arXiv subjects

Christoph Wockel

Publications and source records attributed to Christoph Wockel.

At least 19 recordsLinked to original sources

arXiv2017

Quasi-periodic paths and a string 2-group model from the free loop group

In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group $LSpin$ (or more generally $LG$ for compact simple simply-connected Lie groups $G$). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group $S^1$, which is of crucial importance, for instance in the construction of a (hypothetical) $S^1$-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using $LG$. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.

math.DG↗

arXiv2015

Functorial aspects of the reconstruction of Lie groupoids from their bisections

To a Lie groupoid over a compact base, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing the base). Moreover, it gives rise to an adjunction between the category of Lie groupoids over a fixed base and the category of Lie groups acting on the base. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

math.CT↗

arXiv2015

(Re)constructing Lie groupoids from their bisections and applications to prequantisation

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold.

math.DG↗

arXiv2015

Integrating central extensions of Lie algebras via Lie 2-groups

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial π_2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.

math.DG↗

arXiv2014

The Lie group of bisections of a Lie groupoid

In this article we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.

math.DG↗

arXiv2014

Topological group cohomology of Lie groups and Chern-Weil theory for compact symmetric spaces

In this paper we analyse the topological group cohomology of finite-dimensional Lie groups. We introduce a technique for computing it (as abelian groups) for torus coefficients by the naturally associated long exact sequence. The upshot in there is that certain morphisms in this long exact coefficient sequence can be accessed (at least for semi-simple Lie groups) very conveniently by the Chern-Weil homomorphism of the naturally associated compact dual symmetric space. Since the latter is very well-known, this gives the possibility to compute the topological group cohomology of the classical simple Lie groups. In addition, we establish a relation to characteristic classes of flat bundles.

math.AT↗

arXiv2013

A Cocycle Model for Topological and Lie Group Cohomology

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology.

math.AT↗

arXiv2012

Equivalences of Smooth and Continuous Principal Bundles with Infinite-Dimensional Structure Group

Let K be a a Lie group, modeled on a locally convex space, and M a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.

math.DG↗

arXiv2012

A Smooth Model for the String Group

We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to establish some facts on the homotopy theory of Lie 2-groups. Moreover, we provide an explicit comparison of string structures for the two models and a uniqueness result for Lie 2-group models.

math.AT↗

arXiv2012

Making Lifting Obstructions Explicit

If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class δ_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a \hat K-bundle \hat P over X with P \cong \hat P/Z. In this paper we establish a link between homotopy theoretic data and the obstruction class δ_1(P) which in many cases can be used to calculate this class in explicit terms. Writing \partial_d^P \: π_d(X) \to π_{d-1}(K) for the connecting maps in the long exact homotopy sequence, two of our main results can be formulated as follows. If Z is a quotient of a contractible group by the discrete group Γ, then the homomorphism π_3(X) \to Γinduced by δ_1(P) \in \check H^2(X,\uline Z) \cong H^3_{\rm sing}(X,Γ) coincides with \partial_2^{\hat K} \circ \partial_3^P and if Z is discrete, then δ_1(P) \in \check H^2(X,\uline Z) induces the homomorphism -\partial_1^{\hat K} \circ \partial_2^P \: π_2(X) \to Z. We also obtain some information on obstruction classes defining trivial homomorphisms on homotopy groups.

math.AT↗

arXiv2012

Topological Group Cohomology with Loop Contractible Coefficients

We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.

math.AT↗

arXiv2012

Universal Central Extensions of Gauge Algebras and Groups

We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.

math.DG↗

arXiv2011

Categorified central extensions, étale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras

Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem. The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.

math.GR↗

arXiv2009

Non-integral central extensions of loop groups

It is well-known that the central extensions of the loop group of a compact, simple and 1-connected Lie group are parametrised by their level $k \in Z$. This article concerns the question how much can be said for arbitrary $k \in R$ and we show that for each $k$ there exists a Lie groupoid which has the level $k$ central extension as its quotient if $k \in Z$. By considering categorified principal bundles we show, moreover, that the corresponding Lie groupoid has the expected bundle structure.

math-ph↗

arXiv2009

The diffeomorphism supergroup of a finite-dimensional supermanifold

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of diffeomorphism supergroups.

math.DG↗

arXiv2009

Principal 2-bundles and their gauge 2-groups

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.

math.DG↗

arXiv2009

Central extensions of groups of sections

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.

math.DG↗

arXiv2006

The Topology of Gauge Groups

This paper has been withdrawn by the author. The content of the previous versions is now covered by the more recent papers - math.DG/0610252 (concerning the Lie group structuren on the gauge groups) - math.DG/0612522 (concerning the weak homotopy equivalence) - math.AT/0511404 (concerning the homotopy groups of gauge groups)

math-ph↗