On Derivation of Goldman Bracket

TL;DR

This paper constructs an infinite-dimensional Lie algebra of gauge-invariant observables with Goldman-type brackets for the G2 gauge group on Riemann surfaces, extending known results for classical groups.

Contribution

It introduces a new Goldman bracket derivation for G2 and generalizes the algebra of gauge-invariant observables on Riemann surfaces.

Findings

Established a Lie algebra for G2 gauge group

Provided an alternative derivation for classical groups

Extended Goldman bracket framework to exotic gauge groups

Abstract

In this paper, we obtain an infinite dimensional Lie algebra of exotic gauge invariant observables that is closed under Goldman-type bracket associated with monodromy matrices of flat connections on a compact Riemann surface for $G_{2}$ gauge group. As a by-product, we give an alternative derivation of known Goldman bracket for classical gauge groups $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$, $U(n)$, $SU(n)$, $Sp(2n,\mathbb{R})$ and $SO(n)$.