  We prove some conjectures about vertex algebras which emerge in gauge theory
constructions associated to the geometric Langlands program. In particular, we
present the conjectural kernel vertex algebra for the $S T^2 S$ duality
transformation in $SU(2)$ gauge theory. We find a surprising coincidence, which
gives a powerful hint about the nature of the corresponding duality wall.
  Concretely, we determine the branching rules for the small $N=4$
superconformal algebra at central charge $-9$ as well as for the generic large
$N=4$ superconformal algebra at central charge $-6$. Moreover we obtain the
affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$
superconformal algebra times a free fermion as Quantum Hamiltonian reductions
of the large $N=4$ superconformal algebras at $c=-6$.
