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.

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions, such as timeslicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen's universal algebra. Report no.: ZMP-HH/17-26, Hamburger Beiträge zur Mathematik Nr. 682Keywords: algebraic quantum field theory, locally covariant quantum field theory, colored operads, change of color adjunctions, Fredenhagen's universal algebra MSC 2010: 81Txx, 18D50

Based on a detailed definition of extended homotopy quantum field theories we develop a field-theoretic orbifold construction for these theories when the target space is the classifying space of a finite group G, i.e. for G-equivariant topological field theories. More precisely, we use a recently developed bicategorical version of the parallel section functor to associate to an extended equivariant topological field theory an ordinary extended topological field theory. One main motivation is the 3-2-1-dimensional case where our orbifold construction allows us to describe the orbifoldization of equivariant modular categories by a geometric construction. As an important ingredient of this result, we prove that a 3-2-1-dimensional G-equivariant topological field theory yields a G-multimodular category by evaluation on the circle. The orbifold construction is a special case of a pushforward operation along an arbitrary morphism of finite groups and provides a valuable tool for the construction of extended homotopy quantum field theories.

We prove that the parallel transport of a flat n − 1-gerbe on any given target space gives rise to an n-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent version of twisted and equivariant Dijkgraaf-Witten models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories giving us in the 3-2-1-dimensional case a new class of equivariant modular tensor categories which can be understood as twisted versions of the equivariant modular categories constructed by Maier, Nikolaus and Schweigert.

An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism G −→ H of finite groups assigns in a functorial way to a G-equivariant topological field theory an H-equivariant topological field theory, the pushforward theory. When H is the trivial group, this yields an orbifold construction for G-equivariant topological field theories which unifies and generalizes several known algebraic notions of orbifoldization.