  We study the partition function of both Close-Packed Dimers and the Critical
Ising Model on a square lattice embedded on a genus two surface. Using
numerical and analytical methods we show that the determinants of the Kasteleyn
adjacency matrices have a dependence on the boundary conditions that, for large
lattice size, can be expressed in terms of genus two theta functions. The
period matrix characterizing the continuum limit of the lattice is computed
using a discrete holomorphic structure. These results relate in a direct way
the lattice combinatorics with conformal field theory, providing new insight to
the lattice regularization of conformal field theories on higher genus Riemann
Surfaces.
