Intersection numbers for loaded cycles associated with Selberg-type integrals

Mimachi, Katsuhisa; Ohara, Katsuyoshi; Yoshida, Masaaki

doi:10.2748/tmj/1113246749

Cited by 13 publications

(9 citation statements)

References 5 publications

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“…Intersection theory for twisted cycles was introduced by Kita and Yoshida in 1992 [16], who later developed it further in a series of papers [17][18][19]. Since then, intersection numbers have been evaluated for a large family of different types of hypergeometric functions [80,[94][95][96][97][98][99][100][101][102], including Selberg-type integrals [2,81,85,86,103,104]. For our purposes, intersection numbers of twisted cycles play a central role in the KLT relations by computing entries of the inverse of the KLT kernel.…”

Section: Inverse Klt Kernel As Intersection Numbers Of Twisted Cyclesmentioning

confidence: 99%

Combinatorics and topology of Kawai-Lewellen-Tye relations

Mizera

2017

J. High Energ. Phys.

129

251

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We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the underlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the α -corrected biadjoint scalar amplitudes recently defined by the author [1]. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.

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Section: Inverse Klt Kernel As Intersection Numbers Of Twisted Cyclesmentioning

confidence: 99%

Combinatorics and topology of Kawai-Lewellen-Tye relations

Mizera

2017

J. High Energ. Phys.

129

251

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“…For previous discussion of fibrations of twisted cohomologies in the context of their intersection theory see [142][143][144]. 15…”

Section: Fibred Moduli Spaces and Twisted Cohomologiesmentioning

confidence: 99%

Aspects of Scattering Amplitudes and Moduli Space Localization

Mizera

2019

Preprint

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We elaborate on the recent proposal that intersection numbers of certain cohomology classes on the moduli space of genus-zero Riemann surfaces with n punctures, M 0,n , compute tree-level scattering amplitudes in quantum field theories with a finite spectrum of particles. The relevant cohomology groups are twisted by representations of the fundamental group π 1 (M 0,n ) that describes how punctures braid around each other on the Riemann surface. Such a structure can be used to link the space of Riemann surfaces with the space of kinematic invariants. Intersection numbers of said cohomology classes-whose representatives we call twisted forms-can be shown to fully localize on the boundaries of M 0,n , which are in one-to-one map with Feynman diagrams. In this work we develop systematic approaches towards accessing such boundary information. We prove that when twisted forms are logarithmic, their intersection numbers have a simple expansion in terms of trivalent Feynman diagrams weighted by residues, allowing only for massless propagators on the internal and external lines. It is also known that in the massless limit intersection numbers have a different localization formula on the support of so-called scattering equations. Nevertheless, for physical applications one also needs to study non-logarithmic forms as they are responsible for propagation of massive states. We utilize the natural fibre bundle structure of M 0,n -which allows for a direct access to the boundaries-to introduce recursion relations for intersection numbers that "integrate out" puncture-by-puncture. The resulting recursion involves only linear algebra of certain matrices describing braiding properties of M 0,n and evaluating one-dimensional residues, thus paving a way for explicit analytic computations of scattering amplitudes. Together with the previous reformulation of the tree-level S-matrix of string theory in terms of twisted forms, the results of this work complete a unified geometric framework for studying scattering amplitudes from genus-zero Riemann surfaces. We show that a web of dualities between different homology and cohomology groups allows for deriving a host of identities among various types of amplitudes computed from the moduli space, which in this setup become a consequence of linear algebra. Throughout this work we emphasize that algebraic computations can be supplemented-or indeed replaced-by combinatorial, geometric, and topological ones.

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“…As it is usual in the study of scattering amplitudes, such simplicity hints at the existence of some underlying structure. Perhaps it can be related to the work on planar algebras [48] or intersection matrices associated to Selberg integrals [49,50]. 4 Additionally, the field theory bi-adjoint amplitudes have a CHY representation that allowed for its identification with the inverse KLT kernel in the first place [6].…”

Section: Future Directionsmentioning

confidence: 99%

Inverse of the string theory KLT kernel

Mizera

2017

J. High Energ. Phys.

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Abstract:The field theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the α -corrected bi-adjoint scalar amplitudes that are exact in α . We find compact expressions in terms of graphs, where the standard Feynman propagators 1/p 2 are replaced by either 1/ sin(πα p 2 /2) or 1/ tan(πα p 2 /2), as determined by a recursive procedure. We demonstrate how the same object can be used to conveniently expand open string partial amplitudes in a BCJ basis.

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Intersection numbers for loaded cycles associated with Selberg-type integrals

Abstract: We evaluate the intersection numbers of loaded cycles associated with an n-fold Selberg-type integral. We proceed inductively using high-dimensional local systems.

Cited by 13 publications

References 5 publications

Combinatorics and topology of Kawai-Lewellen-Tye relations

Combinatorics and topology of Kawai-Lewellen-Tye relations

Aspects of Scattering Amplitudes and Moduli Space Localization

Inverse of the string theory KLT kernel

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