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Causally Disjoint Discs: Another $\mathbb{E}_n$-operad

Ryan Grady

TL;DR

This work defines the operad of causally disjoint discs $\mathcal{P}_{\mathsf{CD_n}}$ to encode causal structure in Lorentzian perturbative AQFT and shows it is homotopy equivalent to the $(n-1)$-disc operad $\mathcal{P}_{\mathsf{Disc_{n-1}}}$, hence algebras over $\mathcal{P}_{\mathsf{CD_n}}$ model $\mathbb{E}_{n-1}$-algebras up to homotopy. It constructs the prefactorization framework from Top-enriched orthogonal categories to colored operads and proves functoriality of algebra maps under orthogonal-category morphisms, then establishes an explicit geometric deformation retract via embeddings $\varepsilon_{n,k}$ to realize the equivalence. The work further extends the framework to causally disjoint causal diamonds, showing a unified homotopy equivalence among $\mathcal{P}_{\mathsf{CD_n}}$, $\mathcal{P}_{\mathsf{CDiam_n}}$, and $\mathcal{P}_{\mathsf{Disc_{n-1}}}$, implying that associated algebras reproduce $\mathbb{E}_{n-1}$-structures. Finally, it discusses connections to pAQFT observables through the forgetful map $\omega$ and sketches a potential Wick rotation interpretation in sufficiently topological theories.

Abstract

Motivated by (perturbative) quantum observables in Lorentzian signature we define a new operad: the operad of causally disjoint disks. In order to describe this operad we use the orthogonal categories of Benini, Schenkel, and Woike and the prefactorization functor of Benini, Carmona, Grant-Stuart, and Schenkel. Along the way we extend these constructions to the topological setting, i.e., (multi-)categories enriched over spaces.

Causally Disjoint Discs: Another $\mathbb{E}_n$-operad

TL;DR

This work defines the operad of causally disjoint discs to encode causal structure in Lorentzian perturbative AQFT and shows it is homotopy equivalent to the -disc operad , hence algebras over model -algebras up to homotopy. It constructs the prefactorization framework from Top-enriched orthogonal categories to colored operads and proves functoriality of algebra maps under orthogonal-category morphisms, then establishes an explicit geometric deformation retract via embeddings to realize the equivalence. The work further extends the framework to causally disjoint causal diamonds, showing a unified homotopy equivalence among , , and , implying that associated algebras reproduce -structures. Finally, it discusses connections to pAQFT observables through the forgetful map and sketches a potential Wick rotation interpretation in sufficiently topological theories.

Abstract

Motivated by (perturbative) quantum observables in Lorentzian signature we define a new operad: the operad of causally disjoint disks. In order to describe this operad we use the orthogonal categories of Benini, Schenkel, and Woike and the prefactorization functor of Benini, Carmona, Grant-Stuart, and Schenkel. Along the way we extend these constructions to the topological setting, i.e., (multi-)categories enriched over spaces.
Paper Structure (11 sections, 12 theorems, 8 equations, 3 figures)

This paper contains 11 sections, 12 theorems, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and conventions
  3. (Colored)-Operads
  4. Causal structure
  5. Orthogonal categories and prefactorization
  6. Orthogonal categories
  7. The functor of prefactorization
  8. Causally disjoint discs
  9. An equivalence between $\mathcal{P}_{\mathsf{CD_n}}$ and $\mathcal{P}_{\mathsf{Disc_{n-1}}}$
  10. Disjoint Causal Diamonds
  11. Examples

Key Result

Lemma 2.3

Causal disjointness defines a symmetric and transitive relation on the subsets of $\mathbb R^{1,n-1}$.

Figures (3)

  • Figure 1: $S$ and $T$ are causally disjoint subsets.
  • Figure 2: The subsets $S$ and $T$ are not causally disjoint.
  • Figure 3: The unit disc, its causal envelope, and the standard causal diamond $\Diamond$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 19 more