Abelian gauge theories on compact manifolds and the Gribov ambiguity

TL;DR

This paper investigates the quantization of abelian gauge theories on compact manifolds, highlighting the Gribov ambiguity due to bundle non-triviality, and proposes a modified measure to account for topological effects in calculations.

Contribution

It introduces a modified functional integral measure for abelian gauge theories on compact manifolds that incorporates the Gribov ambiguity and topological effects.

Findings

The Gribov ambiguity arises from non-trivial bundle structures.

The modified measure allows calculation of partition functions and Green's functions.

Topological features lead to non-zero vacuum expectation values.

Abstract

We study the quantization of abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the non-triviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, the Greens functions and the field strength correlating functions in any dimension using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. The Greens functions are shown to be affected by the non-trivial topology, giving rise to non-vanishing vacuum expectation values for the gauge fields.