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A smooth model for the string group. (English) Zbl 1339.22009

String structures and the string group play an important role in algebraic topology, string theory, and geometry. The group \(\mathbf{String}\) is defined to be a \(3\)-connected cover of the spin group. More generally, it is denoted by \(\mathbf{String}_G\) the \(3\)-connected cover of any compact, simple and \(1\)-connected Lie group \(G\). This definition fixes only its homotopy type and makes abstract homotopy theoretic constructions possible. These models are not very well suited for geometric applications, one is rather interested in concrete models that carry, for instance, topological or even Lie group structures. There is a direct cohomological argument showing that \(\mathbf{String}_G\) cannot be a finite CW-complex or a finite-dimensional manifold, so the best thing one can hope for is a model for \(\mathbf{String}_G\) as a topological group or an infinite dimensional Lie group. There have been various constructions of models of \(\mathbf{String}_G\) as \(A_{\infty}\)-spaces or topological groups, but the question whether an infinite-dimensional Lie group model is also possible remains open. One of the main contributions of the present paper is to give an affirmative answer to this question and provide an explicit Lie group model, based on a topological construction of Stolz.
In this paper the authors construct a model for the string group as an infinite-dimensional Lie group. In a second step, they extend this model by a contractible Lie group to a Lie \(2\)-group model. Moreover, they provide an explicit comparison of string structures for the two models and a uniqueness result for Lie \(2\)-group models.
Reviewer: Cenap Özel (Bolu)

MSC:

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E70 Applications of Lie groups to the sciences; explicit representations
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