Abstract
We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to a worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. The rigid and local symmetries of the action, respectively, are equivalent to the Poisson bracket being compatible with the de Rham operator and obeying graded Jacobi identities. We propose that perturbative quantization of the model yields a covariantized differential star product algebra of Kontsevich type. We comment on the resemblance to the topological A model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. V. Fedosov, A simple geometrical construction of deformation quantization, J. Diff. Geom. 40 (1994) 213.
M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157 [q-alg/9709040].
N. Ikeda, Two-dimensional gravity and nonlinear gauge theory, Annals Phys. 235 (1994) 435 [hep-th/9312059] [INSPIRE].
P. Schaller and T. Strobl, Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9 (1994) 3129 [hep-th/9405110] [INSPIRE].
A.S. Cattaneo, G. Felder and L. Tomassini, From local to global deformation quantization of poisson manifolds, math/0012228 [INSPIRE].
M. Alexandrov, M. Kontsevich, A. Schwartz and O. Zaboronsky, The Geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A 12 (1997) 1405 [hep-th/9502010] [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, On the AKSZ formulation of the Poisson σ-model, Lett. Math. Phys. 56 (2001) 163 [math/0102108] [INSPIRE].
C.-S. Chu and P.-M. Ho, Poisson algebra of differential forms, Int. J. Mod. Phys. 12 (1997) 5573 [q-alg/9612031] [INSPIRE].
E.J. Beggs and S. Majid, Semiclassical differential structures, math/0306273 [INSPIRE].
A. Tagliaferro, The star product for differential forms on symplectic Manifolds, arXiv:0809.4717 [INSPIRE].
S. McCurdy and B. Zumino, Covariant star product for exterior differential forms on symplectic manifolds, AIP Conf. Proc. 1200 (2010) 204 [arXiv:0910.0459] [INSPIRE].
M. Chaichian, M. Oksanen, A. Tureanu and G. Zet, Covariant star product on symplectic and Poisson spacetime manifolds, Int. J. Mod. Phys. A 25 (2010) 3765 [arXiv:1001.0503] [INSPIRE].
H. Bursztyn, Poisson vector bundles, contravariant connections and deformations, Prog. Theor. Phys. Suppl. 144 (2001) 26.
J.S. Park, Topological open p-branes, hep-th/0012141.
N. Ikeda, Deformation of BF theories, topological open membrane and a generalization of the star deformation, JHEP 07 (2001) 037 [hep-th/0105286] [INSPIRE].
N. Ikeda, Lectures on AKSZ topological field theories for physicists, arXiv:1204.3714 [INSPIRE].
N. Boulanger, N. Colombo and P. Sundell, A minimal BV action for Vasiliev’s four-dimensional higher spin gravity, JHEP 10 (2012) 043 [arXiv:1205.3339] [INSPIRE].
C. Arias, N. Boulanger, P. Sundell and A. Torres-Gomez, Differential algebras and covariant Hamiltonian dynamics: a primer for physicists, to appear.
L. Baulieu, A.S. Losev and N.A. Nekrasov, Target space symmetries in topological theories. 1, JHEP 02 (2002) 021 [hep-th/0106042] [INSPIRE].
E. Frenkel and A. Losev, Mirror symmetry in two steps: A-I-B, Commun. Math. Phys. 269 (2006) 39 [hep-th/0505131] [INSPIRE].
F. Bonechi and M. Zabzine, Poisson σ-model on the sphere, Commun. Math. Phys. 285 (2009) 1033 [arXiv:0706.3164] [INSPIRE].
E. Witten, Topological sigma models, Commun. Math. Phys. 118 (1988) 411.
L. Baulieu and I.M. Singer, The topological sigma model, Commun. Math. Phys. 125 (1989) 227.
S.Y. Wu, Topological quantum field theories on manifolds with a boundary, Commun. Math. Phys. 136 (1991) 157.
M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3 + 1)-dimensions, Phys. Lett. B 243 (1990) 378 [INSPIRE].
N. Boulanger, E. Sezgin and P. Sundell, 4D higher spin gravity with dynamical two-form as a Frobenius-Chern-Simons gauge theory, arXiv:1505.04957 [INSPIRE].
N. Colombo and P. Sundell, Twistor space observables and quasi-amplitudes in 4D higher spin gravity, JHEP 11 (2011) 042 [arXiv:1012.0813] [INSPIRE].
N. Colombo and P. Sundell, Higher spin gravity amplitudes from zero-form charges, arXiv:1208.3880 [INSPIRE].
V.E. Didenko and E.D. Skvortsov, Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory, JHEP 04 (2013) 158 [arXiv:1210.7963] [INSPIRE].
V.E. Didenko, J. Mei and E.D. Skvortsov, Exact higher-spin symmetry in CFT: free fermion correlators from Vasiliev Theory, Phys. Rev. D 88 (2013) 046011 [arXiv:1301.4166] [INSPIRE].
J. Engquist, P. Sundell and L. Tamassia, On singleton composites in non-compact WZW models, JHEP 02 (2007) 097 [hep-th/0701051] [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: 1503.05625
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
Arias, C., Boulanger, N., Sundell, P. et al. 2D sigma models and differential Poisson algebras. J. High Energ. Phys. 2015, 95 (2015). https://doi.org/10.1007/JHEP08(2015)095
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)095