Hierarchy of Modular Graph Identities

Abstract

The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues for the extended family of dihedral modular graph functions are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues $s(s-1)$ for positive integers $s$ bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.

54 pages, one reference added, minor typos corrected in second version; minor edits and corrections to asymptotics of reference [15] included and implemented in third version

References in corpus (9)

• On the modular structure of the genus-one Type II superstring low energy expansion
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Proving relations between modular graph functions
• Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
• $D^6{R}^4$ amplitudes in various dimensions
• Poisson equation for the Mercedes diagram in string theory at genus one
• Johnson's homomorphisms and the Arakelov-Green function
• Simplifying the one loop five graviton amplitude in type IIB string theory
• Gross--Schoen Cycles and Dualising Sheaves

Cited by in corpus (4)

• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• Poisson equations for elliptic modular graph functions
• A simplifying feature of the heterotic one loop four graviton amplitude
• Zero mode of the Fourier series of some modular graphs from Poincare series