Abstract
We evaluate the path integral of the Poisson sigma model on the sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kähler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.
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Alexandrov M., Kontsevich M., Schwartz A., Zaboronsky O.: The Geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 12, 1405 (1997)
Barannikov S., Kontsevich M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices 4, 201 (1998)
Batalin I.A., Vilkovisky G.A.: Relativistic S matrix of dynamical systems with Boson and Fermion constraints. Phys. Lett. B 69, 309 (1977)
Bergamin L., Grumiller D., Kummer W., Vassilevich D.V.: Classical and quantum integrability of 2D dilaton gravities in Euclidean space. Class. Quant. Grav. 22, 1361 (2005)
Calvo I.: Supersymmetric WZ-Poisson sigma model and twisted generalized complex geometry. Lett. Math. Phys. 77, 53 (2006)
Cattaneo A.S., Felder G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000)
Cattaneo A.S., Felder G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56, 163 (2001)
Cattaneo, A.S.: From Topological Field Theory to Deformation Quantization and Reduction. Proceedings of ICM 2006, Vol. III, Zurich: European Mathematical Society, 2006, pp. 339–365
Dolgushev, V.: The Van den Bergh duality and the modular symmetry of a Poisson variety. http://arXiv.org/list/math/0612288, 2006
Evens S., Lu J.-H.: Poisson harmonic forms, Kostant harmonic forms, and the S 1-equivariant cohomology of K/T. Adv. Math. 142, 171 (1999)
Evens, S., Lu, J.-H., Weinstein, A.: Transverse measures, the modular class, and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. 2. 50(200), 417–436 (1999)
Felder G., Shoikhet B.: Deformation quantization with traces. Lett. Math. Phys. 53(1), 75–86 (2000)
Fiorenza, D.: An introduction to the Batalin-Vilkovisky formalism, Comptes Rendus des Rencontres Mathematiques de Glanon, Edition 2003
Frenkel E., Losev A.: Mirror symmetry in two steps: A-I-B. Commun. Math. Phys. 269, 39 (2007)
Gualtieri, M.: Generalized Complex Geometry. Oxford University, DPhil thesis, 2004, avaible at http://arXiv.org/list/math.DG/0401221, 2004
Gualtieri, M.: Generalized Complex Geometry. http://arXiv.org/list/math.DG/0703298, 2007
Gualtieri, M.: Branes on Poisson varieties. http://arXiv.org/abs/0710.2719v1[math.DG], 2007
Henneaux, M., Teitelboim, C.(1992). Quantization of Gauge Systems. Princeton Series in Physics, Princeton, NJ: Princeton Univ. Press, 1992
Hirshfeld, A.C., Schwarzweller, T.: The partition function of the linear Poisson-sigma model on arbitrary surfaces. http://arXiv.org/list/hep-th/0112086, 2001
Hitchin N.: Generalized Calabi-Yau manifolds. Quart. J. Math. Oxford Ser. 54, 281 (2003)
Hitchin N.: Instantons, Poisson structures and generalized Kaehler geometry. Commun. Math. Phys. 265, 131 (2006)
Hitchin, N.: Bihermitian metrics on Del Pezzo surfaces. http://arXiv.org/list/math.DG/0608213, 2006
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. Providence, RI: Amer Math. Soc., 2003
Ikeda N.: Two-dimensional gravity and nonlinear gauge theory. Annals Phys. 235, 435 (1994)
Kapustin A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49 (2004)
Kapustin, A., Li, Y.: Topological sigma-models with H-flux and twisted generalized complex manifolds. http://arXiv.org/list/hep-th/0407249, 2004
Kosmann-Schwarzbach Y.: Exact Gerstenhaber algebras and Lie bialgebroids. Acta Appl. Math. 41, 153–165 (1995)
Kosmann-Schwarzbach, Y.: Modular vector fields and Batalin-Vilkovisky algebras. In: Poisson geometry (Warsaw, 1998), Banach Center Publ. 51, Warsaw: Polish Acad. Sci., 2000, pp. 109–129
Kosmann-Schwarzbach Y., Monterde J.: Divergence operators and odd Poisson brackets. Ann. Inst. Fourier (Grenoble) 52(2), 419–456 (2002)
Koszul, J.-L.: Crochet de Schouten-Nijenhuis et cohomologie. In: The mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque 1985, Numero Hors Serie, 257–271, (1985)
Krotov, D., Losev, A.: Quantum field theory as effective BV theory from Chern-Simons. http://arXiv.org/list/hep-th/0603201, 2006
Kummer W., Liebl H., Vassilevich D.V.: Exact path integral quantization of generic 2-D dilaton gravity. Nucl. Phys. B 493, 491 (1997)
Laurent-Gengoux, C., Stienon, M., Xu, P.: Holomorphic Poisson Structures and Groupoids. http://arXiv.org/list/0707.4253v4[math.DG], 2007, to appear in Intl. Math. Res. Notices
Li, Y.: On deformations of generalized complex structurs: The generalized Calabi-Yau case. http://arXiv.org/list/hep-th/0508030, 2005
Lu Z.J., Weinstein A., Xu P.: Manin Triples for Lie Bialgebroids. J. Diff. Geom. 45, 547 (1997)
Lyakhovich S.L., Sharapov A.A.: Characteristic classes of gauge systems. Nucl. Phys. B 703, 419 (2004)
Lyakhovich S., Zabzine M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548, 243 (2002)
Manin, Yu.I.: Three constructions of Frobenius manifolds: a comparative study, In:Surveys in differential geometry, 497–554, Surv. Differ. Geom., VII, Somerville, MA: Int. Press, 2000, pp. 497–554
Manin, Yu.I.: Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society Colloquium Publications 47, Providence, RI: American Mathematical Society, 1999
Mnev, P.: Notes on simplicial BF theory. http://arXiv.org/list/hep-th/0610326, 2006
Pestun V.: Topological strings in generalized complex space. Adv. Theor. Math. Phys. 11, 399 (2007)
Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. In: Quantization, Poisson Brackets and Beyond, Theodore Voronov (ed.), Contemp. Math., Vol. 315, Providence, RI: Amer. Math. Soc., 2002
Schaller P., Strobl T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A 9, 3129 (1994)
Schwarz A.S.: Geometry of Batalin-Vilkovisky quantization. Commun. Math. Phys. 155, 249 (1993)
Vaisman, I.: Lectures on the geometry of Poisson manifolds. Progress in Mathematics, 118. Basel: Birkhauser Verlag, 1994
Weinstein A.: The modular automorphism group of Poisson manifolds. J. Geom. Phys. 23, 379 (1997)
Witten E.: Topological Sigma Models. Commun. Math. Phys. 118, 411 (1988)
Witten, E.: Mirror manifolds and topological field theory. http://arXiv.org/list/hep-th/9112056, 1991
Xu P.: Gerstenhaber algebras and BV-algebras in Poisson geometry. Commun. Math. Phys. 200, 545 (1999)
Zabzine M.: Lectures on generalized complex geometry and supersymmetry. Archivum mathematicum (supplement) 42, 119–146 (2006)
Zucchini R.: A sigma model field theoretic realization of Hitchin’s generalized complex geometry. JHEP 0411, 045 (2004)
Zucchini R.: A topological sigma model of biKaehler geometry. JHEP 0601, 041 (2006)
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Communicated by N.A. Nekrasov
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Bonechi, F., Zabzine, M. Poisson Sigma Model on the Sphere. Commun. Math. Phys. 285, 1033–1063 (2009). https://doi.org/10.1007/s00220-008-0615-1
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DOI: https://doi.org/10.1007/s00220-008-0615-1