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

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.