Vertex operator superalgebra/sigma model correspondences: The four-torus case

We propose a correspondence between vertex operator superalgebras and families of sigma models in which the two structures are related by symmetry properties and a certain reflection procedure. The existence of such a correspondence is motivated by previous work on $N=(4,4)$ supersymmetric non-linear sigma models on K3 surfaces, and on a vertex operator superalgebra with Conway group symmetry. Here we present an example of the correspondence for $N=(4,4)$ supersymmetric non-linear sigma models on four-tori, and compare it to the K3 case.