We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of * -algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group, namely the group of vertical principal bundle automorphisms. Properties of our functor are investigated in detail and, similar to earlier works, it is found that due to topological obstructions the locality property of locally covariant quantum field theory is violated. Furthermore, we prove that, for Abelian structure groups containing a nontrivial compact factor, the gauge invariant Borchers-Uhlmann algebra of the vector dual of the bundle of connections is not separating on gauge equivalence classes of principal connections. We introduce a topological generalization of the concept of locally covariant quantum fields. As examples, we construct for the category of principal U (1)-bundles two natural transformations from singular homology functors to the quantum field theory functor that can be interpreted as the Chern class and the electric charge. In this case we also prove that the electric charges can be consistently set to zero, which yields another quantum field theory functor that satisfies all axioms of locally covariant quantum field theory.Keywords: locally covariant quantum field theory, quantum field theory on curved spacetimes, gauge theory on principal bundles MSC 2010: 81T20, 81T05, 81T13, 53Cxx

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U (1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.Keywords: Affine bundles, globally hyperbolic spacetimes, locally covariant quantum field theory, quantum field theory on curved spacetimes, microlocal spectrum condition MSC 2010: 81T20, 53C80, 58J45, 35LxxRemark 2.2. The linear part of an affine map f :Example 2.3. Every vector space V can be regarded as an affine space modeled on itself: Choose A = V (as sets) and define Φ via the abelian group structure + on V . In this case the affine endomorphisms are given by affine transformations, i.e. maps f :a linear map and b ∈ V . The linear part of this map is f V . Example 2.4. Consider a short exact sequence of vector spaces and linear maps 0 / / W 1 f / / W 2 g / / W 3 / / 0 . (2.2)We say that a linear map a : W 3 → W 2 is a splitting of this sequence, if g • a = id W 3 . Let us denote by A := a ∈ Hom R (W 3 , W 2 ) : g • a = id W 3 ⊆ Hom R (W 3 , W 2 ) the set of all splittings and by V := Hom R (W 3 , W 1 ) the vector space of linear maps from W 3 to W 1 . Consider the following mapThe map Φ(a, v) : W 3 → W 2 is linear and satisfies g • Φ(a, v) = id W 3 , for all a ∈ A and v ∈ V . Furthermore, Φ defines an action of the abelian group (V, +) on A, i.e. for all a ∈ A, Φ(a, 0) = a and, for all a ∈ A and v, wSince f is injective, the group action Φ is free. It is also transitive: Let a, a ′ ∈ A be arbitrary, then a ′ − a : W 3 → Ker(g) ⊆ W 2 is a linear map with values in the kernel of the map g : W 2 → W 3 . Since the sequence is exact, we have that Im(f ) = Ker(g) and that f :Hence, the triple (A, V, Φ) is an affine space.Remark 2.5. Let (A, V, Φ) be an affine space. Since V is a finite-dimensional vector space, it comes with a canonical topology induced from an Euclidean norm and with a canonical C ∞structure induced by R n . Fixing any element a ∈ A we obtain a bijection of sets Φ a := Φ(a, · ) :

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.