Geometric quantization and non-perturbative Poisson sigma model

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 acertain integrality condition for the Poisson cohomology class $[\alpha]$. The same condition was considered before by Crainic and Zhu but in a different context. In the case when $[\alpha]$ 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.