Surface defects, flavored modular differential equations and modularity

Abstract

Every 4d $\mathcal{N}=2$ SCFT $\mathcal{T}$ corresponds to an associated VOA $\mathbb{V}(\mathcal{T})$, which is in general non-rational with a more involved representation theory. Null states in $\mathbb{V}(\mathcal{T})$ can give rise to non-trivial 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.

76 pages, 3 figures

References in corpus (9)

• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Vertex operator algebras of Argyres-Douglas theories from M5-branes
• Chiral Algebras for Trinion Theories
• AGT on the S-duality Wall
• sl^(2)_{-1/2}: A Case Study
• On Irregular Singularity Wave Functions and Superconformal Indices
• Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds
• Generalized matrix models and AGT correspondence at all genera
• Needles in a haystack: An algorithmic approach to the classification of 4d $\mathcal{N}=2$ SCFTs