Counting Higgs bundles and type quiver bundles

Abstract

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.