Macdonald Index from VOA and Graded Unitarity

TL;DR

This paper introduces a new VOA-based method to derive a special limit of the Macdonald index for 4d $ ext{N}=2$ SCFTs, extending the understanding of their spectrum and unitarity properties.

Contribution

It provides an intrinsic, assumption-free approach to extract a non-Schur limit of the Macdonald index directly from VOAs for unitary theories.

Findings

Validated the method in various examples.

Extended the approach to theories with surface defects.

Proposed a notion of graded unitarity in the presence of defects.

Abstract

The SCFT/VOA correspondence provides a powerful framework for studying 4d $\mathcal N=2$ superconformal field theories (SCFTs) through the mathematical machinery of 2d vertex operator algebras (VOAs). It captures the Schur operators of the underlying SCFT, whose spectrum is encoded by the Schur index and its refinement, the Macdonald index. While the Schur index is identified with the vacuum character of the associated VOA, a general VOA-based derivation of the Macdonald index has remained elusive. In this letter, we propose a novel and intrinsic method for recovering a special non-Schur limit of the Macdonald index directly from the VOA. The construction requires no additional assumptions and applies whenever the underlying 4d theory is unitary. We test the proposal in a variety of examples, and further extend it to the case with surface defects, suggesting a notion of graded unitarity in the presence of defects. Our method also introduces a new class of series for general VOAs, analogous to but distinct from the conventional character, and potentially useful in broader contexts.