Bohr--Sommerfeld Lagrangians of moduli spaces of Higgs bundles

TL;DR

This paper characterizes the Bohr-Sommerfeld Lagrangians in the moduli space of Higgs bundles on a Riemann surface, showing they are exactly the components of the nilpotent cone, extending to Higgs G-bundles and parabolic cases.

Contribution

It identifies the Bohr-Sommerfeld Lagrangians in Higgs bundle moduli spaces as the irreducible components of the nilpotent cone, generalizing previous results to broader settings.

Findings

Bohr-Sommerfeld Lagrangians are the irreducible components of the nilpotent cone.

The result extends to Higgs G-bundles.

The result also applies to parabolic Higgs bundles.

Abstract

Let $X$ be a compact connected Riemann surface of genus at least two. Let $M_H(r,d)$ denote the moduli space of semistable Higgs bundles on $X$ of rank $r$ and degree $d$. We prove that the compact complex Bohr-Sommerfeld Lagrangians of $M_H(r,d)$ are precisely the irreducible components of the nilpotent cone in $M_H(r,d)$. This generalizes to Higgs $G$-bundles and also to the parabolic Higgs bundles.