A model for the String group

The string group String(n) is the 3-connected cover of Spin(n). Given and compact simply connected group G, we will let StringG be its 3-connected cover. The group StringG is only defined up to homotopy, and various models have appeared in the literature. Stephan Stolz and Peter Teichner [7], [6] have a couple of models of StringG, one of which, inspired by Anthony Wassermann, is an extension of G by the group of projective unitary operators in a particular Von-Neuman algebra. Jean-Luc Brylinski [4] has a model which is a U(1)-gerbe with connection over the group G. More recently, John Baez et al [2] came up with a model of StringG in their quest for a 2-Lie group integrating a given 2-Lie algebra. We show how to produce their model by applying a certain canonical procedure to their 2-Lie algebra. A 2-Lie algebra is a two step L∞-algebra. It consists of two vector spaces V0 and V1, and three brackets [ ], [ , ], [ , , ] acting on V := V0 ⊕ V1. They are of degree -1, 0, and 1 respectively and satisfy various axioms, see [1] for more details. A 2-group is a group object in a 2-category [3]. It has a multiplication μ : G → G, and an associator α : μ ◦ (μ × 1) ≃ μ ◦ (1 × μ) satisfying the pentagon axiom. There are strict and weak versions. If the 2-category is that of C∞ Artin stacks, we get the notion of a 2-Lie group. Since Artin stacks are represented by Lie groupoids, we can think of (strict) 2-Lie group as group objects in Lie groupoids. Equivalently, these are crossed modules in the category of smooth manifolds [3]. It is also good to consider weak 2-groups. The classifying space of a weak 2-group contains (up to homotopy) the same amount of information as the 2group itself. So we will replace 2-Lie groups with their classifying space. This also allows for an easy way to talk about n-Lie groups. The following definition was inspired by discussions with Jacob Lurie: