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.