Combinatorial Aspects of Scattering Amplitudes

2023-12-11 Science

Author : Matteo Parisi
Release : 2023-12-11
Genre : Science
Kind : eBook
Book Rating : 696/5 ( reviews)

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Book Synopsis Combinatorial Aspects of Scattering Amplitudes by : Matteo Parisi

Download or read book Combinatorial Aspects of Scattering Amplitudes written by Matteo Parisi. This book was released on 2023-12-11. Available in PDF, EPUB and Kindle. Book excerpt: This book is a significant contribution within and across High Energy Physics and Algebraic Combinatorics. It is at the forefront of the recent paradigm shift according to which physical observables emerge from geometry and combinatorics. It is the first book on the amplituhedron, which encodes the scattering amplitudes of N=4 Yang-Mills theory, a cousin of the theory of strong interactions of quarks and gluons. Amplituhedra are generalizations of polytopes inside the Grassmannian, and they build on the theory of total positivity and oriented matroids. This book unveils many new combinatorial structures of the amplituhedron and introduces a new important related object, the momentum amplituhedron. Moreover, the work pioneers the connection between amplituhedra, cluster algebras and tropical geometry. Combining extensive introductions with proofs and examples, it is a valuable resource for researchers investigating geometrical structures emerging from physics for some time to come.

Grassmannian Geometry of Scattering Amplitudes

2016-05-05 Science

Author : Nima Arkani-Hamed
Release : 2016-05-05
Genre : Science
Kind : eBook
Book Rating : 645/5 ( reviews)

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Book Synopsis Grassmannian Geometry of Scattering Amplitudes by : Nima Arkani-Hamed

Download or read book Grassmannian Geometry of Scattering Amplitudes written by Nima Arkani-Hamed. This book was released on 2016-05-05. Available in PDF, EPUB and Kindle. Book excerpt: Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang–Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids - in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the broader fields of mathematical physics.

Aspects of Scattering Amplitudes and Moduli Space Localization

2020-09-23 Science

Author : Sebastian Mizera
Release : 2020-09-23
Genre : Science
Kind : eBook
Book Rating : 108/5 ( reviews)

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Book Synopsis Aspects of Scattering Amplitudes and Moduli Space Localization by : Sebastian Mizera

Download or read book Aspects of Scattering Amplitudes and Moduli Space Localization written by Sebastian Mizera. This book was released on 2020-09-23. Available in PDF, EPUB and Kindle. Book excerpt: This thesis proposes a new perspective on scattering amplitudes in quantum field theories. Their standard formulation in terms of sums over Feynman diagrams is replaced by a computation of geometric invariants, called intersection numbers, on moduli spaces of Riemann surfaces. It therefore gives a physical interpretation of intersection numbers, which have been extensively studied in the mathematics literature in the context of generalized hypergeometric functions. This book explores physical consequences of this formulation, such as recursion relations, connections to geometry and string theory, as well as a phenomenon called moduli space localization. After reviewing necessary mathematical background, including topology of moduli spaces of Riemann spheres with punctures and its fundamental group, the definition and properties of intersection numbers are presented. A comprehensive list of applications and relations to other objects is given, including those to scattering amplitudes in open- and closed-string theories. The highlights of the thesis are the results regarding localization properties of intersection numbers in two opposite limits: in the low- and the high-energy expansion. In order to facilitate efficient computations of intersection numbers the author introduces recursion relations that exploit fibration properties of the moduli space. These are formulated in terms of so-called braid matrices that encode the information of how points braid around each other on the corresponding Riemann surface. Numerous application of this approach are presented for computation of scattering amplitudes in various gauge and gravity theories. This book comes with an extensive appendix that gives a pedagogical introduction to the topic of homologies with coefficients in a local system.

Recent Trends in Algebraic Combinatorics

2019-01-21 Mathematics

Author : Hélène Barcelo
Release : 2019-01-21
Genre : Mathematics
Kind : eBook
Book Rating : 412/5 ( reviews)

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Book Synopsis Recent Trends in Algebraic Combinatorics by : Hélène Barcelo

Download or read book Recent Trends in Algebraic Combinatorics written by Hélène Barcelo. This book was released on 2019-01-21. Available in PDF, EPUB and Kindle. Book excerpt: This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.

Combinatorial Aspect of Integrable Systems

2007 Mathematics

Author : Arkady Berenstein
Release : 2007
Genre : Mathematics
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Combinatorial Aspect of Integrable Systems by : Arkady Berenstein

Download or read book Combinatorial Aspect of Integrable Systems written by Arkady Berenstein. This book was released on 2007. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of six papers based on the expository lectures of the workshop "Combinatorial Aspect of Integrable Systems" held at RIMS during July 26-30, 2004, as a part of the Project Research 2004 "Method of Algebraic Analysis in Integrable Systems". The topics range over crystal bases of quantum groups, its algebra-geometric analogue known as geometric crystal, generalizations of Robinson-Schensted type correspondence, fermionic formula related to Bethe ansatz, applications of crystal bases to soliton celluar automata, Yang-Baxter maps, and integrable discrete dynamics. All the papers are friendly written with many illustrative examples and intimately related to each other. This volume will serve as a good guide for researchers and graduate students who are interested in this fascinating subject.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

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Combinatorial Aspects of Scattering Amplitudes

Grassmannian Geometry of Scattering Amplitudes

Aspects of Scattering Amplitudes and Moduli Space Localization

Recent Trends in Algebraic Combinatorics

Combinatorial Aspect of Integrable Systems

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