Poisson equation for the three loop ladder diagram in string theory at genus one

Abstract

The three loop ladder diagram is a graph with six links and four cubic vertices that contributes to the D^{12} R^4 amplitude at genus one in type II string theory. The vertices represent the insertion points of vertex operators on the toroidal worldsheet and the links represent scalar Green functions connecting them. By using the properties of the Green function and manipulating the various expressions, we obtain a modular invariant Poisson equation satisfied by this diagram, with source terms involving one, two and three loop diagrams. Unlike the source terms in the Poisson equations for diagrams at lower orders in the momentum expansion or the Mercedes diagram, a particular source term involves a five point function containing a holomorphic and a antiholomorphic worldsheet derivative acting on different Green functions. We also obtain simple equalities between topologically distinct diagrams, and consider some elementary examples.

23 pages, LaTeX, 29 figures

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Cited by in corpus (23)

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• 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
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Tetrahedral modular graph functions
• 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. II. Iterated integrals of cusp forms
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• 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
• Poisson equations for elliptic 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
• To the cusp and back: Resurgent analysis for modular graph functions
• Relations between elliptic modular graphs
• Zero mode of the Fourier series of some modular graphs from Poincare series
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Poisson equation for genus two string invariants: a conjecture
• Integral of depth zero to three basis of Modular Graph Functions
• Integrating simple genus two string invariants over moduli space