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 a e-print archive: http://arXiv.org/abs/math.sg/0507223 FRANCESCO BONECHI ET AL.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.

On the geometry of prequantization spaces

“…As a by-product, we obtain the prequantization condition for Γ s (P) in terms of period groups on P. Then we show that this condition is automatically satisfied when the Dirac manifold P admits a prequantization circle bundle Q over it. This generalizes some of the results in [8,2]. This paper ends with three appendices.…”

“…We will see in items (4) and (5) of Theorem 4.11 that the prequantizability and integrability of (P, L) imply that Γ s (P) is prequantizable, and that the prequantization bundleΓ c (P) is a groupoid integrating L c , so A(Γ c (P)) ∼ = L c where "A" denotes the functor that takes the Lie algebroid of a Lie groupoid. (In the Poisson case this follows from [8,2]. )…”

“…The prequantizability of (P, L) implies that the period group of any source fiber of Γ s (P) is contained in Z (see Section 3.3 of [2], or Theorem 4.2 below for a straightforward generalization). This last condition is equivalent to saying that the symplectic groupoid Γ s (P) is prequantizable in the sense of [6] (see Prop.…”

“…SinceL is G invariant, doing this at every x ∈ Q we obtain a well defined subbundle of E 1 (Q/G), which however might fail to be smooth. 2 We have a surjective map…”

“…This last condition is equivalent to saying that the symplectic groupoid Γ s (P) is prequantizable in the sense of [6] (see Prop. 2 in [2] or Thm. 3 in [8]).…”