arXiv subjects

Konrad Waldorf

Publications and source records attributed to Konrad Waldorf.

At least 19 recordsLinked to original sources

arXiv2026

Adjusted connections on non-abelian bundle gerbes

Higher gauge theory for non-abelian structure 2-groups faces significant challenges when extending beyond the fake-flat sector, which suffers from limited applicability in physical models. A promising resolution involves equipping 2-groups with additional structure, known as adjustments. We present a comprehensive theory of adjusted connections on non-abelian bundle gerbes, classified by Saemann's adjusted version of non-abelian differential cohomology. This theory enables, in particular, a new coordinate-independent formulation of Tellez-Dominguez' lifting theorem, establishing a correspondence between adjusted connections on non-abelian bundle gerbes and connections on abelian bundle 2-gerbes.

math.DG↗

arXiv2026

Real Twistings are 2-Line Bundles

We construct and study a bicategory of super 2-line bundles over graded Lie groupoids, providing a unified framework for geometric models of twistings of (real) K-theory. The core of our work is to exhibit a wide range of models from the literature as special cases, among them several variants of bundle gerbes (real/equivariant/Jandl), Freed-Moore's twisted groupoid extensions, Freed-Hopkins-Teleman's K-theory twistings, Moutuou's real twistings, Freed's invertible algebra bundles, and Distler-Freed-Moore's orientifold twistings.

math.AT↗

arXiv2026

Categorical symmetries of T-duality

Topological T-duality correspondences are higher categorical objects that can be classified by a strict Lie 2-group. In this article we compute the categorical automorphism group of this 2-group; hence, the higher-categorical symmetries of topological T-duality. We prove that the categorical automorphism group is a non-central categorical extension of the integral split pseudo-orthogonal group. We show that it splits over several subgroups, and that its k-invariant is 2-torsion.

math.AT↗

arXiv2025

Classification of adjustments on central crossed modules

Adjustments are additional structures on crossed modules of Lie groups, serving as a tool in higher gauge theory to circumvent the fake flatness of connections on 2-bundles. In this article, we investigate the existence and classification of adjustments, as well as their covariance under weak equivalences. Our approach is based on a differentiation/integration correspondence with an infinitesimal version of adjustments on the associated crossed module of Lie algebras, which we then study using Lie algebra techniques. Our main result is that infinitesimal adjustments exist if and only if the Kassel-Loday classof the crossed module lies in the image of the (Lie algebraic) Chern-Weil homomorphism.

math.DG↗

arXiv2024

Internal geometry and functors between sites

Locality is implemented in an arbitrary category using Grothendieck topologies. We explore how different Grothendieck topologies on one category can be related, and, more general, how functors between categories can preserve them. As applications of locality, we review geometric objects such as sheaves, groupoids, functors, bibundles, and anafunctors internal to an arbitrary Grothendieck site. We give definitions such that all these objects are invariant under equivalences of Grothendieck topologies and certain functors between sites. As examples of sites, we look at categories of smooth manifolds, diffeological spaces, topological spaces, and sheaves, and we study properties of various functors between those.

math.CT↗

arXiv2024

The stringor bundle

We set up a framework of 2-Hilbert bundles, which allows to rigorously define the "stringor bundle", a higher differential geometric object anticipated by Stolz and Teichner in an unpublished preprint about 20 years ago. Our framework includes an associated bundle construction, allowing us to associate a 2-Hilbert bundle with a principal 2-bundle and a unitary representation of its structure 2-group. We prove that the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle obtained from applying our associated bundle construction to a string structure on a manifold and the stringor representation of the string 2-group that we discovered in earlier work. This establishes a perfect analogy to spin manifolds, representations of the spin groups, and spinor bundles.

math.AT↗

arXiv2023

String structures and loop spaces

We provide a concise and accessible introduction to (geometric) string structures, highlighting their connection to loop spaces and outlining relationships with neighboring topics.

math-ph↗

arXiv2023

A representation of the string 2-group

We construct a representation of the string 2-group on a 2-vector space, aiming to establish it as the categorification of the spinor representation. Our model for 2-vector spaces is based on the Morita bicategory of von Neumann algebras, and we specifically represent the string 2-group on the hyperfinite type III_1 factor.

math.OA↗

arXiv2023

Lie 2-groups from loop group extensions

We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.

math.DG↗

arXiv2023

Pontrjagin duality on multiplicative Gerbes

We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided.

math.AT↗

arXiv2023

Transgression of D-branes

Closed strings can be seen either as one-dimensional objects in a target space or as points in the free loop space. Correspondingly, a B-field can be seen either as a connection on a gerbe over the target space, or as a connection on a line bundle over the loop space. Transgression establishes an equivalence between these two perspectives. Open strings require D-branes: submanifolds equipped with vector bundles twisted by the gerbe. In this paper we develop a loop space perspective on D-branes. It involves bundles of simple Frobenius algebras over the branes, together with bundles of bimodules over spaces of paths connecting two branes. We prove that the classical and our new perspectives on D-branes are equivalent. Further, we compare our loop space perspective to Moore-Segal/Lauda-Pfeiffer data for open-closed 2-dimensional topological quantum field theories, and exhibit it as a smooth family of reflection-positive, colored knowledgable Frobenius algebras.

math-ph↗

arXiv2023

Geometric T-duality: Buscher rules in general topology

The classical Buscher rules describe T-duality for metrics and B-fields in a topologically trivial setting. On the other hand, topological T-duality addresses aspects of non-trivial topology while neglecting metrics and B-fields. In this article we develop a new unifying framework for both aspects.

math.DG↗

arXiv2022

2-vector bundles

We develop a ready-to-use comprehensive theory for (super) 2-vector bundles over smooth manifolds. It is based on the bicategory of (super) algebras, bimodules, and intertwiners as a model for 2-vector spaces. We discuss symmetric monoidal structures and the corresponding notions of dualizability, and we derive a classification in terms of Cech cohomology with values in a crossed module. One important feature of our 2-vector bundles is that they contain bundle gerbes as well as ordinary algebra bundles as full sub-bicategories, and hence provide a unifying framework for these so far distinct objects. We provide several examples of isomorphisms between bundle gerbes and algebra bundles, coming from representation theory, twisted K-theory, and spin geometry.

math.DG↗

arXiv2022

The insidious bicategory of algebra bundles

In this paper we construct a bicategory of (super) algebra bundles over a smooth manifold, where the 1-morphisms are bundles of bimodules. The main point is that naive definitions of bimodule bundles will not lead to a well-defined composition law in such a bicategory, at least not if non-invertible bimodules and non-semisimple algebras are desired. This problem has not been addressed so far in the literature. We develop a complete solution, and also address symmetric monoidal structures as well as the corresponding questions of dualizability.

math.DG↗

arXiv2022

Fusion of implementers for spinors on the circle

We consider the space of odd spinors on the circle, and a decomposition into spinors supported on either the top or on the bottom half of the circle. If an operator preserves this decomposition, and acts on the bottom half in the same way as a second operator acts on the top half, then the fusion of both operators is a third operator acting on the top half like the first, and on the bottom half like the second. Fusion restricts to the Banach-Lie group of restricted orthogonal operators, which supports a central extension of implementers on a Fock space. In this article, we construct a lift of fusion to this central extension. Our construction uses Tomita-Takesaki theory for the Clifford-von Neumann algebras of the decomposed space of spinors. Our motivation is to obtain an operator-algebraic model for the basic central extension of the loop group of the spin group, on which the fusion of implementers induces a fusion product in the sense considered in the context of transgression and string geometry. In upcoming work we will use this model to construct a fusion product on a spinor bundle on the loop space of a string manifold, completing a construction proposed by Stolz and Teichner.

math.OA↗

arXiv2021

Observation of multi-skyrmion objects created by size and density control in Ta/CoFeB/MgO films

Magnetic skyrmions are chiral spin textures with a nontrivial topology that offer a potential for future magnetic memory and storage devices. The controlled formation and adjustment of size and density of magnetic skyrmions in Ta/CoFeB/MgO trilayers is demonstrated. It is the ideal candidate for the use as a bottom electrode integration into CoFeB/MgO/CoFeB magnetic tunnel junctions. Varying the CoFeB thickness close to the out-of-plane to in-plane magnetic phase transition, we find that subtle energy contributions enable the skyrmion formation in a narrow thickness window, corresponding to only around 10 pm variation in CoFeB thickness. Using magneto-optical imaging with quantitative image processing, variations in skyrmion diameter and distribution below the Abbe limit can be analyzed. We demonstrate a high degree of diameter and density control. Zero-field stable skyrmions can be set with proper magnetic field initialization. This demonstrated tunability and degree of comprehension of skyrmion formation, paves the way for future skyrmion based magnetic memory. Moreover, we demonstrate a controlled merging of individual skyrmions to complex topological objects. We compare our results with the baby-Skyrme model, developed to describe the soliton nature, for any topological charge n, and demonstrate the ability to form multi-skyrmion objects. These objects will be interesting for fundamental mathematical studies of the topological behavior of solitons in the future.

cond-mat.mes-hall↗

arXiv2021

Smooth Functorial Field Theories from B-Fields and D-Branes

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz-Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke-Turner-Willerton to include open strings. At the same time, it generalises work of Moore-Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed-Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda-Pfeiffer.

math-ph↗

arXiv2020

Connes fusion of spinors on loop space

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of 2-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle covering the fusion of loops, i.e., the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish a novel relation between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as a part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

math.OA↗