  The low-energy expansion of one-loop amplitudes in type II string theory
generates a series of world-sheet integrals whose integrands can be represented
by world-sheet Feynman diagrams. These integrands are modular invariant and
understanding the structure of the action of the modular Laplacian on them is
important for determining their contribution to string scattering amplitudes.
In this paper we study a particular infinite family of such integrands
associated with three-loop scalar vacuum diagrams of tetrahedral topology and
find closed forms for the action of the Laplacian. We analyse the possible
eigenvalues and degeneracies of the Laplace operator by group- and
representation-theoretic means.
