Analogues elliptiques des nombres multizétas

Enriquez, Benjamin

doi:10.24033/bsmf.2718

Cited by 56 publications

(186 citation statements)

References 2 publications

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“….) are Q[(2πi) −1 ]-linear combinations of MZVs [21,24]. Twisted eMZVs with b j ∈ {0, τ /2} allow for a similar expansion,…”

Section: Emzvs and Twisted Emzvsmentioning

confidence: 99%

“…The notation ξ 0 instructs to pick up the zero th order in the Laurent expansion w.r.t. an auxiliary variable ξ ∈ C. Moreover, we will need the key identity [21,78] (…”

Section: The τ -Derivativementioning

confidence: 99%

“…The desired α -expansions at genus one are generated by solving the differential equations (along with an initial value at τ → i∞) via standard Picard iteration. Accordingly, the accompanying eMZVs will appear as iterated integrals over holomorphic Eisenstein series of SL 2 (Z) [21,24,25], "iterated Eisenstein integrals" for short. In this way, all relations among eMZVs are automatically built in 3 , and the α -expansions of A-cycle integrals are available in a maximally simplified form.…”

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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.

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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“….) are Q[(2πi) −1 ]-linear combinations of MZVs [21,24]. Twisted eMZVs with b j ∈ {0, τ /2} allow for a similar expansion,…”

Section: Emzvs and Twisted Emzvsmentioning

confidence: 99%

“…The notation ξ 0 instructs to pick up the zero th order in the Laurent expansion w.r.t. an auxiliary variable ξ ∈ C. Moreover, we will need the key identity [21,78] (…”

Section: The τ -Derivativementioning

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.

183

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“…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

Schlotterer

2017

J. High Energ. Phys.

106

159

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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.

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“…It turns out that suitable differential forms defining classes of iterated integrals can be identified starting from geometrical considerations: taking the first abelian differential on the simplest genus-zero surface, the Riemann sphere, leads to the class of multiple polylogarithms [1][2][3][4][5] while abelian differentials on a genus-one Riemann surface are the starting point for the elliptic polylogarithms [6,7] to be discussed in this article.…”

Section: Introductionmentioning

confidence: 99%

Functional relations for elliptic polylogarithms

Broedel

Kaderli²

2020

J. Phys. A: Math. Theor.

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Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities.

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Analogues elliptiques des nombres multizétas

Cited by 56 publications

References 2 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

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

Functional relations for elliptic polylogarithms

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