All-order differential equations for one-loop closed-string integrals and modular graph forms

Abstract

We investigate generating functions for the integrals over world-sheet tori appearing in closed-string one-loop amplitudes of bosonic, heterotic and type-II theories. These closed-string integrals are shown to obey homogeneous and linear differential equations in the modular parameter of the torus. We spell out the first-order Cauchy-Riemann and second-order Laplace equations for the generating functions for any number of external states. The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.

54+24 pages, v2: typos corrected, version published in JHEP

References in corpus (52)

• Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes
• Super-Poincare Covariant Quantization of the Superstring
• Writing CFT correlation functions as AdS scattering amplitudes
• Elliptic multiple zeta values and one-loop superstring amplitudes
• Multiloop Amplitudes and Vanishing Theorems using the Pure Spinor Formalism for the Superstring
• Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
• Closed String Amplitudes as Single-Valued Open String Amplitudes
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• On the modular structure of the genus-one Type II superstring low energy expansion
• Non-abelian $Z$-theory: Berends-Giele recursion for the ${\alpha }^{\prime }$-expansion of disk integrals
• Genus-One String Amplitudes from Conformal Field Theory
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• All order alpha'-expansion of superstring trees from the Drinfeld associator
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Multiparticle SYM equations of motion and pure spinor BRST blocks
• Non-Abelian Born-Infeld Action and Type I - Heterotic Duality (II): Nonrenormalization Theorems
• Multiparticle one-loop amplitudes and S-duality in closed superstring theory
• New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level
• On tree-level higher order gravitational couplings in superstring theory
• String scattering in flat space and a scaling limit of Yang-Mills correlators
• Amplitude relations in heterotic string theory and Einstein-Yang-Mills
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• The One-Loop Five-Graviton Amplitude and the Effective Action
• Proving relations between modular graph functions
• String predictions for near future colliders from one-loop scattering amplitudes around D-brane worlds
• From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop
• Eisenstein series and automorphic representations
• Current Algebra on the Torus
• Double-Copy Structure of One-Loop Open-String Amplitudes
• Closed strings as single-valued open strings: A genus-zero derivation
• Exploring transcendentality in superstring amplitudes
• String corrections to AdS amplitudes and the double-trace spectrum of N=4 SYM
• Poisson equation for the Mercedes diagram in string theory at genus one
• String amplitudes from field-theory amplitudes and vice versa
• Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure
• 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
• Monodromy relations from twisted homology
• The One-loop Open Superstring Massless Five-point Amplitude with the Non-Minimal Pure Spinor Formalism
• Tetrahedral modular graph functions
• From maximal to minimal supersymmetry in string loop amplitudes
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• Towards the n-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics
• String Correlators: Recursive Expansion, Integration-by-Parts and Scattering Equations
• Towards the n-point one-loop superstring amplitude I: Pure spinors and superfield kinematics
• Poisson equation for the three loop ladder diagram in string theory at genus one
• Eigenvalue equation for the modular graph $C_{a,b,c,d}$
• Holomorphic subgraph reduction of higher-point modular graph forms
• Simplifying one-loop amplitudes in superstring theory
• Eigenvalue equation for genus two modular graphs
• Low momentum expansion of one loop amplitudes in heterotic string theory
• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series

Cited by in corpus (33)

• Tree-level amplitudes from the pure spinor superstring
• 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
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Loop amplitudes monodromy relations and color-kinematics duality
• All-order alpha'-expansion of one-loop open-string 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
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Modular graph forms from equivariant iterated Eisenstein integrals
• From Little String Free Energies Towards Modular Graph Functions
• 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
• Relations between elliptic modular graphs
• Two dialects for KZB equations: generating one-loop open-string integrals
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Integrating three-loop modular graph functions and transcendentality of string amplitudes
• Basis decompositions of genus-one string integrals
• Symmetric Orbifold Theories from Little String Residues
• Open-string integrals with multiple unintegrated punctures at genus one
• Advanced tools for basis decompositions of genus-one string integrals
• Poisson equation for genus two string invariants: a conjecture
• Integral of depth zero to three basis of Modular Graph Functions
• Transcendentality violation in type IIB string amplitudes
• Type II superstring amplitude at one-loop and transcendentality
• From Modular Graph Forms to Iterated Integrals
• Integrating simple genus two string invariants over moduli space
• Probing Leptophobic Dark Sectors via Gravitational Wave Signatures
• A construction of single-valued elliptic polylogarithms
• Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts