  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.

High Energy Physics - Theory