arXiv subjects

Alexander Schenkel

Publications and source records attributed to Alexander Schenkel.

At least 55 records · Page 3Linked to original sources

arXiv2019

Model-independent comparison between factorization algebras and algebraic quantum field theory on Lorentzian manifolds

This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses, including suitable variants of the local constancy and descent axioms. The main result is an equivalence theorem between (Cauchy constant and additive) algebraic quantum field theories and (Cauchy constant, additive and time-orderable) prefactorization algebras.

math-ph↗

arXiv2019

Higher Structures in Algebraic Quantum Field Theory

A brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.

hep-th↗

arXiv2019

Involutive categories, colored $\ast$-operads and quantum field theory

Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field) theory, we develop the involutive analogs of colored operads and their algebras, named colored $\ast$-operads and $\ast$-algebras. Central to the definition of colored $\ast$-operads is the involutive monoidal category of symmetric sequences, which we obtain from a general product-exponential $2$-adjunction whose right adjoint forms involutive functor categories. For $\ast$-algebras over $\ast$-operads we obtain involutive analogs of the usual change of color and operad adjunctions. As an application, we turn the colored operads for algebraic quantum field theory into colored $\ast$-operads. The simplest instance is the associative $\ast$-operad, whose $\ast$-algebras are unital and associative $\ast$-algebras.

math.CT↗

arXiv2019

Homotopy theory of algebraic quantum field theories

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.

math-ph↗

arXiv2018

Algebraic quantum field theory on spacetimes with timelike boundary

We analyze quantum field theories on spacetimes $M$ with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime $M$ of theories defined only on the interior $\mathrm{int}M$. The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay's F-locality property. Our main result is the following characterization theorem: Every quantum field theory on $M$ that is additive from the interior (i.e.\ generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior $\mathrm{int}M$ and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein-Gordon field.

math-ph↗

arXiv2018

The stack of Yang-Mills fields on Lorentzian manifolds

We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang-Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang-Mills Cauchy problem and show that its well-posedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in [S. Hollander, Israel J. Math. 163, 93-124 (2008)], which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as $\mathrm{B}G_\mathrm{con}$.

math-ph↗

arXiv2017

Quantum field theories on categories fibered in groupoids

We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.

math-ph↗

arXiv2017

Mapping spaces and automorphism groups of toric noncommutative spaces

We develop a sheaf theory approach to toric noncommutative geometry which allows us to formalize the concept of mapping spaces between two toric noncommutative spaces. As an application we study the `internalized' automorphism group of a toric noncommutative space and show that its Lie algebra has an elementary description in terms of braided derivations.

math.QA↗

arXiv2017

Global anomalies on Lorentzian space-times

We formulate an algebraic criterion for the presence of global anomalies on globally hyperbolic space-times in the framework of locally covariant field theory. We discuss some consequences and check that it reproduces the well-known global $SU(2)$ anomaly in four space-time dimensions.

hep-th↗

arXiv2017

Wavefront sets and polarizations on supermanifolds

In this paper we develop the foundations for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super wavefront set for superdistributions which generalizes Dencker's polarization sets for vector-valued distributions to supergeometry. In particular, our super wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory.

math-ph↗

arXiv2017

Non-existence of natural states for Abelian Chern-Simons theory

We give an elementary proof that Abelian Chern-Simons theory, described as a functor from oriented surfaces to C*-algebras, does not admit a natural state. Non-existence of natural states is thus not only a phenomenon of quantum field theories on Lorentzian manifolds, but also of topological quantum field theories formulated in the algebraic approach.

math-ph↗

arXiv2016

Differential cohomology and locally covariant quantum field theory

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the CCR-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of C*-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet-Lie group structure on differential cohomology groups.

hep-th↗

arXiv2016

Noncommutative principal bundles through twist deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

math.QA↗

arXiv2016

Poisson algebras for non-linear field theories in the Cahiers topos

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

math-ph↗

arXiv2016

Abelian duality on globally hyperbolic spacetimes

We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of $C^*$-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.

hep-th↗

arXiv2016

Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.

math.QA↗

arXiv2016

Working with Nonassociative Geometry and Field Theory

We review aspects of our formalism for differential geometry on noncommutative and nonassociative spaces which arise from cochain twist deformation quantization of manifolds. We work in the simplest setting of trivial vector bundles and flush out the details of our approach providing explicit expressions for all bimodule operations, and for connections and curvature. As applications, we describe the constructions of physically viable action functionals for Yang-Mills theory and Einstein-Cartan gravity on noncommutative and nonassociative spaces, as first steps towards more elaborate models relevant to non-geometric flux deformations of geometry in closed string theory.

hep-th↗

arXiv2015

Supergeometry in locally covariant quantum field theory

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions.

math-ph↗