Multiplicative Hitchin Systems and Supersymmetric Gauge Theory

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 hyperkähler 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.

66 pages. Minor edit correcting a misattributed reference

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• On Quiver W-algebras and Defects from Gauge Origami
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• di-Langlands correspondence and extended observables
• A Family of ${\mathrm{GL}}_r$ Multiplicative Higgs Bundles on Rational Base
• Bispectral duality and separation of variables from surface defect transition
• The Kapustin-Witten equations and nonabelian Hodge theory
• Multiplicative Higgs bundles and involutions