Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

Mano, Toshiyuki; Watanabe, Hidenobu

doi:10.1090/s0002-9939-2012-11221-3

Cited by 13 publications

(43 citation statements)

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“…To the best of our knowledge, the closest construction in the literature to the one in this work are the ones by refs. [52,53]. In particular, the authors of ref.…”

Section: Discussionmentioning

confidence: 99%

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Open-string integrals with multiple unintegrated punctures at genus one

Kaderli¹,

Rodríguez²

2022

Preprint

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We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the A-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the N unintegrated punctures and the modular parameter τ . These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in α -the string length squared-in terms of elliptic multiple polylogarithms (eMPLs). In the N -puncture case, the KZB equation reveals a representation of B 1,N , the braid group of N strands on a torus, acting on its solutions. We write the simplest of these braid group elements -the braiding one puncture around another -and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the socalled universal case is written in terms of the genus-one Drinfeld-Kohno algebra t 1,N d, a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.

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“…To the best of our knowledge, the closest construction in the literature to the one in this work are the ones by refs. [52,53]. In particular, the authors of ref.…”

Section: Discussionmentioning

confidence: 99%

“…While ϕ τ and φτ agree on the A-cycle, we note that the holomorphicity of the latter makes it easier to relate to the mathematical literature of KZB equations and twisted (co)homology. See section 6 and refs [52,53]…”

mentioning

confidence: 99%

Open-string integrals with multiple unintegrated punctures at genus one

Kaderli¹,

Rodríguez²

2022

Preprint

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“…The structure of twisted cohomology groups whose coefficients are rank-one local systems on a complex torus was studied in [6] in a more general setting. Results needed in the present paper are special cases in [6].…”

Section: Definition Of the Wirtinger Integral Of Level Nmentioning

confidence: 99%

“…If we consider a family of curves with the base space H whose fiber over τ ∈ H is M, then we can define a natural integrable connection on H 1 (M, L) (see [6]). Writing down this connection with respect to the generators of H 1 (M, L) described in Proposition 2.1, we obtain a system of linear differential equations satisfied by the Wirtinger integrals of level N as follows.…”

Section: Differential Equation For the Wirtinger Integral Of Level Nmentioning

confidence: 99%

“…In addition, we explicitly give the values of characteristic exponents at cusps and determine the spectral types at the regular singularities (Proposition 3.2 and Corollary 3.1). Watanabe and the author studied in [6] the structure of twisted cohomology and homology groups whose coefficients are rank-one local systems on a complex torus in a more general setting than the present paper. The Wirtinger integral of level N is a special case of the integrals treated in [6]; while the integrals in [6] contain configurations of points on complex tori and their periods as independent variables, in the Wirtinger integral of level N the configuration of points is restricted to that of N-torsion points.…”

Section: Introductionmentioning

confidence: 95%

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ON A GENERALIZATION OF WIRTINGER^|^rsquo;S INTEGRAL

Mano¹

2012

Kyushu J. Math.

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Abstract. W. Wirtinger, in 1902, introduced an integral representation for the Gauss hypergeometric function in terms of theta functions. We give a generalization of this integral representation. More concretely, we study a definite integral of a power product of theta functions which has zeros at N-torsion points (N is a natural number greater than one) on a one-dimensional complex torus. We show that this integral gives a solution of a system of Fuchsian differential equations on the modular curve of level N and determine the spectral types of the system at its regular singularities.

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A double copy from twisted (co)homology at genus one

Bhardwaj,

Pokraka,

Ren

et al. 2024

J. High Energ. Phys.

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We study the twisted (co)homology of a family of genus-one integrals — the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated. While not actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [1]. Using intersection numbers — an inner product on the vector space of allowed differential forms — we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [2]. We also use the intersection index — an inner product on the vector space of allowed contours — to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.

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Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

Cited by 13 publications

References 17 publications

Open-string integrals with multiple unintegrated punctures at genus one

Open-string integrals with multiple unintegrated punctures at genus one

ON A GENERALIZATION OF WIRTINGER^|^rsquo;S INTEGRAL

A double copy from twisted (co)homology at genus one

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