Highest weight generating functions for Hilbert series

Amihay Hanany,Rudolph Kalveks

doi:10.1007/jhep10(2014)152

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.

Highlights

Some combination of flavour, colour and/or other symmetry group representations
In SQCD, the theories are without superpotentials (W ≡ 0) and the gauge group structures are based on those specified by the chiral scalar field definitions; in the other cases, such as SUSY quiver theories for instanton moduli spaces, the theories have non-trivial superpotentials W and the vacuum conditions can place F-flatness constraints on the field representations and/or give rise to hidden symmetry groups
We read off the exponents of the fugacities {m, m1}, which give the Dynkin labels of the global SU(2) and Yang-Mills SU(2) representations respectively, and identify the building blocks of the theory according to the irreps in which they transform

Summary

Introduction

Some combination of flavour, colour and/or other symmetry group representations. Such product group structures arise in supersymmetric (“SUSY”) quiver gauge theories. The Plethystics Program deploys plethystic functions and Weyl integration to construct generating functions for the Hilbert series of objects transforming under various representations of classical or exceptional Lie groups or product groups. Such procedures can be used to identify GIOs, which are necessarily singlets of the gauge group, and the results can be arranged into Hilbert series and their generating functions that encode information about the GIOs at a given level of field counting. Hilbert series can be expressed in refined form, with fields described in terms of class functions built from the characters of the irreps of the symmetry group. While refined Hilbert series do fully encode moduli spaces, their generating functions can be cumbersome to deploy, with complicated plethystic procedures being necessary to extract character expansions

Results

Discussion

Conclusion