We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg ? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possess es natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.
KeywordsVertex Operator SuperalgebrasIntertwining OperatorsRiemann SurfacesSzegöKernelModular FormsTheta-FunctionsFrobenius—Fay and Jacobi Product Identities
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