nLab From String structures to Spin structures on loop spaces

This page concerns:

• Alessandra Capotosti:

  
  

  From String structures to Spin structures on loop spaces

  
  

  Ph.D. thesis,

  Università degli Studi Roma Tre, Rome, April 2016

  
  

  thesis pdf, pdf

  talk slides pdf

on differential string structures.

Summary

This thesis derives the transgression map of String structures on an $n$-dimensional smooth manifold $X$ ($n \geq 3$) to Spin structures on its loop space from the existence of a natural morphism of smooth $\infty$-stacks

$$\mathbf{B}Spin \longrightarrow \mathbf{B}^2\big(\mathbf{B}U(1)_{conn}\big)$$

refining the first fractional Pontryagin class (following Fiorenza, Schreiber and Stasheff 2012). This transgression map has been known for a while in the Physics literature, and a completely rigorous proof has been then given by Konrad Waldorf in a series of papers.

Recall that a Spin manifold $X$ is endowed with a String structure if the map $X \overset{f_{T X}}\rightarrow B Spin(n) \overset{\frac{1}{2}p_1}\rightarrow K(\mathbb{Z}, 4)$ is homotopically trivial, where $f_{T X}$ denotes the classifying map for the tangent bundle of $X$. In this case, the map $f_{TX}$ can be lifted to a map $X \to B String(n)$, where $B String(n)$ is defined as the homotopy fiber of $\frac{1}{2}p_1$. By taking the based loop space of $B String(n)$ one obtains the topological String group: $String(n) = \Omega BString(n)$.

Let now $\mathcal{L}X$ be the free loop space of the Spin manifold $X$ and let $\mathcal{L}Spin(n)$ be the loop group of the Spin group. Since the Spin group $Spin(n)$ is connected, the loop space of the Spin manifold $X$ is naturally an $\mathcal{L}Spin(n)$-manifold (an infinite dimensional one), i.e., the tangent bundle $T (\mathcal{L}X)$ is naturally associated with an $\mathcal{L}Spin(n)$-bundle over $\mathcal{L}X$ (see Waldorf 2011, Lemma 5.1 and Spera & Wurzbacher 2007).

The loop group $\mathcal{L}Spin(n)$ has a universal central extension

$$1 \to U (1) \to\widetilde{\mathcal{L}Spin(n)} \to LSpin(n) \to 1$$

and one defines a Spin structure on $\mathcal{L}X$ as a lift of the structure group of the tangent bundle $T (\mathcal{L}X)$ of $\mathcal{L}X$ from $\mathcal{L}Spin(n)$ to $\widetilde{\mathcal{L}Spin(n)}$, i.e., as a lift of the tangent bundle morphism $T (\mathcal{L}X) \,\colon\, \mathcal{L}X \to B\mathcal{L}Spin(n)$ to a morphism

$$\mathcal{L}X \to B\widetilde{\mathcal{L}Spin(n)}.$$

Notice how a Spin structure on a loop space is not, strictly speaking a Spin structure, i.e., it is not a morphism to $BSpin(dim \mathcal{L}X)$. On the other hand, since $\mathcal{L}X$ is infinite dimensional, such a notion would be meaningless.

It is well known in the theoretical physics folklore that a String structure on $X$ is essentially the same thing as a Spin structure on $\mathcal{L}X$. In a series of articles Konrad Waldorf has given a rigorous proof of this statement, proving that there is a natural transgression map

$$\{\text{String structures on}\;X\} \to \{\text{Spin structures on}\;\mathcal{L}X\},$$

which induces a bijection at the level of isomorphism classes

$$\{\text{String structures on}\;X\}/\sim \to \{\text{Spin structures on}\;\mathcal{L}X\}/\sim,$$

as soon as $X$ is compact and simply connected.

In this Thesis it is shown how the transgression map considered by Waldorf is naturally obtained from general constructions in the $\infty$-category of smooth infinity-stack. It should however be stressed that this abstract derivation of the transgression map does not come with a proof that one gets a bijection on equivalence classes when $X$ is compact and simply connected, and the latter result need to be proved by ad hoc methods.

The crucial point in the stacky construction of the transgression map is the existence of a natural morphism of smooth stacks

$$\mathbf{B}Spin \longrightarrow {\mathbf{B}}^2\big(\mathbf{B}U(1)_{conn}\big)$$

refining the first fractional Pontryagin class. This morphism of smooth stacks appears implicitly in Waldorf 2016 in the form of a multiplicative bundle gerbe with connection over the Spin group, and as such plays an essential role in Waldorf’s proof. In particular, the central extension $\widetilde{\mathcal{L}Spin}$ considered in the Thesis is what Waldorf calls a fusion extension of $\mathcal{L}Spin$.

But once the canonical multiplicative gerbe with connection over $B Spin$ is regarded as a morphism of smooth $\infty$-stacks, everything else follows by very general reasoning. For instance, the fact that $\widetilde{\mathcal{L}Spin}$ is a fusion extension of $\mathcal{L}Spin$ is encoded into the natural homotopy fiber sequence of smooth stacks

$$\array{ \mathbf{B}\widetilde{\mathcal{L}Spin} & \rightarrow & \ast \\ \big\downarrow && \big\downarrow \\ \mathbf{B}\mathcal{L}Spin &\rightarrow& \mathbf{B}^2 U(1) }$$

induced by the morphism $\mathbf{B}Spin \to {\mathbf{B}}^2\big({\mathbf{B}}U(1)_{conn}\big)$ and by the holonomy morphism. Also, in the Thesis is provided a direct proof of the existence of the morphism $\mathbf{B}Spin \rightarrow {\mathbf{B}}^2\big({\mathbf{B}}U(1)_{conn}\big)$ which does not rely on Waldorf’s result and is instead based on the differential refinement of the first fractional Pontryagin class

$$\frac{1}{2} \hat{\mathbf{p}}_1 \colon \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn}$$

constructed in Fiorenza, Schreiber and Stasheff 2012.

Related entries

• stringor bundle

category: reference