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

Christopher Schommer-Pries

TL;DR

The paper develops a finite-dimensional, geometric model of the String group as a central extension of smooth 2-groups within the bicategory $\mathrm{Bibun}$ (Lie groupoids, bibundles, and bibundle maps). It shows that central extensions of smooth 2-groups are classified by Segal-Mitchison topological group cohomology, with isomorphism classes corresponding to $H^3_{SM}(G;A)$ and a contractible extension category for $G=Spin(n)$, $A=S^1$. A generator of $H^3_{SM}(Spin(n);S^1)$ yields a central extension whose geometric realization has the homotopy type of $String(n)$, providing a finite-dimensional, unique (up to unique isomorphism) model and a clear path to studying string structures via smooth 2-group extensions. The work links Segal-Mitchison cohomology with smooth gerbe techniques and presents a fully finite-dimensional framework for string-geometry that interacts with $A_\infty$-space structures and potential connections to tmf and related topological phenomena.

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 naïve 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).

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

TL;DR

The paper develops a finite-dimensional, geometric model of the String group as a central extension of smooth 2-groups within the bicategory (Lie groupoids, bibundles, and bibundle maps). It shows that central extensions of smooth 2-groups are classified by Segal-Mitchison topological group cohomology, with isomorphism classes corresponding to and a contractible extension category for , . A generator of yields a central extension whose geometric realization has the homotopy type of , providing a finite-dimensional, unique (up to unique isomorphism) model and a clear path to studying string structures via smooth 2-group extensions. The work links Segal-Mitchison cohomology with smooth gerbe techniques and presents a fully finite-dimensional framework for string-geometry that interacts with -space structures and potential connections to tmf and related topological phenomena.

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 naïve 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 -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).

Paper Structure

This paper contains 19 sections, 29 theorems, 85 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Lie Groupoids and Smooth Stacks
  3. Lie Groupoids
  4. Lie Groupoids
  5. Bibundles and Smooth Stacks
  6. Transversality for Stacks
  7. 2-Groups in Stacks
  8. 2-Groups
  9. 2-Groups in General Bicategories
  10. Abelian Groups in Bicategories
  11. 2-Groups as a Localization
  12. Smooth 2-Groups and Gerbes
  13. Extensions of 2-Groups
  14. Smooth 2-Groups and $A_\infty$-Spaces
  15. A Finite Dimensional String 2-Group
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G$ be a Lie group and $A$ an abelian Lie group, viewed as a trivial $G$-module. Then we have an (unnatural) equivalence of symmetric monoidal bicategories: where $H^i_\text{SM}( G; A)$ denotes the smooth version Segal-Mitchison topological group cohomology Segal70. Moreover, isomorphism classes of central extensions, are in natural bijection with $H^3_\text{SM}(G;A)$.

Figures (10)

  • Figure 1: The 'Pentagon' Identity
  • Figure 2: The 'Triangle' Identity
  • Figure 3: Axiom 1 for 2-Group homomorphisms.
  • Figure 4: Axiom 2 for 2-Group homomorphisms.
  • Figure 5: Axiom 3 for 2-Group homomorphisms.
  • ...and 5 more figures

Theorems & Definitions (122)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Example 4: $G$-Spaces
  • Example 5: Čech Groupoids
  • Example 6: Smooth Maps
  • Example 7: Lie Homomorphisms
  • Example 8
  • Lemma 9
  • Definition 10
  • ...and 112 more