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For all 4d $\mathcal{N} = 4$ SYM theories with simple gauge groups $G$, weshow that the residues of the integrands in the $\mathcal{N} = 4$ Schurindices, which are related to Gukov-Witten type surface defects in thetheories, 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, areadditional solutions to the flavored modular differential equations satisfiedby the original Schur index. The prediction is verified in the $G = SU(2)$case, where an additional logarithmic solution is constructed.

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