On the gauge orbit space stratification (a review)

TL;DR

This review explores the mathematical structure of gauge orbit space stratification, focusing on SU(n) theories in four dimensions, providing methods for classifying orbit types and their orderings, with applications to Chern-Simons theory.

Contribution

It develops a general method for classifying gauge orbit types in SU(n) theories using cohomology and algebraic equations, extending understanding of stratification structures.

Findings

Orbit types classified by cohomology elements.

Partial ordering characterized by algebraic equations.

Application to nodal configurations in Chern-Simons theory.

Abstract

First, we review the basic mathematical structures and results concerning the gauge orbit space stratification. This includes general properties of the gauge group action, fibre bundle structures induced by this action, basic properties of the stratification and the natural Riemannian structures of the strata. In the second part, we study the stratification for theories with gauge group $\rmSU(n)$ in space time dimension 4. We develop a general method for determining the orbit types and their partial ordering, based on the 1-1 correspondence between orbit types and holonomy-induced Howe subbundles of the underlying principal $\rmSU(n)$-bundle. We show that the orbit types are classified by certain cohomology elements of space time satisfying two relations and that the partial ordering is characterized by a system of algebraic equations. Moreover, operations for generating direct successors and direct predecessors are formulated, which allow one to construct the set of orbit types, starting from the principal one. Finally, we discuss an application to nodal configurations in topological Chern-Simons theory.