Macdonald Index and Chiral Algebra

TL;DR

The paper proposes a refined character for the vacuum module of a chiral algebra associated to a 4d ${\cal N}=2$ SCFT to capture the Macdonald index, which refines the Schur index with an extra fugacity. It defines a filtration of the vacuum module and uses associated graded pieces with generator-weight counting to reproduce ${\cal I}(q,t)$ in examples where the chiral algebra is simple, notably Virasoro minimal models for $(A_1,A_{2n})$ and $\widehat{\mathfrak{su}}(2)_k$ for $(A_1,D_{2n+1})$, including careful treatment of null states and Sugawara relations. Across levels, the refined character agrees with Macdonald indices, and a closed-form formula is given in the no-null-state case; the approach also extends to Lagrangian theories via tensor products and gauging. This provides a concrete 2d-analytic framework to access richer 4d ${\cal N}=2$ data, with potential implications for Higgs-branch geometry, defect insertions, and bootstrap analyses.

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.