# Alexander SchenkelUniversity of Nottingham | Notts · School of Mathematical Sciences

Alexander Schenkel

Dr.

## About

80

Publications

6,076

Reads

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821

Citations

Introduction

For my current activities, see my webpage http://www.aschenkel.eu

Additional affiliations

August 2019 - present

October 2016 - present

September 2016 - July 2019

Education

October 2003 - May 2008

## Publications

Publications (80)

This paper develops a concept of relative Cauchy evolution for the class of homotopy algebraic quantum field theories (AQFTs) that are obtained by canonical commutation relation quantization of Poisson chain complexes. The key element of the construction is a rectification theorem proving that the homotopy time-slice axiom, which is a higher catego...

A simple model for the localization of the category $\mathbf{CLoc}_2$ of oriented and time-oriented globally hyperbolic conformal Lorentzian $2$-manifolds at all Cauchy morphisms is constructed. This provides an equivalent description of $2$-dimensional conformal algebraic quantum field theories (AQFTs) satisfying the time-slice axiom in terms of o...

We study the quantization of the canonical unshifted Poisson structure on the derived cotangent stack $T^\ast[X/G]$ of a quotient stack, where $X$ is a smooth affine scheme with an action of a (reductive) smooth affine group scheme $G$. This is achieved through an \'etale resolution of $T^\ast[X/G]$ by stacky CDGAs that allows for an explicit descr...

We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define Green hyperbolic complexes through a generalization of retarded and advanced Green's operators, called retard...

It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag-Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime...

This paper develops a concept of relative Cauchy evolution for the class of homotopy algebraic quantum field theories (AQFTs) that are obtained by canonical commutation relation quantization of Poisson chain complexes. The key element of the construction is a rectification theorem proving that the homotopy time-slice axiom, which is a higher catego...

This paper provides a systematic study of gauge symmetries in the dynamical fuzzy spectral triple models for quantum gravity that have been proposed by Barrett and collaborators. We develop both the classical and the perturbative quantum BV formalism for these models, which in particular leads to an explicit homological construction of the perturba...

This paper provides a detailed study of 4-dimensional Chern-Simons theory on R2×CP1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2\times \mathbb {C}P^1$$\...

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, l...

We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite...

We study the derived critical locus of a function [Formula: see text] on the quotient stack of a smooth affine scheme [Formula: see text] by the action of a smooth affine group scheme [Formula: see text]. It is shown that [Formula: see text] is a derived quotient stack for a derived affine scheme [Formula: see text], whose dg-algebra of functions i...

We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite...

We study the derived critical locus of a function $f:[X/G]\to \mathbb{A}_{\mathbb{K}}^1$ on the quotient stack of a smooth affine scheme $X$ by the action of a smooth affine group scheme $G$. It is shown that $\mathrm{dCrit}(f) \simeq [Z/G]$ is a derived quotient stack for a derived affine scheme $Z$, whose dg-algebra of functions is described expl...

This paper develops a concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructe...

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, l...

This paper provides a detailed study of $4$-dimensional Chern-Simons theory on $\mathbb{R}^2 \times \mathbb{C}P^1$ for an arbitrary meromorphic $1$-form $\omega$ on $\mathbb{C}P^1$. Using techniques from homotopy theory, the behaviour under finite gauge transformations of a suitably regularised version of the action proposed by Costello and Yamazak...

It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein–Gordon and linear Yang–Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green’s operators). Quantization of the associated unshifted Poisson structure determines a unique (up...

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 desc...

We provide an elegant homological construction of the extended phase space for linear Yang–Mills theory on an oriented and time-oriented Lorentzian manifold M with a time-like boundary \(\partial M\) that was proposed by Donnelly and Freidel (JHEP 1609:102, 2016). This explains and formalizes many of the rather ad hoc constructions for edge modes a...

This paper studies geometric structures on noncommutative hypersurfaces within a module-theoretic approach to noncommutative Riemannian (spin) geometry. A construction to induce differential, Riemannian and spinorial structures from a noncommutative embedding space to a noncommutative hypersurface is developed and applied to obtain noncommutative h...

This paper develops a novel concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are cons...

We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we stu...

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold M with a time-like boundary ∂M that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing i...

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold $M$ with a time-like boundary $\partial M$ that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes...

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 conc...

It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green's operators). Quantization of the associated unshifted Poisson structure determines a unique (up...

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.

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 desc...

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.

The $\theta$-deformed Hopf fibration $\mathbb{S}^3_\theta\to \mathbb{S}^2$ over the commutative $2$-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter is...

The θ-deformed Hopf fibration S 3 θ → S 2 over the commutative 2-sphere is compared with its classical counterpart. It is shown that there exists a natural isomorphism between the corresponding associated module functors and that the affine spaces of classical and deformed connections are isomorphic. The latter isomorphism is equivariant under an a...

We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we stu...

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 conc...

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 conc...

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...

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...

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...

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...

We construct a colored operad whose category of algebras is canonically isomorphic to 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...

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 deriv...

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 approa...

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 wel...

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...

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...

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.

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 \({{\mathfrak{A}}}\) :...

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 formu...

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 connect...

By adapting the Cheeger-Simons approach to differential cohomology, we
establish a notion of differential cohomology with compact support. We show
that it is functorial with respect to open embeddings and that it fits into a
natural diagram of exact sequences which compare it to compactly supported
singular cohomology and differential forms with co...

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 Lei...

We study chain complexes of field configurations and observables for Abelian
gauge theory on contractible manifolds, and show that they can be extended to
non-contractible manifolds by using techniques from homotopy theory. The
extension prescription yields functors from a category of manifolds to suitable
categories of chain complexes. The extende...

We systematically study noncommutative and nonassociative algebras A and
their bimodules as algebras and bimodules internal to the representation
category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of
the monoidal category of A-bimodules by internal homomorphisms, and describe
explicitly their evaluation and composition morph...

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 coh...

We provide a detailed analysis of the classical and quantized theory of a
multiplet of inhomogeneous Klein-Gordon fields, which couple to the spacetime
metric and also to an external source term; thus the solutions form an affine
space. Following the formulation of affine field theories in terms of
presymplectic vector spaces as proposed in [Annale...

We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our op...

The aim of this work is to complete our program on the quantization of
connections on arbitrary principal U(1)-bundles over globally hyperbolic
Lorentzian manifolds. In particular, we show that one can assign via a
covariant functor to any such bundle an algebra of observables which separates
gauge equivalence classes of connections. The C*-algebra...

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...

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 pr...

The focus of this PhD thesis is on applications, new developments and
extensions of the noncommutative gravity theory proposed by Julius Wess and his
group.
In part one we propose an extension of the usual symmetry reduction procedure
to noncommutative gravity. We classify in the case of abelian Drinfel'd twists
all consistent deformations of spati...

Let H be a Hopf algebra, A a left H-module algebra and V a left H-module
A-bimodule. We study the behavior of the right A-linear endomorphisms of V
under twist deformation. We in particular construct a bijective quantization
map to the right A_\star-linear endomorphisms of V_\star, with A_\star,V_\star
denoting the usual twist deformations of A,V....

Given a Hopf algebra H, we study modules and bimodules over an algebra A that
carry an H-action, as well as their morphisms and connections. Bimodules
naturally arise when considering noncommutative analogues of tensor bundles.
For quasitriangular Hopf algebras and bimodules with an extra
quasi-commutativity property we induce connections on the te...

We develop a general setting for the quantization of linear bosonic and
fermionic field theories subject to local gauge invariance and show how
standard examples such as linearized Yang-Mills theory and linearized general
relativity fit into this framework. Our construction always leads to a
well-defined and gauge-invariant quantum field algebra, t...

In this article we define and investigate a notion of parallel transport on
finite projective modules over finite matrix algebras. Given a derivation-based
differential calculus on the algebra and a connection on the module, we
construct for every derivation X a module parallel transport, which is a lift
to the module of the one-parameter group of...

In this article we study the quantization and causal properties of a massive
spin 3/2 Rarita-Schwinger field on spatially flat Friedmann-Robertson-Walker
(FRW) spacetimes. We construct Zuckerman's universal conserved current and
prove that it leads to a positive definite inner product on solutions of the
field equation. Based on this inner product,...

We summarize our recently proposed approach to quantum field theory on
noncommutative curved spacetimes. We make use of the Drinfel'd twist deformed
differential geometry of Julius Wess and his group in order to define an action
functional for a real scalar field on a twist-deformed time-oriented, connected
and globally hyperbolic Lorentzian manifo...

We study the quantum field theory (QFT) of a free, real, massless and
curvature coupled scalar field on self-similar symmetric spacetimes, which are
deformed by an abelian Drinfel'd twist constructed from a Killing and a
homothetic Killing vector field. In contrast to deformations solely by Killing
vector fields, such as the Moyal-Weyl Minkowski sp...

We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particu...

We consider an interacting scalar quantum field theory on noncommutative Euclidean space. We implement a family of noncommutative deformations, which -- in contrast to the well known Moyal-Weyl deformation -- lead to a theory with modified kinetic term, while all local potentials are unaffected by the deformation. We show that our models, in partic...

We construct consistent noncommutative (NC) deformations of the Randall-Sundrum spacetime that solve the NC Einstein equations with a non-trivial Poisson tensor depending on the fifth coordinate. In a class of these deformations where the Poisson tensor is exponentially localized on one of the branes (the NC-brane), we study the effects on bulk par...

In this article we construct the quantum field theory of a free real scalar field on a class of noncommutative manifolds, obtained via deformation quantization using triangular Drinfel'd twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green's operators and the fundamental so...

We investigate the infrared (IR) effects of Lorentz violating terms in the gravitational sector using functional renormalization group methods similar to Reuter and collaborators. The model we consider consists of pure quantum gravity coupled to a preferred foliation, described effectively via a scalar field with non-standard dynamics. We find that...

We review the noncommutative gravity of Wess et al. and discuss its physical applications. We define noncommutative symmetry reduction and construct deformed symmetric solutions of the noncommutative Einstein equations. We apply our framework to find explicit deformed cosmological and black hole solutions and discuss their phenomenology. This artic...

We derive noncommutative Einstein equations for abelian twists and their solutions in consistently symmetry reduced sectors, corresponding to twisted FRW cosmology and Schwarzschild black holes. While some of these solutions must be rejected as models for physical spacetimes because they contradict observations, we find also solutions that can be m...

As a preparation for a mathematically consistent study of the physics of symmetric spacetimes in a noncommutative setting, we study symmetry reductions in deformed gravity. We focus on deformations that are given by a twist of a Lie algebra acting on the spacetime manifold. We derive conditions on those twists that allow a given symmetry reduction....

## Projects

Project (1)

Develop a framework for QFT based on higher algebraic structures and apply it to understand quantum gauge theories.