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no. 4
Triangulations of cosmological polytopes
Juhnke, Martina1; Solus, Liam2; Venturello, Lorenzo3
1 Department of Mathematics University of Osnabrück Albrechtstraße 28a 49076 Osnabrück Germany
2 Department of Mathematics KTH Royal Institute of Technology SE-100 44 Stockholm Sweden
3 Department of Mathematics Università di Pisa L.go B. Pontecorvo 5 56127 Pisa Italy
Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 1141-1168.

A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. In this paper, we identify such formulas for the canonical form via algebraic techniques. It is shown that the toric ideal of every cosmological polytope admits a Gröbner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including trees and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. For paths and cycles, these characterizations admit simple enumeration. Hence, we obtain formulas for the normalized volume of these polytopes, extending previous observations of Kühne and Monin.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.430
Classification: 52B20, 05E45, 13F65
Keywords: cosmological polytopes, toric ideals, Gröbner basis

Juhnke, Martina 1; Solus, Liam 2; Venturello, Lorenzo 3

1 Department of Mathematics University of Osnabrück Albrechtstraße 28a 49076 Osnabrück Germany
2 Department of Mathematics KTH Royal Institute of Technology SE-100 44 Stockholm Sweden
3 Department of Mathematics Università di Pisa L.go B. Pontecorvo 5 56127 Pisa Italy
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Juhnke, Martina; Solus, Liam; Venturello, Lorenzo. Triangulations of cosmological polytopes. Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 1141-1168. doi : 10.5802/alco.430. https://alco.centre-mersenne.org/articles/10.5802/alco.430/

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