Parametrization of the moduli space of flat SL(2,R) connections on the torus

TL;DR

This paper explicitly parametrizes the moduli space of flat SL(2,R) connections on a torus, providing a clear classification, canonical forms, and a Hausdorff topology, thus clarifying its structure.

Contribution

It introduces a canonical form and explicit parametrization for the moduli space of flat SL(2,R) connections on the torus, enhancing understanding of its structure.

Findings

Explicit parametrization of the moduli space

Classification based on spectral properties

Hausdorff topology proposal

Abstract

The moduli space of flat SL(2,R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous conjugation by SL(2,R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.