Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group

TL;DR

This paper constructs a finite-dimensional model of the String 2-group as a central extension of Lie 2-groups, providing a geometric and algebraic framework that generalizes previous models and ensures the correct homotopy type.

Contribution

It introduces a finite-dimensional, geometric model of the String 2-group as a central extension of Lie 2-groups within Lie groupoids, unifying and extending prior approaches.

Findings

Provides a classification of central extensions via topological group cohomology.

Constructs a nerve that yields a simplicial manifold comparable to existing models.

Establishes a unique, finite-dimensional model with the correct homotopy type of String(n).

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 na\"ive 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-Lauda. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by G. Segal, 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 with the model of A. Henriques. 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 unique central extension of Spin(n).