Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Universe Fan

Hadleigh Frost, Felix Lotter

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.

The Universe Fan

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.
Paper Structure (29 sections, 35 theorems, 156 equations, 16 figures)

This paper contains 29 sections, 35 theorems, 156 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Nested sets, nestable sets, and causal regions
  3. Causal paths and properties of $\Psi$
  4. Refinements and polytopes
  5. Lattices and Amplitudes
  6. Building sets and Nestoids
  7. Matroid amplitudes
  8. Factorization
  9. The Universe Fan and Wavefunctions
  10. The universe fan
  11. Links and factorization
  12. The maximal cones and cosmological polytopes
  13. The cones of the universe fan
  14. Trees, tubes, and boolean building sets
  15. Cuts of $T_N$ and the canonical projection
  16. ...and 14 more sections

Key Result

Theorem 1

The residues of $\Psi(L,\mathcal{ G })$ on its poles are given by for two nestable sets $\mathcal{ G }_1 \subset [\hat{0},r_*]$ and $\mathcal{ G }_2 \subset L$, defined by $R$. Here, $A^\text{sh}([\hat{0},r_*],\mathcal{ G }_1)$ is the amplitude $A([\hat{0},r_*],\mathcal{ G }_1)$ after a redefinition of variables. As an important special case, $\Psi(L,\mathcal{ G }

Figures (16)

  • Figure 1: The star graph (A), and the associated building set (B) given by the graph's connected subgraphs, where $\hat{1} = 1234$ denotes the whole graph.
  • Figure 2: A nested set $\{b,ab,\hat{1}\}$ of the building set $\mathcal{ G }$ (B) corresponds to a tubing of the graph (A).
  • Figure 3: The generators of the universe fan $\mathcal{ U }_\mathcal{ G }$ correspond to regions cut out by tubes on the graph. (A), (B), (C) show the regions for $ab^+$, $ab^+ + b^-$, and $\hat{1}^+ + b^-$, respectively.
  • Figure 4: The poset $\mathcal{G}$ for the bowtie graph.
  • Figure 5: The three different types of Hasse diagrams for nested sets associated to the bowtie graph
  • ...and 11 more figures

Theorems & Definitions (138)

  • Example 1.1
  • Example 1.2
  • Definition : Definition \ref{['dfn:UG']}
  • Theorem : Theorem \ref{['thm:universe-links']}, Corollary \ref{['cor:wavefunction-res']}, Corollary \ref{['cor:tot-energy-res']}
  • Theorem : \ref{['lem:induced-refinement']}, \ref{['prop:coolbijection']}, \ref{['cor:local-nest-refinement']}
  • Definition 2.1: Building sets
  • Proposition 2.2: backman2024
  • Definition 2.3: Nestable sets
  • Definition 2.4: Nested sets
  • Definition 2.5
  • ...and 128 more