Abstract
We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.
References
P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9 (1994) 3129 [hep-th/9405110] [INSPIRE].
N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435 [hep-th/9312059] [INSPIRE].
A.S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212 (2000) 591 [math/9902090] [INSPIRE].
A.S. Cattaneo and G. Felder, Poisson sigma models and deformation quantization, Mod. Phys. Lett. A 16 (2001) 179 [hep-th/0102208] [INSPIRE].
T. Klosch and T. Strobl, Classical and quantum gravity in (1+1)-Dimensions. Part 1: A unifying approach, Class. Quant. Grav. 13 (1996) 965 [Erratum ibid. 14 (1997) 825] [gr-qc/9508020] [INSPIRE].
D. Grumiller, W. Kummer and D.V. Vassilevich, Dilaton gravity in two-dimensions, Phys. Rept. 369 (2002) 327 [hep-th/0204253] [INSPIRE].
A.Y. Alekseev, P. Schaller and T. Strobl, The topological G/G WZW model in the generalized momentum representation, Phys. Rev. D 52 (1995) 7146 [hep-th/9505012] [INSPIRE].
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 [hep-th/9502010] [INSPIRE].
N. Ikeda, Lectures on AKSZ sigma models for Physicists, in Workshop on Strings, Membranes and Topological Field Theory, pp. 79–169, WSPC, (2017), DOI [arXiv:1204.3714] [INSPIRE].
A. Kotov, P. Schaller and T. Strobl, Dirac sigma models, Commun. Math. Phys. 260 (2005) 455 [hep-th/0411112] [INSPIRE].
T. Courant, Dirac manifolds, Trans. Am. Math. Soc. 319 (1990) 631.
Z.-J. Liu, A. WEinstein and P. Xu, Manin Triples for Lie Bialgebroids, J. Diff. Geom. 45 (1997) 547 [dg-ga/9508013] [INSPIRE].
M. Gualtieri, Generalized complex geometry, Ph.D. Thesis, Oxford University, (2003), math/0401221 [INSPIRE].
C.M. Hull and B.J. Spence, The gauged nonlinear sigma model with Wess-Zumino term, Phys. Lett. B 232 (1989) 204 [INSPIRE].
C.M. Hull and B.J. Spence, The geometry of the gauged sigma model with Wess-Zumino term, Nucl. Phys. B 353 (1991) 379 [INSPIRE].
V. Salnikov and T. Strobl, Dirac sigma models from gauging, JHEP 11 (2013) 110 [arXiv:1311.7116] [INSPIRE].
E. Plauschinn, T-duality revisited, JHEP 01 (2014) 131 [arXiv:1310.4194] [INSPIRE].
A. Chatzistavrakidis, A. Deser, L. Jonke and T. Strobl, Beyond the standard gauging: gauge symmetries of Dirac sigma models, JHEP 08 (2016) 172 [arXiv:1607.00342] [INSPIRE].
A. Chatzistavrakidis, A. Deser, L. Jonke and T. Strobl, Strings in Singular Space-Times and their Universal Gauge Theory, Annales Henri Poincaré 18 (2017) 2641 [arXiv:1608.03250] [INSPIRE].
P. Ševera and T. Strobl, Transverse generalized metrics and 2d sigma models, J. Geom. Phys. 146 (2019) 103509 [arXiv:1901.08904] [INSPIRE].
N. Ikeda and T. Strobl, On the relation of Lie algebroids to constrained systems and their BV/BFV formulation, Annales Henri Poincaré 20 (2019) 527 [arXiv:1803.00080] [INSPIRE].
N. Ikeda, Momentum sections in Hamiltonian mechanics and sigma models, SIGMA 15 (2019) 076 [arXiv:1905.02434] [INSPIRE].
P. Bouwknegt, Lectures on cohomology, T-duality, and generalized geometry, Lect. Notes Phys. 807 (2010) 261 [INSPIRE].
A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].
P. Ševera, Letters to Alan WEinstein about Courant algebroids, arXiv:1707.00265 [INSPIRE].
H. Bursztyn, G.R. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007) 726 [math/0509640] [INSPIRE].
Z. Chen, M. Stiénon and P. Xu, On regular Courant algebroids, J. Symplectic Geom. 11 (2013) 1.
P. Ševera, Poisson-Lie T-duality and Courant Algebroids, Lett. Math. Phys. 105 (2015) 1689 [arXiv:1502.04517] [INSPIRE].
K. Wright, Generalised contact geometry as reduced generalised complex geometry, J. Geom. Phys. 130 (2018) 331 [arXiv:1708.09550] [INSPIRE].
A. Kotov and T. Strobl, Gauging without initial symmetry, J. Geom. Phys. 99 (2016) 184 [arXiv:1403.8119] [INSPIRE].
C. Klimčík and T. Strobl, WZW — Poisson manifolds, J. Geom. Phys. 43 (2002) 341 [math/0104189] [INSPIRE].
P. Ševera and A. WEinstein, Poisson geometry with a 3 form background, Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133] [INSPIRE].
N. Ikeda and T. Strobl, BV and BFV for the H-twisted Poisson sigma model, arXiv:1912.13511 [INSPIRE].
A. Lichnerowicz, Les variétés de Jacobi et leurs algébres de Lie associées, J. Diff. Geom. 12 (1977) 253.
M. Crainic and C. Zhu, Integrability of Jacobi structures, Annales Inst. Fourier 57 (2007) 1181.
I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser Verlag (1994).
J.M. Nunes da Costa and F. Petalidou, Twisted Jacobi Manifolds, Twisted Dirac-Jacobi Structures and Quasi-Jacobi Bialgebroids, J. Phys. A 39 (2006) 10449.
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
D. Iglesias and J.C. Marrero, Generalized Lie bialgebroids and Jacobi structures, J. Geom. Phys. 40 (2001) 176.
J. Grabowski and G. Marmo, Jacobi structures revisited, J. Phys. A 34 (2001) 10975 [math/0111148] [INSPIRE].
M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press (1992).
A. Chatzistavrakidis, A. Deser, L. Jonke and T. Strobl, Gauging as constraining: the universal generalised geometry action in two dimensions, PoS CORFU2016 (2017) 087 [arXiv:1705.05007] [INSPIRE].
F. Bourgeois, A survey of contact homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lect. Notes 49 (2009) 45.
A. Kirillov, Local Lie Algebras, Russ. Math. Surv. 31 (1976) 55.
C.-M. Marle, On Jacobi Manifolds and Jacobi Bundles, in P. Dazord and A. Weinstein eds. Symplectic Geometry, Groupoids, and Integrable Systems, Mathematical Sciences Research Institute Publications, volume 20, Springer, New York, U.S.A. (1991).
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].
N. Ikeda, Chern-Simons gauge theory coupled with BF theory, Int. J. Mod. Phys. A 18 (2003) 2689 [hep-th/0203043] [INSPIRE].
A. Deser, M.A. Heller and C. Sämann, Extended Riemannian Geometry II: Local Heterotic Double Field Theory, JHEP 04 (2018) 106 [arXiv:1711.03308] [INSPIRE].
A. Chatzistavrakidis, L. Jonke, F.S. Khoo and R.J. Szabo, Double Field Theory and Membrane sigma-models, JHEP 07 (2018) 015 [arXiv:1802.07003] [INSPIRE].
O. Hohm, W. Siegel and B. Zwiebach, Doubled α′-geometry, JHEP 02 (2014) 065 [arXiv:1306.2970] [INSPIRE].
O. Hohm and B. Zwiebach, Green-Schwarz mechanism and α′-deformed Courant brackets, JHEP 01 (2015) 012 [arXiv:1407.0708] [INSPIRE].
O.A. Bedoya, D. Marqués and C. Núñez, Heterotic α’-corrections in Double Field Theory, JHEP 12 (2014) 074 [arXiv:1407.0365] [INSPIRE].
W.H. Baron, E. Lescano and D. Marqués, The generalized Bergshoeff-de Roo identification, JHEP 11 (2018) 160 [arXiv:1810.01427] [INSPIRE].
A. Bravetti, H. Cruz and D. Tapias, Contact Hamiltonian mechanics, Annals Phys. 376 (2017) 17.
F. Bascone, F. Pezzella and P. Vitale, Jacobi σ-models, arXiv:2007.12543 [INSPIRE].
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Chatzistavrakidis, A., Šimunić, G. Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures. J. High Energ. Phys. 2020, 173 (2020). https://doi.org/10.1007/JHEP11(2020)173
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DOI: https://doi.org/10.1007/JHEP11(2020)173
Keywords
- Gauge Symmetry
- Sigma Models
- Differential and Algebraic Geometry
- Topological Field Theories