The "Dual" Variables Of Yang-Mills Theory And Local Gauge Invariant Variables

TL;DR

This paper develops a method to express Yang-Mills theories in terms of local gauge invariant variables, reproducing known solutions in 2D and proposing gravity-like formulations in higher dimensions, with potential simplifications.

Contribution

It introduces a novel approach to rewrite Yang-Mills actions using auxiliary fields, leading to local gauge invariant variables and simplified models in various dimensions.

Findings

Reproduces 2D $SU(N)$ solutions using local variables

Derives a gravity-like theory in 3D for $SU(2)$

Expresses 4D $SU(2)$ theory with only six degrees of freedom

Abstract

After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge invariant variables. This method reproduces the known solution of the two dimensional $SU(N)$ theory. In more than two dimensions the action splits into a topological part and a part proportional to $\alpha_s$. We demonstrate the procedure for $SU(2)$ in three dimensions where we reproduce a gravity-like theory. We discuss the four dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional $SU(2)$ theory can be expressed in terms of a local action with six degrees of freedom only.