Modular Constraints on Superconformal Field Theories

Abstract

We constrain the spectrum of $\mathcal{N}=(1,1)$ and $\mathcal{N}=(2,2)$ superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the ${\Gamma }_{\theta }$ congruence subgroup of the full modular group $SL(2,\mathbb{Z})$. We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without $U(1)$ charges. Especially, the upper bounds on the twist gap for the non-current primaries exhibit interesting peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to ${\Gamma }_{\theta }$. Some of the candidate theories have been discussed by Höhn in the context of self-dual extremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS${}_3$.

50 pages, 16 figures