Positive Geometries for Scattering Amplitudes in Momentum Space

Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form - the canonical form. There are known examples where the canonical form of some positive geometry, defined in some kinematic space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of the positive geometry are in bijection with the physical singularities of the scattering amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical positive geometry. It lives in momentum twistor space and describes tree-level (and the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric Yang-Mills theory. In this dissertation, we study three positive geometries defined in on-shell momentum space: the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron, the Momentum Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level scattering amplitudes for different theories in different spacetime dimensions. The three positive geometries share a series of interrelations in terms of their boundary posets and canonical forms. We review these relationships in detail, highlighting the author's contributions. We study their boundary posets, classifying all boundaries and hence all physical singularities at the tree level. We develop new combinatorial results to derive rank-generating functions which enumerate boundaries according to their dimension. These generating functions allow us to prove that the Euler characteristics of the three positive geometries are one. In addition, we discuss methods for manipulating canonical forms using ideas from computational algebraic geometry.Comment: PhD Dissertatio

The Grassmannian for celestial superamplitudes

© The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), https://creativecommons.org/licenses/by/4.0/Recently, scattering amplitudes in four-dimensional Minkowski spacetime have been interpreted as conformal correlation functions on the two-dimensional celestial sphere, the so-called celestial amplitudes. In this note we consider tree-level scattering amplitudes in N = 4 super Yang-Mills theory and present a Grassmannian formulation of their celestial counterparts. This result paves the way towards a geometric picture for celestial superamplitudes, in the spirit of positive geometries.Peer reviewedFinal Published versio

Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron

© 2023 by the author(s). This work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher’s result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions for contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. It is postulated by Livia Ferro, Tomasz Łukowski and Robert Moerman (2020) that contracted Grassmannian forests are in bijection with boundary strata of the momentum amplituhedron, an object encoding the tree-level S-matrix of maximally supersymmetric Yang–Mills theory. With this assumption, our results give a rank generating function for the boundary strata of the momentum amplituhedron, and imply that the Euler characteristic of the momentum amplituhedron is 1.Peer reviewe

Pushforwards via scattering equations with applications to positive geometries

© 2022 The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), https://creativecommons.org/licenses/by/4.0/In this paper we explore and expand the connection between two modern descriptions of scattering amplitudes, the CHY formalism and the framework of positive geometries, facilitated by the scattering equations. For theories in the CHY family whose S-matrix is captured by some positive geometry in the kinematic space, the corresponding canonical form can be obtained as the pushforward via the scattering equations of the canonical form of a positive geometry defined in the CHY moduli space. In order to compute these canonical forms in kinematic spaces, we study the general problem of pushing forward arbitrary rational differential forms via the scattering equations. We develop three methods which achieve this without ever needing to explicitly solve any scattering equations. Our results use techniques from computational algebraic geometry, including companion matrices and the global duality of residues, and they extend the application of similar results for rational functions to rational differential forms.Peer reviewe

On the geometry of the orthogonal momentum amplituhedron

© The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0). https://creativecommons.org/licenses/by/4.0/In this paper we focus on the orthogonal momentum amplituhedron Ok, a recently introduced positive geometry that encodes the tree-level scattering amplitudes in ABJM theory. We generate the full boundary stratification of Ok for various k and conjecture that its boundaries can be labelled by so-called orthogonal Grassmannian forests (OG forests). We determine the generating function for enumerating these forests according to their dimension and show that the Euler characteristic of the poset of these forests equals one. This provides a strong indication that the orthogonal momentum amplituhedron is homeomorphic to a ball. This paper is supplemented with the Mathematica package orthitroids which contains useful functions for exploring the structure of the positive orthogonal Grassmannian and the orthogonal momentum amplituhedron.Peer reviewe

On the geometry of the orthogonal momentum amplituhedron

© The Authors. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0). https://creativecommons.org/licenses/by/4.0/In this paper we focus on the orthogonal momentum amplituhedron Ok, a recently introduced positive geometry that encodes the tree-level scattering amplitudes in ABJM theory. We generate the full boundary stratification of Ok for various k and conjecture that its boundaries can be labelled by so-called orthogonal Grassmannian forests (OG forests). We determine the generating function for enumerating these forests according to their dimension and show that the Euler characteristic of the poset of these forests equals one. This provides a strong indication that the orthogonal momentum amplituhedron is homeomorphic to a ball. This paper is supplemented with the Mathematica package orthitroids which contains useful functions for exploring the structure of the positive orthogonal Grassmannian and the orthogonal momentum amplituhedron.Peer reviewe

Kleiss-Kuijf relations from momentum amplituhedron geometry

44 pages, 19 figuresAbstract: In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.Peer reviewe

Positive Geometries for Scattering Amplitudes in Momentum Space

Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form—the canonical form. There are known examples where the canonical form of some positive geometry, defined in some kinematic space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of the positive geometry are in bijection with the physical singularities of the scattering amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical positive geometry. It lives in momentum twistor space and describes tree-level (and the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric Yang-Mills theory. In this dissertation, we study three positive geometries defined in on-shell momentum space: the Arkani-Hamed–Bai–He–Yan (ABHY) associahedron, the Momentum Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level scattering amplitudes for different theories in different spacetime dimensions. The three positive geometries share a series of interrelations in terms of their boundary posets and canonical forms. We review these relationships in detail, highlighting the author’s contributions. We study their boundary posets, classifying all boundaries and hence all physical singularities at the tree level. We develop new combinatorial results to derive rank-generating functions which enumerate boundaries according to their dimension. These generating functions allow us to prove that the Euler characteristics of the three positive geometries are one. In addition, we discuss methods for manipulating canonical forms using ideas from computational algebraic geometry

The Informational Role of the Media in Private Lending

ABSTRACT We investigate whether a borrower's media coverage influences the syndicated loan origination and participation decisions of informationally disadvantaged lenders, loan syndicate structures, and interest spreads. In syndicated loan deals, information asymmetries can exist between lenders that have a relationship with a borrower and less informed, nonrelationship lenders competing to serve as lead arranger on a syndicated loan, and also between lead arrangers and less informed syndicate participants. Theory suggests that the aggressiveness with which less informed lenders compete for a loan deal increases in the sentiment of public information signals about a borrower. We extend this theory to syndicated loans and hypothesize that the likelihood of less informed lenders serving as the lead arranger or joining a loan syndicate is increasing in the sentiment of media‐initiated, borrower‐specific articles published prior to loan origination. We find that as media sentiment increases (1) outside, nonrelationship lenders have a higher probability of originating loans; (2) syndicate participants are less likely to have a previous relationship with the borrower or lead bank; (3) lead banks retain a lower percentage of loans; and (4) loan spreads decrease

The Informational Role of the Media in Private Lending

We investigate whether a borrower’s media coverage influences the syndicated loan origination and participation decisions of informationally disadvantaged lenders, loan syndicate structures, and interest spreads. In syndicated loan deals, information asymmetries can exist between lenders that have a relationship with a borrower and less informed, nonrelationship lenders competing to serve as lead arranger on a syndicated loan, and also between lead arrangers and less informed syndicate participants. Theory suggests that the aggressiveness with which less informed lenders compete for a loan deal increases in the sentiment of public information signals about a borrower. We extend this theory to syndicated loans and hypothesize that the likelihood of less informed lenders serving as the lead arranger or joining a loan syndicate is increasing in the sentiment of mediaâ initiated, borrowerâ specific articles published prior to loan origination. We find that as media sentiment increases (1) outside, nonrelationship lenders have a higher probability of originating loans; (2) syndicate participants are less likely to have a previous relationship with the borrower or lead bank; (3) lead banks retain a lower percentage of loans; and (4) loan spreads decrease.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/136338/1/joar12131_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/136338/2/joar12131.pd