A + B(\kappa + \epsilon ) < 0$|⁠ , contradicting the semistability assumption. To summarize, we must have $$\begin{align*} & \textrm{E}_{\theta}^{\textrm{NA}}(\Phi_{C,\kappa}) = A_C + B_C\kappa> 0 \end{align*}$$ for all |$C \subseteq X$| and all |$\kappa \in (0,\bar{\kappa }_{C})$|⁠ . In particular, the infimum in ( 16 ) is not realized and |$(X,[\omega ])$| thus satisfies the condition for |$\textrm{J}^{\theta }$| -stability across all relatively Kähler test configurations of the form |$\Phi _{C,\kappa }$| for |$C \subseteq X$| irreducible curves. Finally, suppose more generally that |$\Delta _{\theta }^{\textrm{alg}}(\omega ) = -R$| for some |$R \geq 0$|⁠ . Then |$[\theta _{R}]:= [\theta ] + R[\omega ]$| is a Kähler class, and moreover $$\begin{align}& \frac{\textrm{E}_{\theta_{R}}^{\textrm{NA}}(\Phi_{C,\kappa})}{||\Phi_{C,\kappa}||} = \frac{\textrm{E}_{\theta}^{\textrm{NA}}(\Phi_{C,\kappa})}{||\Phi_{C,\kappa}||} + R\end{align}$$ (17) for every test configuration |$\Phi _{C,\kappa }$|⁠ . Moreover, |$ \Delta ^{\textrm{alg}}(\theta _{R}, \omega ) = \Delta ^{\textrm{alg}}(\theta , \omega ) + R = 0, $| reducing the argument to the previous case. Indeed, by the above argument no test configuration realizes the infimum defining |$\Delta ^{\textrm{alg}}(\theta _{R}, \omega )$|⁠ , and by ( 17 ) the same conclusion holds also for the infimum defining |$\Delta ^{\textrm{alg}}(\theta ,\omega )$|⁠ , concluding the proof. In other words, since the inequality in the statement of Lemma 5.11 is strict, these test configurations never optimally destabilize (although in the limit |$\kappa \rightarrow 0$| the right-hand side in Lemma 5.11 converges to the left-hand side, for suitably chosen curves |$C \subseteq X$|⁠ ). To summarize, if one restricts to normal and relatively Kähler test configurations of the special form |$\Phi _{C,\kappa }$| coming from deformation to the normal cone , then (restricted) J-semistability would be equivalent to (restricted) J-stability, and these are both closed conditions in the Kähler cone (the fact that J-semistability is a closed condition is well-known; see, e.g., [ 43 ]). Moreover, it is known that uniform J-stability is equivalent to existence of solutions to the J-equation (by [ 7 , 9 ]), and these conditions are in general not equivalent to J-stability.

Using a recent result of Hattori [ 31 ] we now prove that most of the above picture remains true over the larger set of all normal and relatively Kähler test configurations:

  Proposition 5.12. Suppose that |$H$| and |$L$| are ample |$\mathbb{Q}$| -line bundles on a smooth projective surface |$X$|⁠ . Then |$(X,L)$| is |$\textrm{J}^{\textrm{H}}$| -semistable if and only if it is |$\textrm{J}^{\textrm{H}}$| -stable. Moreover, if |$[\theta ]$| and |$[\omega ]$| are rational |$(1,1)$| -classes on |$X$| such that there is no solution to ( 14 ), then there does not exist any optimally destabilizing relatively Kähler test configuration for |$([\theta ],[\omega ])$|⁠ , that is, $$\begin{align*} & \Delta^{\textrm{alg}}(\theta,\omega) < \frac{\textrm{E}_{\theta}^{\textrm{NA}}(\Phi)}{||\Phi||} \end{align*}$$ for all normal and relatively Kähler test configurations |$\Phi \in \mathcal{H}^{\textrm{NA}}(X,[\omega ])$|⁠ .

  Proof. Suppose that |$(X,L)$| is |$\textrm{J}^{\textrm{H}}$| -semistable. By Lemma 2.6 (originally due to [ 35 , Proposition 13] based on computations in [ 39 ]) it is elementary to see that the |$\mathbb{Q}$| -line bundle $$\begin{align*} & 2\frac{H.L}{L^2}L - H \end{align*}$$ is nef (if not, we could find a test configuration |$\Phi _{C,\kappa }$| with |$\kappa $| small enough such that |$\textrm{J}^{\textrm{NA}}(\Phi _{C,\kappa }) < 0$|⁠ , thus contradicting semistability). The same is true replacing |$(H,L)$| by |$(rH,rL)$| where |$r> 0$| is chosen such that |$rc_{1}(H),rc_{1}(L) \in H^{1,1}(X,\mathbb{Z})$|⁠ . One may then apply [ 31 , Proposition 7.4], which in turn implies that |$(X,rL)$| is |$\textrm{J}^{\textrm{rH}}$| -stable (which is equivalent to |$(X,L)$| being |$\textrm{J}^{\textrm{H}}$| -stable), proving the first part. Conversely, stability implies semistability by definition. For the final part of the statement, suppose that |$\Delta ^{\textrm{alg}}(\theta , \omega ) = -R$| for some |$R \geq 0$|⁠ . Then |$[\theta _{R}]:= [\theta ] + R[\omega ]$| is a Kähler class, and moreover |$ \Delta ^{\textrm{alg}}(\theta _{R}, \omega ) = \Delta ^{\textrm{alg}}(\theta , \omega ) + R = 0, $| so that |$(X,[\omega ])$| is |$\textrm{J}^{\theta _{R}}$| -semistable. Moreover, it follows from [ 43 , Theorem 1] and Theorem 5.3 that $$\begin{align*} & R = 2\frac{[\theta].[\omega]}{[\omega]^2} - \frac{\int_C \theta}{\int_C \omega} \end{align*}$$ for some irreducible curve |$C \subseteq X$|⁠ . Since |$R \in \mathbb{Q}$| the |$(1,1)$| -class |$[\theta _{R}]$| is rational, and we may apply the first part of the proof to conclude that |$(X,[\omega ])$| is also |$\textrm{J}^{\theta _{R}}$| -stable. By definition of stability we thus have $$\begin{align*} & \frac{\textrm{E}_{\theta_R}^{\textrm{NA}}(\Phi)}{||\Phi||}> 0 = \Delta^{\textrm{alg}}(\theta_R,\omega) \end{align*}$$ for all |$\Phi \in \mathcal{H}^{\textrm{NA}}(X,[\omega ])$|⁠ . Moreover, $$\begin{align*} & \frac{\textrm{E}_{\theta}^{\textrm{NA}}(\Phi)}{||\Phi||} = \frac{\textrm{E}_{\theta_R}^{\textrm{NA}}(\Phi)}{||\Phi||} - R> \Delta^{\textrm{alg}}(\theta_R, \omega) - R = \Delta^{\textrm{alg}}(\theta, \omega), \end{align*}$$ concluding the proof. Emphasizing the relationship between test configurations for |$(X,[\omega ])$| with |$[\omega ] = c_{1}(L)$| for some ample line bundle on |$X$|⁠ , and test configurations for |$(X,L)$| (see [ 45 ]), and that optimally destabilizing test configurations can by definition only exist if the J-equation is not solvable, the above result in particular yields the following:

Let |$X$| be a smooth projective surface with |$L$| and |$H$| two |$\mathbb{Q}$| -line bundles on |$X$|⁠ . Then no optimally destabilizing normal and relatively Kähler test configuration exists for |$(X,L,H)$|⁠ , in the sense of ( 4 ).

Suppose that |$X$| is a smooth projective Kähler surface admitting at least one negative curve. Then there exist ample |$\mathbb{Q}$| -line bundles |$L$| and |$H$| on |$X$| such that |$(X,L)$| is |$J^{H}$| -stable but not uniformly |$J^{H}$| -stable.

If by contrast |$X$| admits no negative curves, then by [ 43 ] the |$J^{H}$| -equation can always be solved for |$(X,L)$| for every pair of ample line bundles |$(L,H)$| on |$X$|⁠ , and by [ 14 , 25 , 35 ], |$(X,L)$| is then always both |$J^{H}$| -stable and uniformly |$J^{H}$| -stable, for all such pairs. Together with Theorem 5.14 above, this gives a satisfactory answer in all possible cases when |$X$| is a projective surfaces.

  Proof. Proof of Theorem 5.14 First note that if |$X$| is projective and admits at least one negative curve, then one can always find a rational nef and big, but not Kähler, class |$[\omega ^{\prime}] \in H^{1,1}(X,\mathbb{Q})$|⁠ . Indeed, since |$X$| is projective, pick an ample divisor |$A$|⁠ , and since |$X$| admits a curve of negative self-intersection, pick a curve |$C$| such that |$C^{2} <0$|⁠ . One can moreover check that $$\begin{align*} & \sup\left\{t>0 \; | \; A + tC \textrm{is nef.}\right\} = -\frac{A\cdot C}{C^2} \in \mathbb Q. \end{align*}$$ Then the |$\mathbb{Q}$| -divisor |$ B:= A-\frac{A\cdot C}{C^{2}}C $| is pseudoeffective and nef, but not ample. Because $$\begin{align*} & B^2 = A^2 -\frac{(A\cdot C)^2}{C^2}> 0, \end{align*}$$ the class |$B \in \mathcal{B}_{X}$|⁠ , that is, |$B$| is also big. We therefore pick such a nef and big, but not Kähler, |$[\omega ^{\prime}] \in H^{1,1}(X,\mathbb{Q})$| and without loss of generality normalize |$[\omega ^{\prime}]$| and |$[\theta ]$| such that |$[\omega ^{\prime}]^{2} = [\theta ]^{2}$|⁠ . Let |$\textrm{Js}^{\theta }$| (resp. |$\textrm{UJs}^{\theta }$|⁠ ) denote the set of Kähler class |$[\omega ]$| such that |$(X,[\omega ])$| is |$\textrm{J}^{\theta }$| -stable (resp. uniformly |$\textrm{J}^{\theta }$| -stable). Now if |$X$| admits a negative curve, then there exists at least one pair |$([\theta ],[\omega ]) \in \mathcal{C}_{X} \times \mathcal{C}_{X}$| of Kähler classes on |$X$| such that |$(X,[\omega ])$| is not |$\textrm{J}^{\theta }$| -semistable (see [ 43 , Theorem 6]). By continuity (of |$\Delta _{\textrm{NM}}(\theta ,\omega )$|⁠ , see [ 43 , Theorem 1]) one can then argue that there exists a |$\textrm{J}^{\theta }$| -semistable class which is not uniformly |$\textrm{J}^{\theta }$| -stable, and it remains to see that under the above hypotheses there exists a rational such class |$[\omega ]$|⁠ . To see this, note that since |$[\omega ^{\prime}]$| and |$[\theta ]$| are both |$\mathbb{Q}$| -classes by construction, then so is $$\begin{align*} & [\omega_t]:= (1-t)[\omega] + t[\theta], \end{align*}$$ for every |$t \in \mathbb{Q}$|⁠ . By a direct computation identical to that in [ 43 , Proposition 16 and Equation (14)] we moreover see that |$[\omega _{t}] \in \textrm{Js}^{\theta } \setminus \textrm{UJs}^{\theta } \neq \emptyset $| if and only if $$\begin{align*} & 2\frac{[\theta].[\omega_t]}{[\omega_t]^2}[\omega_t] - t^{-1} = 0. \end{align*}$$ which happens precisely for |$t = 1/2$| (by the normalization assumed at the beginnning of the proof; this is moreover of course equivalent to showing that |$\tau _{\theta ,\omega _{t}}$| is nef but not Kähler precisely when |$t = 1/2$|⁠ ). Since |$1/2 \in \mathbb{Q}$|⁠ , |$[\omega _{1/2}] \in H^{1,1}(X,\mathbb{Q})$|⁠ , and Proposition 5.12 shows that |$(X,[\omega _{1/2}])$| is |$J^{\theta }$| -stable but not uniformly |$J^{\theta }$| -stable, concluding the proof.

This sheds additional light on a recent example due to Hattori [ 31 ], of a polarized smooth surface |$(X,L)$| and an ample line bundle |$H$| such that |$(X,L)$| is |$J^{H}$| -stable but not uniformly. Indeed, the above result shows that such examples exist in abundance, namely on any compact Kähler surface with at least one negative curve, for well chosen |$\mathbb{Q}$| -line bundles.

5.4 Real algebraic boundary

  Proposition 5.17. Let |$K \subseteq \mathcal{C}_{X} \times \mathcal{C}_{X}$| be any compact subset whose image under |$\Psi $| is contained in the convex hull of |$k$| pseudoeffective classes, and consider the |$(1,1)$| -class $$\begin{align*} & \tau_{\theta,\omega}:= 2([\theta].[\omega])[\omega] - [\omega]^2[\theta] \in H^{1,1}(X,\mathbb{R}). \end{align*}$$ Then the set $$\begin{align*} & \left(\textrm{Jss} \setminus \textrm{UJs}\right)_{\vert K} = \left\{([\theta],[\omega]) \in K: (X,[\omega]) \; \textrm{is} \; \textrm{J}^{\theta}\textrm{-semistable} \; \textrm{but} \; \textrm{not} \; \textrm{uniformly} \; \textrm{J}^{\theta}\textrm{-stable} \right\} \end{align*}$$ is real algebraic. More explicitly, there is a finite set of curves |$\left \{C_{1}, \dots , C_{\ell } \right \}$| of cardinality |$\ell \leq k\rho (X)$|⁠ , such that $$\begin{align*} & \left(\textrm{Jss} \setminus \textrm{UJs}\right)_{\vert K} = \left\{([\theta],[\omega]) \in K: \prod_{i = 1}^{\ell} \int_{C_i} \tau_{\theta,\omega} = 0 \right\}. \end{align*}$$

  Proof. First note that uniform |$J^{\theta }$| -stability is equivalent to |$\int _{C} \tau _{\theta ,\omega }> 0$|⁠ , and |$J^{\theta }$| -semistability is equivalent to |$\int _{C} \tau _{\theta ,\omega } \geq 0$|⁠ , for all curves |$C \subseteq X$|⁠ . Indeed, the former equivalence is due to [ 7 ], and to prove the second we first note that |$J^{\theta }$| -semistability implies that |$\int _{C} \tau _{\theta ,\omega } \geq 0$|⁠ . If |$X$| is projective this is a direct consequence of Lemma 2.6 . In general, it follows from the standard formula $$\begin{align*} &\textrm{E}^{\textrm{NA}}_{\theta + \epsilon\omega,\omega} = \textrm{E}^{\textrm{NA}}_{\theta,\omega} + \epsilon||.||,\end{align*}$$ that |$J^{\theta }$| -semistability implies uniform |$J^{\theta + \epsilon \omega }$| -stability for any |$\epsilon> 0$|⁠ . This in turn implies that |$\tau _{\theta + \epsilon \omega ,\omega } = \tau _{\theta ,\omega } + \epsilon \omega $| is Kähler for all |$\epsilon>0$|⁠ , by [ 7 ]. Hence |$\tau _{\theta ,\omega }$| must be nef. Moreover, it is immediate from the main Theorem 5.3 to see that both sets $$\begin{align}& \textrm{Jss}_{\vert K} = \left\{([\theta],[\omega]) \in K: (X,[\omega])\;\textrm{is}\;J^{\theta}-\textrm{semistable} \right\}\end{align}$$ (18) and $$\begin{align}& \left\{([\theta],[\omega]) \in K: \int_{C} \tau_{\theta,\omega} \geq 0 \right\}.\end{align}$$ (19) are realized as the smallest closed sets containing $$\begin{align*} & \textrm{UJs}_{\vert K} = \left\{([\theta],[\omega]) \in K: (X,[\omega])\;\textrm{is}\; \textrm{uniformly}\;J^{\theta}-\textrm{stable} \right\} = \left\{([\theta],[\omega]) \in K: \int_C \tau_{\theta,\omega}> 0 \right\}. \end{align*}$$ (Where for the last equality we have used [ 16 , 47 ]. Combined with Theorem 5.3 this entails that there exists a collection of curves |$\left \{C_{i}\right \}$| of cardinality |$\ell \leq k\rho (X)$| such that $$\begin{align*} & \left(\textrm{Jss} \setminus \textrm{UJs}\right)_{\vert K} = \prod_{i = 1}^{\ell} \int_{C_i} \tau_{\theta,\omega} = 0, \end{align*}$$ defining a real algebraic set in |$K$|⁠ . This is what we wanted to prove.

The conditions |$\int _{C_{i}} \tau _{\theta ,\omega } = 0$| define codimension |$1$| subsets by a similar argument to Proposition 3.17 .

In this section we consider certain well-known theorems regarding geometric flows associated with the J-equation and the deformed Hermitian Yang–Mills equation, and remark on a possible relationship between the singularities that develop in each of these cases.

6.1 Remarks on a J-flow result of Song-Weinkove

  Theorem 6.1 (Song-Weinkove, [ 46 ] Theorem 1.4). Let |$X$| be a compact Kähler surface and let |$\alpha , \beta $| be Kähler classes on |$X$|⁠ . Suppose the J-equation ( 22 ) does not admit a solution in |$\alpha $|⁠ . Then, there exists an effective |${\mathbb{R}}$| -divisor $$\begin{align*} & D= \sum_{i=1}^\ell a_i E_i \end{align*}$$ on |$X$| such that the class |$\tau _{\textrm{J}} (\beta ,\alpha )-D$| is Kähler, where $$\begin{align*} & \tau_{\textrm{J}} (\beta,\alpha):= C_{\beta,\alpha}\alpha - \beta. \end{align*}$$ The irreducible components |$E_{i}$| of |$D$| are all curves of negative self-intersection. Moreover, the J-flow |$\varphi (\cdot ,t)$| remains bounded (uniformly in |$t$|⁠ ) in the set |$X\setminus S(D)$| where $$\begin{align*} & S(D):= \bigcup_{i=1}^\ell E_i. \end{align*}$$ Let |$\tilde S$| denote the set given by the intersection of all the sets |$S(D)$| as |$D$| varies over the linear system |$|D|$|⁠ . Then, there exists a sequence |$(x_{j},t_{j})$| of points and times in |$X\times [0,\infty )$| such that, as |$j \to \infty $|⁠ , |$t_{j}\to \infty $| and the points |$x_{j}$| get arbitrarily close to |$\tilde S$|⁠ , and $$\begin{align*} & (|\varphi| + |\triangle_\theta \varphi|)(x_j,t_j) \to \infty. \end{align*}$$

In fact, using the framework of the Zariski decomposition, we can make the above theorem a bit more precise as follows.

  Proposition 6.2. In the above Theorem 6.1 , we can always take |$D$| to be $$\begin{align*} & D = N(\tau_{\textrm{J}} (\beta,\alpha) - \varepsilon\beta), \end{align*}$$ the negative part of the Zariski decomposition of |$\tau _{\textrm{J}}(\beta ,\alpha ) - \varepsilon \beta $| for any |$\varepsilon> 0$| small enough. In particular, we may always take |$\ell \leq \rho (X)$| and |$\tilde S = S(D)$|⁠ . If |$X$| is moreover projective, then |$\ell \leq \rho (X) - 1$|⁠ .

  Proof. The proof of Theorem 6.1 (Theorem 1.4 in [ 46 ]) shows that in fact, the conclusion of the theorem holds for any effective |${\mathbb{R}}$| -divisor such that the class |$\tau _{\textrm{J}}(\beta ,\alpha ) - D$| is a Kähler class. However, recall that the class |$\tau _{\textrm{J}}(\beta ,\alpha ) \in \mathcal P^{+}_{X}$| and so for every |$\varepsilon>0$| small enough, |$\tau _{\textrm{J}} (\beta ,\alpha ) - \varepsilon \beta $| is also in |$\mathcal P^{+}_{X}$|⁠ , and clearly $$\begin{align*} & \tau_{\textrm{J}} (\beta,\alpha) - N(\tau_{\textrm{J}} (\beta,\alpha)-\varepsilon\beta) = Z(\tau_{\textrm{J}} (\beta,\alpha)-\varepsilon\beta) + \varepsilon\beta \end{align*}$$ is a Kähler class for every |$\varepsilon>0$| small enough. Writing $$\begin{align*} & D = N(\tau_{\textrm{J}} (\beta,\alpha)-\varepsilon\beta) = \sum_{i=0}^\ell a_i E_i \end{align*}$$ for irreducible curves |$E_{i}$| on |$X$|⁠ , we see that we can always take |$\ell \leq \rho (X)$|⁠ , because the intersection matrix of the negative part of the Zariski decomposition is always negative-definite. Moreover, according to [ 3 , Proposition 3.13], the class represented by the negative part of the Zariski decomposition always admits precisely one positive current, so we see that |$|D|$| admits the unique element |$D$| in it. This means $$\begin{align*} &\tilde S = S(D) = \bigcup_{i=1}^\ell E_i.\end{align*}$$ If |$X$| is projective, then |$NS(X)_{\mathbb{R}}$| contains a positive eigenvector of the intersection pairing, and we get |$\ell \leq \rho (X) - 1$|⁠ .

6.2 Remarks on the line bundle mean curvature flow and a result of Takahashi

  Theorem 6.3 (Takahashi, [ 49 , Theorem 1.1]). Suppose the initial data |$\omega _{0}, \theta $|⁠ , and |$\hat \Theta = \hat \Theta (\beta ,\alpha )$| satisfy $$\begin{align*} & \tilde\tau_{\textrm{dHYM}}(\theta,\omega_0):= \omega_0 + \cot(\hat\Theta)\theta \end{align*}$$ is a semi-positive form, and we have $$\begin{align}& \arctan(\lambda_{1}(0)) + \arctan(\lambda_{2}(0))>\frac{\pi}{2}\end{align}$$ (25) and |$\hat \Theta> \frac{\pi }{2}$|⁠ , then there exist finitely many curves |$\tilde E_{1},\cdots ,\tilde E_{k}$| of negative self-intersection such that the flow |$\varphi (\cdot ,t)$| converges to a function |$\varphi _{\infty }$| which is smooth on the complement of the set $$\begin{align*} & S = \bigcup_{i=1}^k \tilde E_i. \end{align*}$$ Moreover, the current defined by $$\begin{align*} & F_\infty = \omega_0 + \sqrt{-1} \partial\bar\partial{\varphi_\infty} \end{align*}$$ satisfies the deformed Hermitian Yang–Mills equation in the sense of currents on all of |$X$|⁠ . Here, we make the simple observation that the curves supplied by the above Theorem 6.3 of Takahashi in fact comprise a subset of those supplied by Theorem 6.1 due to Song–Weinkove. More precisely, we have the following.

In the setup of the above Theorem 6.3 , we can take the set of curves |$\left \{\tilde E_{1}, \cdots , \tilde E_{k}\right \}$| to be a subset of the set of curves |$\left \{E_{1},\cdots , E_{\ell }\right \}$| given by Theorem 6.1 . In particular, |$k \leq \ell \leq \rho (X)$|⁠ .

  Proof. Set $$\begin{align*} & \tau_{\textrm{dHYM}}(\beta,\alpha):= \alpha + \cot(\hat\Theta(\beta,\alpha))\beta. \end{align*}$$ The proof of Theorem 6.3 (see [ 49 , Theorem 1.1] or [ 29 , Theorem 1.4]) shows that the curves |$\tilde E_{1},\cdots , \tilde E_{k}$| can be taken to be the irreducible components of any effective |${\mathbb{R}}$| -divisor |$\tilde D = \sum _{i=1}^{k} \tilde a_{i} \tilde E_{i}$| such that $$\begin{align*} & \tau_{\textrm{dHYM}}(\beta,\alpha) - \tilde D \end{align*}$$ is a Kähler class. Just like in Proposition 6.2 , we can take |$\tilde D$| to be $$\begin{align*} & \tilde D = N(\tau_{\textrm{dHYM}}(\beta,\alpha) - \varepsilon\beta), \end{align*}$$ the negative part of the Zariski decomposition of the class |$\tau _{\textrm{dHYM}}(\beta ,\alpha ) - \varepsilon \beta $| for any |$\varepsilon>0 $| small enough. But now observe that $$\begin{align*} & \tau_{\textrm{dHYM}}(\beta,\alpha) = \frac{\int_X\alpha^2}{2\int_X \beta\cdot\alpha} \tau_{\textrm{J}} (\beta,\alpha) + \frac{\left(\int_X \alpha^2\right)\left(\int_X \beta^2\right)}{2 \int_X \beta\cdot\alpha}\beta \end{align*}$$ and so $$\begin{align*} N(\tau_{\textrm{dHYM}}(\beta,\alpha) - \varepsilon\beta) &= N\left(\frac{\int_{X}\alpha^{2}}{2\int_{X} \beta\cdot\alpha} \tau_{\textrm{J}} (\beta,\alpha) - \varepsilon\beta + \frac{\left(\int_{X} \alpha^{2}\right)\left(\int_{X} \beta^{2}\right)}{2 \int_{X} \beta\cdot\alpha}\beta\right)\\ &\leq \frac{\int_{X}\alpha^{2}}{2\int_{X} \beta\cdot\alpha} N\left(\tau_{\textrm{J}}(\beta,\alpha)-\varepsilon C_{\beta,\alpha}\beta\right)+N\left(\frac{\left(\int_{X} \alpha^{2}\right)\left(\int_{X} \beta^{2}\right)}{2 \int_{X} \beta\cdot\alpha}\beta\right)\\ &\leq \frac{\int_{X}\alpha^{2}}{2\int_{X} \beta\cdot\alpha} N\left(\tau_{\textrm{J}}(\beta,\alpha)-\varepsilon C_{\beta,\alpha}\beta\right), \end{align*}$$ by the convexity of the negative part of the Zariski decomposition (see [ 3 , Proposition 3.9 (i)]) and that fact that the negative part of a Kähler class is zero. The proof is now completed by taking |$\varepsilon> 0$| small enough.

6.3 Remarks on a dHYM flow result of Fu-Yau-Zhang

An analogous result holds also in the case of the dHYM flow studied recently by Fu-Yau-Zhang [ 29 ], with the same proof:

In [ 29 , Theorem 1.4], the finitely many curves |$E_{i}$| of negative self-intersection can be taken to be a subset of the set of curves |$\left \{E_{1},\cdots , E_{\ell }\right \}$| given by Theorem 6.1 . In particular the number of such curves is bounded above by |$\rho (X)$|⁠ .

We are grateful to Jacopo Stoppa, Ruadhaí Dervan, Lars Martin Sektnan, Sébastien Boucksom, and the anonymous referees for their feedback and many helpful comments and suggestions. The second named author is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754513 and Aarhus University Research Foundation.

Bayer , A. “ Polynomial Bridgeland stability conditions and the large volume limit .” Geom. Topol. 13 , no. 4 ( 2009 ): 2389 – 425 . https://doi.org/10.2140/gt.2009.13.2389 .

Google Scholar

Blum , H. , Y. Liu , and C. Zhou . “ Optimal destabilization of K-unstable Fano varieties via stability thresholds .” To appear in Geom. Topol ., arXiv:1907.05399 [math.AG] .

Boucksom , S. Cônes positifs des variétés complexes compactes . PhD thesis, Mathématiques [math] . Université Joseph-Fourier-Grenoble I , 2002 tel-00002268 .

Google Preview

Boucksom , S. , T. Hisamoto , and M. Jonsson . “ Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs .” Université de Grenoble. Annales de l’Institut Fourier. Univ. Grenoble I 67 , no. 2 ( 2017 ): 743 – 841 . https://doi.org/10.5802/aif.3096 .

Bridgeland , T. “ Stability conditions on triangulated categories .” Ann. of Math. (2) 166 , no. 2 ( 2007 ): 317 – 45 .

Bridgeland , T. “ Stability conditions on K3 surfaces .” Duke Math. J. 141 , no. 2 ( 2008 ): 241 – 91 .

Chen , G. “ The J-equation and the supercritical deformed Hermitian Yang–Mills equation .” Invent. Math. 225 , no. 2 ( 2021 ): 529 – 602 . https://doi.org/10.1007/s00222-021-01035-3 .

Calabi , E. “ The space of Kähler metrics .” Proc. Int. Congress Math. 1954 , no. 2 ( 1954 ): 206 .

Chen , X. X. “ On the lower bound of the Mabuchi K-energy and its application .” Int. Math. Res. Notices 2000 ( 2000 ): 607 – 23 . https://doi.org/10.1155/S1073792800000337 .

Chu , J. , T. Collins , and M. Lee . “ The space of almost calibrated |$\left (1,1\right )$| forms on a compact Kähler manifold .” Geom. Topol. 25 , no. 5 ( 2021 ): 2573 – 619 . https://doi.org/10.2140/gt.2021.25.2573 .

Chu , J. , M. Lee , and R. Takahashi . “ A Nakai-Moishezon type criterion for supercritical deformed Hermitian Yang–Mills equation .” To appear in J. Differential Geom . Preprint arxiv:2105.10725v3 [math.DG] .

Collins , T. , A. Jacob , and S.-T. Yau . “ |$\left (1,1\right )$| forms with specified Lagrangian phase: a priori estimates and algebraic obstructions .” Camb. J. Math. 8 ( 2020 ): 407 – 52 . https://doi.org/10.4310/CJM.2020.v8.n2.a4 .

Collins , T. , and Y. Shi . “ Stability and the deformed Hermitian Yang–Mills equation .” Preprint arXiv:2004.04831 [math.DG] .

Collins , T. , and G. Székelyhidi . “ Convergence of the J-flow on toric manifolds .” J. Differential Geom. 107 , no. 1 ( 2017 ): 47 – 81 .

Collins , T. , D. Xie , and S.-T. Yau . “ The Deformed Hermitian Yang–Mills Equation in Geometry and Physics .” Geometry and Physics , vol. I , 69 – 90 . Oxford Univ. Press , Oxford , 2018 , https://doi.org/10.1093/oso/9780198802013.003.0004 .

Datar , V. , and V. Pingali . “ A numerical criterion for generalised Monge-Ampère equations on projective manifolds .” Geom. Funct. Anal. 31 , no. 4 ( 2021 ): 767 – 814 . https://doi.org/10.1007/s00039-021-00577-1 .

Delcroix , T. , and S. Jubert . “ An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons .” Preprint arXiv:2202.02996v2 [math.DG] .

Delcroix , T. “ K-stability of Fano spherical varieties .” Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 53 , no. 3 ( 2020 ): 615 – 62 .

Delcroix , T. “ Uniform K-stability of polarized spherical varieties .” Preprint arXiv:2009.06463v2 [math.AG] .

Demailly , J.-P. , and M. Păun . “ Numerical characterization of the Kähler cone of a compact Kähler manifold .” Ann. of Math. (2) 159 , no. 3 ( 2004 ): 1247 – 74 .

Dervan , R. “ Uniform stability of twisted constant scalar curvature Kähler metrics .” Int. Math. Res. Not. IMRN 2016 ( 2016 ): 4728 – 83 . https://doi.org/10.1093/imrn/rnv291 .

Dervan , R. “ Stability conditions in geometric invariant theory .” Preprint arXiv:2207.04766v1 [math.AG] .

Dervan , R. “ K-semistability of optimal degenerations .” Quart. J. Math. Oxford Ser. (2) 71 , no. 3 ( 2020 ): 989 – 95 . https://doi.org/10.1093/qmathj/haaa012 .

Dervan , R. , J.B. McCarthy , and L.M. Sektnan . “ Z -critical connections and Bridgeland stability conditions .” Preprint arXiv:2012.10426v3 [math.DG] .

Dervan , R. , and J. Ross . “ K-stability for Kähler manifolds .” Math. Res. Lett. 24 , no. 3 ( 2017 ): 689 – 739 . https://doi.org/10.4310/MRL.2017.v24.n3.a5 .

Donaldson , S. K. “ Moment maps and diffeomorphisms .” Asian J. Math. 3 , no. 1 ( 1999 ): 1 – 16 . https://doi.org/10.4310/AJM.1999.v3.n1.a1 .

Donaldson , S. K. “ Scalar curvature and stability of toric varieties .” J. Differential Geom. 62 , no. 2 ( 2002 ): 289 – 349 .

Donaldson , S. K. “ Lower bounds on the Calabi functional .” J. Differential Geom. 70 , no. 3 ( 2005 ): 453 – 72 . https://doi.org/10.4310/jdg/1143642909 .

Fu , J. , S.-T. Yau , and D. Zhang . “ A deformed Hermitian Yang–Mills flow .” Preprint arXiv:2105.13576v3 [math.DG] .

Futaki , A. “ An obstruction to the existence of Einstein-Kähler metrics .” Invent. Math. 73 , no. 3 ( 1983 ): 437 – 43 . https://doi.org/10.1007/BF01388438 .

Hattori , M. “ A decomposition formula for J-stability and its applications .” Preprint arXiv:2103.04603v2 [math.AG] .

Hisamoto , T. “ Geometric flow, multiplier ideal sheaves and optimal destabilizer for Fano manifold .” Preprint arXiv:1901.08480v3 [math.DG] .

Jacob , A. , and S.-T. Yau . “ A special Lagrangian type equation for holomorphic line bundles .” Math. Ann. 369 , no. 1-2 ( 2017 ): 869 – 98 . https://doi.org/10.1007/s00208-016-1467-1 .

Lamari , A. “ Le cône kählérien d’une surface .” J. Math. Pures Appl. (9) 78 , no. 3 ( 1999 ): 249 – 63 .

Lejmi , M. , and G. Székelyhidi . “ The J-flow and stability .” Adv. Math. 274 ( 2015 ): 404 – 31 . https://doi.org/10.1016/j.aim.2015.01.012 .

Mabuchi , T. “ Some symplectic geometry on compact Kähler manifolds I .” Osaka J. Math. 24 , no. 2 ( 1987 ): 227 – 52 .

McCarthy , J. B. Stability conditions and canonical metrics . PhD thesis , 2022 .

Pingali , P. “ The deformed Hermitian Yang–Mills equation on three-folds .” Anal. PDE 15 , no. 4 ( 2022 ): 921 – 35 . https://doi.org/10.2140/apde.2022.15.921 .

Ross , J. , and R. Thomas . “ An obstruction to the existence of constant scalar curvature Kähler metrics .” J. Differential Geom. 72 , no. 3 ( 2006 ): 429 – 66 .

Sektnan , L. M. , and C. Tipler . “ Analytic K-semistability and wall-crossing .” Preprint arXiv:2212.08383 [math.DG] .

Székelyhidi , G. “ Optimal test-configurations for toric varieties .” J. Differential Geom. 80 , no. 3 ( 2008 ): 501 – 23 .

Sjöström Dyrefelt , Z. “ K-semistability of cscK manifolds with transcendental cohomology class .” J. Geom. Anal. 28 , no. 4 ( 2018 ): 2927 – 60 . https://doi.org/10.1007/s12220-017-9942-9 .

Sjöström Dyrefelt , Z. “ Optimal lower bounds for Donaldson’s J-functional .” Adv. Math. 374 ( 2020 ): 107271 , 37 pp . https://doi.org/10.1016/j.aim.2020.107271 .

SjöströmDyrefelt , Z. “ Openness of uniform K-stability in the Kähler cone .” Preprint arXiv:2011,14806v1 [math.DG] .

Sjöström Dyrefelt , Z. K-stabilité et variétés kähleriennes avec classe transcendante . PhD thesis, Université de Toulouse , 2017 . http://thesesups.ups-tlse.fr/3577/ .

Song , J. , and B. Weinkove . “ On the convergence and singularities of the J-flow with applications to the Mabuchi energy .” Comm. Pure Appl. Math. 61 , no. 2 ( 2008 ): 210 – 29 . https://doi.org/10.1002/cpa.20182 .

Song , J. “ Nakai-Moishezon criterions for complex Hessian equations .” Preprint arXiv:2012.07956v1 [math.DG] .

Stoppa , J. “ Twisted constant scalar curvature Kähler metrics and Kähler slope stability .” J. Differential Geom. 83 , no. 3 ( 2009 ): 663 – 91 . https://doi.org/10.4310/jdg/1264601038 .

Takahashi , R. “ Collapsing of the line bundle mean curvature flow on Kähler surfaces .” Calc. Var. Partial Differential Equations 60 , no. 1 ( 2021 ): paper no. 27, 18 pp .

Takahashi , R. “ The Kähler-Ricci flow and quantitative bounds for Donaldson-Futaki invariants of optimal degenerations .” Proc. Amer. Math. Soc. 148 , no. 8 ( 2020 ): 3527 – 36 . https://doi.org/10.1090/proc/15004 .

Tian , G. “ Kähler-Einstein metrics with positive scalar curvature .” Invent. Math. 130 , no. 1 ( 1997 ): 1 – 37 . https://doi.org/10.1007/s002220050176 .

Yau , S.-T. “ On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I .” Comm. Pure Appl. Math. 31 , no. 3 ( 1978 ): 339 – 411 . https://doi.org/10.1002/cpa.3160310304 .

Zariski , O. “ The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface .” Ann. of Math. (2) 76 ( 1962 ): 560 – 615 .

Zhang , Z. “ Optimal destabilizing centers and equivariant K-stability .” Invent. Math. 226 , no. 1 ( 2021 ): 195 – 223 . https://doi.org/10.1007/s00222-021-01046-0 .

Zhou , C. “ On the shape of K-semistable domain and wall crossing for K-stability .” Preprint arXiv:2302.13503 [math.AG] .

Zhou , C. “ Chamber decomposition for K-semistable domains and VGIT .” Preprint arXiv:2303.10963 [math.AG] .

Email alerts

Citing articles via.

• Recommend to your Library

Affiliations

• Online ISSN 1687-0247
• Print ISSN 1073-7928
• Copyright © 2024 Oxford University Press
• About Oxford Academic
• Publish journals with us
• University press partners
• What we publish
• New features  
• Open access
• Institutional account management
• Rights and permissions
• Get help with access
• Accessibility
• Advertising
• Media enquiries
• Oxford University Press
• Oxford Languages
• University of Oxford

Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide

• Copyright © 2024 Oxford University Press
• Cookie settings
• Cookie policy
• Privacy policy
• Legal notice

This Feature Is Available To Subscribers Only

Sign In or Create an Account

This PDF is available to Subscribers Only

For full access to this pdf, sign in to an existing account, or purchase an annual subscription.

• OU Homepage
• The University of Oklahoma

OU Researcher Receives $3.1M Grant for Clean Hydrogen Technologies

The project is part of $750M in funding through President Biden’s Bipartisan Infrastructure Law granted to reduce the cost of clean hydrogen.

NORMAN, OKLA.  – Hanping Ding, Ph.D., an assistant professor in the School of Aerospace and Mechanical Engineering at the University of Oklahoma, has been awarded a $3.1 million grant from the Hydrogen and Fuel Cell Technologies Office in the Department of Energy through the Bipartisan Infrastructure Law to further research in clean hydrogen production. The funding is part of a $750 million effort in President Biden’s Investing in American agenda. The money from the Department of Energy will go to 52 projects across 24 states to position the United States as a global leader in the clean hydrogen industry.

The combined outcomes of the 52 projects should allow the U.S. to produce enough technology per year to power 15% of medium- and heavy-duty trucks sold each year, produce an extra 1.3 million tons of clean hydrogen annually, and support more than 1,500 new jobs.

Ding’s three-year project will address the technical challenges of proton-conducting solid oxide electrolysis cell stacks, a type of technology that splits water into hydrogen and oxygen gases using electricity. By enabling this process, the stacks allow for the efficient conversion of electrical energy into chemical energy, producing hydrogen as a clean and renewable fuel source. Hydrogen produced through this method can result in zero greenhouse gas emissions. The goal of Ding’s project is to develop the technology to be suitable for real-world use. 

“This project will advance the technology maturity of [the technology] and, from a bigger picture, promote the green hydrogen applications of the state of Oklahoma,” said Ding.

Finding a way to store and convert energy is necessary to make renewable and sustainable energy more feasible. Clean hydrogen is a way for industries to reduce emissions while continuing to provide services needed for modern life. Ding’s  Advanced Materials and Clean Energy Laboratory researches technological improvements to reach net-zero emissions. The lab specializes in materials research, development and prototype system demonstration for fuel cells, hydrogen production and electrochemical processing.

Under this grant, OU will collaborate with researchers at Massachusetts Institute of Technology, Kansas State University, and Chemtronergy LLC to deliver this advanced electrolysis technology. The Idaho National Laboratory and Lawrence Livermore National Laboratory also support the research.

Ding’s project is well aligned with the goals of the Oklahoma Hydrogen Roadmap from the Hydrogen Production, Transportation and Infrastructure Task Force report, which includes a near-term goal of hydrogen storage and innovative technologies and long-term goals of low carbon hydrogen and equipment manufacturing.

About the project

The project, “Development of Readily Manufactured and Interface Engineered Proton-Conducting Solid Oxide Electrolysis Cells with High Efficiency and Durability,” is funded through the Department of Energy grant DE-FOA-0002922.

About the University of Oklahoma

Founded in 1890, the University of Oklahoma is a public research university located in Norman, Oklahoma. As the state’s flagship university, OU serves the educational, cultural, economic and health care needs of the state, region and nation. OU was named the state’s highest-ranking university in  U.S. News & World Report’s  most recent Best Colleges list .  For more information about the university, visit  ou.edu .

Recent News

University of oklahoma joins u.s. space command’s academic engagement enterprise.

The University of Oklahoma has been selected to join the U.S. Space Command's Academic Engagement Enterprise. This program fosters collaboration between the Space Command’s enterprise and academic institutions to cultivate space-focused research, innovation and education.

Researchers Discover Cell ‘Crosstalk’ That Triggers Cancer Cachexia

New research from the University of Oklahoma reveals a previously unknown chain of events sparking the development of cancer cachexia, a debilitating muscle-wasting condition that almost always occurs in people diagnosed with pancreatic cancer. The research, led by Min Li, Ph.D., a professor in the OU College of Medicine, is published in the journal Cancer Cell.

Hanping Ding, Ph.D., an assistant professor in the School of Aerospace and Mechanical Engineering at the University of Oklahoma, has been awarded a $3.1 million grant from the Hydrogen and Fuel Cell Technologies Office in the Department of Energy through the Bipartisan Infrastructure Law to further research in clean hydrogen production. The funding is part of a $750 million effort in President Biden’s Investing in American agenda.

• Accessibility
• Sustainability
• OU Job Search
• Legal Notices
• Resources and Offices
• OU Report It!

• Campus News

University releases new details ahead of July 1 implementation of Campus Self-Defense Act, Campus Conversation planned

Friday, April 12, 2024

Following Friday’s (April 12) Board of Governors approval of BOG Rule 5.14 — Deadly Weapons, Dangerous Objects and the West Virginia Campus Self-Defense Act — the University is moving into the next phase of the campus carry implementation process. Passed by the West Virginia Legislature and signed into law by Gov. Jim Justice in 2023, the Campus Self-Defense Act allows a person to carry a concealed pistol or revolver on the grounds of an institution of higher education, with some exceptions, if that person has a current and valid license to carry a concealed deadly weapon. The University must comply with the law and BOG Rule 5.14 details what the law will look like in practice on WVU Campuses once the law goes into effect on July 1.

Watch a video providing additional information about campus carry at WVU. To continue the discussion with the University community, a Campus Conversation is scheduled for 11 a.m. Thursday (April 18) via Zoom. Join the Campus Conversation. Presenters include:     • Corey Farris , dean of students and co-chair of the Campus Safety Steering Group ,     • Travis Mollohan , associate vice president for government relations and collaboration, and chair of the Campus Carry Sub-Group     • Kevin Cimino , deputy general counsel

Zoom webinar capacity is limited to 1,000 attendees. Overflow viewing will be available via the University’s YouTube channel . The Campus Conversation will supplement scheduled in-person meetings focused on campus with faculty, staff, students and other University stakeholders. Read more from Friday’s (April 12) BOG meeting. Find additional information on the campus carry webpage , part of the Safety and Wellness website .  

New resources include:

·      Updated FAQ

·      Inside WVUToday interview with Campus Carry Sub-group Chair Travis Mollohan

Learn more about University Police training opportunities. Questions not addressed through the listed resources can be directed to [email protected] . 

• Short report
• Open access
• Published: 12 April 2024

A modified action framework to develop and evaluate academic-policy engagement interventions

• Petra Mäkelä   ORCID: orcid.org/0000-0002-0938-1175 1 ,
• Annette Boaz   ORCID: orcid.org/0000-0003-0557-1294 2 &
• Kathryn Oliver   ORCID: orcid.org/0000-0002-4326-5258 1  

Implementation Science volume  19 , Article number:  31 ( 2024 ) Cite this article

231 Accesses

17 Altmetric

Metrics details

There has been a proliferation of frameworks with a common goal of bridging the gap between evidence, policy, and practice, but few aim to specifically guide evaluations of academic-policy engagement. We present the modification of an action framework for the purpose of selecting, developing and evaluating interventions for academic-policy engagement.

We build on the conceptual work of an existing framework known as SPIRIT (Supporting Policy In Health with Research: an Intervention Trial), developed for the evaluation of strategies intended to increase the use of research in health policy. Our aim was to modify SPIRIT, (i) to be applicable beyond health policy contexts, for example encompassing social, environmental, and economic policy impacts and (ii) to address broader dynamics of academic-policy engagement. We used an iterative approach through literature reviews and consultation with multiple stakeholders from Higher Education Institutions (HEIs) and policy professionals working at different levels of government and across geographical contexts in England, alongside our evaluation activities in the Capabilities in Academic Policy Engagement (CAPE) programme.

Our modifications expand upon Redman et al.’s original framework, for example adding a domain of ‘Impacts and Sustainability’ to capture continued activities required in the achievement of desirable outcomes. The modified framework fulfils the criteria for a useful action framework, having a clear purpose, being informed by existing understandings, being capable of guiding targeted interventions, and providing a structure to build further knowledge.

The modified SPIRIT framework is designed to be meaningful and accessible for people working across varied contexts in the evidence-policy ecosystem. It has potential applications in how academic-policy engagement interventions might be developed, evaluated, facilitated and improved, to ultimately support the use of evidence in decision-making.

Peer Review reports

Contributions to the literature

There has been a proliferation of theories, models and frameworks relating to translation of research into practice. Few specifically relate to engagement between academia and policy.

Challenges of evidence-informed policy-making are receiving increasing attention globally. There is a growing number of academic-policy engagement interventions but a lack of published evaluations.

This article contributes a modified action framework that can be used to guide how academic-policy engagement interventions might be developed, evaluated, facilitated, and improved, to support the use of evidence in policy decision-making.

Our contribution demonstrates the potential for modification of existing, useful frameworks instead of creating brand-new frameworks. It provides an exemplar for others who are considering when and how to modify existing frameworks to address new or expanded purposes while respecting the conceptual underpinnings of the original work.

Academic-policy engagement refers to ways that Higher Education Institutions (HEIs) and their staff engage with institutions responsible for policy at national, regional, county or local levels. Academic-policy engagement is intended to support the use of evidence in decision-making and in turn, improve its effectiveness, and inform the identification of barriers and facilitators in policy implementation [ 1 , 2 , 3 ]. Challenges of evidence-informed policy-making are receiving increasing attention globally, including the implications of differences in cultural norms and mechanisms across national contexts [ 4 , 5 ]. Although challenges faced by researchers and policy-makers have been well documented [ 6 , 7 ], there has been less focus on actions at the engagement interface. Pragmatic guidance for the development, evaluation or comparison of structured responses to the challenges of academic-policy engagement is currently lacking [ 8 , 9 ].

Academic-policy engagement exists along a continuum of approaches from linear (pushing evidence out from academia or pulling evidence into policy), relational (promoting mutual understandings and partnerships), and systems approaches (addressing identified barriers and facilitators) [ 4 ]. Each approach is underpinned by sets of beliefs, assumptions and expectations, and each raises questions for implementation and evaluation. Little is known about which academic-policy engagement interventions work in which settings, with scarce empirical evidence to inform decisions about which interventions to use, when, with whom, or why, and how organisational contexts can affect motivation and capabilities for such engagement [ 10 ]. A deeper understanding through the evaluation of engagement interventions will help to identify inhibitory and facilitatory factors, which may or may not transfer across contexts [ 11 ].

The intellectual technologies [ 12 ] of implementation science have proliferated in recent decades, including models, frameworks and theories that address research translation and acknowledge difficulties in closing the gap between research, policy and practice [ 13 ]. Frameworks may serve overlapping purposes of describing or guiding processes of translating knowledge into practice (e.g. the Quality Implementation Framework [ 14 ]); or helping to explain influences on implementation outcomes (e.g. the Theoretical Domains Framework [ 15 ]); or guiding evaluation (e.g. the RE-AIM framework [ 16 , 17 ]. Frameworks can offer an efficient way to look across diverse settings and to identify implementation differences [ 18 , 19 ]. However, the abundance of options raises its own challenges when seeking a framework for a particular purpose, and the use of a framework may mean that more weight is placed on certain aspects, leading to a partial understanding [ 13 , 17 ].

‘Action frameworks’ are predictive models that intend to organise existing knowledge and enable a logical approach for the selection, implementation and evaluation of intervention strategies, thereby facilitating the expansion of that knowledge [ 20 ]. They can guide change by informing and clarifying practical steps to follow. As flexible entities, they can be adapted to accommodate new purposes. Framework modification may include the addition of constructs or changes in language to expand applicability to a broader range of settings [ 21 ].

We sought to identify one organising framework for evaluation activities in the Capabilities in Academic-Policy Engagement (CAPE) programme (2021–2023), funded by Research England. The CAPE programme aimed to understand how best to support effective and sustained engagement between academics and policy professionals across the higher education sector in England [ 22 ]. We first searched the literature and identified an action framework that was originally developed between 2011 and 2013, to underpin a trial known as SPIRIT (Supporting Policy In health with Research: an Intervention Trial) [ 20 , 23 ]. This trial evaluated strategies intended to increase the use of research in health policy and to identify modifiable points for intervention.

We selected the SPIRIT framework due to its potential suitability as an initial ‘road map’ for our evaluation of academic-policy interventions in the CAPE programme. The key elements of the original framework are catalysts, organisational capacity, engagement actions, and research use. We wished to build on the framework’s embedded conceptual work, derived from literature reviews and semi-structured interviews, to identify policymakers’ views on factors that assist policy agencies’ use of research [ 20 ]. The SPIRIT framework developers defined its “locus for change” as the policy organisation ( [ 20 ], p. 151). They proposed that it could offer the beginning of a process to identify and test pathways in policy agencies’ use of evidence.

Our goal was to modify SPIRIT to accommodate a different locus for change: the engagement interface between academia and policy. Instead of imagining a linear process in which knowledge comes from researchers and is transmitted to policy professionals, we intended to extend the framework to multidirectional relational and system interfaces. We wished to include processes and influences at individual, organisational and system levels, to be relevant for HEIs and their staff, policy bodies and professionals, funders of engagement activities, and facilitatory bodies. Ultimately, we seek to address a gap in understanding how engagement strategies work, for whom, how they are facilitated, and to improve the evaluation of academic-policy engagement.

We aimed to produce a conceptually guided action framework to enable systematic evaluation of interventions intending to support academic-policy engagement.

We used a pragmatic combination of processes for framework modification during our evaluation activities in the CAPE programme [ 22 ]. The CAPE programme included a range of interventions: seed funding for academic and policy professional collaboration in policy-focused projects, fellowships for academic placements in policy settings, or for policy professionals with HEI staff, training for policy professionals, and a range of knowledge exchange events for HEI staff and policy professionals. We modified the SPIRIT framework through iterative processes shown in Table  1 , including reviews of literature; consultations with HEI staff and policy professionals across a range of policy contexts and geographic settings in England, through the CAPE programme; and piloting, refining and seeking feedback from stakeholders in academic-policy engagement.

A number of characteristics of the original SPIRIT framework could be applied to academic-policy engagement. While keeping the core domains, we modified the framework to capture dynamics of engagement at multiple academic and policy levels (individuals, organisations and system), extending beyond the original unidirectional focus on policy agencies’ use of research. Components of the original framework, the need for modifications, and their corresponding action-oriented implications are shown in Table  2 . We added a new domain, ‘Impacts and Sustainability’, to consider transforming and enduring aspects at the engagement interface. The modified action framework is shown in Fig.  1 .

SPIRIT Action Framework Modified for Academic-Policy Engagement Interventions (SPIRIT-ME), adapted with permission from the Sax Institute. Legend: The framework acknowledges that elements in each domain may influence other elements through mechanisms of action and that these do not necessarily flow through the framework in a ‘pipeline’ sequence. Mechanisms of action are processes through which engagement strategies operate to achieve desired outcomes. They might rely on influencing factors, catalysts, an aspect of an intervention action, or a combination of elements

Identifying relevant theories or models for missing elements

Catalysts and capacity.

Within our evaluation of academic-policy interventions, we identified a need to develop the original domain of catalysts beyond ‘policy/programme need for research’ and ‘new research with potential policy relevance’. Redman et al. characterised a catalyst as “a need for information to answer a particular problem in policy or program design, or to assist in supporting a case for funding” in the original framework (p. 149). We expanded this “need for information” to a perceived need for engagement, by either HEI staff or policy professionals, linking to the potential value they perceived in engaging. Specifically, there was a need to consider catalysts at the level of individual engagement, for example HEI staff wanting research to have real-world impact, or policy professionals’ desires to improve decision-making in policy, where productive interactions between academic and policy stakeholders are “necessary interim steps in the process that lead to societal impact” ( [ 24 ], p. 214). The catalyst domain expands the original emphasis on a need for research, to take account of challenges to be overcome by both the academic and policy communities in knowing how, and with whom, to engage and collaborate with [ 25 ].

We used a model proposing that there are three components for any behaviour: capability, opportunity and motivation, which is known as the COM-B model [ 26 ]. Informed by CAPE evaluation activities and our discussions with stakeholders, we mapped the opportunity and motivation constructs into the ‘catalysts’ domain of the original framework. Opportunity is an attribute of the system that can facilitate engagement. It may be a tangible factor such as the availability of seed funding, or a perceived social opportunity such as institutional support for engagement activities. Opportunity can act at the macro level of systems and organisational structures. Motivation acts at the micro level, deriving from an individual’s mental processes that stimulate and direct their behaviours; in this case, taking part in academic-policy engagement actions. The COM-B model distinguishes between reflective motivation through conscious planning and automatic motivation that may be instinctive or affective [ 26 ].

We presented an early application of the COM-B model to catalysts for engagement at an academic conference, enabling an informal exploration of attendees’ subjective views on the clarity and appropriateness, when developing the framework. This application introduces possibilities for intervention development and support by highlighting ‘opportunities’ and ‘motivations’ as key catalysts in the modified framework.

Within the ‘capacity’ domain, we retained the original levels of individuals, organisations and systems. We introduced individual capability as a construct from the COM-B model, describing knowledge, skills and abilities to generate behaviour change as a precursor of academic-policy engagement. This reframing extends the applicability to HEI staff as well as policy professionals. It brings attention to different starting conditions for individuals, such as capabilities developed through previous experience, which can link with social opportunity (for example, through training or support) as a catalyst.

Engagement actions

We identified a need to modify the original domain ‘engagement actions’ to extend the focus beyond the use of research. We added three categories of engagement actions described by Best and Holmes [ 27 ]: linear, relational, and systems. These categories were further specified through a systematic mapping of international organisations’ academic-policy engagement activities [ 5 ]. This framework modification expands the domain to encompass: (i) linear ‘push’ of evidence from academia or ‘pull’ of evidence into policy agencies; (ii) relational approaches focused on academic-policy-maker collaboration; and (iii) systems’ strategies to facilitate engagement for example through strategic leadership, rewards or incentives [ 5 ].

We retained the elements in the original framework’s ‘outcomes’ domain (instrumental, tactical, conceptual and imposed), which we found could apply to outcomes of engagement as well as research use. For example, discussions between a policy professional and a range of academics could lead to a conceptual outcome by considering an issue through different disciplinary lenses. We expanded these elements by drawing on literature on engagement outcomes [ 28 ] and through sense-checking with stakeholders in CAPE. We added capacity-building (changes to skills and expertise), connectivity (changes to the number and quality of relationships), and changes in organisational culture or attitude change towards engagement.

Impacts and sustainability

The original framework contained endpoints described as: ‘Better health system and health outcomes’ and ‘Research-informed health policy and policy documents’. For modification beyond health contexts and to encompass broader intentions of academic-policy engagement, we replaced these elements with a new domain of ‘Impacts and sustainability’. This domain captures the continued activities required in achievement of desirable outcomes [ 29 ]. The modification allows consideration of sustainability in relation to previous stages of engagement interventions, through the identification of beneficial effects that are sustained (or not), in which ways, and for whom. Following Borst [ 30 ], we propose a shift from the expectation that ‘sustainability’ will be a fixed endpoint. Instead, we emphasise the maintenance work needed over time, to sustain productive engagement.

Influences and facilitators

We modified the overarching ‘Policy influences’ (such as public opinion and media) in the original framework, to align with factors influencing academic-policy engagement beyond policy agencies’ use of research. We included influences at the level of the individual (for example, individual moral discretion [ 31 ]), the organisation (for example, managerial practices [ 31 ]) and the system (for example, career incentives [ 32 ]). Each of these processes takes place in the broader context of social, policy and financial environments (that is, potential sources of funding for engagement actions) [ 29 ].

We modified the domain ‘Reservoir of relevant and reliable research’ underpinning the original framework, replacing it with ‘Reservoir of people skills’, to emphasise intangible facilitatory work at the engagement interface, in place of concrete research outputs. We used the ‘Promoting Action on Research Implementation in Health Services’ (PARiHS) framework [ 33 , 34 ], which gives explicit consideration to facilitation mechanisms for researchers and policy-makers [ 13 ] . Here, facilitation expertise includes mechanisms that focus on particular goals (task-oriented facilitation) or enable changes in ways of working (holistic-oriented facilitation). Task-orientated facilitation skills might include, for example, the provision of contacts, practical help or project management skills, while holistic-oriented facilitation involves building and sustaining partnerships or support skills’ development across a range of capabilities. These conceptualisations aligned with our consultations with facilitators of engagement in CAPE. We further extended these to include aspects identified in our evaluation activities: strategic planning, contextual awareness and entrepreneurial orientation.

Piloting and refining the modified framework through stakeholder engagement

We piloted an early version of the modified framework to develop a survey for all CAPE programme participants. During this pilot stage, we sought feedback from the CAPE delivery team members across HEI and policy contexts in England. CAPE delivery team members are based at five collaborating universities with partners in the Parliamentary Office for Science and Technology (POST) and Government Office for Science (GO-Science), and Nesta (a British foundation that supports innovation). The HEI members include academics and professional services knowledge mobilisation staff, responsible for leading and coordinating CAPE activities. The delivery team comprised approximately 15–20 individuals (with some fluctuations according to individual availabilities).

We assessed appropriateness and utility, refined terminology, added domain elements and explored nuances. For example, stakeholders considered the multi-layered possibilities within the domain ‘capacity’, where some HEI or policy departments may demonstrate a belief that it is important to use research in policy, but this might not be the perception of the organisation as a whole. We also sought stakeholders’ views on the utility of the new domains, for example, the identification of facilitator expertise such as acting as a knowledge broker or intermediary; providing training, advice or guidance; facilitating engagement opportunities; creating engagement programmes; and sustainability of engagement that could be conceptualised at multiple levels: personally, in processes or through systems.

Testing against criteria for useful action framework

The modified framework fulfils the properties of a useful action framework [ 20 ]:

It has a clearly articulated purpose: development and evaluation of academic-policy engagement interventions through linear, relational and/or system approaches. It has identified loci for change, at the level of the individual, the organisation or system.

It has been informed by existing understandings, including conceptual work of the original SPIRIT framework, conceptual models identified from the literature, published empirical findings, understandings from consultation with stakeholders, and evaluation activities in CAPE.

It can be applied to the development, implementation and evaluation of targeted academic-policy engagement actions, the selection of points for intervention and identification of potential outcomes, including the work of sustaining them and unanticipated consequences.

It provides a structure to build knowledge by guiding the generation of hypotheses about mechanisms of action in academic-policy engagement interventions, or by adapting the framework further through application in practice.

The proliferation of frameworks to articulate processes of research translation reveals a need for their adaptation when applied in specific contexts. The majority of models in implementation science relate to translation of research into practice. By contrast, our focus was on engagement between academia and policy. There are a growing number of academic-policy engagement interventions but a lack of published evaluations [ 10 ].

Our framework modification provides an exemplar for others who are considering how to adapt existing conceptual frameworks to address new or expanded purposes. Field et al. identified the multiple, idiosyncratic ways that the Knowledge to Action Framework has been applied in practice, demonstrating its ‘informal’ adaptability to different healthcare settings and topics [ 35 ]. Others have reported on specific processes for framework refinement or extension. Wiltsey Stirman et al. adopted a framework that characterised forms of intervention modification, using a “pragmatic, multifaceted approach” ( [ 36 ], p.2). The authors later used the modified version as a foundation to build a further framework to encompass implementation strategies in a range of settings [ 21 ]. Oiumet et al. used the approach of borrowing from a different disciplinary field for framework adaptation, by using a model of absorptive capacity from management science to develop a conceptual framework for civil servants’ absorption of research knowledge [ 37 ].

We also took the approach of “adapting the tools we think with” ( [ 38 ], p.305) during our evaluation activities on the CAPE programme. Our conceptual modifications align with the literature on motivation and entrepreneurial orientation in determining policy-makers’ and researchers’ intentions to carry out engagement in addition to ‘usual’ roles [ 39 , 40 ]. Our framework offers an enabler for academic-policy engagement endeavours, by providing a structure for approaches beyond the linear transfer of information, emphasising the role of multidirectional relational activities, and the importance of their facilitation and maintenance. The framework emphasises the relationship between individuals’ and groups’ actions, and the social contexts in which these are embedded. It offers additional value by capturing the organisational and systems level factors that influence evidence-informed policymaking, incorporating the dynamic features of contexts shaping engagement and research use.

Conclusions

Our modifications extend the original SPIRIT framework’s focus on policy agencies’ use of research, to encompass dynamic academic-policy engagement at the levels of individuals, organisations and systems. Informed by the knowledge and experiences of policy professionals, HEI staff and knowledge mobilisers, it is designed to be meaningful and accessible for people working across varied contexts and functions in the evidence-policy ecosystem. It has potential applications in how academic-policy engagement interventions might be developed, evaluated, facilitated and improved, and it fulfils Redman et al.’s criteria as a useful action framework [ 20 ].

We are testing the ‘SPIRIT-Modified for Engagement’ framework (SPIRIT-ME) through our ongoing evaluation of academic-policy engagement activities. Further empirical research is needed to explore how the framework may capture ‘additionality’, that is, to identify what is achieved through engagement actions in addition to what would have happened anyway, including long-term changes in strategic behaviours or capabilities [ 41 , 42 , 43 ]. Application of the modified framework in practice will highlight its strengths and limitations, to inform further iterative development and adaptation.

Availability of data and materials

Not applicable.

Stewart R, Dayal H, Langer L, van Rooyen C. Transforming evidence for policy: do we have the evidence generation house in order? Humanit Soc Sci Commun. 2022;9(1):1–5.

Article   Google Scholar  

Sanderson I. Complexity, ‘practical rationality’ and evidence-based policy making. Policy Polit. 2006;34(1):115–32.

Lewin S, Glenton C, Munthe-Kaas H, Carlsen B, Colvin CJ, Gülmezoglu M, et al. Using Qualitative Evidence in Decision Making for Health and Social Interventions: An Approach to Assess Confidence in Findings from Qualitative Evidence Syntheses (GRADE-CERQual). PLOS Med. 2015;12(10):e1001895.

Article   PubMed   PubMed Central   Google Scholar  

Bonell C, Meiksin R, Mays N, Petticrew M, McKee M. Defending evidence informed policy making from ideological attack. BMJ. 2018;10(362):k3827.

Hopkins A, Oliver K, Boaz A, Guillot-Wright S, Cairney P. Are research-policy engagement activities informed by policy theory and evidence? 7 challenges to the UK impact agenda. Policy Des Pract. 2021;4(3):341–56.

Google Scholar  

Head BW. Toward More “Evidence-Informed” Policy Making? Public Adm Rev. 2016;76(3):472–84.

Walker LA, Lawrence NS, Chambers CD, Wood M, Barnett J, Durrant H, et al. Supporting evidence-informed policy and scrutiny: A consultation of UK research professionals. PLoS ONE. 2019;14(3):e0214136.

Article   CAS   PubMed   PubMed Central   Google Scholar  

Graham ID, Tetroe J, Group the KT. Planned action theories. In: Knowledge Translation in Health Care. John Wiley and Sons, Ltd; 2013. p. 277–87. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118413555.ch26 Cited 2023 Nov 1

Davies HT, Powell AE, Nutley SM. Mobilising knowledge to improve UK health care: learning from other countries and other sectors – a multimethod mapping study. Southampton (UK): NIHR Journals Library; 2015. (Health Services and Delivery Research). Available from: http://www.ncbi.nlm.nih.gov/books/NBK299400/ Cited 2023 Nov 1

Oliver K, Hopkins A, Boaz A, Guillot-Wright S, Cairney P. What works to promote research-policy engagement? Evid Policy. 2022;18(4):691–713.

Nelson JP, Lindsay S, Bozeman B. The last 20 years of empirical research on government utilization of academic social science research: a state-of-the-art literature review. Adm Soc. 2023;28:00953997231172923.

Bell D. Technology, nature and society: the vicissitudes of three world views and the confusion of realms. Am Sch. 1973;42:385–404.

Milat AJ, Li B. Narrative review of frameworks for translating research evidence into policy and practice. Public Health Res Pract. 2017; Available from: https://apo.org.au/sites/default/files/resource-files/2017-02/apo-nid74420.pdf Cited 2023 Nov 1

Meyers DC, Durlak JA, Wandersman A. The quality implementation framework: a synthesis of critical steps in the implementation process. Am J Community Psychol. 2012;50(3–4):462–80.

Article   PubMed   Google Scholar  

Cane J, O’Connor D, Michie S. Validation of the theoretical domains framework for use in behaviour change and implementation research. Implement Sci. 2012;7(1):37.

Glasgow RE, Battaglia C, McCreight M, Ayele RA, Rabin BA. Making implementation science more rapid: use of the RE-AIM framework for mid-course adaptations across five health services research projects in the veterans health administration. Front Public Health. 2020;8. Available from: https://www.frontiersin.org/articles/10.3389/fpubh.2020.00194 Cited 2023 Jun 13

Nilsen P. Making sense of implementation theories, models and frameworks. Implement Sci IS. 2015 Apr 21 10. Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4406164/ Cited 2020 May 4

Sheth A, Sinfield JV. An analytical framework to compare innovation strategies and identify simple rules. Technovation. 2022;1(115):102534.

Birken SA, Powell BJ, Shea CM, Haines ER, Alexis Kirk M, Leeman J, et al. Criteria for selecting implementation science theories and frameworks: results from an international survey. Implement Sci. 2017;12(1):124.

Redman S, Turner T, Davies H, Williamson A, Haynes A, Brennan S, et al. The SPIRIT Action Framework: A structured approach to selecting and testing strategies to increase the use of research in policy. Soc Sci Med. 2015;136:147–55.

Miller CJ, Barnett ML, Baumann AA, Gutner CA, Wiltsey-Stirman S. The FRAME-IS: a framework for documenting modifications to implementation strategies in healthcare. Implement Sci. 2021;16(1):36.

CAPE. CAPE. 2021. CAPE Capabilities in Academic Policy Engagement. Available from: https://www.cape.ac.uk/ Cited 2021 Aug 3

CIPHER Investigators. Supporting policy in health with research: an intervention trial (SPIRIT)—protocol for a stepped wedge trial. BMJ Open. 2014;4(7):e005293.

Spaapen J, Van Drooge L. Introducing ‘productive interactions’ in social impact assessment. Res Eval. 2011;20(3):211–8.

Williams C, Pettman T, Goodwin-Smith I, Tefera YM, Hanifie S, Baldock K. Experiences of research-policy engagement in policymaking processes. Public Health Res Pract. 2023. Online early publication. https://doi.org/10.17061/phrp33232308 .

Michie S, van Stralen MM, West R. The behaviour change wheel: a new method for characterising and designing behaviour change interventions. Implement Sci. 2011;6(1):42.

Best A, Holmes B. Systems thinking, knowledge and action: towards better models and methods. Evid Policy J Res Debate Pract. 2010;6(2):145–59.

Edwards DM, Meagher LR. A framework to evaluate the impacts of research on policy and practice: A forestry pilot study. For Policy Econ. 2020;1(114):101975.

Scheirer MA, Dearing JW. An agenda for research on the sustainability of public health programs. Am J Public Health. 2011;101(11):2059–67.

Borst RAJ, Wehrens R, Bal R, Kok MO. From sustainability to sustaining work: What do actors do to sustain knowledge translation platforms? Soc Sci Med. 2022;1(296):114735.

Zacka B. When the state meets the street: public service and moral agency. Harvard university press; 2017. Available from: https://books.google.co.uk/books?hl=en&lr=&id=3KdFDwAAQBAJ&oi=fnd&pg=PP1&dq=zacka+when+the+street&ots=x93YEHPKhl&sig=9yXKlQiFZ0XblHrbYKzvAMwNWT4 Cited 2023 Nov 28

Torrance H. The research excellence framework in the United Kingdom: processes, consequences, and incentives to engage. Qual Inq. 2020;26(7):771–9.

Rycroft-Malone J. The PARIHS framework—a framework for guiding the implementation of evidence-based practice. J Nurs Care Qual. 2004;19(4):297–304.

Stetler CB, Damschroder LJ, Helfrich CD, Hagedorn HJ. A guide for applying a revised version of the PARIHS framework for implementation. Implement Sci. 2011;6(1):99.

Field B, Booth A, Ilott I, Gerrish K. Using the knowledge to action framework in practice: a citation analysis and systematic review. Implement Sci. 2014;9(1):172.

Wiltsey Stirman S, Baumann AA, Miller CJ. The FRAME: an expanded framework for reporting adaptations and modifications to evidence-based interventions. Implement Sci. 2019;14(1):58.

Ouimet M, Landry R, Ziam S, Bédard PO. The absorption of research knowledge by public civil servants. Evid Policy. 2009;5(4):331–50.

Martin D, Spink MJ, Pereira PPG. Multiple bodies, political ontologies and the logic of care: an interview with Annemarie Mol. Interface - Comun Saúde Educ. 2018;22:295–305.

Sajadi HS, Majdzadeh R, Ehsani-Chimeh E, Yazdizadeh B, Nikooee S, Pourabbasi A, et al. Policy options to increase motivation for improving evidence-informed health policy-making in Iran. Health Res Policy Syst. 2021;19(1):91.

Athreye S, Sengupta A, Odetunde OJ. Academic entrepreneurial engagement with weak institutional support: roles of motivation, intention and perceptions. Stud High Educ. 2023;48(5):683–94.

Bamford D, Reid I, Forrester P, Dehe B, Bamford J, Papalexi M. An empirical investigation into UK university–industry collaboration: the development of an impact framework. J Technol Transf. 2023 Nov 13; Available from: https://doi.org/10.1007/s10961-023-10043-9 Cited 2023 Dec 20

McPherson AH, McDonald SM. Measuring the outcomes and impacts of innovation interventions assessing the role of additionality. Int J Technol Policy Manag. 2010;10(1–2):137–56.

Hind J. Additionality: a useful way to construct the counterfactual qualitatively? Eval J Australas. 2010;10(1):28–35.

Download references

Acknowledgements

We are very grateful to the CAPE Programme Delivery Group members, for many discussions throughout this work. Our thanks also go to the Sax Institute, Australia (where the original SPIRIT framework was developed), for reviewing and providing helpful feedback on the article. We also thank our reviewers who made very constructive suggestions, which have strengthened and clarified our article.

The evaluation of the CAPE programme, referred to in this report, was funded by Research England. The funding body had no role in the design of the study, analysis, interpretation or writing the manuscript.

Author information

Authors and affiliations.

Department of Health Services Research and Policy, Faculty of Public Health and Policy, London School of Hygiene and Tropical Medicine, 15-17 Tavistock Place, Kings Cross, London, WC1H 9SH, UK

Petra Mäkelä & Kathryn Oliver

Health and Social Care Workforce Research Unit, The Policy Institute, Virginia Woolf Building, Kings College London, 22 Kingsway, London, WC2B 6LE, UK

Annette Boaz

You can also search for this author in PubMed   Google Scholar

Contributions

PM conceptualised the modification of the framework reported in this work. All authors made substantial contributions to the design of the work. PM drafted the initial manuscript. AB and KO contributed to revisions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Petra Mäkelä .

Ethics declarations

Ethics approval and consent to participate.

Ethics approval was granted for the overarching CAPE evaluation by the London School of Hygiene and Tropical Medicine Research Ethics Committee (reference 26347).

Consent for publication

Competing interests.

The authors declare that they have no competing interests.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . The Creative Commons Public Domain Dedication waiver ( http://creativecommons.org/publicdomain/zero/1.0/ ) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

Reprints and permissions

About this article

Cite this article.

Mäkelä, P., Boaz, A. & Oliver, K. A modified action framework to develop and evaluate academic-policy engagement interventions. Implementation Sci 19 , 31 (2024). https://doi.org/10.1186/s13012-024-01359-7

Download citation

Received : 09 January 2024

Accepted : 20 March 2024

Published : 12 April 2024

DOI : https://doi.org/10.1186/s13012-024-01359-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Evidence-informed policy
• Academic-policy engagement
• Framework modification

Implementation Science

ISSN: 1748-5908

• Submission enquiries: Access here and click Contact Us
• General enquiries: [email protected]