Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Geometric BV for twisted Courant sigma models and the BRST power finesse (2401.00425v2)

Published 31 Dec 2023 in hep-th, math-ph, and math.MP

Abstract: We study twisted Courant sigma models, a class of topological field theories arising from the coupling of 3D 0-/2-form BF theory and Chern-Simons theory and containing a 4-form Wess-Zumino term. They are examples of theories featuring a nonlinearly open gauge algebra, where products of field equations appear in the commutator of gauge transformations, and they are reducible gauge systems. We determine the solution to the master equation using a technique, the BRST power finesse, that combines aspects of the AKSZ construction (which applies to the untwisted model) and the general BV-BRST formalism. This allows for a geometric interpretation of the BV coefficients in the interaction terms of the master action in terms of an induced generalised connection on a 4-form twisted (pre-)Courant algebroid, its Gualtieri torsion and the basic curvature tensor. It also produces a frame independent formulation of the model. We show, moreover, that the gauge fixed action is the sum of the classical one and a BRST commutator, as expected from a Schwarz type topological field theory.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (60)
  1. N. Ikeda, “Chern-Simons gauge theory coupled with BF theory,” Int. J. Mod. Phys. A 18, 2689-2702 (2003) doi:10.1142/S0217751X03015155 [arXiv:hep-th/0203043 [hep-th]].
  2. C. Hofman and J. S. Park, “BV quantization of topological open membranes,” Commun. Math. Phys. 249 (2004), 249-271 doi:10.1007/s00220-004-1106-7 [arXiv:hep-th/0209214 [hep-th]].
  3. C. Hofman and J. S. Park, “Topological open membranes,” [arXiv:hep-th/0209148 [hep-th]].
  4. D. Roytenberg, “AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories,” Lett. Math. Phys. 79 (2007), 143-159 doi:10.1007/s11005-006-0134-y [arXiv:hep-th/0608150 [hep-th]].
  5. D. Roytenberg, “Courant algebroids, derived brackets and even symplectic supermanifolds,” PhD Thesis, arXiv:math/9910078
  6. D. Roytenberg, “On the structure of graded symplectic supermanifolds and Courant algebroids,” Quantization, Poisson Brackets and Beyond, Theodore Voronov (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, [arXiv:math/0203110 [math.SG]].
  7. N. Halmagyi, “Non-geometric String Backgrounds and Worldsheet Algebras,” JHEP 07 (2008), 137 doi:10.1088/1126-6708/2008/07/137 [arXiv:0805.4571 [hep-th]].
  8. D. Mylonas, P. Schupp and R. J. Szabo, “Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds,” JHEP 09 (2012), 012 doi:10.1007/JHEP09(2012)012 [arXiv:1207.0926 [hep-th]].
  9. A. Chatzistavrakidis, L. Jonke and O. Lechtenfeld, “Sigma models for genuinely non-geometric backgrounds,” JHEP 11 (2015), 182 doi:10.1007/JHEP11(2015)182 [arXiv:1505.05457 [hep-th]].
  10. T. Bessho, M. A. Heller, N. Ikeda and S. Watamura, “Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids,” JHEP 04 (2016), 170 doi:10.1007/JHEP04(2016)170 [arXiv:1511.03425 [hep-th]].
  11. M. Hansen and T. Strobl, “First Class Constrained Systems and Twisting of Courant Algebroids by a Closed 4-form,” doi:10.1142/9789814277839_0008 [arXiv:0904.0711 [hep-th]].
  12. C. Klimcik and T. Strobl, “WZW - Poisson manifolds,” J. Geom. Phys. 43 (2002), 341-344 doi:10.1016/S0393-0440(02)00027-X [arXiv:math/0104189 [math.SG]].
  13. I. Vaisman, “Transitive Courant algebroids,” Int. J. Math. Math. Sci. 2005 (2005), 1737-1758 doi:10.1155/IJMMS.2005.1737 [arXiv:math/0407399 [math.DG]].
  14. I. A. Batalin and G. A. Vilkovisky, “Gauge Algebra and Quantization,” Phys. Lett. B 102 (1981), 27-31 doi:10.1016/0370-2693(81)90205-7
  15. M. Henneaux and C. Teitelboim, “Quantization of Gauge Systems,” Princeton University Press, 1994, ISBN 978-0-691-03769-1, 978-0-691-21386-6
  16. M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich, “The Geometry of the master equation and topological quantum field theory,” Int. J. Mod. Phys. A 12 (1997), 1405-1429 doi:10.1142/S0217751X97001031 [arXiv:hep-th/9502010 [hep-th]].
  17. E. Witten, “A Note on the Antibracket Formalism,” Mod. Phys. Lett. A 5 (1990), 487 doi:10.1142/S0217732390000561
  18. A. S. Schwarz, “Geometry of Batalin-Vilkovisky quantization,” Commun. Math. Phys. 155 (1993), 249-260 doi:10.1007/BF02097392 [arXiv:hep-th/9205088 [hep-th]].
  19. N. Ikeda and T. Strobl, “BV and BFV for the H-twisted Poisson sigma model,” Annales Henri Poincare 22 (2021) no.4, 1267-1316 doi:10.1007/s00023-020-00988-0 [arXiv:1912.13511 [hep-th]].
  20. A. Chatzistavrakidis, L. Jonke, T. Strobl and G. Šimunić, “Topological Dirac sigma models and the classical master equation,” J. Phys. A 56 (2023) no.1, 015402 doi:10.1088/1751-8121/acb09a [arXiv:2206.14258 [hep-th]].
  21. A. Chatzistavrakidis, N. Ikeda and G. Šimunić, “The BV action of 3D twisted R-Poisson sigma models,” JHEP 10 (2022), 002 doi:10.1007/JHEP10(2022)002 [arXiv:2206.03683 [hep-th]].
  22. M. Jotz Lean, R. A. Mehta, T. Papantonis, “Modules and representations up to homotopy of Lie n-algebroids,” Journal of Homotopy and Related Structures (2023), arXiv:2001.01101
  23. A. Chatzistavrakidis, T. Kodžoman, Z. Škoda, “Brane mechanics, gapped Lie n-algebroids and representations up to homotopy,” in preparation
  24. M. Gualtieri, “Branes on Poisson varieties,” doi:10.1093/acprof:oso/9780199534920.003.0018 [arXiv:0710.2719 [math.DG]].
  25. A. Chatzistavrakidis and L. Jonke, “Basic curvature and the Atiyah cocycle in gauge theory,” [arXiv:2302.04956 [hep-th]].
  26. A. D. Blaom, “Geometric structures as deformed infinitesimal symmetries,” Trans. Am. Math. Soc. 358 (2006) 3651.
  27. C. A. Abad and M. Crainic, “Representations up to homotopy of Lie algebroids,” J. reine angew. Math. 663 (2012), p. 91—126, DOI:10.1515/CRELLE.2011.095
  28. M. Crainic and R.  L. Fernandes, “Secondary Characteristic Classes of Lie Algebroids,” In: Carow-Watamura, U., Maeda, Y., Watamura, S. (eds) Quantum Field Theory and Noncommutative Geometry. Lecture Notes in Physics, vol 662. Springer, Berlin, Heidelberg. https://doi.org/10.1007/113427869
  29. A. Gracia-Saz and R. A. Mehta, “Lie algebroid structures on double vector bundles and representation theory of Lie algebroids,” Adv. Math., 223 no.4 (2010), 1236–1275 doi:10.1016/j.aim.2009.09.010 [arXiv:0810.0066 [math.DG]]
  30. A. S. Cattaneo and F. Schaetz, “Introduction to supergeometry,” Rev. Math. Phys. 23 (2011), 669-690 doi:10.1142/S0129055X11004400 [arXiv:1011.3401 [math-ph]].
  31. J. Qiu and M. Zabzine, “Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications,” Archivum Math. 47 (2011), 143-199 [arXiv:1105.2680 [math.QA]].
  32. N. Ikeda, “Lectures on AKSZ Sigma Models for Physicists,” doi:10.1142/9789813144613_0003 [arXiv:1204.3714 [hep-th]].
  33. A. Yu. Vaintrob, “Lie algebroids and homological vector fields,” Russ. Math. Surv. 52 428 (1997)
  34. A. Chatzistavrakidis, “Topological field theories induced by twisted R-Poisson structure in any dimension,” JHEP 09, 045 (2021) doi:10.1007/JHEP09(2021)045 [arXiv:2106.01067 [hep-th]].
  35. P. Ševera and A. Weinstein, “Poisson geometry with a 3 form background,” Prog. Theor. Phys. Suppl. 144 (2001), 145-154 doi:10.1143/PTPS.144.145 [arXiv:math/0107133 [math.SG]].
  36. Zhangju Liu, Yunhe Sheng, Xiaomeng Xu, “The Pontryagin class for pre-Courant algebroids,” Journal of Geometry and Physics, Volume 104, 2016, Pages 148-162, ISSN 0393-0440, https://doi.org/10.1016/j.geomphys.2016.02.007.
  37. T. Voronov, “Graded manifolds and drinfeld doubles for Lie bialgebroids,” [arXiv:math/0105237 [math.DG]].
  38. N. Ikeda, “Higher Dimensional Lie Algebroid Sigma Model with WZ Term,” Universe 7 (2021) no.10, 391 [arXiv:2109.02858 [hep-th]].
  39. M. Grützmann and T. Strobl, “General Yang–Mills type gauge theories for p𝑝pitalic_p-form gauge fields: From physics-based ideas to a mathematical framework or From Bianchi identities to twisted Courant algebroids,” Int. J. Geom. Meth. Mod. Phys. 12 (2014), 1550009 doi:10.1142/S0219887815500097 [arXiv:1407.6759 [hep-th]].
  40. N. Ikeda and T. Strobl, “From BFV to BV and spacetime covariance,” JHEP 12 (2020), 141 doi:10.1007/JHEP12(2020)141 [arXiv:2007.15912 [hep-th]].
  41. Z. Liu, A. Weinstein and P. Xu, “Manin Triples for Lie Bialgebroids,” J. Diff. Geom. 45 (1997) no.3, 547-574 [arXiv:dg-ga/9508013 [math.DG]].
  42. E. Boffo and P. Schupp, “Deformed graded Poisson structures, Generalized Geometry and Supergravity,” JHEP 01 (2020), 007 doi:10.1007/JHEP01(2020)007 [arXiv:1903.09112 [hep-th]].
  43. A. Kotov and T. Strobl, “Lie algebroids, gauge theories, and compatible geometrical structures,” Rev. Math. Phys. 31, no.04, 1950015 (2018) doi:10.1142/S0129055X19500156 [arXiv:1603.04490 [math.DG]].
  44. C. J. Grewcoe and L. Jonke, “Courant Sigma Model and L∞subscript𝐿L_{\infty}italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-algebras,” Fortsch. Phys. 68 (2020) no.6, 2000021 doi:10.1002/prop.202000021 [arXiv:2001.11745 [hep-th]].
  45. D. Roytenberg, A. Weinstein, “Courant Algebroids and Strongly Homotopy Lie Algebras,” Letters in Mathematical Physics 46, 81–93 (1998). https://doi.org/10.1023/A:1007452512084
  46. O. Hohm and B. Zwiebach, “L∞subscript𝐿L_{\infty}italic_L start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT Algebras and Field Theory,” Fortsch. Phys. 65 (2017) no.3-4, 1700014 doi:10.1002/prop.201700014 [arXiv:1701.08824 [hep-th]].
  47. M. Grigoriev and D. Rudinsky, “Notes on the L∞\infty∞-approach to local gauge field theories,” J. Geom. Phys. 190 (2023), 104863 doi:10.1016/j.geomphys.2023.104863 [arXiv:2303.08990 [hep-th]].
  48. A. Chatzistavrakidis, L. Jonke, F. S. Khoo and R. J. Szabo, “Double Field Theory and Membrane Sigma-Models,” JHEP 07 (2018), 015 doi:10.1007/JHEP07(2018)015 [arXiv:1802.07003 [hep-th]].
  49. J. M. Figueroa-O’Farrill and N. Mohammedi, “Gauging the Wess-Zumino term of a sigma model with boundary,” JHEP 08 (2005), 086 doi:10.1088/1126-6708/2005/08/086 [arXiv:hep-th/0506049 [hep-th]].
  50. C. Mayer and T. Strobl, “Lie Algebroid Yang Mills with Matter Fields,” J. Geom. Phys. 59 (2009), 1613-1623 doi:10.1016/j.geomphys.2009.07.018 [arXiv:0908.3161 [hep-th]].
  51. M. Bojowald, A. Kotov and T. Strobl, “Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries,” J. Geom. Phys. 54 (2005), 400-426 doi:10.1016/j.geomphys.2004.11.002 [arXiv:math/0406445 [math.DG]].
  52. N. Ikeda and X. Xu, “Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries,” J. Math. Phys. 55 (2014), 113505 doi:10.1063/1.4900834 [arXiv:1301.4805 [math.SG]].
  53. A. S. Cattaneo and G. Felder, “A Path integral approach to the Kontsevich quantization formula,” Commun. Math. Phys. 212 (2000), 591-611 doi:10.1007/s002200000229 [arXiv:math/9902090 [math]].
  54. F. Bonechi, P. Mnev and M. Zabzine, “Finite dimensional AKSZ-BV theories,” Lett. Math. Phys. 94 (2010), 197-228 doi:10.1007/s11005-010-0423-3 [arXiv:0903.0995 [hep-th]].
  55. N. Ikeda, “Deformation of BF theories, topological open membrane and a generalization of the star deformation,” JHEP 07 (2001), 037 [arXiv:hep-th/0105286 [hep-th]].
  56. I. A. Batalin and G. A. Vilkovisky, “Quantization of Gauge Theories with Linearly Dependent Generators,” Phys. Rev. D 28 (1983), 2567-2582 [erratum: Phys. Rev. D 30 (1984), 508] doi:10.1103/PhysRevD.28.2567
  57. F. Bonechi, A. S. Cattaneo and M. Zabzine, “Towards equivariant Yang-Mills theory,” J. Geom. Phys. 189 (2023), 104836 doi:10.1016/j.geomphys.2023.104836 [arXiv:2210.00372 [hep-th]].
  58. D. Birmingham, M. Blau, M. Rakowski and G. Thompson, “Topological field theory,” Phys. Rept. 209 (1991), 129-340 doi:10.1016/0370-1573(91)90117-5
  59. M. Grigoriev and A. Kotov, “Presymplectic AKSZ formulation of Einstein gravity,” JHEP 09 (2021), 181 doi:10.1007/JHEP09(2021)181 [arXiv:2008.11690 [hep-th]].
  60. M. Grigoriev, “Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles,” [arXiv:2212.11350 [math-ph]].
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets