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Triangulations of cosmological polytopes
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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 Gr\"obner 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 K\"uhne and Monin.
Forward citations
Cited by 2 Pith papers
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Euler Discriminant of Complements of Hyperplanes
The Euler discriminant of families of hyperplane complements is the zero set of an explicit product of determinants indexed by connected square subgraphs.
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Ternary relations and their polytopes
Cosmological polytopes and Graev polytopes are unified through ternary relations on 2-dimensional simplicial posets, yielding new facet families for some root-system polytopes.
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