Relations between elliptic multiple zeta values and a special derivation algebra

2020

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183

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless n-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter τ which is known from the A-elliptic Knizhnik-Zamolodchikov-Bernard associator. The expressions for their τderivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension α . In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the α -expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at τ → i∞ is expressed in terms of their genus-zero analogues -(n+2)-point Parke-Taylor integrals over disk boundaries. Our results yield a compact formula for α -expansions of n-point integrals over boundaries of cylinder-or Möbius-strip worldsheets, where any desired order is accessible from elementary operations. Contents 7 Formal properties 59 7.1 Uniform transcendentality 59 7.2 Coaction 60 8 Conclusions 63 A Resolving cycles of Kronecker-Eisenstein series 65 A.1 Two points 65 A.2 Three points 66 A.3 Higher points 67 B The non-planar Green function at the cusp 67 C More on the τ -derivatives of A-cycle integrals 68 C.1 The 6 × 6 representation of the derivations at four points 68 C.2 The τ -derivative at five points in a 24-element basis 69 D Transformation matrices between twisted cycles 69 D.1 Four-point example 70 D.2 Weighted combinations 70 E Examples of α -expansions 71 E.1 Three points: integrating f (1) 12 f (3) 23 71 E.2 Four points: MZVs for the initial values 71 F Fourier expansion of A-cycle graph functions 72 1 The notion of single-valued MZVs was introduced in [6, 7]. 2 See for instance [13-16] for earlier work on tree-level α -expansions at n ≤ 7 points, in particular [14, 15, 17, 18] for synergies with hypergeometric-function representations. 3 This relies on the linear-independence result of [26] on iterated Eisenstein integrals.-2 -trices in [34]. As a genus-one generalization, the A-cycle integrals under investigation are shown to obey the same type of differential equation in τ as the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) associator [35][36][37]. In particular, n-point A-cycle integrals induce (n−1)!×(n−1)! matrix representations of certain derivations dual to Eisenstein series [38] which accompany the iterated Eisenstein integrals in the α -expansions.• Third, only (n−3)! choices for n-point disk integrands are inequivalent under integration by parts, i.e. the so-called twisted cohomology at genus zero has dimensio...

Section: Summary Of the Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

One-loop open-string integrals from differential equations: all-order α′-expansions at n points

Mafra

2020

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183

“…In fact, the entire τ -dependence in the η j -expansion of D τ η is carried by holomorphic Eisenstein series, see Section 6 for further details. That is why (1.4) manifests the appearance of iterated Eisenstein integrals in the α -expansion of open-string integrals, a canonical representation of eMZVs exposing all their relations over Q, MZVs and (2πi) −1 [38,41].…”

Section: Open-string Integrals and Differential Equationsmentioning

confidence: 99%

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

Gerken

Kleinschmidt

2020

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140

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 secondorder 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. arXiv:1911.03476v2 [hep-th] 21 Jan 2020 F Derivation of component equations at three points 65 F.1 General Cauchy-Riemann component equations at three points 66 F.2 Examples of Cauchy-Riemann component equations at three points 67 F.3 Further examples for Laplace equations at three points 68 -ii -10The subscript of ∇DG aims to avoid confusion with the differential operators of Sections 3.1 and 3.2 and refers to the authors D'Hoker and Green of [9] which initiated the systematic study of relations between modular graph forms via repeated action of (2.59).

“…It might be rewarding to approach the low-energy expansion of superstring loop amplitudes at higher multiplicity with Berends-Giele methods. At the one-loop order, this concerns annulus integrals involving elliptic multiple zeta values [85][86][87] and torus integrals involving modular graph functions [88][89][90][91][92][93][94][95][96].…”

Section: Further Directionsmentioning

confidence: 99%

Non-abelian Z-theory: Berends-Giele recursion for the α ′-expansion of disk integrals

Mafra

2017

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106

159

We present a recursive method to calculate the α -expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as Z-theory, we pinpoint the equation of motion of Z-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order α 7 is made available on the website http://repo.or.cz/BGap.git.