Abstract
The worldvolume theory of D3-branes probing four D7-branes and an O7-plane on $\mathbb{C}^2/\mathbb{Z}_2$ is given by a supersymmetric USp x USp gauge theory. We demonstrate that, at least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space. This allows us to express the moduli space of SO(8) instantons on the smooth ALE space as a hyperk\"ahler quotient of a flat space times the Higgs branch of a class S theory. We also discuss a generalization to $\mathbb{C}^2/\mathbb{Z}_{2n}$, and speculate how to extend the analysis to bigger SO groups and to ALE spaces of other types.
Highlights
When p = 2, the gauge theory is three dimensional with N = 4 supersymmetry
At least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space
At least for SO(8) instantons with a particular holonomy at infinity, we can go to a dual description of the original gauge theory, where we can add appropriate FI terms
Summary
Let us first recall the quiver description of the moduli space of SO(N ) instantons on the orbifold C2/Z2, with the holonomy at infinity given by diag(+ + · · · + − − · · · −). Let us consider the orthogonal instantons on the blown-up ALE space C2/Z2, with the holonomy at infinity given by (2.1). The second Stiefel-Whitney class of the bundle is determined by the holonomy at infinity, and does not give additional topological data. When N− is a multiple of four, we see that the dimensions of the moduli spaces on the orbifold and the smooth ALE space agree when we take k+ = k, k− = k + N−/4. On the contrary, the difference k+ − k− does not correspond to any data of the gauge configuration on the smooth ALE space
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