The affine vertex superalgebra of D(2,1;-v/w) at level 1

Abstract

The affine vertex superalgebra A=L^1(D(2,1;-v/w)) plays a key role as the geometric Langlands kernel VOA for SVOAs associated to so(3), osp(1|2) and other rank one Lie superalgbras. Since D(2,1;a) is an extension of the direct sum of 3 copies of sl(2), A can be naturally realized as an extension of L_1=L^k(sl(2))x L^l(sl(2))x L^1(sl(2)) for admissible levels k=u/v-2 and l=u/w-2. Here, I use constructions of gluing VOAs to realize A as an L_1 extension, and the theory of VOA extensions to classify irreducible modules in A-wtmod_>=0. Using the Adamovic procedure', an alternate realization of A is given as a subalgebra of L^(k-1)(sl(2))x B^l, where the SVOA B^l is constructed from a half-lattice' and L^1(sl(2)). This allows calculation of modular S-matrices for A modules induced from relaxed highest weight L_1 modules.