Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series

Abstract

Modular graph functions are $SL(2,\mathbb{Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to ${\Gamma }_{\infty }\backslash PSL(2,\mathbb{Z})$. The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under $\tau \to -\bar{\tau }$ are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space ${\mathfrak{A}}_w$ of odd two-loop modular graph functions of weight $w$. For $w\le 11$ the bound is saturated and we exhibit a basis for ${\mathfrak{A}}_w$.

arXiv comment88 pages (of which 32 pages are appendices); proof of convergence of Poincare series; example calculation of Petersson inner product; and Lemma 7.5 on the integral of cuspidal functions added in revised version

References in corpus (29)

• The Multiple Zeta Value Data Mine
• Polylogarithms, Multiple Zeta Values and Superstring Amplitudes
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Motivic Multiple Zeta Values and Superstring Amplitudes
• The low energy expansion of the one-loop type II superstring amplitude
• Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
• On the modular structure of the genus-one Type II superstring low energy expansion
• Closed Superstring Amplitudes, Single-Valued Multiple Zeta Values and Deligne Associator
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• The Box Graph In Superstring Theory
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• Proving relations between modular graph functions
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus
• Single-valued periods and multiple zeta values
• Poisson equation for the Mercedes diagram in string theory at genus one
• Analytic Computing Methods for Precision Calculations in Quantum Field Theory
• Simplifying the one loop five graviton amplitude in type IIB string theory
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Tetrahedral modular graph functions
• A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces
• Single-valued hyperlogarithms, correlation functions and closed string amplitudes
• Holomorphic subgraph reduction of higher-point modular graph forms
• Eigenvalue equation for genus two modular graphs
• Low momentum expansion of one loop amplitudes in heterotic string theory
• Rankin-Selberg methods for closed string amplitudes
• A simplifying feature of the heterotic one loop four graviton amplitude
• String theory integrands and supergravity divergences
• Supergravity limit of genus two modular graph functions in the worldline formalism

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• Harmonic analysis of 2d CFT partition functions
• Exploring transcendentality in superstring amplitudes
• 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
• 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
• Modular graph forms from equivariant iterated Eisenstein integrals
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Discreteness and Integrality in Conformal Field Theory
• Elliptic modular graph forms I: Identities and generating series
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• To the cusp and back: Resurgent analysis for modular graph functions
• Zero mode of the Fourier series of some modular graphs from Poincare series
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Absence of irreducible multiple zeta-values in melon modular graph functions
• Integral of depth zero to three basis of Modular Graph Functions
• Integral of two-loop modular graph functions
• Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
• A construction of single-valued elliptic polylogarithms