Macdonald index from 3d TQFT

Abstract

We propose a new fermionic sum formula for the Macdonald index of a class of Argyres-Douglas theories. The formula arises naturally from a three-dimensional topological field theory obtained via a twisted dimensional reduction of the 4d theory. Such a reduction often gives rise to a 3d ${\mathcal N}=2$ abelian Chern-Simons matter theory, which is expected to flow to an ${\mathcal N}=4$ superconformal fixed point. After performing a topological twist, we obtain a 3d TFT admitting boundary conditions that support the vertex operator algebra associated with the original 4d theory. In this framework, the Macdonald index appears as a half-index of the 3d gauge theory, with the Macdonald grading determined by a distinguished $U(1)_A$ symmetry in the infrared ${\mathcal N}=4$ superconformal algebra. We present a general procedure to identify this $U(1)_A$ symmetry and, whenever possible, show that it reproduces the refined character of the associated vertex operator algebra, thereby recovering the Macdonald index. Our construction also gives a hint towards the IR formula for the Macdonald index in terms of 4d BPS particles on the Coulomb branch.