Applications of the Liouville symplectic form on the cotangent bundle of a loop group

TL;DR

This paper demonstrates that the Liouville symplectic form on the cotangent bundle of a loop group underpins the symplectic structures on various moduli spaces of principal Higgs bundles and connections on a Riemann surface, revealing a unified origin.

Contribution

It establishes that the symplectic structures on moduli stacks and spaces of framed Higgs bundles and connections derive from the Liouville form on the cotangent bundle of the loop group, unifying previous constructions.

Findings

Liouville symplectic form induces structures on moduli spaces

Unified origin of different symplectic structures

Connections to principal Higgs bundles and framed connections

Abstract

Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^*G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures -- on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing -- were constructed earlier by various authors. Our results show that they all have a common origin.