The Genus Two Partition Function for Free Bosonic and Lattice Vertex Operator Algebras

Understand

We define the $n$-point function for a vertex operator algebra on a genus two Riemann surface in two separate sewing schemes where either two tori are sewn together or a handle is sewn to one torus.

• We explicitly obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and lattice vertex operator algebras in both sewing schemes.
• We prove that the partition functions are holomorphic in the sewing parameters on given suitable domains and describe their modular properties.
• Finally, we show that the partition functions cannot be equal in the neighborhood of a two-tori degeneration point where they can be explicitly compared.