On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra

Abstract

We compute the Schur index of Argyres-Douglas theories of type (A N −1 , A M −1 ) with surface operators inserted, via the Higgsing prescription proposed by D. Gaiotto, L. Rastelli and S. S. Razamat. These surface operators are obtained by turning on position-dependent vacuum expectation values of operators in a UV theory which can flow to the Argyres-Douglas theories. We focus on two series of (A N −1 , A M −1 ) theories; one with gcd(N, M ) = 1 and the other with M = N (k − 1) for an integer k ≥ 2. Our results are identified with the characters of non-vacuum modules of the associated 2d chiral algebras, which explicitly confirms a remarkable correspondence recently discovered

Testing Macdonald index as a refined character of chiral algebra

“…as the "wavefunction" of the inserted surface operator. As shown in [21], these surface operators have a clear one-to-one correspondence with primary operators (non-singular modules) in minimal models when we consider (A 1 , A 2k ) theories. We can then consider the situation that we add two regular punctures to the original theory, and generate two surface operators at the origin of the chiral algebra plane via the Higgsing method.…”

“…Let us review the Higgsing prescription first formulated in [23] and then put forward in [21] to fit into the class S picture, which is a useful method to generate surface operators in 4d N = 2 gauge theories with no Lagrangian description.…”

“…As shown in [21], when p = n, n ′ i 's become trivial, and n i is related to the label of surface operator through…”

“…We would like to try to compute the Macdonald index with surface operator for (A n−1 , A m−1 ) AD theories with gcd(n, m) = 1 in this article, and compare our results with the corresponding refined character. The tool we will use in the gauge theory side is the TQFT picture of AD theories and the Higgsing method that generates a surface operator extending in the orthogonal directions to the chiral algebra plane, developed in [23] and rewritten in terms of the class S construction in [21].…”

An infrared bootstrap of the Schur index with surface defects

“…Upon flowing to the IR, some nontrivial degrees of freedom survive at the origin z = 0, which can be interpreted as a surface defect [31]. This is confirmed by considering Lagrangian theories coupled to two-dimensional N = (2, 2) theories in [71] (see also [36,37,106]).…”