The Coulomb branch indices of Argyres-Douglas theories on $L(k,1)\times S^{ 1}$ are recently identified with matrix elements of modular transforms of certain $2d$ vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of $(2N+3,2)$ minimal models are proposed to compute the full Coulomb branch index of $(A_{ 1},A_{ 2N})$ Argyres-Douglas theories on the same space. Morever, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.
Categories: hep-th math-ph math.AG math.MP math.QA math.RT