We provide a model of the String group as a central extension of finite-dimensional
–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
–category of Lie
groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth
–group subsumes the
notion of Lie –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
–group is a
special case of such extensions. There is a nerve construction which can be applied to these
–groups
to obtain a simplicial manifold, allowing comparison with the model
of Henriques [arXiv:math/0603563]. The geometric realization is an
–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).
Keywords
$2$–group, stack, central extension, string group, gerbe
