Poisson Sigma Model with branes and hyperelliptic Riemann surfaces

TL;DR

This paper derives explicit superpropagators for the Poisson Sigma Model with boundary conditions involving multiple branes, utilizing hyperelliptic Riemann surfaces and theta functions, extending previous work on deformation quantization.

Contribution

It generalizes the superpropagator construction to multiple branes using hyperelliptic Riemann surfaces and theta functions, advancing the understanding of boundary conditions in the Poisson Sigma Model.

Findings

Explicit superpropagators for arbitrary number of branes derived.

Use of hyperelliptic Riemann surfaces and theta functions in the construction.

Identification of zero mode contributions in higher genus cases.

Abstract

We derive the explicit form of the superpropagators in presence of general boundary conditions (coisotropic branes) for the Poisson Sigma Model. This generalizes the results presented in Cattaneo and Felder's previous works for the Kontsevich angle function used in the deformation quantization program of Poisson manifolds without branes or with at most two branes. The relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons P_n with particular "alternating" boundary conditions. In 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 modes contributions.