On the Schwinger Model on Riemann Surfaces

TL;DR

This paper provides an exact solution of the massless Schwinger model on Riemann surfaces, deriving key functional quantities and revealing its equivalence to a generalized Thirring model, with implications for quantum field theory in curved space.

Contribution

It presents the first exact quantization of the Schwinger model on Riemann surfaces and establishes its equivalence to a nonlocal integrable model, expanding understanding of quantum field theories in curved spaces.

Findings

Explicit partition function and correlation function generating functional derived.

Schwinger model shown to be equivalent to a nonlocal integrable model.

Method applicable to other abelian gauge theories on Riemann surfaces.

Abstract

In this paper, the massless Schwinger model or two dimensional quantum electrodynamics is exactly solved on a Riemann surface. The partition function and the generating functional of the correlation functions involving the fermionic currents are explicitly derived using a method of quantization valid for any abelian gauge field theory and explained in the recent references [F. Ferrari, {\it Class. Quantum Grav.} {\bf 10} (1993), 1065], [F. Ferrari, hep-th 9310024]. In this sense, the Schwinger model is one of the few examples of interacting and nontopological field theories that are possible to quantize on a Riemann surface. It is also shown here that the Schwinger model is equivalent to a nonlocal integrable model which represents a generalization of the Thirring model. Apart from the possible applications in string theory and integrable models, we hope that this result can be also useful in the study of quantum field theories in curved space-times.