Homotopy theory of algebraic quantum field theories

• Published in: Letters in mathematical physics Vol. 109; no. 7; pp. 1487 - 1532
• Main Authors: Benini, Marco, Schenkel, Alexander, Woike, Lukas
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2019; Springer Nature B.V
• Subjects: Algebra; Complex Systems; Field theory; Gauge theory; Geometry; Group Theory and Generalizations; Homology; Homotopy theory; Mathematical and Computational Physics; Physics; Physics and Astronomy; Quantum field theory; Quantum theory; Theoretical; Gauge theory; 55U35; 18D50; Model categories; Homotopy algebras; stacks; BRST/BV formalism; 81Txx; algebras; Colored operads; Algebraic quantum field theory; 18G55
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-018-01151-x

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 ∞ -stacks, and (2) fiber-wise groupoid cohomology of a category fibered in groupoids with coefficients in a strict quantum field theory.