Topological Aspects of Gauge Fixing Yang-Mills Theory on S4

TL;DR

This paper explores the topological structure of gauge fixing in Yang-Mills theory on the four-sphere, revealing how topological invariants influence gauge orbit contributions and the physical implications of gauge dependence.

Contribution

It establishes a connection between topological quantum field theory and gauge fixing in Yang-Mills theory on S4, highlighting the role of topological invariants in gauge orbit contributions.

Findings

Euler character of gauge orbit space varies with topological sector.

Gribov copies affect multiplicity factors in correlation functions.

Topological trivial sectors do not contribute in certain gauges.

Abstract

For an $S_4$ space-time manifold global aspects of gauge-fixing are investigated using the relation to Topological Quantum Field Theory on the gauge group. The partition function of this TQFT is shown to compute the regularized Euler character of a suitably defined space of gauge transformations. Topological properties of the space of solutions to a covariant gauge conditon on the orbit of a particular instanton are found using the $SO(5)$ isometry group of the $S_4$ base manifold. We obtain that the Euler character of this space differs from that of an orbit in the topologically trivial sector. This result implies that an orbit with Pontryagin number $\k=\pm1$ in covariant gauges on $S_4$ contributes to physical correlation functions with a different multiplicity factor due to the Gribov copies, than an orbit in the trivial $\k=0$ sector. Similar topological arguments show that there is no contribution from the topologically trivial sector to physical correlation functions in gauges defined by a nondegenerate background connection. We discuss possible physical implications of the global gauge dependence of Yang-Mills theory.