Restricted representations of the twisted N = 2 superconformal algebra

Tensor product decompositions of the N=2 superconformal algebras are examined to determine when they are finite or multiplicity-free, using various combinatorial identities. Using similar techniques, branching rules for the winding subalgebras of the N=2 superconformal algebras are investigated.

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 $c=-6$.