Complex $K$-theory of moduli spaces of Higgs bundles

Abstract

We establish an isomorphism of complex $K$-theory of the moduli space $\check{\mathcal{M}}$ of “${\mathrm{SL}}_n$”-Higgs bundles of degree $d$ and rank $n$ (in the sense of Hausel–Thaddeus) and twisted complex $K$-theory of the orbifold $\hat{\mathcal{M}}$ of ${\mathrm{PGL}}_n$-Higgs bundles of degree $e$, where $(n,d)=(n,e)=1$. Along the way, we prove the vanishing of torsion for $H^{\ast }(\check{\mathcal{M}})$ and certain twisted complex $K$-theory groups of $\hat{\mathcal{M}}$. We also extend Arinkin’s autoduality of compactified Jacobian to a derived equivalence between ${\mathrm{SL}}_n$- and ${\mathrm{PGL}}_n$-Hitchin systems over the elliptic locus. In the appendix, we develop a formalism of $G$-sheaves of spectra, generalising equivariant homotopy theory to a relative setting.