Bohr--Sommerfeld Lagrangians of moduli spaces of 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.

Final version; Journal of Geometry and Physics (to appear)

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• Mapping class group invariant unitarity of the Hitchin connection over Teichmüller space
• A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications