Zero velocity Lagrangians in the Hitchin system

Abstract

Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the $\mathbb{C}^*$-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.