Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison

Abstract

Let $X$ be the moduli space of stable trace-free rank $r$ Higgs bundles over a smooth projective curve $C$ with fixed determinant $Λ\in\Pic(C)$ and $\gcd(r,\degΛ)=1$. We prove that the $\cc^*$-equivariant quantum product of a divisor on $X$ is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in $X\times X$. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to $X$ under evaluation map. For $Γ=\Pic^0(C)[r]$, we give a character decomposition for $Γ$-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For $\widehat X=[X/Γ]$, which is the moduli space of stable $\PGL_r$-Higgs bundles over $C$. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory $\KU^*(X)$ and twisted K-theory $\KU^*([X/Γ],α)$ by the lifting $μ_r$-gerbe $α$ of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.