Homotopy theory of algebraic quantum field theories

TL;DR

This paper develops a homotopy-theoretic framework for algebraic quantum field theories, incorporating gauge theory, operadic methods, and derived constructions, with applications to BRST/BV formalism and homotopy-coherent causality.

Contribution

It introduces a model structure on chain complex valued quantum field theories and extends algebraic QFT with homotopy-coherent causality concepts, providing new tools for gauge theories.

Findings

Established a canonical model structure for chain complex valued theories.

Provided a derived universal algebra construction relevant to BRST/BV formalism.

Presented examples of homotopy-coherent theories using $$-stacks and fibered groupoid cohomology.

Abstract

Motivated by gauge theory, we develop a general framework for chain complex valued algebraic quantum field theories. Building upon our recent operadic approach to this subject, we show that the category of such theories carries a canonical model structure and explain the important conceptual and also practical consequences of this result. As a concrete application we provide a derived version of Fredenhagen's universal algebra construction, which is relevant e.g. for the BRST/BV formalism. We further develop a homotopy theoretical generalization of algebraic quantum field theory with a particular focus on the homotopy-coherent Einstein causality axiom. We provide examples of such homotopy-coherent theories via (1) smooth normalized cochain algebras on $\infty$-stacks, and (2) fiber-wise groupoid cohomology of a category fibered in groupoids with coefficients in a strict quantum field theory.