Highest Weight Generating Functions for Hilbert Series

TL;DR

This paper introduces a novel highest weight generating function (HWG) approach for efficiently encoding and analyzing Hilbert series of various moduli spaces in supersymmetric gauge theories, leveraging group representation theory.

Contribution

The paper develops a new HWG method based on Dynkin labels, plethystic functions, and Weyl integration, providing explicit character generating functions for classical and exceptional groups.

Findings

HWGs enable efficient encoding of Hilbert series.

Explicit character generating functions for low-rank groups are provided.

HWGs reveal how gauge invariant operators interact under symmetrisation.

Abstract

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions (HWGs) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons. We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group. In order to calculate HWGs, we derive and tabulate character generating functions for low rank groups by a variety of methods, including a general character generating function, based on the Weyl Character Formula, for any classical or exceptional group.