Homotopy theory of algebraic quantum field theories

Marco Benini, Alexander Schenkel, Lukas Woike
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.
open access link
@article{Benini:2018oeh, author = "Benini, Marco and Schenkel, Alexander and Woike, Lukas", title = "{Homotopy theory of algebraic quantum field theories}", year = "2018", eprint = "1805.08795", archivePrefix = "arXiv", primaryClass = "math-ph", reportNumber = "ZMP-HH/18-11, Hamburger Beitraege zur Mathematik Nr. 738, ZMP-HH-18-11, HAMBURGER-BEITRAEGE-ZUR-MATHEMATIK-NR.-738", SLACcitation = "%%CITATION = ARXIV:1805.08795;%%" }

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