Tetrahedral modular graph functions

Understand

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.