Abstract
The moduli space of instantons on C^2 for any simple gauge group is studied using the Coulomb branch of N=4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.
Highlights
The ADHM construction exists only for classical gauge groups and, until recently, the instanton partition functions for exceptional gauge groups were only possible by means of superconformal indices [8,9,10] of theories obtained by wrapping M 5branes on punctured Riemann surfaces [11] as in [12] for E6,7,8 instantons, by extrapolating the blow-up equations of [13, 14] as in [15], or by utilizing the generating function of holomorphic functions on the moduli space as in [16,17,18]
The D5-brane U(1) symmetry acts as a flavor group for the worldvolume theory on the D3-branes: it attaches a square node to the extended node of the affine Dynkin diagram, as in [29, 30, 36]
In this paper we have proposed a simple formula for the Hilbert series of moduli spaces of pure Yang-Mills instantons, which arise as Coulomb branches of three-dimensional N = 4 generalized quiver gauge theories whose quiver diagrams are given by over-extended Dynkin diagrams
Summary
We summarize various brane constructions for moduli spaces of instantons of classical gauge groups [4, 5, 36,37,38,39]. The Higgs branch of these theories is achieved when the D2-branes are inside the D6branes; the Coulomb branch is realized when the D2-branes are away from the D6-branes It is the Higgs branch of these quiver gauge theories that reproduces the moduli space of G-instantons, where G is the flavor symmetry group of the quiver. The D5-brane U(1) symmetry acts as a flavor group for the worldvolume theory on the D3-branes: it attaches a square node to the extended node of the affine Dynkin diagram, as in [29, 30, 36] Even though this U(1) node appears naturally as a flavor node in the brane construction, it is useful to treat it on the same footing as the other gauge nodes, and ungauge an overall diagonal U(1) gauge symmetry under which no matter fields are charged.
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