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Binary geometries from pellytopes

Lara Bossinger, Máté L. Telek, Hannah Tillmann-Morris

Abstract

Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell's numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron.

Binary geometries from pellytopes

Abstract

Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell's numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron.

Paper Structure

This paper contains 7 sections, 6 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Binary geometries
  3. Pellytopes
  4. Pellspace
  5. The character lattice and bounded characters
  6. The $u$-equations for the Pellspace
  7. Relationship between the Pellspace and $\widetilde{\mathcal{M}}_{0,n}$

Key Result

lemma 1

Let $\Delta$ be a pure flag complex on $[n]$, and let $\widetilde{U}$ be an affine algebraic variety $\widetilde{U} \subset \mathbb{C}^n$ cut out by $n$ equations of the form as in eq:u-equations. For $S \in \Delta$, the subvariety $\widetilde{U}_S = \widetilde{U} \cap \{ u \in \mathbb{C}^n \mid u_k Moreover, the following hold:

Figures (5)

  • Figure 1: A square and its inner normal fan, which is isomorphic to the flag complex in Example \ref{['Ex:Sec2:Run1']}(b).
  • Figure 2: (a) The pellytope $\mathcal{P}_2$. (b) The inner normal fan of $\mathcal{P}_2$, which is the common refinement of the inner normal fans of $P_1,\, P_2$ and $Q_1$.
  • Figure : (a) $F \subset \mathcal{A}_3$ and its adjacent faces.
  • Figure : (a) $F \subset \mathcal{A}_3$ and its adjacent faces.
  • Figure : (b) $E \subset \mathcal{P}_3$ and its adjacent faces.

Theorems & Definitions (16)

  • lemma 1
  • proof
  • proof : Proof of Proposition \ref{['Prop:PellyFan']}
  • lemma 2
  • proof
  • lemma 3
  • proof
  • lemma 4
  • proof
  • proof
  • ...and 6 more