nLab From String structures to Spin structures on loop spaces

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  • 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.


This thesis derives the transgression map of String structures on an nn-dimensional smooth manifold XX (n3n \geq 3) to Spin structures on its loop space from the existence of a natural morphism of smooth \infty -stacks

BSpinB 2(BU(1) conn) \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 XX is endowed with a String structure if the map Xf TXBSpin(n)12p 1K(,4)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 TXf_{T X} denotes the classifying map for the tangent bundle of XX. In this case, the map f TXf_{TX} can be lifted to a map XBString(n)X \to B String(n), where BString(n)B String(n) is defined as the homotopy fiber of 12p 1\frac{1}{2}p_1. By taking the based loop space of BString(n)B String(n) one obtains the topological String group: String(n)=ΩBString(n)String(n) = \Omega BString(n).

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

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

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

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

XBSpin(n)˜. \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(dimX)BSpin(dim \mathcal{L}X). On the other hand, since X\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 XX is essentially the same thing as a Spin structure on X\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

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

which induces a bijection at the level of isomorphism classes

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

as soon as XX 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 XX 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

BSpinB 2(BU(1) conn) \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 Spin˜\widetilde{\mathcal{L}Spin} considered in the Thesis is what Waldorf calls a fusion extension of Spin\mathcal{L}Spin.

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

BSpin˜ * BSpin B 2U(1) \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 BSpinB 2(BU(1) conn)\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 BSpinB 2(BU(1) conn)\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

12p^ 1:BSpin connB 3U(1) conn \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.

category: reference

Last revised on February 15, 2024 at 17:36:19. See the history of this page for a list of all contributions to it.