A tentative proposal towards an equivariant mirror symmetry for Hitchin systems

TL;DR

This paper proposes extending equivariant mirror symmetry concepts from Coulomb branches of gauge theories to Hitchin systems, suggesting the mirror should be a Landau-Ginzburg model with twisted masses and exploring the additive-multiplicative dichotomy.

Contribution

It introduces a novel framework for equivariant mirror symmetry of Hitchin systems, building on Aganagic's ideas and proposing new dualities involving Landau-Ginzburg models.

Findings

Proposes the equivariant mirror of Hitchin systems as a Landau-Ginzburg model.

Suggests considering the additive-multiplicative dichotomy in Hitchin system mirror symmetry.

Provides conceptual groundwork for future mathematical and physical investigations.

Abstract

Motivated by Aganagic's equivariant mirror symmetry for certain Coulomb branches of a $3d$ $\mathcal{N}= 4$ gauge quiver theory, we would like to propose a set of ideas towards an extension of Aganagic's proposal to Hitchin systems. At the end, there are two main points in our proposal; namely, that the equivariant mirror of the Hitchin systems should be a Landau-Ginzburg model (with twisted masses) and that the dichotomy between additive and multiplicative varieties in the context of mirror symmetry for Nakajima quiver varieties should be considered in the case of Hitchin systems.