Complete Intersection Moduli Spaces in N=4 Gauge Theories in Three Dimensions

TL;DR

This paper analyzes the moduli spaces of 3D N=4 gauge theories, focusing on complete intersection hyperKahler spaces, their Hilbert series, and the relations defining these spaces, using brane realizations and mirror symmetry.

Contribution

It introduces a detailed study of complete intersection hyperKahler moduli spaces in N=4 theories, computing Hilbert series and explicitly describing generators and relations.

Findings

Hilbert series computed for three classes of spaces

Relations match the defining equations of complete intersections

Mirror symmetry exchanges partitions and simplifies analysis

Abstract

We study moduli spaces of a class of three dimensional N=4 gauge theories which are in one-to-one correspondence with a certain set of ordered pairs of integer partitions. It was found that these theories can be realised on brane intervals in Type IIB string theory and can therefore be described using linear quiver diagrams. Mirror symmetry was known to act on such a theory by exchanging the partitions in the corresponding ordered pair, and hence the quiver diagram of the mirror theory can be written down in a straightforward way. The infrared Coulomb branch of each theory can be studied using moment map equations for a hyperKahler quotient of the Higgs branch of the mirror theory. We focus on three infinite subclasses of these singular hyperKahler spaces which are complete intersections. The Hilbert series of these spaces are computed in order to count generators and relations, and they turn out to be related to the corresponding partitions of the theories. For each theory, we explicitly discuss the generators of such a space and relations they satisfy in detail. These relations are precisely the defining equations of the corresponding complete intersection space.