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 Hohn 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$_3$.
Download