# Multiplicative Hitchin Systems and Supersymmetric Gauge Theory

## TL;DR

This paper explores multiplicative Hitchin systems, their connections to supersymmetric gauge theory, and their quantization, establishing new equivalences and analyzing their geometric and algebraic properties.

## Contribution

It proves the hyperk"ahler equivalence of multiplicative Higgs bundles and periodic monopoles, and discusses the quantization leading to Yangian modules.

## Key findings

- Established hyperk"ahler space equivalence
- Analyzed twistor rotation in the multiplicative Hitchin system
- Connected quantization to Yangian modules

## Abstract

Multiplicative Hitchin systems are analogues of Hitchin's integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where C = CP1 with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperk\"ahler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. We also discuss quantization of these moduli spaces, yielding the modules for the Yangian Y(g) discovered by Gerasimov, Kharchev, Lebedev and Oblezin.