Vertex operator algebras of Argyres-Douglas theories from M5-branes

Abstract

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2,0) theory of type J on a punctured sphere. We denote the AD theories as (Jb[k], Y), where Jb[k] and Y represent an irregular and a regular singularity respectively. We restrict to the ‘minimal’ case where Jb[k] has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra Wk2d(J,Y), where k2d=−h+bb+k with h being the dual Coxeter number of J. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b = h. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.