The motivic coaction of multiple zeta values and multiple polylogarithms encodes both structural insights on and computational methods for scattering amplitudes in a variety of quantum field theories and in string theory. In this work, we propose coaction formulae for iterated integrals over holomorphic Eisenstein series that arise from configuration-space integrals at genus one. Our proposal is motivated by formal similarities between the motivic coaction and the single-valued map of multiple polylogarithms at genus zero that are exposed in their recent reformulations via zeta generators. The genus-one coaction of this work is then proposed by analogies with the construction of single-valued iterated Eisenstein integrals via zeta generators at genus one. We show that our proposal exhibits the expected properties of a coaction and deduce $\textit{f}$-alphabet decompositions of the multiple modular values obtained from regularized limits.

Germany

Belgium

Axel Kleinschmidt

Sweden

Franziska Porkert

Oliver Schlotterer

Towards motivic coactions at genus one from zeta generators

Loop amplitudes in string theories reduce to those of gauge theories and (super)gravity in their worldline description as the inverse string tension $\textit{α}$′ tends to zero. The appearance of reducible diagrams in these $\textit{α}$′ $\textit{→}$ 0 limits is determined through so-called pinching rules in the worldline literature. In this work, we extend these pinching rules to the chiral-splitting description of one-loop superstring amplitudes where left- and right-moving degrees of freedom decouple at fixed loop momentum. Starting from six points, the Kronecker-Eisenstein integrands of chiral amplitudes introduce subtleties into the pinching rules and integration-by-parts simplifications. Resolutions of these subtleties are presented and applied to produce a new superspace representation of the six-point one-loop amplitude of type IIA/B supergravity. The worldline computations and their subtleties are compared with the ambitwistor-string approach to one-loop field-theory amplitudes where integration-by-parts manipulations are shown to be more flexible. Throughout this work, the homology invariance of loop-momentum dependent correlation functions on the torus is highlighted as a consistency condition of $\textit{α}$′ $\textit{→}$ 0 limits and their comparison with ambitwistor methods.

Italy

Filippo Balli

United States

Alex Edison

Pinching rules in the chiral-splitting description of one-loop string amplitudes

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

United Kingdom

Daniele Dorigoni

Mehregan Doroudiani

Joshua Drewitt

Switzerland

Martijn Hidding

France

Leila Schneps

Bram Verbeek

Non-holomorphic modular forms from zeta generators

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szegö kernels. Here we show that cyclic products of any number of Szegö kernels on a Riemann surface of arbitrary genus may be decomposed into linear combinations of modular tensors on moduli space that carry all the dependence on the spin structure <math><mi>δ</mi></math>. The <math><mi>δ</mi></math>-independent coefficients in these combinations carry all the dependence on the marked points and are composed of the integration kernels of higher-genus polylogarithms. We determine the antiholomorphic moduli derivatives of the <math><mi>δ</mi></math>-dependent modular tensors.

Eric D’Hoker

Cyclic Products of Higher-Genus Szegö Kernels, Modular Tensors, and Polylogarithms

One-loop scattering amplitudes in string theories involve configuration-space integrals over genus-one surfaces with coefficients of Kronecker-Eisenstein series in the integrand. A conjectural genus-one basis of integrands under Fay identities and integration by parts was recently constructed out of chains of Kronecker-Eisenstein series. In this work, we decompose a variety of more general genus-one integrands into the conjectural chain basis. The explicit form of the expansion coefficients is worked out for infinite families of cases where the Kronecker-Eisenstein series form cycles. Our results can be used to simplify multiparticle amplitudes in supersymmetric, heterotic and bosonic string theories and to investigate loop-level echoes of the field-theory double-copy structures of string tree-level amplitudes. The multitude of basis reductions in this work strongly validate the recently proposed chain basis and stimulate mathematical follow-up studies of more general configuration-space integrals with additional marked points or at higher genus.

Carlos Rodriguez

Canada

China

Yong Zhang

Basis decompositions of genus-one string integrals

The summation over spin structures, which is required to implement the GSO projection in the RNS formulation of superstring theories, often presents a significant impediment to the explicit evaluation of superstring amplitudes. In this paper we discover that, for Riemann surfaces of genus two and even spin structures, a collection of novel identities leads to a dramatic simplification of the spin structure sum. Explicit formulas for an arbitrary number of vertex points are obtained in two steps. First, we show that the spin structure dependence of a cyclic product of Szegö kernels (i.e. Dirac propagators for worldsheet fermions) may be reduced to the spin structure dependence of the four-point function. Of particular importance are certain trilinear relations that we shall define and prove. In a second step, the known expressions for the genus-two even spin structure measure are used to perform the remaining spin structure sums. The dependence of the spin summand on the vertex points is reduced to simple building blocks that can already be identified from the two-point function. The hyper-elliptic formulation of genus-two Riemann surfaces is used to derive these results, and its SL(2, ℂ) covariance is employed to organize the calculations and the structure of the final formulas. The translation of these results into the language of Riemann ϑ-functions, and applications to the evaluation of higher-point string amplitudes, are relegated to subsequent companion papers.

Cyclic products of Szegö kernels and spin structure sums. Part I. Hyper-elliptic formulation

We take a major step towards computing D-dimensional one-loop amplitudes in general gauge theories, compatible with the principles of unitarity and the color-kinematics duality. For n-point amplitudes with either supersymmetry multiplets or generic non-supersymmetric matter in the loop, simple all-multiplicity expressions are obtained for the maximal cuts of kinematic numerators of n-gon diagrams. At n = 6, 7 points with maximal supersymmetry, we extend the cubic-diagram numerators to encode all contact terms, and thus solve the long-standing problem of simultaneously realizing the following properties: color-kinematics duality, manifest locality, optimal power counting of loop momenta, quadratic rather than linearized Feynman propagators, compatibility with double copy as well as all graph symmetries. Color-kinematics dual representations with similar properties are presented in the half-maximally supersymmetric case at n = 4, 5 points. The resulting gauge-theory integrands and their supergravity counterparts obtained from the double copy are checked to reproduce the expected ultraviolet divergences.

Song He

Henrik Johansson

Fei Teng

Perfecting one-loop BCJ numerators in SYM and supergravity

We present a method to compute the integrands of one-loop Einstein-Yang-Mills amplitudes for any number of external gauge and gravity multiplets. Our construction relies on the double-copy structure of Einstein-Yang-Mills as (super-)Yang-Mills with the so-called YM+ϕ 3 theory — pure Yang-Mills coupled to bi-adjoint scalars — which we implement via one-loop Cachazo-He-Yuan formulae. The YM+ϕ 3 building blocks are obtained from forward limits of tree-level input in external gluons and scalars, and we give the composition rules for any number of traces and orders in the couplings g and κ. On the one hand, we spell out supersymmetry- and dimension-agnostic relations that reduce loop integrands of Einstein-Yang-Mills to those of pure gauge theories. On the other hand, we present four-point results for maximal and half-maximal supersymmetry where all supersymmetry cancellations are exposed. In the half-maximal case, we determine six-dimensional anomalies due to chiral hypermultiplets in the loop.

One-loop amplitudes in Einstein-Yang-Mills from forward limits

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown’s alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown’s conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown’s construction fully explicit to all orders.

Denmark

Nils Matthes

Modular graph forms from equivariant iterated Eisenstein integrals

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.