Abstract
We show that in the context of two-dimensional sigma models minimal coupling of an ordinary rigid symmetry Lie algebra g leads naturally to the appearance of the “generalized tangent bundle” \( \mathbb{T} \) M≡TM⊕T * M by means of composite fields. Gauge transformations of the composite fields follow the Courant bracket, closing upon the choice of a Dirac structure D ⊂ \( \mathbb{T} \) M (or, more generally, the choide of a “small Dirac-Rinehart sheaf” \( \mathcal{D} \)), in which the fields as well as the symmetry parameters are to take values. In these new variables, the gauge theory takes the form of a (non-topological) Dirac sigma model, which is applicable in a more general context and proves to be universal in two space-time dimensions: a gauging of g of a standard sigma model with Wess-Zumino term exists, iff there is a prolongation of the rigid symmetry to a Lie algebroid morphism from the action Lie algebroid M × \( \mathfrak{g} \) → M into D → M (or the algebraic analogue of the morphism in the case of \( \mathcal{D} \)). The gauged sigma model results from a pullback by this morphism from the Dirac sigma model, which proves to be universal in two-spacetime dimensions in this sense.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Kotov, P. Schaller and T. Strobl, Dirac σ-models, Commun. Math. Phys. 260 (2005) 455 [hep-th/0411112] [INSPIRE].
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].
S.P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Sov. Math. Dokl. 24 (1981) 222.
E. Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 455.
A. Kotov and T. Strobl, Gauging without initial symmetry, arXiv:1403.8119 [INSPIRE].
A. Alekseev and T. Strobl, Current algebras and differential geometry, JHEP 03 (2005) 035 [hep-th/0410183] [INSPIRE].
V. Salnikov and T. Strobl, Dirac σ-models from gauging, JHEP 11 (2013) 110 [arXiv:1311.7116] [INSPIRE].
M. Bojowald, A. Kotov and T. Strobl, Lie algebroid morphisms, Poisson σ-models and off-shell closed gauge symmetries, J. Geom. Phys. 54 (2005) 400 [math/0406445] [INSPIRE].
C.M. Hull, A. Karlhede, U. Lindström and M. Roček, Nonlinear σ models and their gauging in and out of superspace, Nucl. Phys. B 266 (1986) 1 [INSPIRE].
C.M. Hull and B.J. Spence, The gauged nonlinear σ model with Wess-Zumino term, Phys. Lett. B 232 (1989) 204 [INSPIRE].
J.M. Figueroa-O’Farrill and S. Stanciu, Equivariant cohomology and gauged bosonic σ-models, hep-th/9407149 [INSPIRE].
C.L. Rogers, L-infinity algebras from multisymplectic geometry, Lett. Math. Phys. 100 (2012) 29 [arXiv:1005.2230] [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].
A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles, arXiv:0711.4106 [INSPIRE].
A. Cannas da Silva and A. Weinstein, Geometric models for noncommutative algebras, AMS Berkeley Mathematics Lecture Notes series, American Mathematical Society, U.S.A. (1999).
K. Gawedzki and A. Kupiainen, G/H conformal field theory from gauged WZW model, Phys. Lett. B 215 (1988) 119.
A. Kotov and T. Strobl, Generalizing geometry — Algebroids and σ-models, in Handbook on pseudo-Riemannian geometry and supersymmetry, V. Cortes, European Mathematical Society (2010), arXiv:1004.0632 [INSPIRE].
M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University press, Princeton U.S.A. (1991).
I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27.
J.-S. Park, Topological open p-branes, hep-th/0012141 [INSPIRE].
G. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc. 108 (1963) 195.
U. Lindström, M. Roček, R. von Unge and M. Zabzine, Generalized Kähler manifolds and off-shell supersymmetry, Commun. Math. Phys. 269 (2007) 833 [hep-th/0512164] [INSPIRE].
M. Zabzine, Lectures on generalized complex geometry and supersymmetry, Archivum Math. 42 (2006) 119 [hep-th/0605148] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1407.5439
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Kotov, A., Salnikov, V. & Strobl, T. 2d gauge theories and generalized geometry. J. High Energ. Phys. 2014, 21 (2014). https://doi.org/10.1007/JHEP08(2014)021
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)021