Homotopy theory of algebraic quantum field theories

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

Higher Structures in Algebraic Quantum Field Theory

“…The second point will be discussed in detail in Section 4. For details on the material presented below we refer to []. A generalization to other types of field theories (e.g.…”

Coloring Operads for Algebraic Field Theory

“…Theories on that are determined by the extension of their restriction are the j ‐local theories; these are exactly the theories that live in the essential image of . As was shown in [], is a generalization of Fredenhagen's universal algebra construction …”

“…In this section we take some first steps to refine the results from the previous sections to a model categorical setting. Homotopy AQFT has earlier been studied in [] and []. We will mostly give a sketch of our results, again referring to [] for details.…”

“…The crucial step is to construct an operad that encodes the Einstein causality property. To do this, we follow the strategy set out in . We generalize the operads constructed therein, allowing for more general field theories that are described by other algebras.…”

“…In these proceedings we will not consider *‐structures, or more generally involutive categories. For a treatment of these in the operadic framework we refer to …”