Abstract
The moduli space of instantons on an ALE space is studied using the moduli space of \( \mathcal{N}=4 \) field theories in three dimensions. For instantons in a simple gauge group G on \( {\mathrm{\mathbb{C}}}^2/{\mathrm{\mathbb{Z}}}_n \), 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 \( {\mathrm{\mathbb{C}}}^2/\widehat{G} \), 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 A 2n−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 D N .
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Mekareeya, N. The moduli space of instantons on an ALE space from 3d \( \mathcal{N}=4 \) field theories. J. High Energ. Phys. 2015, 1–30 (2015). https://doi.org/10.1007/JHEP12(2015)174
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DOI: https://doi.org/10.1007/JHEP12(2015)174