Positive Geometries and Differential Forms with Non-Logarithmic Singularities I

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 polytope 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 dimensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.

arXiv comment47 pages, figures in Tikz

References in corpus (15)

• The Amplituhedron
• On graviton non-Gaussianities during inflation
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• Positive Geometries and Canonical Forms
• Positive Amplitudes In The Amplituhedron
• Toric reduction and a conjecture of Batyrev and Materov
• A Note on Polytopes for Scattering Amplitudes
• The Momentum Amplituhedron
• Gravity On-shell Diagrams
• Stokes Polytopes : The positive geometry for ${\varphi }^4$ interactions
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• Notes on Scattering Amplitudes as Differential Forms
• Biadjoint scalar tree amplitudes and intersecting dual associahedra
• Amplituhedron meets Jeffrey-Kirwan Residue
• On-shell diagrams and the geometry of planar N < 4 SYM theories

Cited by in corpus (13)

• Cosmological Bootstrap in Slow Motion
• Amplitudes meet Cosmology: A (Scalar) Primer
• Steinmann Relations and the Wavefunction of the Universe
• Positive geometry, local triangulations, and the dual of the Amplituhedron
• On one-loop corrections to the Bunch-Davies wavefunction of the universe
• The Geometry of Cosmological Correlators
• Amplituhedra, and Beyond
• Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes
• Amplituhedron-like geometries
• The Asymptotic Structure of Cosmological Integrals
• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• $\mathcal{N}=7$ On-Shell Diagrams and Supergravity Amplitudes in Momentum Twistor Space
• Cosmological Infrared Subtractions & Infrared-Safe Computables