# 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 »

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences
- Princeton Univ., NJ (United States). Dept. of Physics
- 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:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- 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)},

issn = {1029-8479},

number = 11,

volume = 2017,

place = {United States},

year = {2017},

month = {11}

}