Eigenvalue equation for genus two modular graphs

Abstract

We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under the variation of the Beltrami differentials. This equation involves seven distinct graphs, three of which appear in the integrand of the $D^8{\mathcal{R}}^4$ term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory.

67 pages, LaTex; 27 figures

References in corpus (4)

• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Tetrahedral modular graph functions
• Low momentum expansion of one loop amplitudes in heterotic string theory
• A simplifying feature of the heterotic one loop four graviton amplitude

Cited by in corpus (15)

• Tree-level amplitudes from the pure spinor superstring
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Eigenvalue equation for the modular graph $C_{a,b,c,d}$
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Modular graph forms from equivariant iterated Eisenstein integrals
• Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• Relations between elliptic modular graphs
• On the multiloop soft theorem of the dilaton in the bosonic string
• Poisson equation for genus two string invariants: a conjecture
• Integrating simple genus two string invariants over moduli space