(BAA)-branes from higher Teichm\"uller theory

TL;DR

This paper constructs moduli stacks related to higher Teichmüller theory and Higgs bundles, extending known correspondences and exploring their mirror symmetry under S-duality.

Contribution

It introduces new moduli stacks from the relative Langlands program and extends the Cayley correspondence to a Lagrangian morphism over the Hitchin moduli stack.

Findings

Construction of moduli stacks from holomorphic Lagrangians

Extension of Cayley correspondence to a Lagrangian morphism

Initiation of hyperholomorphic mirror partner study under S-duality

Abstract

Interpreting certain holomorphic Lagrangians that arise from the relative Langlands program, we construct moduli stacks underlying the generalized Slodowy categories of Collier--Sanders and $G^\mathbf{R}$-Higgs bundles over a Riemann surface. Furthermore, we extend the Cayley correspondence of Bradlow--Collier--Garc\'ia-Prada--Gothen--Oliveira to a morphism of Lagrangians over the Hitchin moduli stack, and initiate the study of its hyperholomorphic mirror partner under $S$-duality.