Abstract
Positive geometries encode the physics of scattering amplitudes in flat space- time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the poly- tope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher di- mensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.
Highlights
In the context of scalar scattering in the form of the ABHY associahedron [6, 7] for the bi-adjoint cubic interactions, and Stokes polytopes [8, 9] for planar quartic interactions.1 Even more surprisingly, they appeared in cosmology, where the canonical form of the socalled cosmological polytopes encodes the wavefunction of the universe [10], which is the relevant quantum mechanical observable, for a large class of toy models
They appeared in cosmology, where the canonical form of the socalled cosmological polytopes encodes the wavefunction of the universe [10], which is the relevant quantum mechanical observable, for a large class of toy models. It was recently introduced an extension of canonical forms for general polytopes, named stringy canonical form, which depends on a certain deformation parameter and, when applied to the ABHY associahedron return the Koba-Nielsen integral known in string theory [11]
The definition of cosmological polytopes as generated from a space of triangles embedded in projective space by intersecting them in the midpoints of their sides and taking the convex hull of their vertices, can be generalised by including a collection of segments in the building blocks allowing to get intersected in their only midpoint; a specific limit of the canonical form of the polytopes constructed in this way returns a differential form with higher order poles whose coefficient represents the correct wavefunction of the universe for certain scalar states in cosmology
Summary
We briefly review the definition as well as the salient features of positive geometries and the associated canonical forms This allows us to set the notation and make our discussion self-contained. The residue of the canonical form with respect to such a simple pole is nothing but the codimension one differential form ω(yj) which depends only on the collective local coordinates yj and it constitutes the canonical form of the (codimension-one) boundary component (C(j), C≥(j0) ): ResC(j) {ω(X, X≥0)} = Reshj = 0 {ω(X, X≥0)} = ω(yj ) = ω(C(j), C≥(j0) ),. The canonical form ω(X, X≥0) provides a characterisation of the positive geometry (X, X≥0), associating the boundary components {(C , (j) C≥(j0) )} of (X, X≥0) to its singularities
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