Central extensions of smooth $2$–groups and a finite-dimensional string $2$–group

Abstract

We provide a model of the String group as a central extension of finite-dimensional $2$–groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive $2$–category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth $2$–group subsumes the notion of Lie $2$–group introduced by Baez and Lauda [Theory Appl. Categ. 12 (2004) 423–491]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London (1970) 377–387], and our String $2$–group is a special case of such extensions. There is a nerve construction which can be applied to these $2$–groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [arXiv:math/0603563]. The geometric realization is an $A_{\infty }$–space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models, our construction takes place entirely within the framework of finite-dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin(n).