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We define the partition and $n$-point functions for a vertex operator algebraon a genus two Riemann surface formed by sewing two tori together. We obtainclosed formulas for the genus two partition function for the Heisenberg freebosonic string and for any pair of simple Heisenberg modules. We prove that thepartition function is holomorphic in the sewing parameters on a given suitabledomain and describe its modular properties for the Heisenberg and latticevertex operator algebras and a continuous orbifolding of the rank two fermionvertex operator super algebra. We compute the genus two Heisenberg vector$n$-point function and show that the Virasoro vector one point functionsatisfies a genus two Ward identity for these theories.

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I