Geometric quantization and non-perturbative Poisson sigma model

Bonechi, F; Cattaneo, A S; Zabzine, M (2006). Geometric quantization and non-perturbative Poisson sigma model. Advances in Theoretical and Mathematical Physics, 10(5):683-712.

Abstract

In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class [α]. The same condition was considered before by Crainic and Zhu but in a diﬀerent context. In the case when [α] is in the image of the sharp map we reproduce the Vaisman’s condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show, with a diﬀerent procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end we discuss the case of a generic coisotropic D-brane.

Abstract

In this note we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a certain integrality condition for the Poisson cohomology class [α]. The same condition was considered before by Crainic and Zhu but in a diﬀerent context. In the case when [α] is in the image of the sharp map we reproduce the Vaisman’s condition for prequantizable Poisson manifolds. For integrable Poisson manifolds we show, with a diﬀerent procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end we discuss the case of a generic coisotropic D-brane.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2006 07 Jan 2010 15:03 06 Dec 2017 20:44 International Press 1095-0753 http://www.intlpress.com/ATMP/ATMP-issue_10_5.php http://arxiv.org/abs/math/0507223