Tri-vertices and SU(2)'s

TL;DR

This paper studies a class of N=2 supersymmetric gauge theories represented by graphs of tri-vertices and lines, analyzing their hypermultiplet moduli spaces through Hilbert series to explore dualities and classify their properties.

Contribution

It introduces a graph-based framework for N=2 theories and derives a general formula for the Hilbert series of their hypermultiplet moduli spaces.

Findings

The Hilbert series depends only on genus and external legs.

Theories with same genus and external legs have identical Hilbert series.

The results support the conjecture of S-duality relating these theories.

Abstract

We examine a class of N=2 supersymmetric gauge theories in (3+1) dimensions whose Lagrangians are determined by graphs consisting of two building blocks, namely a tri-vertex and a line. A line represents an SU(2) gauge group and a tri-vertex represents a matter field in the trifundamental representation of SU(2)^3. These graphs can be topologically classified by the genus and the number of external legs. This paper focuses on the hypermultiplet moduli spaces of the aforementioned theories. We compute the Hilbert series which count all chiral operators on the hypermultiplet moduli space. Several examples show that theories corresponding to different graphs with the same genus and the same number of external legs possess the same Hilbert series. This is in agreement with the conjecture that such theories are related to each other by S-duality. We also give a general expression for the Hilbert series for the graph with any genus and any number of external legs.