Weak Gauge PDEs
Gauge PDEs generalise the AKSZ construction when dealing with generic local gauge theories. Despite being very flexible and invariant, these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2025 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2025
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/214187 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Weak Gauge PDEs. Maxim Grigoriev and Dmitry Rudinsky. SIGMA 21 (2025), 096, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862720424800944128 |
|---|---|
| author | Grigoriev, Maxim Rudinsky, Dmitry |
| author_facet | Grigoriev, Maxim Rudinsky, Dmitry |
| citation_txt | Weak Gauge PDEs. Maxim Grigoriev and Dmitry Rudinsky. SIGMA 21 (2025), 096, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Gauge PDEs generalise the AKSZ construction when dealing with generic local gauge theories. Despite being very flexible and invariant, these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion of a weak gauge PDE in which the nilpotency of the BRST differential is relaxed in a controllable way. In this approach, a nontopological local gauge theory can be described in terms of a finite-dimensional geometrical object. Moreover, among the equivalent weak gauge PDEs describing a given system, a minimal one can usually be found and is unique in a certain sense. In the case of a Lagrangian system, the respective weak gauge PDE naturally arises from its weak presymplectic formulation. We prove that any weak gauge PDE determines the standard jet-bundle Batalin-Vilkovisky formulation of the underlying gauge theory, giving an unambiguous physical interpretation of these objects. The formalism is illustrated by a few examples, including the non-Lagrangian self-dual Yang-Mills theory and a finite jet bundle. We also discuss possible applications of the approach to the characterisation of those infinite-dimensional gauge PDEs that correspond to local theories.
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| first_indexed | 2026-03-21T02:22:34Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214187 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T02:22:34Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grigoriev, Maxim Rudinsky, Dmitry 2026-02-20T07:58:36Z 2025 Weak Gauge PDEs. Maxim Grigoriev and Dmitry Rudinsky. SIGMA 21 (2025), 096, 22 pages 1815-0659 2020 Mathematics Subject Classification: 35B06; 58A50; 37K06; 81T70; 70S15 arXiv:2408.08287 https://nasplib.isofts.kiev.ua/handle/123456789/214187 https://doi.org/10.3842/SIGMA.2025.096 Gauge PDEs generalise the AKSZ construction when dealing with generic local gauge theories. Despite being very flexible and invariant, these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion of a weak gauge PDE in which the nilpotency of the BRST differential is relaxed in a controllable way. In this approach, a nontopological local gauge theory can be described in terms of a finite-dimensional geometrical object. Moreover, among the equivalent weak gauge PDEs describing a given system, a minimal one can usually be found and is unique in a certain sense. In the case of a Lagrangian system, the respective weak gauge PDE naturally arises from its weak presymplectic formulation. We prove that any weak gauge PDE determines the standard jet-bundle Batalin-Vilkovisky formulation of the underlying gauge theory, giving an unambiguous physical interpretation of these objects. The formalism is illustrated by a few examples, including the non-Lagrangian self-dual Yang-Mills theory and a finite jet bundle. We also discuss possible applications of the approach to the characterisation of those infinite-dimensional gauge PDEs that correspond to local theories. We wish to thank K. Druzhkov, I. Krasil’shchik, A. Mamekin, M. Markov, A. Verbovetsky, and especially I. Dneprov and A. Kotov for fruitful discussions. The authors are also grateful to the anonymous referees for their valuable suggestions and comments that helped to improve the manuscript. Maxim Grigoriev, supported by the ULYSSE Incentive Grant for Mobility in Scientific Research [MISU] F.6003.24, F.R.S.-FNRS, Belgium, also at Lebedev Physical Institute and Institute for Theoretical and Mathematical Physics, Lomonosov MSU. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Weak Gauge PDEs Article published earlier |
| spellingShingle | Weak Gauge PDEs Grigoriev, Maxim Rudinsky, Dmitry |
| title | Weak Gauge PDEs |
| title_full | Weak Gauge PDEs |
| title_fullStr | Weak Gauge PDEs |
| title_full_unstemmed | Weak Gauge PDEs |
| title_short | Weak Gauge PDEs |
| title_sort | weak gauge pdes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214187 |
| work_keys_str_mv | AT grigorievmaxim weakgaugepdes AT rudinskydmitry weakgaugepdes |