G-bundles on Abelian surfaces, hyperkahler manifolds, and stringy Hodge numbers

Abstract

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified group. M(G,A) has the structure of a hyperkahler orbifold. We show that when G is Sp(n) or SU(n), M(G,A) has a natural hyperkahler desingularization which we exhibit as a moduli space of G_C-bundles with an altered stability condition. In this way, we obtain the two known families of hyperkahler manifolds, the Hilbert scheme of points on a K3 surface and the generalized Kummer varieties. We show that for G not Sp(n) or SU(n), the moduli space M(G,A) does not admit a hyperkahler resolution, in fact, it does not have a crepant resolution. Inspired by the physicists Vafa and Zaslow, Batyrev and Dais define ``stringy Hodge numbers'' for certain orbifolds. These numbers are conjectured to agree with the Hodge numbers of a crepant resolution (when it exists). We compute the stringy Hodge numbers of M(SU(n),A) and M(Sp(n),A) and verify the conjecture in these cases.