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 1 -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/. 57T10, 22A22, 53C08; 18D10

Division algebras and supersymmetry III

“…We choose, however, to focus on α. This simplifies our later work, and because Lie 2-algebras based on j have already been the subject of much scrutiny [3,10,31,47], it should be possible to combine what we do here with the work of other authors to arrive at a more complete picture.…”

“…First defined by Baez-Crans [3], it is so-named because it turned out to be intimately related to the string group, String(n), the topological group obtained from SO(n) by killing the 1st and 3rd homotopy groups. For a description of this relationship, as well as the construction of Lie 2-groups which integrate string(n), see the papers of Baez-Crans-Schreiber-Stevenson [10], Henriques [31], and Schommer-Pries [47].…”

“…The real lesson of the string Lie 2-algebra is that, once again, our notion of Lie 2-group is not general enough. By generalizing the concept of Lie 2-group, various authors, like Baez-CransSchreiber-Stevenson [10], Henriques [31], and Schommer-Pries [47], were successful in integrating string(n).…”

“…While this definition is known to fall short in important ways, it has the virtue of being fairly simple. Ultimately, one should use an alternative definition, like that of Henriques [31] or Schommer-Pries [47], which weakens the notion of product on a group: rather than an algebraic operation in which there is a unique product of any two group elements, 'the' product is defined only up to equivalence.…”

“…There are more sophisticated approaches to integrating the string Lie 2-algbera, like those due to Baez, Crans, Schreiber and Stevenson [10] or Schommer-Pries [47], and a general technique to integrate any Lie n-algebra due to Henriques [31], which Schreiber [50] has in turn generalized to handle Lie n-superalgebras and more. All three techniques involve generalizing the notion of Lie 2-group (or Lie n-group, for Henriques and Schreiber) away from the world of finite-dimensional manifolds, and the latter three generalize the notion of 2-group to one in which products are defined only up to equivalence.…”