Chiral Rings, Futaki Invariants, Plethystics, and Groebner Bases

TL;DR

This paper advances the understanding of chiral rings in 4d supersymmetric theories by refining computational methods for K-stability using Hilbert series, Gr"obner bases, and plethystics, enabling analysis of more complex vacuum moduli spaces.

Contribution

It introduces a modified numerator approach for Hilbert series calculations and applies Gr"obner basis and plethystic techniques to analyze non-complete intersection moduli spaces.

Findings

Corrected Hilbert series method for K-stability determination

Expanded analysis to non-complete intersection moduli spaces

Provided numerous examples illustrating the techniques

Abstract

We study chiral rings of 4d $\mathcal{N}=1$ supersymmetric gauge theories via the notion of K-stability. We show that when using Hilbert series to perform the computations of Futaki invariants, it is not enough to only include the test symmetry information in the former's denominator. We discuss a way to modify the numerator so that K-stability can be correctly determined, and a rescaling method is also applied to simplify the calculations involving test configurations. All of these are illustrated with a host of examples, by considering vacuum moduli spaces of various theories. Using Gr\"obner basis and plethystic techniques, many non-complete intersections can also be addressed, thus expanding the list of known theories in the literature.