Defects, modular differential equations, and free field realization of N = 4 VOAs

TL;DR

This paper links residues of 4d N=4 SYM Schur indices with vacuum characters of free field VOAs, revealing new solutions to modular differential equations and providing explicit examples.

Contribution

It establishes a novel connection between Schur index residues and free field VOA characters for all simple gauge groups in 4d N=4 SYM theories.

Findings

Residues match vacuum characters of free field VOAs.

Residues satisfy modular differential equations.

Explicit logarithmic solution constructed for SU(2).

Abstract

For all 4d $\mathcal{N} = 4$ SYM theories with simple gauge groups $G$, we show that the residues of the integrands in the $\mathcal{N} = 4$ Schur indices, which are related to Gukov-Witten type surface defects in the theories, equal the vacuum characters of rank$G$ copies of $bc \beta \gamma$ systems that provide the free field realization of associated $\mathcal{N} = 4$ VOAs. This result predicts that these residues, as module characters, are additional solutions to the flavored modular differential equations satisfied by the original Schur index. The prediction is verified in the $G = SU(2)$ case, where an additional logarithmic solution is constructed.