Superconformal indices of three-dimensional theories related by mirror symmetry

Abstract

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories

Down the rabbit hole with theories of class S $$ \mathcal{S} $$

“…These expressions can be shown to be equal [43]. We can see this explicitly in the Higgs/Coulomb limit.…”

T ρ σ (G) theories and their Hilbert series

“…For some of the theories of our interest in this paper, namely good or ugly T σ ρ (G) theories with G a classical group, the standard formula [17][18][19][20] for the superconformal index may be written down. Involving an infinite sum over magnetic charges of integrals over the gauge group, this formula is not of simple evaluation.…”

Factorization of the 3d superconformal index

“…As will be shown in the subsection 4.1, the superconformal 7 We use the refined vertex formalism to write down the topological string partition function. For the notation, please refer to [31].…”