Lattice Green functions and Calabi-Yau differential equations

By making the connection between four-dimensional lattice Green functions (LGFs) and Picard-Fuchs ordinary differential equations of Calabi-Yau manifolds, we have given explicit forms for the coefficients of the four-dimensional LGFs on the simple-cubic and body-centred cubic lattices, in terms of finite sums of products of binomial coefficients, and have shown that the corresponding four-dimensional face-centred cubic LGF satisfies a fourth-order ODE of degree 7, which we have found but not solved. The fact that the Picard-Fuchs equations that appear here are similar to those that appear in string theory perhaps makes the first connection between these two apparently unrelated topics. (fast track communication)