Surface defects, flavored modular differential equations, and modularity

Abstract

Every 4D $\mathcal{N}=2$ superconformal field theory $\mathcal{T}$ corresponds to an associated vertex operator algebra $\mathbb{V}(\mathcal{T})$, which is in general nonrational with a more involved representation theory. Null states in $\mathbb{V}(\mathcal{T})$ can give rise to nontrivial flavored modular differential equations, which must be satisfied by the refined/flavored character of all the $\mathbb{V}(\mathcal{T})$ modules. Taking some ${A}_{1}$ theories ${\mathcal{T}}_{g,n}$ of class $\mathcal{S}$ as examples, we construct the flavored modular differential equations satisfied by the Schur index. We show that three types of surface defect indices give rise to common solutions to these differential equations and therefore are sources of $\mathbb{V}(\mathcal{T})$-module characters. These equations transform almost covariantly under modular transformations, ensuring the presence of logarithmic solutions which may correspond to characters of logarithmic modules.