Aspects of Scattering Amplitudes and Moduli Space Localization

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

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A note on scattering amplitudes on the moduli space of ABJM

Journal of High Energy Physics ◽

10.1007/jhep06(2015)194 ◽

2015 ◽

Vol 2015 (6) ◽

Author(s):

Marco S. Bianchi

Keyword(s):

Scattering Amplitudes ◽

Moduli Space

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ON THE EVALUATION OF COMPTON SCATTERING AMPLITUDES IN STRING THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x98002237 ◽

1998 ◽

Vol 13 (27) ◽

pp. 4717-4757 ◽

Cited By ~ 1

Author(s):

ANDREA PASQUINUCCI ◽

MICHELA PETRINI

Keyword(s):

Scattering Amplitudes ◽

String Theory ◽

Moduli Space ◽

Modular Forms ◽

String Model ◽

World Sheet ◽

Electron Positron ◽

The World ◽

Spin Fields ◽

Compton Amplitude

We consider the Compton amplitude for the scattering of a photon and an (massless) "electron/positron" at one loop (i.e. genus 1) in a four-dimensional fermionic heterotic string model. Starting from the bosonization of the world sheet fermions needed to explicitly construct the spin fields representing the space–time fermions, we present all the steps of the computation which leads to the explicit form of the amplitude as an integral of modular forms over the moduli space.

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BORDERED RIEMANN SURFACES, SCHOTTKY GROUPS AND OFF-SHELL STRING AMPLITUDES

International Journal of Modern Physics A ◽

10.1142/s0217751x91000927 ◽

1991 ◽

Vol 06 (10) ◽

pp. 1719-1747 ◽

Cited By ~ 5

Author(s):

M.A. MARTÍN-DELGADO ◽

J. RAMÍREZ MITTELBRUNN

Keyword(s):

Scattering Amplitudes ◽

Partition Function ◽

Moduli Space ◽

Riemann Surfaces ◽

Mass Shell ◽

Tree Level ◽

Schottky Groups ◽

Functional Integrals ◽

String Amplitudes ◽

Closed Bosonic String

We propose an off-shell extension of the closed bosonic string scattering amplitudes as functional integrals over bordered Riemann surfaces. The tree level off-shell N-scalars amplitude is handled with the help of Schottky groups and the Burnside θ-series. Using these tools and the asymptotic behaviour of the partition function at the boundary of the moduli space, we show that the off-shell amplitudes exhibit the tachyon mass-shell poles in the external momenta. In addition, their residues are shown to be the semi-off-shell amplitudes and the well-known Koba-Nielsen amplitudes.

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Tevelev degrees and Hurwitz moduli spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000670 ◽

2021 ◽

pp. 1-32

Author(s):

A. CELA ◽

R. PANDHARIPANDE ◽

J. SCHMITT

Keyword(s):

Scattering Amplitudes ◽

Exact Solutions ◽

Projective Geometry ◽

Moduli Space ◽

Moduli Spaces ◽

Intersection Theory ◽

Dyck Paths ◽

Simple Recursion ◽

Almost All ◽

Two Parameter

Abstract We interpret the degrees which arise in Tevelev’s study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalisation). We find exact solutions which specialise to Tevelev’s formula in his cases and connect to the projective geometry of lines and Castelnuovo’s classical count of $g^1_d$ ’s in other cases. For almost all values, the calculation of the two parameter generalisation of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

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Topology of the quantum modified moduli space

Surveys In High Energy Physics ◽

10.1080/01422410108228800 ◽

2001 ◽

Vol 15 (4) ◽

pp. 279-289

Author(s):

S. L. Dubovsky

Keyword(s):

Moduli Space

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Improving Space Localization Properties of the Discrete Wavelet Transform

Informatica ◽

10.15388/informatica.2013.416 ◽

2013 ◽

Vol 24 (4) ◽

pp. 657-675

Author(s):

Jonas Valantinas ◽

Deividas Kančelkis ◽

Rokas Valantinas ◽

Gintarė Viščiūtė

Keyword(s):

Wavelet Transform ◽

Discrete Wavelet Transform ◽

Discrete Wavelet ◽

Space Localization

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A Primer on Mapping Class Groups (PMS-49)

10.23943/princeton/9780691147949.001.0001 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Graduate Students ◽

Moduli Space ◽

Riemann Surfaces ◽

Class Group ◽

Mapping Class ◽

Torelli Group ◽

Surface Bundles ◽

Surface Homeomorphisms ◽

Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

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