### Abstract:

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 $V_L$. 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 $V_L$ \emph{are} compatible, each being identified with the genus two theta function of $L$.