Higgs, Coulomb, and Hall-Littlewood

TL;DR

This paper investigates the relationship between the Higgs branch Hilbert series and Hall-Littlewood index in 4d $ ext{SCFTs}$, revealing counterexamples to their conjectured equivalence in certain class $ ext{S}$ theories with twisted punctures.

Contribution

The authors identify specific class $ ext{S}$ theories where the Higgs branch Hilbert series and Hall-Littlewood index differ, providing new insights into their relationship and constructing 3d mirrors for analysis.

Findings

Counterexamples in theories with four or more $ ext{Z}_2$-twisted punctures.

Construction of 3d mirrors and analysis of Coulomb branch Hilbert series.

Evidence that $a=c$ theories with non-empty Higgs branch have different Hall-Littlewood index and Higgs branch Hilbert series.

Abstract

The Higgs branch of 4d $\mathcal{N}=2$ SCFTs can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class $\mathcal{S}$ theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class $\mathcal{S}$ theory with four or more $\mathbb{Z}_2$-twisted punctures, they do not match. We construct 3d mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4d theory. We further construct $a=c$ theories in class $\mathcal{S}$ using the twisted punctures, and these theories, which includes the $\hat{D}_4(SU(2n+1))$ theories, have Hall--Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all $a=c$ theories with non-empty Higgs branch, including $\mathcal{N}\ge 3$ SCFTs.