On conjectural fermionic formulas for the Macdonald index in Argyres-Douglas theories

TL;DR

This paper proves a duality relation for the Macdonald index in specific Argyres-Douglas theories, leading to a new fermionic formula and confirming a conjecture by Kim, Kim, and Song.

Contribution

It introduces a new conjugate Bailey pair and establishes a fermionic-bosonic duality for the Macdonald index in certain theories, advancing the understanding of their index formulas.

Findings

Proved a duality relation for the Macdonald index in (A_1, D_{2k+1}) theories.

Derived a new fermionic formula from the duality.

Confirmed a conjectured sum-like expression for the index.

Abstract

We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate Bailey pair to be established using techniques from orthogonal polynomials and basic hypergeometric series. In addition, this fermionic formula implies another sum-like expression for the Macdonald index conjectured by Kim, Kim, and Song.