The Universe Fan

TL;DR

This paper introduces a new class of wavefunctions associated with lattices, connecting combinatorial structures like fans and polytopes to quantum cosmology, and generalizing existing models such as cosmological polytopes.

Contribution

It defines the universe fan as a Laplace transform of a polyhedral fan, linking combinatorial geometry with wavefunctions in quantum field theory and cosmology.

Findings

Universe fan encodes causality and positivity in lattice wavefunctions

Wavefunctions recover matroid amplitudes as residues

Refinements relate to blow-ups of matroid polytopes and cosmohedron generalizations

Abstract

The wavefunction of the universe, as studied in perturbative quantum field theory, is a rational function whose singularities and factorization properties encode a rich underlying combinatorial structure. We define and study a broad generalization of such wavefunctions that can be associated to any lattice. We obtain these wavefunctions as the Laplace transform of a polyhedral fan, the universe fan, whose cones are defined by positivity conditions reflecting a notion of causality in the lattice, and we describe its face lattice. In the matroid case, the universe fan projects to the nested set fan, and the wavefunctions we define recover the matroid amplitudes introduced by Lam as residues. Moreover, in the case relevant for physics, the positivity conditions give a novel way to study the wavefunction, and we show how it is related to the cosmological polytopes of Arkani-Hamed, Benincasa, Postnikov. Finally, we study refinements of the universe fan induced by piecewise linear (tropical) functions. The resulting subdivisions project to refinements of the nested set fan and correspond dually to blow-ups of matroid polytopes, generalizing the cosmohedron polytope.