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 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.

Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals

“…In [1], N. Arkani-Hamed, T. Lam and one of the authors introduced stringy canonical forms, which are string-like integrals as α ′ -deformation of canonical form [2] for general polytopes. Let's briefly review the idea: we consider an integral over R d >0 , which depends on a collection of polynomials, p I , with positive coefficients:…”

Unwinding the amplituhedron in binary

“…So for k = 1, this space can be identified with the dual of the (cyclic) polytope coming from the external data. The obvious extension to general k gives one natural working definition for a dual of the amplituhedron, as described in [10,17]. This definition can naturally be extended to loops when m = 4.…”

“…As described in [10,17], there are further motivations to find a dual amplituhedron; by analogy with the well-understood case of k = 1, we can hope for a direct and intrinsic definition of the canonical form with logarithmic singularities on the amplituhedron expressed as an integral over the dual geometry. As already described in [17] for the simplest case of G + (2, 4), a direct extension of the analogy with k = 1 already involves novel features not seen for polytopes. It will be interesting to see if the definition of the dual amplituhedron we have given will nonetheless end up playing an important role in determining the canonical form for both tree and loops.…”

“…This physics and mathematics has been explored from a variety of perspectives in the past few years (see e.g. [6][7][8][9][10][11][12][13][14][15][16]), and a systematic mathematical exploration of the notion of "positive geometries" has recently been initiated in [17].…”

Prescriptive unitarity

“…Most of these discoveries have been fueled by direct computation -pushing the limits of our theoretical reach (often for toy models) to uncover unanticipated, simplifying structures in the formulae that result, and using these insights to build more powerful tools. The lessons learned through such investigations include the (BCFW) on-shell recursion relations at tree-and loop-level, [7,8] and [9]; the discovery of a hidden dual conformal invariance [10][11][12] as well as the duality to Wilson loops and correlation functions [13][14][15][16][17][18][19][20]; the connection to Grassmannian geometry [21][22][23][24][25][26][27] and the amplituhedron [28][29][30][31][32][33][34][35][36][37][38][39]; various bootstrap methods [40][41][42][43][44][45][46][47][48][49][50][51]; the twistor…”