\(G\)-bundles on abelian surfaces, hyperkähler manifolds, and stringy Hodge numbers (Q2719827)

Let \(G\) be a compact, simple, connected, simply connected Lie group, and let \(M_G(A)\) be the moduli space of flat \(G\) connections on an Abelian surface \(A\). If \(G^{\mathbb C}\) denotes the complexification of \(G\), then \(M_G(A)\) is the coarse moduli space of \(s\)-equivalence classes of semi-stable holomorphic \(G^{\mathbb C}\)-bundles on \(A\) with trivial Chern classes. In this paper, the authors prove that \(M_G(A)\) has a hyperKähler resolution if and only if \(G\) is \(SU(n)\) or \(Sp(n)\) and that the resolution is realized as a certain moduli space of \(G\)-bundles in those cases. They also show that the stringy Hodge numbers of \(M_{SU(n)} (A)\) and \(M_{Sp(n)} (A)\) coincide with the ordinary Hodge numbers of their corresponding hyperKähler resolutions.