Perturbed geodesics on the moduli space of flat connections and Yang-Mills theory

TL;DR

This paper establishes a bijective correspondence between perturbed closed geodesics and perturbed Yang-Mills connections on the moduli space of flat SO(3)-connections over a surface, preserving Morse index.

Contribution

It proves the bijection between perturbed geodesics and Yang-Mills connections, linking geometric and gauge-theoretic structures in a novel way.

Findings

Bijection between perturbed geodesics and Yang-Mills connections.

Preservation of Morse index in the correspondence.

Applicable to moduli space of flat SO(3)-bundles over surfaces.

Abstract

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang-Mills connections with the same Morse index.