A tentative proposal towards an equivariant mirror symmetry for Hitchin systems
John Alexander Cruz Morales
TL;DR
This work outlines a speculative program to extend Aganagic's equivariant mirror symmetry from Coulomb branches to Hitchin systems associated with class $\mathcal{S}$ theories. It argues that the natural B-model partner should be a Landau-Ginzburg model with twisted masses and emphasizes carrying over the additive–multiplicative dichotomy to Hitchin settings, guided by hypertoric insights. The piece surveys background on Coulomb/Higgs branches, ADE quiver theories, and both standard and multiplicative Hitchin systems, then proposes concrete steps toward an equivariant Hitchin mirror, including an upstairs/downstairs framework and a plan to develop equivariant genus-0 Gromov–Witten theory to obtain semisimple Frobenius manifolds. The proposed program aims to connect mirror symmetry for Hitchin systems with broader questions in class $\mathcal{S}$, geometric Langlands, and moduli of bundles, providing a scaffold for future rigorous developments and potential applications in mathematical physics.
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.