Macdonald Index and Chiral Algebra

TL;DR

This paper proposes a method to derive the Macdonald index of 4d N=2 SCFTs from their associated 2d chiral algebras, extending the understanding of operator counting beyond the Schur index.

Contribution

It introduces a conjecture for obtaining the Macdonald index via a refined character of the chiral algebra's vacuum module, tested on specific Argyres-Douglas theories.

Findings

The refined character reproduces the Macdonald index for tested theories.

The prescription works for chiral algebras with Virasoro and affine Kac-Moody structures.

Additional generators require knowledge from the 4d theory.

Abstract

For any 4d N=2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type $(A_1, A_{2n})$ and $(A_1, D_{2n+1})$ where the chiral algebras are given by Virasoro and su(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.