On the Gribov Problem for Generalized Connections

TL;DR

This paper studies the structure of the space of generalized connections in gauge theories, revealing the presence of the Gribov problem in non-abelian cases and analyzing the bundle structure of different strata.

Contribution

It proves that each stratum of the space forms a locally trivial fiber bundle and characterizes the generic stratum's structure group, highlighting the Gribov problem for non-abelian theories.

Findings

Generic stratum is a principal fiber bundle with a specific structure group.

For abelian theories, the generic stratum is globally trivial.

Non-abelian theories like SU(N) exhibit a nontrivial generic stratum, indicating the Gribov problem.

Abstract

The bundle structure of the space $\Ab$ of Ashtekar's generalized connections is investigated in the compact case. It is proven that every stratum is a locally trivial fibre bundle. The only stratum being a principal fibre bundle is the generic stratum. Its structure group equals the space $\Gb$ of all generalized gauge transforms modulo the constant center-valued gauge transforms. For abelian gauge theories the generic stratum is globally trivial and equals the total space $\Ab$. However, for a certain class of non-abelian gauge theories -- e.g., all SU(N) theories -- the generic stratum is nontrivial. This means, there are no global gauge fixings -- the so-called Gribov problem. Nevertheless, there is a covering of the generic stratum by trivializations each having total induced Haar measure 1.