Complexification of Gauge Theories

TL;DR

This paper investigates the conditions under which gauge theory reductions can be simplified by complexification, with applications to Yang-Mills theories and the use of holomorphic variables.

Contribution

It establishes criteria for when the physical phase space of gauge systems can be obtained via complexification, linking geometric reduction to complexified gauge groups.

Findings

Conditions for equivalence of reduction processes in finite dimensions.

Extension of conditions to infinite-dimensional Yang-Mills theories.

Discussion on using holomorphic Wilson loops as global coordinates.

Abstract

For the case of a first-class constrained system with an equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations $G$ is equivalent to the single process of dividing out the initial phase space by the complexification $G_C$ of $G$. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold in the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.