Hilbert Series for Moduli Spaces of Instantons on C^2/Z_n

TL;DR

This paper computes Hilbert Series for moduli spaces of G-instantons on A-type ALE spaces, revealing algebraic and geometric properties, and confirms consistency across different quiver descriptions.

Contribution

It provides explicit Hilbert Series calculations for instanton moduli spaces on ALE spaces for various classical groups, connecting gauge theories, brane configurations, and geometric structures.

Findings

Hilbert Series encode moduli space dimensions and generators

Explicit calculations for a wide class of instantons

Consistency of Hilbert Series across different quiver descriptions

Abstract

We study chiral gauge-invariant operators on moduli spaces of G instantons for any classical group G on A-type ALE spaces using Hilbert Series (HS). Moduli spaces of instantons on an ALE space can be realized as Higgs branches of certain quiver gauge theories which appear as world-volume theories on Dp branes in a Dp-D(p+4) system with the D(p+4) branes (with or without O(p+4) planes) wrapping the ALE space. We study in detail a list of quiver gauge theories which are related to G-instantons of arbitrary ranks and instanton numbers on a generic A_{n-1} ALE space and discuss the corresponding brane configurations. For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. In a large number of examples involving lower rank instantons, we demonstrate that HS for equivalent instantons of isomorphic gauge groups but very different quiver descriptions do indeed agree, as expected.