Positive Geometries and Canonical Forms

TL;DR

The paper formalizes the notions of positive geometries and canonical forms, providing precise recursive axioms, and develops two robust mechanisms to relate complex spaces to simpler ones: triangulation and push-forward maps. It surveys a rich landscape of examples including polytopes, Grassmannians, toric, cluster, and flag varieties, and demonstrates multiple strategies to compute canonical forms, from direct pole-zero constructions to integral representations. A central theme is the Amplituhedron program, connecting positivity to scattering amplitudes, while also developing a general mathematical framework that encompasses nonpolynomial cases, dual constructions, and positivity notions such as positive convexity. The work lays a foundation for future exploration of canonical structures in broad geometric settings, with implications for both mathematics and high-energy physics.

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.

Figures (12)

• Figure 1: Canonical forms of (a) a triangle, (b) a quadrilateral, (c) a segment of the unit disk with $y\geq 1/10$, (d) a sector of the unit disk with central angle $2\pi/3$ symmetric about the $y$-axis, and (e) the unit disk. The form is identically zero for the unit 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 $\pm 1$ at zero-dimensional boundaries and zero elsewhere, and the form is positively oriented on the interior.
• Figure 2: A triangle $X_{\geq 0}$ triangulated by three smaller triangles $X_{i,\geq 0}$ for $i=1,2,3$. The vertices along the vertical mid-line are denoted $P,Q$ and $R$.
• Figure 3: (a) A non-degenerate vs. (b) a degenerate elliptic curve. The former does not provide a valid embedding space for a positive geometry, while the shaded "tear-drop" is a valid (non-normal) positive geometry.
• Figure 4: The Cayley cubic curve. The plane separating the translucent and solid parts of the surface is given by $x_0=0$.
• Figure 5: An "ampersand" curve with boundary given by a quartic polynomial. The shaded "teardrop" is a positive geometry.

Theorems & Definitions (53)

• Example 2.1
• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4
• Example 6.1
• Example 6.2
• Conjecture 6.3
• Example 7.1
• Example 7.2