Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and n-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szego{double acute} kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all n-point functions in terms of a genus two Szego{double acute} kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.