SL(2,R) Yang-Mills theory on a circle

TL;DR

This paper explores the unique topological and quantization challenges of SL(2,R) Yang-Mills theory on a circle, highlighting non-Hausdorff topology and fixed points affecting the physical configuration space.

Contribution

It provides a detailed analysis of the non-Hausdorff topology and fixed points in SL(2,R) Yang-Mills theory on a circle, revealing complexities in canonical quantization.

Findings

Gauge transformations have hyperbolic fixed points

Configuration space has non-Hausdorff topology

Quantization ambiguity is more pronounced

Abstract

The kinematics of SL(2,R) Yang-Mills theory on a circle is considered, for reasons that are spelled out. The gauge transformations exhibit hyperbolic fixed points, and this results in a physical configuration space with a non-Hausdorff "network" topology. The ambiguity encountered in canonical quantization is then much more pronounced than in the compact case, and can not be resolved through the kind of appeal made to group theory in that case.