Tetrahedral modular graph functions

Abstract

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.

40 pages. v2: reference added. Version published in JHEP

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• Higher genus modular graph functions, string invariants, and their exact asymptotics
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Exploring transcendentality in superstring amplitudes
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
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• Eigenvalue equation for the modular graph $C_{a,b,c,d}$
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• Holomorphic subgraph reduction of higher-point modular graph forms
• Elliptic modular graph forms I: Identities and generating series
• Eigenvalue equation for genus two modular graphs
• Low momentum expansion of one loop amplitudes in heterotic string theory
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• To the cusp and back: Resurgent analysis for modular graph functions
• Relations between elliptic modular graphs
• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series
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• Poisson equation for genus two string invariants: a conjecture
• Type II superstring amplitude at one-loop and transcendentality
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• Integral of depth zero to three basis of Modular Graph Functions
• From Modular Graph Forms to Iterated Integrals
• Integrating simple genus two string invariants over moduli space