On the Configuration Space of Gauge Theorie

TL;DR

This paper explores the geometric and topological structure of the configuration space in gauge theories, revealing convexity properties, singularities, and their relation to simpler models like electrodynamics.

Contribution

It provides a detailed analysis of the configuration space structure, including the description of singularities and the isomorphism to simpler gauge theories, advancing understanding of gauge theory geometry.

Findings

Irreducible orbits form a geodesically convex stratum.

Singularities in SU(2) theories are conical and form a Z_2 orbifold.

Configuration space of electrodynamics is related to SU(2) space via orbifold.

Abstract

We investigate the structure of the configuration space of gauge theories and its description in terms of the set of absolute minima of certain Morse functions on the gauge orbits. The set of absolute minima that is obtained when the background connection is a pure gauge is shown to be isomorphic to the orbit space of the pointed gauge group. We also show that the stratum of irreducible orbits is geodesically convex, i.e. there are no geometrical obstructions to the classical motion within the main stratum. An explicit description of the singularities of the configuration space of SU(2) theories on a topologically simple space-time and on the lattice is obtained; in the continuum case we find that the singularities are conical and that the singular stratum is isomorphic to a Z_2 orbifold of the configuration space of electrodynamics.