Poisson sigma model with branes and hyperelliptic Riemann surfaces

Abstract

We derive the explicit form of the superpropagators in the presence of general boundary conditions (coisotropic branes) for the Poisson sigma model. This generalizes the results presented by Cattaneo and Felder [“A path integral approach to the Kontsevich quantization formula,” Commun. Math. Phys. 212, 591 (2000)] and Cattaneo and Felder [“Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model,” Lett. Math. Phys. 69, 157 (2004)] for Kontsevich’s angle function [Kontsevich, M., “Deformation quantization of Poisson manifolds I,” e-print arXiv:hep.th/0101170] used in the deformation quantization program of Poisson manifolds. The relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons Pn with particular “alternating” boundary conditions. In the presence of more than three branes we use first order Riemann theta functions with odd singular characteristics on the Jacobian variety of a hyperelliptic Riemann surface (canonical setting). In genus g the superpropagators present g zero mode contributions.