arXiv subjects

Konrad Waldorf

Publications and source records attributed to Konrad Waldorf.

At least 37 records · Page 2Linked to original sources

arXiv2020

Smooth Fock bundles, and spinor bundles on loop space

We address the construction of smooth bundles of fermionic Fock spaces, a problem that appears frequently in fermionic gauge theories. Our main motivation is the spinor bundle on the free loop space of a string manifold, a structure anticipated by Killingback, with a construction outlined by Stolz-Teichner. We develop a general framework for constructing smooth Fock bundles, and obtain as an application a complete and well-founded construction of spinor bundles on loop space.

math.RT↗

arXiv2019

Higher geometry for non-geometric T-duals

We investigate topological T-duality in the framework of non-abelian gerbes and higher gauge groups. We show that this framework admits the gluing of locally defined T-duals, in situations where no globally defined ("geometric") T-duals exists. The gluing results into new, higher-geometrical objects that can be used to study non-geometric T-duals, alternatively to other approaches like non-commutative geometry.

math.AT↗

arXiv2018

Non-abelian gerbes and some applications in string theory

We review a systematic construction of the 2-stack of bundle gerbes via descent, and extend it to non-abelian gerbes. We review the role of non-abelian gerbes in orientifold sigma models, for the anomaly cancellation in supersymmetric sigma models, and in a geometric description of so-called non-geometric T-duals.

hep-th↗

arXiv2017

A global perspective to connections on principal 2-bundles

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen-Messing. This provides a consistent, global perspective to higher gauge theory.

math.DG↗

arXiv2017

Parallel transport in principal 2-bundles

A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we construct explicitly the parallel transport of a connection on a principal 2-bundle. Parallel transport along a path is a Morita equivalence between the fibres over the end points, and parallel transport along a surface is an intertwiner between Morita equivalences. We prove that our constructions fit into the general axiomatic framework for categorified parallel transport and surface holonomy.

math.DG↗

arXiv2016

Transgressive loop group extensions

A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry over the Lie group. We show how transgressive central extensions can be characterized in a loop-group theoretical way, in terms of loop fusion and thin homotopy equivariance.

math.DG↗

arXiv2015

String geometry vs. spin geometry on loop spaces

We introduce various versions of spin structures on free loop spaces of smooth manifolds, based on a classical notion due to Killingback, and additionally coupled to two relations between loops: thin homotopies and loop fusion. The central result of this article is an equivalence between these enhanced versions of spin structures on the loop space and string structures on the manifold itself. The equivalence exists in two settings: in a purely topological one and a in geometrical one that includes spin connections and string connections. Our results provide a consistent, functorial, one-to-one dictionary between string geometry and spin geometry on loop spaces.

math.DG↗

arXiv2015

A Loop Space Formulation for Geometric Lifting Problems

We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.

math.DG↗

arXiv2014

Parallel Transport and Functors

Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport.

math.DG↗

arXiv2013

Connections on non-abelian Gerbes and their Holonomy

We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group. Upon choosing certain examples of Lie 2-groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: non-abelian differential cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These relationships convey a well-defined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to non-abelian gerbes. Several new features of surface holonomy are exposed under its extension to non-abelian gerbes; for example, it carries an action of the mapping class group of the surface.

math.DG↗

arXiv2013

Four Equivalent Versions of Non-Abelian Gerbes

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.

math.DG↗

arXiv2013

Lifting Problems and Transgression for Non-Abelian Gerbes

We discuss various lifting and reduction problems for bundles and gerbes in the context of a strict Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen's long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces - in case of a Lie 2-group with a single object - to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structure on the loop space.

math.AT↗

arXiv2013

Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps

We prove that isomorphism classes of principal bundles over a diffeological space are in bijection to certain maps on its free loop space, both in a setup with and without connections on the bundles. The maps on the loop space are smooth and satisfy a "fusion" property with respect to triples of paths. Our bijections are established by explicit group isomorphisms: transgression and regression. Restricted to smooth, finite-dimensional manifolds, our results extend previous work of J. W. Barrett.

math.DG↗

arXiv2013

Local Theory for 2-Functors on Path 2-Groupoids

In this article we discuss local aspects of 2-functors defined on the path 2-groupoid of a smooth manifold; in particular, local trivializations and descent data. This is a contribution to a project that provides an axiomatic formulation of connections on (possibly non-abelian) gerbes in terms of 2-functors. The main result of this paper establishes the first part of this formulation: we prove an equivalence between the globally defined 2-functors and their locally defined descent data. The second part appears in a separate publication; there we prove equivalences between descent data, on one side, and various existing versions of gerbes with connection on the other side.

math.GN↗

arXiv2013

Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection

We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit functors in both directions: transgression and regression. We describe applications to geometric lifting problems and loop group extensions.

math.DG↗

arXiv2013

Spin structures on loop spaces that characterize string manifolds

Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well-known that only the if-part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string, as desired. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.

math.AT↗

arXiv2013

String Connections and Chern-Simons Theory

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string connections: it enables us to prove that every string structure admits a string connection and that the possible choices form an affine space. Further we discover a new relation between string connections, 3-forms on the base manifold, and degree three differential cohomology. We also discuss in detail the relation between our formulation of string connections and the original version of Stolz and Teichner.

math.DG↗

arXiv2012

Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles

We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor called regression, and complements a similar equivalence for bundles and gerbes equipped with connections, derived previously in Part II of this series of papers. The two equivalences provide a complete loop space formulation of the geometry of gerbes; functorial, monoidal, natural in the base manifold, and consistent with passing from the setting "with connections" to the one "without connections". We discuss an application to lifting problems, which provides in particular loop space formulations of spin structures, complex spin structures, and spin connections.

math.DG↗