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Alexander Schenkel

Publications and source records attributed to Alexander Schenkel.

At least 19 recordsLinked to original sources
arXiv2026

Computational homological methods for integrable field theories

We develop explicit computational tools for the recent homological approach to the construction of $2$-dimensional integrable field theories on $\Sigma$ from $4$-dimensional semi-holomorphic Chern-Simons theory on $\Sigma \times C$. In this framework, the operation of integrating out the spectral curve $C$ is realized by homotopy transfer of a cyclic $L_\infty$-algebra associated with the $4$-dimensional theory with prescribed singularities and boundary conditions. We construct explicit strong deformation retracts for divisor-twisted Dolbeault complexes on $C=\mathbb{C}P^1$ and use them to make the transferred $L_\infty$-structure computationally accessible. As an application, we study the choice of meromorphic $1$-form corresponding to the principal chiral model with a Wess-Zumino term. We compute the transferred Maurer-Cartan action and the associated Lax connection, showing that the former resums to the standard principal chiral model action with a Wess-Zumino term and that the latter reproduces the usual Lax connection.

hep-th↗
arXiv2026

On the structure of higher-dimensional integrable field theories

We propose a general framework for integrable field theories in arbitrary spacetime dimension $d+1$ which is based on $d$-term $L_\infty$-algebras. Specifically, we introduce cyclic $L_\infty$-algebras describing topological-holomorphic higher Chern-Simons theories on $M \times \mathbb{C}P^1$ with suitable singularity structures and boundary conditions, controlled by a meromorphic $1$-form on $\mathbb{C}P^1$. Using homological perturbation theory and homotopy transfer, we construct weakly equivalent models describing $(d+1)$-dimensional field theories on $M$. Their integrability is witnessed by a natural map to an $L_\infty$-algebra describing higher Lax connections, yielding conserved charges associated with higher-dimensional cycles in $M$. The resulting theories admit natural action functionals and recover the Costello-Yamazaki construction in $2$ dimensions.

hep-th↗
arXiv2026

Prefactorization algebras of superselection sectors

This paper revisits the theory of superselection sectors in algebraic quantum field theory from the modern perspective of prefactorization algebras. Under the standard assumptions of Haag duality and a locally faithful vacuum representation, it is shown that every AQFT defined over a filtered orthogonal category of spacetime regions, satisfying some mild additional geometric hypotheses, has an associated locally constant $C^\ast$-categorical prefactorization algebra of superselection sectors over the same orthogonal category. In the case of double cones in the $(n\geq 2)$-dimensional Minkowski spacetime, our approach provides a conceptual explanation for the well-known $\mathbb{E}_n$-monoidal structure on the $C^\ast$-category of superselection sectors as the combination, through Dunn-Lurie additivity $\mathbb{E}_n\simeq \mathbb{E}_1\otimes \mathbb{E}_{n-1}$, of the familiar $\mathbb{E}_1$-monoidal structure from Haag duality and an $\mathbb{E}_{n-1}$-monoidal structure from Lorentzian geometry. A refinement of our results to equivariant contexts under a discrete group $G$ is also provided.

math-ph↗
arXiv2026

The homological algebra of 2d integrable field theories

This article provides a detailed and rigorous study of $4d$ semi-holomorphic Chern-Simons theories and their associated $2d$ integrable field theories from the homological perspective of $L_\infty$-algebras. Through the use of homotopy transfer techniques, it is shown precisely how both the integrable field theory and its corresponding Lax connection emerge from the $4d$ theory, which results in a novel perspective on Lax connections in terms of $L_\infty$-morphisms.

hep-th↗
arXiv2025

$C^\ast$-categorical prefactorization algebras for superselection sectors and topological order

This paper presents a conceptual and efficient geometric framework to encode the algebraic structures on the category of superselection sectors of an algebraic quantum field theory on the $n$-dimensional lattice $\mathbb{Z}^n$. It is shown that, under the typical assumption of Haag duality, the monoidal $C^\ast$-categories of localized superselection sectors carry the structure of a locally constant prefactorization algebra over the category of cone-shaped subsets of $\mathbb{Z}^n$. Employing techniques from higher algebra, one extracts from this datum an underlying locally constant prefactorization algebra defined on open disks in the cylinder $\mathbb{R}^1\times\mathbb{S}^{n-1}$. While the sphere $\mathbb{S}^{n-1}$ arises geometrically as the angular coordinates of cones, the origin of the line $\mathbb{R}^1$ is analytic and rooted in Haag duality. The usual braided (for $n=2$) or symmetric (for $n\geq 3$) monoidal $C^\ast$-categories of superselection sectors are recovered by removing a point of the sphere $\mathbb{R}^1\times(\mathbb{S}^{n-1}\setminus\mathrm{pt}) \cong\mathbb{R}^n$ and using the equivalence between $\mathbb{E}_n$-algebras and locally constant prefactorization algebras defined on open disks in $\mathbb{R}^n$. The non-trivial homotopy groups of spheres induce additional algebraic structures on these $\mathbb{E}_n$-monoidal $C^\ast$-categories, which in the case of $\mathbb{Z}^2$ is given by a braided monoidal self-equivalence arising geometrically as a kind of `holonomy' around the circle $\mathbb{S}^1$. The locally constant prefactorization algebra structures discovered in this work generalize, under some mild geometric conditions, to other discrete spaces and thereby provide a clear link between the geometry of the localization regions and the algebraic structures on the category of superselection sectors.

math-ph↗
arXiv2025

An equivalence theorem for algebraic and functorial QFT

This paper develops a novel approach to functorial quantum field theories (FQFTs) in the context of Lorentzian geometry. The key challenge is that globally hyperbolic Lorentzian bordisms between two Cauchy surfaces cannot change the topology of the Cauchy surface. This is addressed and solved by introducing a more flexible concept of bordisms which provide morphisms from tuples of causally disjoint partial Cauchy surfaces to a later-in-time full Cauchy surface. They assemble into a globally hyperbolic Lorentzian bordism pseudo-operad, generalizing the geometric bordism pseudo-categories of Stolz and Teichner. The associated FQFTs are defined as pseudo-multifunctors into a symmetric monoidal category of unital associative algebras. The main result of this paper is an equivalence theorem between such globally hyperbolic Lorentzian FQFTs and algebraic quantum field theories (AQFTs), both subject to the time-slice axiom and a mild descent condition called additivity.

math-ph↗
arXiv2024

Shifted Poisson structures on higher Chevalley-Eilenberg algebras

This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'.

math.QA↗
arXiv2024

On the equivalence of AQFTs and prefactorization algebras

This paper revisits the equivalence problem between algebraic quantum field theories and prefactorization algebras defined over globally hyperbolic Lorentzian manifolds. We develop a radically new approach whose main innovative features are 1.) a structural implementation of the additivity property used in earlier approaches and 2.) a reduction of the global equivalence problem to a family of simpler spacetime-wise problems. When applied to the case where the target category is a symmetric monoidal $1$-category, this yields a generalization of the equivalence theorem from [Commun. Math. Phys. 377, 971 (2019)]. In the case where the target is the symmetric monoidal $\infty$-category of cochain complexes, we obtain a reduction of the global $\infty$-categorical equivalence problem to simpler, but still challenging, spacetime-wise problems. The latter would be solved by showing that certain functors between $1$-categories exhibit $\infty$-localizations, however the available detection criteria are inconclusive in our case.

math-ph↗
arXiv2024

Derived algebraic geometry of 2d lattice Yang-Mills theory

A derived algebraic geometric study of classical $\mathrm{GL}_n$-Yang-Mills theory on the $2$-dimensional square lattice $\mathbb{Z}^2$ is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets $V =[a,b]\times [c,d]\subseteq \mathbb{Z}^2$ with both sides of length $\geq 2$ is extracted. A locally constant dg-category-valued prefactorization algebra on $\mathbb{Z}^2$ is constructed from the dg-categories of quasi-coherent complexes on the derived stacks of local data.

math-ph↗
arXiv2024

Infinitesimal 2-braidings from 2-shifted Poisson structures

It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $\hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, To\"en, Vaqui\'e and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a Lie $N$-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.

math.QA↗
arXiv2024

5d 2-Chern-Simons theory and 3d integrable field theories

The $4$-dimensional semi-holomorphic Chern-Simons theory of Costello and Yamazaki provides a gauge-theoretic origin for the Lax connection of $2$-dimensional integrable field theories. The purpose of this paper is to extend this framework to the setting of $3$-dimensional integrable field theories by considering a $5$-dimensional semi-holomorphic higher Chern-Simons theory for a higher connection $(A,B)$ on $\mathbb{R}^3 \times \mathbb{C}P^1$. The input data for this theory are the choice of a meromorphic $1$-form $\omega$ on $\mathbb{C}P^1$ and a strict Lie $2$-group with cyclic structure on its underlying Lie $2$-algebra. Integrable field theories on $\mathbb{R}^3$ are constructed by imposing suitable boundary conditions on the connection $(A,B)$ at the $3$-dimensional defects located at the poles of $\omega$ and choosing certain admissible meromorphic solutions of the bulk equations of motion. The latter provides a natural notion of higher Lax connection for $3$-dimensional integrable field theories, including a $2$-form component $B$ which can be integrated over Cauchy surfaces to produce conserved charges. As a first application of this approach, we show how to construct a generalization of Ward's $(2+1)$-dimensional integrable chiral model from a suitable choice of data in the $5$-dimensional theory.

hep-th↗
arXiv2024

Haag-Kastler stacks

This paper provides an alternative implementation of the principle of general local covariance for algebraic quantum field theories (AQFTs) which is more flexible than the original one by Brunetti, Fredenhagen and Verch. This is realized by considering the $2$-functor $\mathsf{HK} : \mathbf{Loc}^\mathrm{op} \to \mathbf{CAT}$ which assigns to each Lorentzian manifold $M$ the category $\mathsf{HK}(M)$ of Haag-Kastler-style AQFTs over $M$ and to each embedding $f:M\to N$ a pullback functor $f^\ast = \mathsf{HK}(f) : \mathsf{HK}(N) \to \mathsf{HK}(M)$ restricting theories from $N$ to $M$. Locally covariant AQFTs are recovered as the points of the $2$-functor $\mathsf{HK}$. The main advantages of this new perspective are: 1.) It leads to technical simplifications, in particular with regard to the time-slice axiom, since global problems on $\mathbf{Loc}$ become families of simpler local problems on individual Lorentzian manifolds. 2.) Some aspects of the Haag-Kastler framework which previously got lost in locally covariant AQFT, such as a relative compactness condition on the open subsets in a Lorentzian manifold $M$, are reintroduced. 3.) It provides a radically new perspective on descent conditions in AQFT, i.e. local-to-global conditions which allow one to recover a global AQFT on a Lorentzian manifold $M$ from its local data in an open cover $\{U_i \subseteq M\}$.

math-ph↗
arXiv2023

Obstructions to the existence of M{\o}ller maps

M{\o}ller maps are identifications between the observables of a perturbatively interacting physical system and the observables of its underlying free (i.e. non-interacting) system. This work studies and characterizes obstructions to the existence of such identifications. The main results are existence and importantly also non-existence theorems, which in particular imply that M{\o}ller maps do not exist for non-Abelian Chern-Simons and Yang-Mills theories on globally hyperbolic Lorentzian manifolds. These results are obtained through homological algebra techniques which are of independent interest in the analysis of classical field theories.

math-ph↗
arXiv2023

Lorentzian bordisms in algebraic quantum field theory

It is shown that every algebraic quantum field theory has an underlying functorial field theory which is defined on a suitable globally hyperbolic Lorentzian bordism pseudo-category. This means that globally hyperbolic Lorentzian bordisms between Cauchy surfaces arise naturally in the context of algebraic quantum field theory. The underlying functorial field theory encodes the time evolution of the original theory, but not its spatially local structure. As an illustrative application of these results, the algebraic and functorial descriptions of a free scalar quantum field are compared in detail.

math-ph↗
arXiv2023

Green hyperbolic complexes on Lorentzian manifolds

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 retarded and advanced Green's homotopies, which are shown to be unique up to a contractible space of choices. We prove homological generalizations of the most relevant features of Green hyperbolic operators, namely that (1) the retarded-minus-advanced cochain map is a quasi-isomorphism, (2) a differential pairing (generalizing the usual fiber-wise metric) on a Green hyperbolic complex leads to covariant and fixed-time Poisson structures and (3) the retarded-minus-advanced cochain map is compatible with these Poisson structures up to homotopy.

math-ph↗
arXiv2023

Universal first-order Massey product of a prefactorization algebra

This paper studies the universal first-order Massey product of a prefactorization algebra, which encodes higher algebraic operations on the cohomology. Explicit computations of these structures are carried out in the locally constant case, with applications to factorization envelopes on $\mathbb{R}^m$ and a compactification of linear Chern-Simons theory on $\mathbb{R}^2\times \mathbb{S}^1$.

math-ph↗
arXiv2023

The linear CS/WZW bulk/boundary system in AQFT

This paper constructs in the framework of algebraic quantum field theory (AQFT) the linear Chern-Simons/Wess-Zumino-Witten system on a class of $3$-manifolds $M$ whose boundary $\partial M$ is endowed with a Lorentzian metric. It is proven that this AQFT is equivalent to a dimensionally reduced AQFT on a $2$-dimensional manifold $B$, whose restriction to the $1$-dimensional boundary $\partial B$ is weakly equivalent to a chiral free boson.

math-ph↗