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 [5]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [56], 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 [23]. The geometric realization is an A∞-space, and in the case of our model, has the correct homotopy type of String.(n). Unlike all previous models [58; 60; 33; 23; 7] our construction takes place entirely within the framework of finitedimensional 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).