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From Loop Groups to 2-Groups

John C. Baez, Alissa S. Crans, Danny Stevenson, Urs Schreiber

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If $G$ is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras $\mathfrak{g}_k$ each having $\mathrm{Lie}(G)$ as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on $G$. There appears to be no Lie 2-group having $\mathfrak{g}_k$ as its Lie 2-algebra, except when $k = 0$. Here, however, we construct for integral k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to $\mathfrak{g}_k$. The objects of this 2-group are based paths in $G$, while the automorphisms of any object form the level-$k$ Kac-Moody central extension of the loop group of $G$. This 2-group is closely related to the $k$th power of the canonical gerbe over $G$. Its nerve gives a topological group that is an extension of $G$ by $K(\mathbb{Z},2)$. When $k = \pm 1$, this topological group can also be obtained by killing the third homotopy group of $G$. Thus, when $G = \mathrm{Spin}(n)$, it is none other than $\mathrm{String}(n)$.

From Loop Groups to 2-Groups

Abstract

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group . A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras each having as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on . There appears to be no Lie 2-group having as its Lie 2-algebra, except when . Here, however, we construct for integral k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to . The objects of this 2-group are based paths in , while the automorphisms of any object form the level- Kac-Moody central extension of the loop group of . This 2-group is closely related to the th power of the canonical gerbe over . Its nerve gives a topological group that is an extension of by . When , this topological group can also be obtained by killing the third homotopy group of . Thus, when , it is none other than .

Paper Structure

This paper contains 14 sections, 24 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Review of Lie 2-Algebras and Lie 2-Groups
  3. Lie 2-algebras
  4. $L_\infty$-algebras
  5. The Lie 2-Algebra ${\mathfrak g}_k$
  6. The Lie 2-Algebra of a Fréchet Lie 2-Group
  7. Review of Loop Groups
  8. Definitions and Basic Properties
  9. The Kac--Moody group $\widehat{\Omega_k G}$
  10. The Lie 2-Group ${\cal P}_kG$
  11. Constructing ${\cal P}_kG$
  12. The Topology of $|{\cal P}_kG|$
  13. The Equivalence Between ${\cal P}_k{\mathfrak g}$ and ${\mathfrak g}_k$
  14. Conclusions

Key Result

Theorem 1.1

. Let $G$ be a simply-connected compact simple Lie group. For any $k \in {\mathbb Z}$, there is a Fréchet Lie 2-group ${\cal P}_kG$ whose Lie 2-algebra ${\cal P}_k{\mathfrak g}$ is equivalent to ${\mathfrak g}_k$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 25 more