The ADHM-like constructions for instantons on [formula omitted] and three-dimensional gauge theories

Noppadol Mekareeya,Diego Rodríguez-Gómez

doi:10.1016/j.nuclphysb.2014.12.009

Abstract

We study the moduli spaces of self-dual instantons on CP2 in a simple group G. When G is a classical group, these instanton solutions can be realized using ADHM-like constructions which can be naturally embedded into certain three-dimensional quiver gauge theories with four supercharges. The topological data for such instanton bundles and their relations to the quiver gauge theories are described. Based on such gauge theory constructions, we compute the Hilbert series of the moduli spaces of instantons that correspond to various configurations. The results turn out to be equal to the Hilbert series of their counterparts on C2 upon an appropriate mapping. We check the former against the Hilbert series derived from the blowup formula for the Hirzebruch surface F1 and find an agreement. The connection between the moduli spaces of instantons on such two spaces is explained in detail.

Highlights

The preferred orientation induced by the complex structure, is taken to be self-dual
String theory provides a nice perspective on ADHM, as from the point of view of the D(p+4) brane dissolving Dp branes is done by turning on a worldvolume instanton on the 4 transverse coordinates to the Dp inside the D(p + 4)
In the case of CP2, the ADHM-like construction of instantons in the unitary group was developed by King in [4] based on previous work by Buchdal in [1,2,3], while the constructions for special orthogonal and symplectic groups were subsequently developed by Bryan and Sanders in [5]

Summary

Gauge theory for the King construction

Note that the description of this branch of the moduli space shares some similarity to that of the Higgs branch of the standard ADHM construction It is parametrized by the the gauge invariant quantities constructed from the massless chiral fields {A2, B1, B2, Q, q} subject to constraints from the F -terms. In order to obtain the Hilbert series that precisely matches that of instantons on C2, it is crucial to use a suitable grading associated with the fugacity t that is compatible with the correct superconformal R-symmetry

Properties of the instanton moduli space

Explicit examples

The blowup formula

G2 instantons

Conclusions