Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes

Broedel, Johannes; Matthes, Nils; Richter, Gregor; Schlotterer, Oliver

doi:10.1088/1751-8121/aac601

Cited by 48 publications

(102 citation statements)

References 61 publications

(257 reference statements)

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“…On top of the structural insights provided by (1.7), it has practical advantages in the explicit evaluation of α -expansions. In contrast to earlier expansion methods for A-cycle integrals [19,20], there is no need to rearrange elliptic iterated integrals via so-called "z-removal" techniques 6 when integrating over one puncture after the other. Moreover, all the kinematic poles of the A-cycle integrals such as s −1 12 are determined by the initial value Z i∞ η ( * |1, B) and do not require any subtraction scheme [25] when integrating over the punctures.…”

Section: Summary Of the Main Resultsmentioning

confidence: 99%

“…matrix representations of the derivations are linear in α , and that the initial value at τ → i∞ is built from uniformly transcendental genuszero integrals. Our open-string results are complemented by recent investigations of transcendentality properties of one-loop amplitudes of heterotic strings [49] and type-II superstrings [50].• No twisted eMZVs in non-planar amplitudes: The earlier expansion method for non-planar open-string amplitudes [20,25] introduces twisted eMZVs or cyclotomic analogues of eMZVs in intermediate steps. To the α -orders considered, however, the end results were found to be expressible in terms of conventional (i.e.…”

mentioning

confidence: 79%

“…When the shifts a j of the elliptic iterated integrals (2.8) are taken to be rational points on the torus, evaluation at z = 1 gives rise to twisted eMZVs [20]…”

Section: Emzvs and Twisted Emzvsmentioning

confidence: 99%

“…• No twisted eMZVs in non-planar amplitudes: The earlier expansion method for non-planar open-string amplitudes [20,25] introduces twisted eMZVs or cyclotomic analogues of eMZVs in intermediate steps. To the α -orders considered, however, the end results were found to be expressible in terms of conventional (i.e.…”

mentioning

confidence: 99%

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

Mafra

Schlotterer

2020

J. High Energ. Phys.

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183

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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...

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Section: Summary Of the Main Resultsmentioning

confidence: 99%

mentioning

confidence: 79%

“…When the shifts a j of the elliptic iterated integrals (2.8) are taken to be rational points on the torus, evaluation at z = 1 gives rise to twisted eMZVs [20]…”

Section: Emzvs and Twisted Emzvsmentioning

confidence: 99%

mentioning

confidence: 99%

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

Mafra

Schlotterer

2020

J. High Energ. Phys.

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183

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“…• The α -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 world-sheet. 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 α -expansions [39,40] is based on generating functions of Kronecker-Eisenstein type, similar to the ones we shall introduce in a closed-string setting.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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).

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Modular and Holomorphic Graph Functions from Superstring Amplitudes

Zerbini¹

2019

Texts &Amp; Monographs in Symbolic Computation

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In these notes, based on a talk given at the conference Elliptic integrals, elliptic functions and modular forms in quantum field theory held at DESY-Zeuthen in October 2017, we compare two classes of functions arising from genus-one superstring amplitudes: modular and holomorphic graph functions. We focus on their analytic properties, we recall the known asymptotic behaviour of modular graph functions and we refine the formula for the asymptotic behaviour of holomorphic graph functions. Moreover, we give new evidence of a conjecture appeared in [4] which relates these two asymptotic expansions. arXiv:1807.04506v1 [math-ph]

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Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes

Cited by 48 publications

References 61 publications

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

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

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

Modular and Holomorphic Graph Functions from Superstring Amplitudes

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