# Positive Geometries and Canonical Forms

## TL;DR

This paper explores positive geometries and canonical forms, establishing their definitions, methods for constructing forms, and providing numerous examples across various mathematical spaces, linking geometry with physics.

## Contribution

It introduces a rigorous mathematical framework for positive geometries and canonical forms, expanding their study beyond physics applications.

## Key findings

- Defined positive geometries and canonical forms precisely.
- Developed methods to derive forms for complex geometries from simpler ones.
- Presented examples in projective spaces, Grassmannians, and other varieties.

## 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. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.