# Homotopy theory of algebraic quantum field theories

May 22, 2018

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.

Keywords:

algebraic quantum field theory, Gauge theory, BRST/BV formalism, model categories, colored operads, homotopy algebras, E_infinity-algebras, stacks