Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 3

Argyres–Douglas theories, modularity of minimal models and refined Chern–Simons

Pages: 643 – 672

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n3.a3

Authors

Can Kozçaz (Department of Physics, Boğaziçi University, Istanbul, Turkey; and Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Shamil Shakirov (Society of Fellows, Harvard University, Cambridge, Massachusetts, U.S.A.; Mathematical Sciences Research Institute, Berkeley, California, U.S.A.; and Institute for Information Transmission Problems, Moscow, Russia)

Wenbin Yan (Yau Mathematical Science Center, Tsinghua University, Beijing, China; and Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

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. Moreover, 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.

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Published 22 February 2023