Liam Solus: Ehrhart Theory for Cosmological Polytopes

ABSTRACT: Arkani-Hamed, Benincasa and Postnikov defined a cosmological polytope associated to a Feynman diagram in their study of the wavefunctions associated to certain cosmological models. By computing the canonical form of this polytope, one computes the contribution of the Feynman diagram to the wavefunction of interest. The theory of positive geometries tells us that one way to compute this canonical form is as a sum of the canonical forms of the facets of a subdivision of the polytope. For simple examples of Feynman diagrams, it is known that specific triangulations correspond to classical physical theories for these computations, but a general theory of triangulations of cosmological polytopes was left as future work. In this talk, we will discuss an algebraic approach to this theory based on Gröbner bases. We will see that every cosmological polytope admits families of regular unimodular triangulations providing one method for computing the canonical form of an arbitrary Feynman diagram. In general, the presentation of these triangulations is in terms of their non-faces. In special cases, we can characterize the facets of these triangulations for certain families of Feynman diagrams, including trees, cycles and more recently certain non-simple graphs. In addition to providing possible new physical theories for the computations of the associated wavefunctions, these results also allow us to recover combinatorial information about these polytopes, such as normalized volumes, extending some recent results of Kühne and Monin. In ongoing work, we have more recently computed formulas for the h*-polynomials of some cosmological polytopes, revealing in special cases desirable properties such as real-rootedness. This talk is based on joint work with Martina Juhnke-Kubitzke and Lorenzo Venturello as well as ongoing work with Justus Bruckamp, Martina Juhnke-Kubitzke and Erik Landin.