Twisted Correlation Functions on Self-sewn Riemann Surfaces via Generalized Vertex Algebra of Intertwiners

Abstract

We review our recent results on computation of the partition and n-point “intertwined” functions for modules of vertex operator superalgebras with formal parameter associated to local parameters on Riemann surfaces obtained by self-sewing of a lower genus Riemann surface. We introduce the torus intertwined n-point functions containing two intertwining operators in the supertrace. Then we define the partition and n-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by self-sewing of the torus. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from the original torus Szegő kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possess natural modular properties. We describe modularity of the generating function for all n-point correlation functions in terms of a genus two Szegő kernel determinant.