Genus two Zhu theory for vertex operator algebras
Gilroy, Thomas Patrick
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In this thesis we consider the recursive properties of correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We derive a system of formal recursive identities, which allow us to express an arbitrary genus two n-point correlation function in terms of (n-1)-point functions. This generalises Zhu reduction for genus one correlation functions. We apply these recursive identities to compute the genus two Heisenberg n-point correlation functions, the genus two Virasoro n-point functions and the genus two Ward Identity. We define a formal differential operator with respect to the sewing parameters and derive differential equations for holomorphic 1-forms, the normalised 2-form, the Heisenberg partition function and the partition function for the Virasoro (2,5)-minimal model. We prove that this formal differential operator is defined on an open subset of the sewing domain.