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

Published for SISSA by Springer Received: November 17, 2019 Accepted: December 26, 2019 Published: January 13, 2020 Jan E. Gerken,a Axel Kleinschmidta,b and Oliver Schlottererc a Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, DE-14476 Potsdam, Germany b International Solvay Institutes ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium c Department of Physics and Astronomy, Uppsala University, SE-75108 Uppsala, Sweden E-mail: , , 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. Keywords: Scattering Amplitudes, Superstrings and Heterotic Strings, Conformal Field Models in String Theory ArXiv ePrint: 1911.03476 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2020)064 JHEP01(2020)064 All-order differential equations for one-loop closed-string integrals and modular graph forms Contents 1 3 4 5 5 6 2 Basics of generating functions for one-loop string integrals 2.1 Kronecker-Eisenstein series 2.1.1 Derivatives of Kronecker-Eisenstein series 2.1.2 Fay identities 2.1.3 Lattice sums and modular transformations 2.2 Koba-Nielsen factor 2.3 Introducing generating functions for world-sheet integrals 2.3.1 Component integrals and string amplitudes 2.3.2 Relations between component integrals 2.4 Modular graph forms 2.4.1 Dihedral examples 2.4.2 Differential equations of modular graph forms 2.4.3 More general graph topologies 2.5 Low-energy expansion of component integrals 2.5.1 Two-point examples 2.5.2 Three-point examples 2.5.3 From modular graph forms to component integrals 6 7 7 8 9 11 12 13 14 15 16 17 18 18 20 22 23 3 Modular differential operators, Cauchy-Riemann- and Laplace equations 23 3.1 Maaß raising and lowering operators 23 3.2 Differential operators on generating series 25 3.3 Two-point warm-up for differential equations 26 3.3.1 Cauchy-Riemann equation 27 3.3.2 Laplace equation 27 3.4 Two-point warm-up for component integrals 28 3.4.1 Cauchy-Riemann equation 28 3.4.2 Laplace equation 29 3.4.3 Lessons for modular graph forms 30 4 Cauchy-Riemann differential equations 4.1 Cauchy-Riemann differential equation at n points 4.2 Three-point examples 4.2.1 Lessons for modular graph forms 4.3 Four-point examples –i– 31 31 35 37 38 JHEP01(2020)064 1 Introduction 1.1 Summary of main results 1.1.1 Open-string integrals and differential equations 1.1.2 Cauchy-Riemann equations 1.1.3 Laplace equations 1.2 Outline 38 38 41 43 6 Towards all-order α0 -expansions of closed-string one-loop amplitudes 6.1 The open-string analogues 6.2 An improved form of closed-string differential equations 6.3 A formal all-order solution to closed-string α0 -expansion 44 44 45 47 7 Summary and outlook 48 A Identities for Ω(z, η, τ ) 49 B Identities between modular graph forms B.1 Topological simplifications B.2 Factorization B.3 Momentum conservation B.4 Holomorphic subgraph reduction B.5 Verifying two-point Cauchy-Riemann equations 51 51 52 52 53 53 C Component integrals versus n-point string amplitudes 54 D Kinematic poles D.1 Subtraction scheme for a two-particle channel D.2 Integration by parts at three points 55 56 57 E Proof of sij -form of product of Kronecker-Eisenstein series E.1 s1n -form at n points E.2 Extending left and right 58 60 61 F Derivation of component equations at three points F.1 General Cauchy-Riemann component equations at three points F.2 Examples of Cauchy-Riemann component equations at three points F.3 Further examples for Laplace equations at three points 62 64 64 65 1 Introduction The low-energy expansion of string amplitudes has become a rewarding subject that carries valuable input on string dualities and that has opened up fruitful connections with number theory and particle phenomenology. The key challenge of low-energy expansions resides in the integrals over punctured world-sheets that are characteristic for string amplitudes and typically performed order-by-order in the inverse string tension α0 . The α0 -expansion of such string integrals then generates large classes of special numbers and functions — the periods of the moduli space Mg,n of n-punctured genus-g surfaces. –1– JHEP01(2020)064 5 Laplace equations 5.1 Laplace equation at n points 5.2 Three-point examples 5.3 n-point examples • The α0 -expansion of one-loop open-string integrals can be expressed via functions that depend on the modular parameter τ of the cylinder or Möbius-strip worldsheet. These functions need to be integrated over τ to obtain the full one-loop string amplitude and were identified [36, 37] as Enriquez’ elliptic multiple zeta values (eMZVs) [38]. A systematic all-order method to generate the eMZVs in open-string α0 -expansions [39, 40] is based on generating functions of Kronecker-Eisenstein type, similar to the ones we shall introduce in a closed-string setting. 1 Higher-loop generalizations of modular graph forms have been studied in [30–34]. –2– JHEP01(2020)064 In particular, one-loop closed-string amplitudes are governed by world-sheets with the topology of a torus and the associated modular group SL2 (Z). The α0 -expansion of such torus integrals introduces a fascinating wealth of non-holomorphic modular forms known as (genus-one) “modular graph forms” which have been studied from a variety of perspectives [1–29].1 These modular graph forms satisfy an intricate web of differential equations with respect to the modular parameter τ of the torus, namely: first-order CauchyRiemann equations relating modular graph forms to holomorphic Eisenstein series [9, 13, 17, 21] and (in-)homogeneous Laplace eigenvalue equations [4, 10, 14, 27]. As the main result of this work, we derive homogeneous Cauchy-Riemann and Laplace equations for generating series of n-point one-loop closed-string integrals and the associated modular graph forms. In contrast to earlier approaches in the literature, our results are valid to all orders in α0 and do not pose any restrictions on the topology of the defining graphs. As part of our construction, we also propose a basis of closed-string integrals in one-loop amplitudes of the bosonic, heterotic and type-II theories. More specifically, we study torus integrals over doubly-periodic Kronecker-Eisenstein series an (...truncated)