Macdonald Indices for Four-dimensional $\mathcal N=3$ Theories

TL;DR

This paper computes vacuum characters for specific vertex operator algebras related to crystallographic complex reflection groups and connects them to the Macdonald limit of superconformal indices for certain four-dimensional  theories, confirming some existing predictions.

Contribution

It provides explicit evaluations of vacuum characters for  VOAs associated with complex reflection groups and links these to the Macdonald indices of  superconformal theories, extending previous conjectures.

Findings

Vacuum characters match conjectured Macdonald indices for specific  theories.

Agreement with known predictions in the Schur index limit.

Explicit calculations support the proposed VOA and index correspondence.

Abstract

We brute-force evaluate the vacuum character for $\mathcal N=2$ vertex operator algebras labelled by crystallographic complex reflection groups $G(k,1,1)=\mathbb Z_k$, $k=3,4,6$, and $G(3,1,2)$. For $\mathbb Z_{3,4}$ and $G(3,1,2)$ these vacuum characters have been conjectured to respectively reproduce the Macdonald limit of the superconformal index for rank one and rank two S-fold $\mathcal N=3$ theories in four dimensions. For the $\mathbb Z_3$ case, and in the limit where the Macdonald index reduces to the Schur index, we find agreement with predictions from the literature.