Reducibility and Gribov Problem in Topological Quantum Field Theory

TL;DR

This paper addresses fundamental issues in topological quantum field theory related to reducible gauge configurations and Gribov ambiguities, proposing a geometric framework to resolve these problems and applying it to the Donaldson--Witten model.

Contribution

It introduces a geometric method to handle reducibility and Gribov problems in topological QFT, ensuring gauge group actions are free and gauge fixing is unambiguous.

Findings

The formalism renders the gauge group action free on the augmented configuration space.

The approach yields a local, potentially renormalizable quantum field theory.

The Gribov problem is inherently linked to localization and can be managed in an equivariant setting.

Abstract

In spite of its simplicity and beauty, the Mathai-Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: $i$) the existence of reducible field configurations on which the action of the gauge group is not free and $ii$) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai--Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt with in a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson--Witten model.