We continue our program to define and study n-point correlation functions
for a vertex operator algebra V on a higher genus compact Riemann surface
obtained by sewing surfaces of lower genus. Here we consider Riemann surfaces
of genus 2 obtained by attaching a handle to a torus. We obtain closed formulas
for the genus two partition function for free bosonic theories and lattice
vertex operator algebras VLβ. We prove that the partition function is
holomorphic in the sewing parameters on a given suitable domain and describe
its modular properties. We also compute the genus two Heisenberg vector
n-point function and show that the Virasoro vector one point function
satisfies a genus two Ward identity. We compare our results with those obtained
in the companion paper, when a pair of tori are sewn together, and show that
the partition functions are not compatible in the neighborhood of a two-tori
degeneration point. The \emph{normalized} partition functions of a lattice
theory VLβ \emph{are} compatible, each being identified with the genus two
theta function of L.Comment: 51 pages, 3 figure