Spacetimes categories and disjointness for algebraic quantum field theory

TL;DR

This paper introduces disjointness relations in categories to better define the domain of algebraic quantum field theories, extending to chiral conformal field theories and ensuring compatibility with existing formulations.

Contribution

It generalizes orthogonality relations to disjointness relations, identifying suitable subcategories for AQFT and chiral CFTs, and compares new frameworks with established formulations.

Findings

Disjointness relations help define suitable categories for AQFT.

The subcategory $ extsf{D}_ extsf{C}$ aligns with $ extsf{Loc}_{d+1}$ for hyperbolic spacetimes.

Chiral CFTs can be formulated within the new categorical framework.

Abstract

An algebraic quantum field theory (AQFT) may be expressed as a functor from a category of spacetimes to a category of algebras of observables. However, a generic category $\mathsf{C}$ whose objects admit interpretation as spacetimes is not necessarily viable as the domain of an AQFT functor; often, additional constraints on the morphisms of $\mathsf{C}$ must be imposed. We introduce disjointness relations, a generalisation of the orthogonality relations of Benini, Schenkel and Woike (arXiv:1709.08657). In any category $\mathsf{C}$ equipped with a disjointness relation, we identify a subcategory $\mathsf{D}_\mathsf{C}$ which is suitable as the domain of an AQFT. We verify that when $\mathsf{C}$ is the category of all globally hyperbolic spacetimes of dimension $d+1$ and all local isometries, equipped with the disjointness relation of spacelike separation, the specified subcategory $\mathsf{D}_\mathsf{C}$ is the commonly-used domain $\mathsf{Loc}_{d+1}$ of relativistic AQFTs. By identifying appropriate chiral disjointness relations, we construct a category $\chi\mathsf{Loc}$ suitable as domain for chiral conformal field theories (CFTs) in two dimensions. We compare this to an established AQFT formulation of chiral CFTs, and show that any chiral CFT expressed in the established formulation induces one defined on $\chi\mathsf{Loc}$.