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Title: Positive geometries and canonical forms

Abstract

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects — the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra — which have been loosely referred to as “positive geometries”. The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of “positive geometries” and their associated “canonical forms” as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones — via “triangulation” on the one hand, and “push-forward” maps between geometries on the other. We present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flagmore » varieties, both for the simplest “simplex-like” geometries and the richer “polytope-like” ones. We also illustrate a number of strategies for computing canonical forms for large classes of positive geometries, ranging from a direct determination exploiting knowledge of zeros and poles, to the use of the general triangulation and push-forward methods, to the representation of the form as volume integrals over dual geometries and contour integrals over auxiliary spaces. These methods yield interesting representations for the canonical forms of wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.« less

Authors:
 [1];  [2];  [3]
  1. Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
  2. Princeton Univ., NJ (United States). Dept. of Physics
  3. Univ. of Michigan, Ann Arbor, MI (United States). Dept. of Mathematics
Publication Date:
Research Org.:
Inst. for Advanced Study, Princeton, NJ (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1502489
Grant/Contract Number:  
SC0009988
Resource Type:
Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 11; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Differential and Algebraic Geometry; Scattering Amplitudes; Supersymmetric Gauge Theory

Citation Formats

Arkani-Hamed, Nima, Bai, Yuntao, and Lam, Thomas. Positive geometries and canonical forms. United States: N. p., 2017. Web. doi:10.1007/jhep11(2017)039.
Arkani-Hamed, Nima, Bai, Yuntao, & Lam, Thomas. Positive geometries and canonical forms. United States. doi:10.1007/jhep11(2017)039.
Arkani-Hamed, Nima, Bai, Yuntao, and Lam, Thomas. Wed . "Positive geometries and canonical forms". United States. doi:10.1007/jhep11(2017)039. https://www.osti.gov/servlets/purl/1502489.
@article{osti_1502489,
title = {Positive geometries and canonical forms},
author = {Arkani-Hamed, Nima and Bai, Yuntao and Lam, Thomas},
abstractNote = {Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects — the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra — which have been loosely referred to as “positive geometries”. The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of “positive geometries” and their associated “canonical forms” as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones — via “triangulation” on the one hand, and “push-forward” maps between geometries on the other. We present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest “simplex-like” geometries and the richer “polytope-like” ones. We also illustrate a number of strategies for computing canonical forms for large classes of positive geometries, ranging from a direct determination exploiting knowledge of zeros and poles, to the use of the general triangulation and push-forward methods, to the representation of the form as volume integrals over dual geometries and contour integrals over auxiliary spaces. These methods yield interesting representations for the canonical forms of wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.},
doi = {10.1007/jhep11(2017)039},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2017,
place = {United States},
year = {2017},
month = {11}
}

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Cited by: 28 works
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Figures / Tables:

Figure 1 Figure 1: Canonical forms of (a) a triangle, (b) a quadrilateral, (c) a segment of the unit disk with $y$ ≥ 1/10, (d) a sector of the unit disk with central angle 2$π$/3 symmetric about the $y$-axis, and (e) the unit disk. The form is identically zero for the unitmore » disk because there are no zero-dimensional boundaries. For each of the other figures, the form has simple poles along each boundary component, all leading residues are ±1 at zero-dimensional boundaries and zero elsewhere, and the form is positively oriented on the interior.« less

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