Hilbert Series of Bipartite Field Theories

TL;DR

This paper analyzes the algebraic structure of mesonic moduli spaces in bipartite field theories using Hilbert series, revealing generators, relations, and symmetry enhancements, and introduces new families of theories with complete intersection toric Calabi-Yau 3-folds.

Contribution

It computes Hilbert series for bipartite field theories, identifying their generators, relations, and symmetry properties, and introduces new theory families with specific geometric properties.

Findings

Identified generators and relations of mesonic moduli spaces.

Discovered symmetry enhancements via refined Hilbert series.

Constructed families of theories with complete intersection toric Calabi-Yau 3-folds.

Abstract

We study the algebraic structure of the mesonic moduli spaces of bipartite field theories by computing the Hilbert series. Bipartite field theories form a large family of 4d N=1 supersymmetric gauge theories that are defined by bipartite graphs on Riemann surfaces with boundaries. By calculating the Hilbert series, we are able to identify the generators and defining generator relations of the mesonic moduli spaces of these theories. Moreover, we show that certain bipartite field theories exhibit enhanced global symmetries which can be identified through the computation of the corresponding refined Hilbert series. As part of our study, we introduce two one-parameter families of bipartite field theories defined on cylinders whose mesonic moduli spaces are all complete intersection toric Calabi-Yau 3-folds.