Homotopical Quantum Field Theory

TL;DR

This paper develops a homotopy-theoretic framework for algebraic quantum field theory and prefactorization algebras, introducing new structures and comparison tools to unify these approaches.

Contribution

It defines homotopy algebraic quantum field theories and prefactorization algebras using a novel coend construction, and establishes a comparison between these two frameworks.

Findings

Homotopy algebraic quantum field theories form homotopy coherent diagrams of A-infinity-algebras.

Homotopy prefactorization algebras are structured as homotopy coherent diagrams of E-infinity-modules.

A comparison morphism between the operads for these theories is constructed and analyzed.

Abstract

Algebraic quantum field theory and prefactorization algebra are two mathematical approaches to quantum field theory. In this monograph, using a new coend definition of the Boardman-Vogt construction of a colored operad, we define homotopy algebraic quantum field theories and homotopy prefactorization algebras and investigate their homotopy coherent structures. Homotopy coherent diagrams, homotopy inverses, A-infinity-algebras, E-infinity-algebras, and E-infinity-modules arise naturally in this context. In particular, each homotopy algebraic quantum field theory has the structure of a homotopy coherent diagram of A-infinity-algebras and satisfies a homotopy coherent version of the causality axiom. When the time-slice axiom is defined for algebraic quantum field theory, a homotopy coherent version of the time-slice axiom is satisfied by each homotopy algebraic quantum field theory. Over each topological space, every homotopy prefactorization algebra has the structure of a homotopy coherent diagram of E-infinity-modules over an E-infinity-algebra. To compare the two approaches, we construct a comparison morphism from the colored operad for (homotopy) prefactorization algebras to the colored operad for (homotopy) algebraic quantum field theories and study the induced adjunctions on algebras.