Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 096, 22 pages      arXiv:2408.08287      https://doi.org/10.3842/SIGMA.2025.096

Weak Gauge PDEs

Maxim Grigoriev a and Dmitry Rudinsky bc
a) Service de Physique de l'Univers, Champs et Gravitation, Université de Mons, 20 place du Parc, 7000 Mons, Belgium
b) Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
c) Lebedev Physical Institute, 53 Leninsky Ave., 119991 Moscow, Russia

Received March 17, 2025, in final form November 03, 2025; Published online November 13, 2025

Abstract
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.

Key words: local gauge theories; gauge PDEs; Batalin-Vilkovisky formalsim; geometry of PDE; differential graded geometry.

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References

  1. Alexandrov M., Schwarz A., Zaboronsky O., Kontsevich M., The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12 (1997), 1405-1429, arXiv:hep-th/9502010.
  2. Alkalaev K.B., Grigoriev M., Frame-like Lagrangians and presymplectic AKSZ-type sigma model, Int. J. Mod. Phys. A 29 (2014), 1450103, 33 pages, arXiv:1312.5296.
  3. Anderson I., The variational bicomplex, Technical Report, Department of Mathematics, Utah State University, 1989.
  4. Barnich G., Brandt F., Henneaux M., Local BRST cohomology in Einstein-Yang-Mills theory, Nuclear Phys. B 455 (1995), 357-408, arXiv:hep-th/9505173.
  5. Barnich G., Brandt F., Henneaux M., Local BRST cohomology in the antifield formalism. I. General theorems, Comm. Math. Phys. 174 (1995), 57-91, arXiv:hep-th/9405109.
  6. Barnich G., Brandt F., Henneaux M., Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000), 439-569, arXiv:hep-th/0002245.
  7. Barnich G., Grigoriev M., Hamiltonian BRST and Batalin-Vilkovisky formalisms for second quantization of gauge theories, Comm. Math. Phys. 254 (2005), 581-601, arXiv:hep-th/0310083.
  8. Barnich G., Grigoriev M., BRST extension of the non-linear unfolded formalism, Bulg. J. Phys. 33 (2006), no. s1, 547-556, arXiv:hep-th/0504119.
  9. Barnich G., Grigoriev M., A Poincaré lemma for sigma models of AKSZ type, J. Geom. Phys. 61 (2011), 663-674, arXiv:0905.0547.
  10. Barnich G., Grigoriev M., First order parent formulation for generic gauge field theories, J. High Energy Phys. 2011 (2011), no. 1, 122, 36 pages, arXiv:1009.0190.
  11. Barnich G., Grigoriev M., Semikhatov A., Tipunin I., Parent field theory and unfolding in BRST first-quantized terms, Comm. Math. Phys. 260 (2005), 147-181, arXiv:hep-th/0406192.
  12. Batalin I., Marnelius R., Superfield algorithms for topological field theories, in Multiple Facets of Quantization and Supersymmetry, World Scientific Publishing, River Edge, NJ, 2002, 233-251, arXiv:hep-th/0110140.
  13. Batalin I.A., Vilkovisky G.A., Gauge algebra and quantization, Phys. Lett. B 102 (1981), 27-31.
  14. Batalin I.A., Vilkovisky G.A., Feynman rules for reducible gauge theories, Phys. Lett. B 120 (1983), 166-170.
  15. Bekaert X., Grigoriev M., Higher-order singletons, partially massless fields, and their boundary values in the ambient approach, Nuclear Phys. B 876 (2013), 667-714, arXiv:1305.0162.
  16. Bekaert X., Grigoriev M., Notes on the ambient approach to boundary values of AdS gauge fields, J. Phys. A 46 (2013), 214008, 23 pages, arXiv:1207.3439.
  17. Bekaert X., Grigoriev M., Skvortsov E.D., Higher spin extension of Fefferman-Graham construction, Universe 4 (2018), no. 2, 17, 26 pages, arXiv:1710.11463.
  18. Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations of mathematical physics, Transl. Math. Monogr., Vol. 182, American Mathematical Society, Providence, RI, 1999.
  19. Bonavolontà G., Kotov A., On the space of super maps between smooth supermanifold, arXiv:1304.0394.
  20. Bonavolontà G., Kotov A., Local BRST cohomology for AKSZ field theories: A global approach, in Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., Springer, Cham, 2015, 325-341, arXiv:1310.0245.
  21. Bonavolontà G., Poncin N., On the category of Lie $n$-algebroids, J. Geom. Phys. 73 (2013), 70-90, arXiv:1207.3590.
  22. Bonechi F., Cattaneo A.S., Zabzine M., Towards equivariant Yang-Mills theory, J. Geom. Phys. 189 (2023), 104836, 17 pages, arXiv:2210.00372.
  23. Bonechi F., Mnëv P., Zabzine M., Finite-dimensional AKSZ-BV theories, Lett. Math. Phys. 94 (2010), 197-228, arXiv:0903.0995.
  24. Boulanger N., Kessel P., Skvortsov E., Taronna M., Higher spin interactions in four-dimensions: Vasiliev versus Fronsdal, J. Phys. A 49 (2016), 095402, 52 pages, arXiv:1508.04139.
  25. Cattaneo A.S., Felder G., A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611, arXiv:math.QA/9902090.
  26. Cattaneo A.S., Felder G., On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56 (2001), 163-179.
  27. Cattaneo A.S., Mnev P., Reshetikhin N., Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 (2014), 535-603, arXiv:1201.0290.
  28. Cattaneo A.S., Mnev P., Reshetikhin N., Perturbative quantum gauge theories on manifolds with boundary, Comm. Math. Phys. 357 (2018), 631-730, arXiv:1507.01221.
  29. Chekmenev A., Grigoriev M., Boundary values of mixed-symmetry massless fields in AdS space, Nuclear Phys. B 913 (2016), 769-791, arXiv:1512.06443.
  30. Costello K., Renormalization and effective field theory, Math. Surveys Monogr., Vol. 170, American Mathematical Society, Providence, RI, 2011.
  31. Didenko V.E., Gelfond O.A., Korybut A.V., Vasiliev M.A., Limiting shifted homotopy in higher-spin theory and spin-locality, J. High Energy Phys. 2019 (2019), 086, 49 pages, arXiv:1909.04876.
  32. Dneprov I., Grigoriev M., Presymplectic BV-AKSZ formulation of conformal gravity, Eur. Phys. J. C 83 (2023), 6, 20 pages, arXiv:2208.02933.
  33. Dneprov I., Grigoriev M., Gritzaenko V., Presymplectic minimal models of local gauge theories, J. Phys. A 57 (2024), 335402, 29 pages, arXiv:2402.03240.
  34. Gelfond O.A., Vasiliev M.A., Spin-locality of higher-spin theories and star-product functional classes, J. High Energy Phys. 2020 (2020), no. 3, 002, 52 pages, arXiv:1910.00487.
  35. Gomis J., París J., Samuel S., Antibracket, antifields and gauge-theory quantization, Phys. Rep. 259 (1995), 145, arXiv:hep-th/9412228.
  36. Grigoriev M., Off-shell gauge fields from BRST quantization, arXiv:hep-th/0605089.
  37. Grigoriev M., Presymplectic structures and intrinsic Lagrangians, arXiv:1606.07532.
  38. Grigoriev M., Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles, in The Diverse World of PDEs—Geometry and Mathematical Physics, Contemp. Math., Vol. 788, American Mathematical Society, Providence, RI, 2023, 111-133, arXiv:2212.11350.
  39. Grigoriev M., Damgaard P.H., Superfield BRST charge and the master action, Phys. Lett. B 474 (2000), 323-330, arXiv:hep-th/9911092.
  40. Grigoriev M., Kotov A., Gauge PDE and AKSZ-type sigma models, Fortschr. Phys. 67 (2019), 1910007, 13 pages, arXiv:1903.02820.
  41. Grigoriev M., Kotov A., Presymplectic AKSZ formulation of Einstein gravity, J. High Energy Phys. 2021 (2021), no. 9, 181, 23 pages, arXiv:2008.11690.
  42. Grigoriev M., Mamekin A., Presymplectic BV-AKSZ for $N=1$, $D=4$ supergravity, arXiv:2503.04559.
  43. Grigoriev M., Markov M., Asymptotic symmetries of gravity in the gauge PDE approach, Classical Quantum Gravity 41 (2024), 135009, 27 pages, arXiv:2310.09637.
  44. Grigoriev M., Rudinsky D., Notes on the $L_\infty$-approach to local gauge field theories, J. Geom. Phys. 190 (2023), 104863, 21 pages, arXiv:2303.08990.
  45. Henneaux M., Spacetime locality of the BRST formalism, Comm. Math. Phys. 140 (1991), 1-13.
  46. Henneaux M., Teitelboim C., Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992.
  47. Ikeda N., Lectures on AKSZ sigma models for physicists, in Noncommutative Geometry and Physics. 4, World Scientific Publishing, Hackensack, NJ, 2017, 79-169, arXiv:1204.3714.
  48. Kazinski P.O., Lyakhovich S.L., Sharapov A.A., Lagrange structure and quantization, J. High Energy Phys. 2005 (2005), no. 7, 076, 42 pages, arXiv:hep-th/0506093.
  49. Kobayashi E.T., A remark on the Nijenhuis tensor, Pacific J. Math. 12 (1962), 963-977, Errarum, Pacific J. Math. 12 (1962), 1467.
  50. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, arXiv:q-alg/9709040.
  51. Kotov A., Strobl T., Characteristic classes associated to $Q$-bundles, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550006, 26 pages, arXiv:0711.4106.
  52. Krasil'shchik I.S., Lychagin V.V., Vinogradov A.M., Geometry of jet spaces and nonlinear partial differential equations, Adv. Stud. Contemp. Math, Vol. 1, Gordon and Breach Science Publishers, New York, 1986.
  53. Krasil'shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys. 61 (2011), 1633-1674, arXiv:1002.0077.
  54. Lee J.M., Introduction to smooth manifolds, 2nd ed., Grad. Texts Math., Vol. 218, Springer, New York, 2012.
  55. Lyakhovich S.L., Sharapov A.A., Quantization of Donaldson-Uhlenbeck-Yau theory, Phys. Lett. B 656 (2007), 265-271, arXiv:0705.1871.
  56. Mehta R.A., $Q$-algebroids and their cohomology, J. Symplectic Geom. 7 (2009), 263-293, arXiv:math.DG/0703234.
  57. Misuna N., Unfolded formulation of $4 d$ Yang-Mills theory, Phys. Lett. B 870 (2025), 139882, 6 pages, arXiv:2408.13212.
  58. Piguet O., Sorella S.P., Algebraic renormalization. Perturbative renormalization, symmetries and anomalies, Lecture Notes in Phys. New Ser. m Monogr., Vol. 28, Springer, Berlin, 1995.
  59. Roytenberg D., On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, American Mathematical Society, Providence, RI, 2002, 169-185, arXiv:math.SG/0203110.
  60. Roytenberg D., AKSZ-BV formalism and Courant algebroid-induced topological field theories, Lett. Math. Phys. 79 (2007), 143-159, arXiv:hep-th/0608150.
  61. Schwarz A., Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys. 155 (1993), 249-260, arXiv:hep-th/9205088.
  62. Severa P., Some title containing the words ''homotopy'' and ''symplectic'', e.g. this one, in Travaux Mathématiques. Fasc. XVI, Trav. Math., Vol. 16, Université du Luxembourg, Luxembourg, 2005, 121-137, arXiv:math.SG/0105080.
  63. Sharapov A., Skvortsov E., Higher spin gravities and presymplectic AKSZ models, Nuclear Phys. B 972 (2021), 115551, 56 pages, arXiv:2102.02253.
  64. Sharapov A., Skvortsov E., Sukhanov A., Van Dongen R., Minimal model of chiral higher spin gravity, J. High Energy Phys. 2022 (2022), no. 9, 134, 31 pages, arXiv:2205.07794.
  65. Sharapov A., Skvortsov E., Van Dongen R., Chiral higher spin gravity and convex geometry, SciPost Phys. 14 (2023), 162, 15 pages, arXiv:2209.01796.
  66. Sharapov A.A., Variational tricomplex, global symmetries and conservation laws of gauge systems, SIGMA 12 (2016), 098, 24 pages, arXiv:1607.01626.
  67. Skvortsov E., Taronna M., On locality, holography and unfolding, J. High Energy Phys. 2015 (2015), no. 11, 044, 36 pages, arXiv:1508.04764.
  68. Vaintrob A.Yu., Lie algebroids and homological vector fields, Russian Math. Surveys 52 (1997), 428-429.
  69. Vasiliev M.A., Equations of motion of interacting massless fields of all spins as a free differential algebra, Phys. Lett. B 209 (1988), 491-497.
  70. Vasiliev M.A., Higher spin gauge theories: Star-product and AdS space, in The Many Faces of the Superworld, World Scientific Publishing, 2000, 533-610, arXiv:hep-th/9910096.
  71. Vasiliev M.A., Nonlinear equations for symmetric massless higher spin fields in $({\rm A}){\rm dS}_d$, Phys. Lett. B 567 (2003), 139-151, arXiv:hep-th/0304049.
  72. Vasiliev M.A., Actions, charges and off-shell fields in the unfolded dynamics approach, Int. J. Geom. Methods Mod. Phys. 3 (2006), 37-80, arXiv:hep-th/0504090.
  73. Vinogradov A.M., Geometry of nonlinear differential equations, J. Math. Sci. 17 (1981), 1624-1649.
  74. Voronov T.T., On a non-abelian Poincaré lemma, Proc. Amer. Math. Soc. 140 (2012), 2855-2872, arXiv:0905.0287.

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