The moduli space of instantons on an ALE space from 3d(formulapresented.)field theories

Abstract

The moduli space of instantons on an ALE space is studied using the moduli space of (formula presented.) field theories in three dimensions. For instantons in a simple gauge group G on (formula presented.), the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the affine Dynkin diagram of G with flavour nodes of unitary groups attached to various nodes of the Dynkin diagram. We provide a simple prescription to determine the ranks and the positions of these flavour nodes from the order of the orbifold n and from the residual subgroup of G that is left unbroken by the monodromy of the gauge field at infinity. For G a simply laced group of type A, D or E, the Higgs branch of such a quiver describes the moduli space of SU(n) instantons on orbifold (formula presented.), where Ĝ is the discrete group that is in McKay correspondence to G. Moreover, we present the quiver whose Coulomb branch is the moduli space of SO(2N) instantons on a smooth ALE space of type A2n−1 with a certain monodromy of the gauge field at infinity. The Higgs branch of such a quiver is conjectured to be the moduli space of SU(2n) instantons on a smooth ALE space of type DN.