Positive geometries and differential forms with non-logarithmic singularities. Part I

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Positive geometries and differential forms with non-logarithmic singularities. Part I. / Benincasa, Paolo; Parisi, Matteo.

In: Journal of High Energy Physics, Vol. 2020, No. 8, 23, 05.08.2020.

Research output: Contribution to journalJournal articlepeer-review

Harvard

Benincasa, P & Parisi, M 2020, 'Positive geometries and differential forms with non-logarithmic singularities. Part I', Journal of High Energy Physics, vol. 2020, no. 8, 23. https://doi.org/10.1007/JHEP08(2020)023

APA

Benincasa, P., & Parisi, M. (2020). Positive geometries and differential forms with non-logarithmic singularities. Part I. Journal of High Energy Physics, 2020(8), [23]. https://doi.org/10.1007/JHEP08(2020)023

Vancouver

Benincasa P, Parisi M. Positive geometries and differential forms with non-logarithmic singularities. Part I. Journal of High Energy Physics. 2020 Aug 5;2020(8). 23. https://doi.org/10.1007/JHEP08(2020)023

Author

Benincasa, Paolo ; Parisi, Matteo. / Positive geometries and differential forms with non-logarithmic singularities. Part I. In: Journal of High Energy Physics. 2020 ; Vol. 2020, No. 8.

Bibtex

@article{4b8af571fa984275b8c7f450055ff88c,
title = "Positive geometries and differential forms with non-logarithmic singularities. Part 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 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.",
keywords = "Differential and Algebraic Geometry, Scattering Amplitudes",
author = "Paolo Benincasa and Matteo Parisi",
year = "2020",
month = aug,
day = "5",
doi = "10.1007/JHEP08(2020)023",
language = "English",
volume = "2020",
journal = "Journal of High Energy Physics (Online)",
issn = "1126-6708",
publisher = "Springer",
number = "8",

}

RIS

TY - JOUR

T1 - Positive geometries and differential forms with non-logarithmic singularities. Part I

AU - Benincasa, Paolo

AU - Parisi, Matteo

PY - 2020/8/5

Y1 - 2020/8/5

N2 - 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.

AB - 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.

KW - Differential and Algebraic Geometry

KW - Scattering Amplitudes

U2 - 10.1007/JHEP08(2020)023

DO - 10.1007/JHEP08(2020)023

M3 - Journal article

VL - 2020

JO - Journal of High Energy Physics (Online)

JF - Journal of High Energy Physics (Online)

SN - 1126-6708

IS - 8

M1 - 23

ER -

ID: 247982599