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2023 | Book

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Combinatorial Aspects of Scattering Amplitudes

Amplituhedra, T-duality, and Cluster Algebras

Author: Matteo Parisi

Publisher: Springer Nature Switzerland

Book Series : Springer Theses

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Among the physical observables in a Quantum Field Theory (QFT), Scattering Amplitudes are central in fundamental physics: they encode the probabilities of any interaction among elementary particles. Thanks to the development of the On-Shell Methods, an ‘S-Matrix Program Reloaded’ initiated at the beginning of the new millennium. Among these, recursion relations and generalized unitarity are nowadays cornerstones of modern amplitude computations. One of the most studied QFT, where all these methods have been applied with great success and main subject of our research, is the maximally supersymmetric Yang-Mills theory (\(\mathcal {N}=4\) SYM) in four dimensions. In this chapter, after providing the motivation and the overall plan of our work, we introduce: \(\mathcal {N}=4\) SYM theory, colour decomposition techniques for gauge theories, the super-space formalism, on-shell variables (spinor helicity and momentum twistors) and recursion relations. These allowed the computation of all tree-level amplitudes and loop-level integrands in \(\mathcal {N}=4\) SYM, and the discovery of their analytical properties and hidden symmetries.
Matteo Parisi
Chapter 2. The Amplituhedron
Abstract
In 2013, building on previous Grassmannian formulations and on Hodges’ ideas in scattering amplitudes, Arkani-Hamed and Trnka defined the amplituhedron \(\mathcal {A}_{n,k,m}\) as the image of the positive Grassmannian \(\textrm{Gr}^{\ge 0}_{k,n}\) under a totally positive linear map. Regarded as a non-linear generalization of (cyclic) polytopes inside the Grassmannian, it is a semialgebraic set with beautiful and rich combinatorics. The \(m=4\) amplituhedron \(\mathcal {A}_{n,k,4}\) encodes tree-level amplitudes in \(\mathcal {N}=4\) SYM. The \(m=2\) amplituhedron \(\mathcal {A}_{n,k,2}\), toy-model for the \(m=4\) case, it also enters the geometry of some loop amplitudes, tree-level form factors of half-BPS operators and correlators in planar \(\mathcal {N}=4\) SYM. In this chapter, after a review on the positive Grassmannian, we introduce and prove new properties of the amplituhedron. We define the amplituhedron-analogue of the matroid stratification of the Grassmannian, which we call sign stratification. We classify all positroid tiles of the amplituhedron \(\mathcal {A}_{n,k,2}\) – full-dimensional images of positroid cells of \({{\,\textrm{Gr}\,}}^{\ge 0}_{k,n}\) on which the amplituhedron map is injective—and we give them an intrinsic sign-characterization in the amplituhedron. We use this result to prove Arkani-Hamed–Thomas–Trnka’s sign-flip conjecture on \(\mathcal {A}_{n,k,2}\). Finally, we review how to extract tree-level amplitudes of \(\mathcal {N}=4\) SYM from the \(m=4\) amplituhedron.
Matteo Parisi
Chapter 3. The Hypersimplex
Abstract
In 1987, the foundational work of Gelfand-Goresky-MacPherson-Serganova initiated the study of the Grassmannian and torus orbits in the Grassmannian via the moment map and matroid polytopes, which arise as moment map images of (closures of) torus orbits. The moment map image of the (positive) Grassmannian \({{\,\textrm{Gr}\,}}_{k+1,n}\) is the \((n-1)\)-dimensional hypersimplex \(\Delta _{k+1,n} \subseteq \mathbb {R}^n\), the convex hull of the indicator vectors \(e_I\in \mathbb {R}^n\) where \(I \in {[n] \atopwithdelims ()k+1}\). In this chapter we introduce and present new results about the hypersimplex and positroid polytopes—(closures of) images of positroid cells of \({{\,\textrm{Gr}\,}}^{\ge 0}_{k+1,n}\) under the moment map. We give a full characterization of positroid polytopes which are images of positroid cells where the moment map is injective. In particular, the full-dimensional ones— we call positroid tiles—are in bijection with plabic trees. We then consider the problem of finding positroid tilings—collections of positroid tiles whose interiors are pair-wise disjoint and cover the hypersimplex. We show that the positive tropical Grassmannain \({{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}\) is the secondary fan for regular subdivisions of \(\Delta _{k+1,n}\) into positroid polytopes. In particular, maximal cones of \({{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}\) are in bijection with regular positroid tilings of \(\Delta _{k+1,n}\).
Matteo Parisi
Chapter 4. T-Duality: The Hypersimplex Versus the Amplituhedron
Abstract
In this chapter we discover a duality—which we call T-duality—between two seemingly unrelated objects. The first is the hypersimplex \(\Delta _{k+1,n}\)—an \((n-1)\)-dimensional polytope in \(\mathbb {R}^n\) which is the moment map image of the positive Grassmannian \(\text{ Gr}^{\ge 0}_{k+1,n}\) and has nice combinatorics in connection with matroid theory and tropical geometry. The second is the \(m=2\) amplituhedron \(\mathcal {A}_{n,k,2}(Z)\)—a 2k-dimensional semialgebraic set in \(\text{ Gr}_{k,k+2}\) which is the image of the positive Grassmannian \(\text{ Gr}^{\ge 0}_{k,n}\) under a linear map induced by a totally positive matrix Z and has been defined by physicists studying scattering amplitudes. After defining T-duality on decorated permutations and plabic graphs labelling positroid cells, we use it to provide many paralles between positroid polytopes in \(\Delta _{k+1,n}\) and Grasstopes in \(\mathcal {A}_{n,k,2}\). Among these, T-duality provides a bijection between positroid tiles of \(\Delta _{k+1,n}\) and \(\mathcal {A}_{n,k,2}\), and the bijection extends to their defining inequalities and the facets. This connects to the combinatorics of plabic tilings and separable permutations on \([n-1]\). Moreover, T-duality provides a bijection between w-simplices—giving the well-known triangulation of \(\Delta _{k+1,n}\) into Weyl alcoves—and w-chambers—giving a decomposition of \(\mathcal {A}_{n,k,2}\) into a subset of its sign strata related to oriented uniform matroids. They are both labelled by permutations w on \([n-1]\) with k descents, hence enumerated by Eulerian numbers \(E_{n-1,k}\).
Matteo Parisi
Chapter 5. Positroid Tilings
Abstract
The positive Grassmannian \({{\,\textrm{Gr}\,}}_{r,n}^{\ge 0}\) has a beautiful decomposition into positroid cells. Given any surjective map \(\phi : {{\,\textrm{Gr}\,}}_{r,n}^{\ge 0} \rightarrow X\), it is natural to try to decompose X using images \(\overline{\phi (S_\pi )}\) of positroid cells \(S_\pi \) under \(\phi \). If such images have pair-wise disjoint interior, cover X and are full-dimensional, then we call their collection \(\{\overline{\phi (S_\pi )}\}\) a positroid dissection. If \(\phi \) is also injective on each cell \(S_\pi \), then \(\{\overline{\phi (S_\pi )}\}\) is a positroid tiling, and each \(\overline{\phi (S_\pi )}\) is a positroid tile. When \(\phi \) is the moment map, then \(\overline{\phi (S_\pi )}\) are positroid polytopes, so a positroid tiling of the hypersimplex is a decomposition into positroid polytopes. When \(\phi \) is the amplituhedron map, \(\overline{\phi (S_\pi )}\) are Grasstopes, so they are the building blocks of positroid tilings of the amplituhedron—which provide expressions for \(\mathcal {N}=4\) SYM scattering amplitudes. In this chapter we prove that T-duality provides a bijection between positroid tilings of the hypersimplex \(\Delta _{k+1,n}\) and of the \(m=2\) amplituhedron \(\mathcal {A}_{n,k,2}\). We also conjecture the same holds true for positroid dissections. We show how to obtain different types of positroid tilings and dissections from BCFW recursion relations and from the positive tropical Grassmannian \({{\,\textrm{Trop}\,}}^+{{\,\textrm{Gr}\,}}_{k+1,n}\)—laying the groundwork to realize the tantalizing possibility of a notion of secondary polytope-like structures for Grasstopes.
Matteo Parisi
Chapter 6. The Momentum Amplituhedron
Abstract
Scattering Amplitudes in planar \(\mathcal {N}=4\) SYM have a dual formulation, due to the Amplitude/Wilson Loop duality which has roots in a certain T-duality from String Theory. On one side, tree-level scattering amplitudes in momentum space can be obtained from a contour integral over collections of \((2n-4)\)-dimensional positroid cells of \({{\,\textrm{Gr}\,}}^{\ge 0}_{k+2,n}\). On the other side, in momentum twistor space, one has collections of 4k-dimensional positroid cells of \({{\,\textrm{Gr}\,}}^{\ge 0}_{k,n}\). While the latter collections were conjectured to give positroid tilings of the \(m=4\) amplituhedron, no object was known to be tiled by the former collections of cells in \({{\,\textrm{Gr}\,}}^{\ge 0}_{k+2,n}\). In this chapter we introduce such an object—the momentum amplituhedron \(\mathcal {M}_{n,k',m}\), defined as the image of the positive Grassmannian \(\text{ Gr}^{\ge 0}_{k',n}\) under a linear map \(\Phi _{\Lambda ,\tilde{\Lambda }}\) induced by two totally positive matrices \(\Lambda ^\perp ,\tilde{\Lambda }\). After discussing the sign stratification and sign-flips for \(\mathcal {M}_{n,k',m}\), we describe its facets in the \(m=4\) case, in connection with factorization channels of amplitudes. We then conjecture that positroid tilings of the \(m=4\) amplituhedron \(\mathcal {A}_{n,k,4}\) and of \(\mathcal {M}_{n,k+2,4}\) are in bijection via T-duality. Finally, we explain how to extract tree-level scattering amplitudes in the spinor helicity space from \(\mathcal {M}_{n,k',m=4}\). As the momentum amplituhedron is formulated in momentum space (as opposed to the amplituhedron), it opens ways to extend its construction to many other theories.
Matteo Parisi
Chapter 7. Cluster Algebras and Amplituhedra
Abstract
Introduced by Fomin and Zelevinsky in 2002 to study total positivity and Lie theory, cluster algebras are commutative rings with a set of distinguished generators (cluster variables) and remarkable combinatorics. Many well-known varieties—such as the Grassmannian \(\text{ Gr}_{m,n}\)—have a cluster structure. In physics, Golden et al. established in 2013 that singularities of amplitudes of planar \(\mathcal {N}=4\) SYM can be described by the \(\text{ Gr}_{4,n}\) cluster algebra. Later, Drummond et al. observed the phenomena of cluster adjacencies, related to compatibility of cluster variables in \(\text{ Gr}_{4,n}\). Since then, cluster algebras have been playing a cutting-edge role in both understanding and computing scattering amplitudes. In this chapter we pioneer the connection between cluster algebras and the amplituhedron. In particular, we formulate and generalize the cluster adjacency conjectures in \(\text{ Gr}_{m,n}\) at tree-level in terms of the amplituhedron \(\mathcal {A}_{n,k,m}\). We then prove them for the \(m=2\) amplituhedron \(\mathcal {A}_{n,k,2}\) and surprisingly find that each positroid tile of \(\mathcal {A}_{n,k,2}\) is the totally positive part of a cluster variety. Finally, we make a new conjecture about cluster structures relating Leading and Landau singularities of amplitudes in planar \(\mathcal {N}=4\) SYM—which we call LL-cluster adjacency. At one loop, we prove it for NMHV amplitudes and test it for N\(^2\)MHV up to 9-points amplitudes using the geometry of the loop amplituhedron. We end by commenting possible implications for the bootstrap program.
Matteo Parisi
Chapter 8. Conclusions
Abstract
Building on Grassmannian formulations for scattering amplitudes in planar \(\mathcal {N}=4\) SYM – introduced by Arkani-Hamed et al. and Bullimore et al.—and on Hodges’ idea that amplitudes are ‘volumes’ of some geometric object, Arkani-Hamed and Trnka arrived at the definition of the amplituhedron in 2013. The above geometrisation programme—formalized in the framework of positive geometries—hinges on the idea that quantum observables in particle physics and cosmology come from underlying (novel) mathematical objects. Physical properties (e.g. locality and unitarity) purely emerge from combinatorics and geometry. Understanding this process advances our grasp of the basic principles of Quantum Field Theory and allows us to perform calculations which were previously beyond reach. Crucially, it also cross-fertilises ideas in pure mathematics, such as in algebriac combinatorics. Our work concerns understanding the combinatorics of tilings, T-duality and the cluster structures of ‘amplituhedra’—the positive geometries relevant for scattering amplitudes of \(\mathcal {N}=4\) SYM (and beyond). In this chapter we review the motivation and the results of our work, and present future directions.
Matteo Parisi
Backmatter
Metadata
Title
Combinatorial Aspects of Scattering Amplitudes
Author
Matteo Parisi
Copyright Year
2023
Electronic ISBN
978-3-031-41069-7
Print ISBN
978-3-031-41068-0
DOI
https://doi.org/10.1007/978-3-031-41069-7

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