cohomology

spin geometry

string geometry

# Contents

A string structure on a manifold is a higher version of a spin structure. A string structure on a manifold with spin structure given by a Spin group-principal bundle to which the tangent bundle is associated is a lift $\hat g$ of the classifying map $g : X \to \mathcal{B} Spin(n)$ through the third nontrivial step $\mathcal{B}String(n) \to \mathcal{B}Spin(n)$ in the Whitehead tower of $BO(n)$ to a String group-principal bundle:

$\array{ && \mathcal{B}String(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B}Spin(n) }$

A lift one further step through the Whitehead tower is a Fivebrane structure.

This has generalizations to the smooth context, where instead of the topological String-group one uses the String Lie 2-group.

Let $X$ be an $n$-dimensional topological manifold.

Its tangent bundle is canonically associated to a $O(n)$-principal bundle, which is in turn classified by a continuous function

$X \to B O(n)$

from $X$ to the classifying space of the orthogonal group $O(n)$.

• A String structure on $X$ is the choice of a lift of this map a few steps through the Whitehead tower of $BO(n)$.

• The manifold “is string” if such a lift exists.

This means the following:

• there is a canonical map $w_1 : B O(n) \to B\mathbb{Z}_2$ from the classifying space of $O(n)$ to that of $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$ that represents the generator of the cohomology $H^1(B O(n), \mathbb{Z}_2)$. The classifying space of the group $SO(n)$ is the homotopy pullback

$\array{ B SO(n) &\to& {*} \\ \downarrow && \downarrow \\ B O(n) &\stackrel{w_1}{\to}& \mathbb{B}\mathbb{Z}_2 }$

Namely using the homotopy hypothesis (which is a theorem, recall), we may identify $B O(n)$ with the one object groupoid whose space of morphisms is $O(n)$ and similarly for $B \mathbb{Z}_2$. Then the map in question is the one induced from the group homomorphism that sends orientation preserving elements in $O(n)$ to the identity and orientation reversing elements to the nontrivial element in $\mathbb{Z}_2$.

• an orientation on $X$ is a choice of lift of the structure group through $B SO(n) \to B O(n)$
$\array{ && B SO(n) \\ & {}^{orientation}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B O(n) } \,.$
• there is a canonical map $w_2 : B SO(n) \to B^2 \mathbb{Z}_2$ representing the generator of $H^2(B SO(n), \mathbb{Z}_2)$. The classifying space of the group $Spin(n)$ is the homotopy pullback

$\array{ B Spin(n) &\to& {*} \\ \downarrow && \downarrow \\ B SO(n) &\stackrel{w_2}{\to}& \mathbb{B}^2\mathbb{Z}_2 }$
• a spin structure on an oriented manifold $X$ is a choice of lift of the structure group through $B Spin(n) \to B SO(n)$
$\array{ && B Spin(n) \\ & {}^{spin structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B SO(n) } \,.$
• there is a canonical map $B Spin(n) \to B^3 U(1)$ The classifying space of the group $String(n)$ is the homotopy pullback

$\array{ B String(n) &\to& {*} \\ \downarrow && \downarrow \\ B Spin(n) &\stackrel{\frac{1}{2}p_1}{\to}& \mathbb{B}^3 U(1) }$
• a string structure on an oriented manifold $X$ is a choice of lift of the structure group through $B String(n) \to B Spin(n)$
$\array{ && B String(n) \\ & {}^{string structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B Spin(n) } \,.$
• there is a canonical map $B String(n) \to B^7 U(1)$ The classifying space of the group $Fivebrane(n)$ is the homotopy pullback

$\array{ B Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ B String(n) &\stackrel{\frac{1}{6}p_2}{\to}& \mathbb{B}^7 U(1) }$
• a fivebrane structure on an string manifold $X$ is a choice of lift of the structure group through $B Fivebrane(n) \to B String(n)$
$\array{ && B Fivebrane(n) \\ & {}^{fivebrane structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B String(n) } \,.$

## Definition

### Topological and smooth string structures

Let the ambient (∞,1)-topos by $\mathbf{H} =$ ETop∞Grpd or Smooth∞Grpd. Write $X$ for a topological manifold or smooth manifold of dimension $n$, respectively.

Write $String(n)$ for the string 2-group, a 1-truncated ∞-group object in $\mathbf{H}$.

###### Definition

The 2-groupoid of (topological or smooth) string structures on $X$ is the hom-space of cocycles $X \to \mathbf{B}String(n)$, or equivalently that of (topological or smooth) $String(n)$-principal 2-bundles:

$String(X) := String(n) Bund(X) \simeq \mathbf{X}(X,\mathbf{B}String) \,.$

NikolajK: Should that be bold H instead of bold X? If not, what is bold X?

Write $\frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin(n) \to \mathbf{B}^3 U(1)$ in $\mathbf{H}$ for the topological or smooth refinement of the first fractional Pontryagin class (see differential string structure for details on this).

###### Observation

The 2-groupoid of string structure on $X$ is the homotopy fiber of $\frac{1}{2}\mathbf{p}_1^X$: the (∞,1)-pullback

$\array{ String(X) &\to& * \\ \downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}Spin(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,.$
###### Proof

By definition of the string 2-group we have the fiber sequence $\mathbf{B} String \to \mathbf{B}Spin \stackrel{\frac{1}{2}} \mathbf{p_1}{\to} \mathbf{B}^3 U(1)$. The hom-functor $\mathbf{H}(X,-)$ preserves every (∞,1)-limit, hence preserves this fiber sequence.

###### Definition

Given a spin structure $S : X \to \mathbf{B} Spin(n)$ we say that the string structures extending this spin-structure is the homotopy fiber $String_S(X)$ of the projection $String(X) \to Spin(X)$ from observation 1:

### Twisted and differential string structures

(…)

The 2-groupoid of string structures is the homotopy fiber of

$\frac{1}{2}p_1 : Top(X, \mathcal{B}Spin) \to Top(X, \mathcal{B}^4 \mathbb{Z})$

over the trivial cocycle. Followowing the general logic of twisted cohomology the 2-groupoids over a nontrivial cocycle $c : X \to \mathcal{B}^4 \mathbb{Z}$ may be thought of as that of twisted string structures.

The Pontryagin class $\frac{1}{2}p_1$ refines to the smooth first fractional Pontryagin class $\frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)$. That leads to differential string structures.

(…)

## Properties

### Choices of string structures

###### Observation

The space of choices of string structures extending a given spin structure $S$ are as follows

• if $[\frac{1}{2}\mathbf{p}_1(S)] \neq 0$ it is empty: $String_S(X) \simeq \emptyset$;

• if $[\frac{1}{2}\mathbf{p}_1(S)] = 0$ it is $String_S(X) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1))$.

In particular the set of equivalence classes of string structures lifting $S$ is the cohomology set

$\pi_0 String_S(X) \simeq H^3(X, \mathbb{Z}) \,.$
###### Proof

Apply the pasting law for (∞,1)-pullbacks on the diagram

$\array{ String_S(X) &\to& String(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{S}{\to}& \mathbf{H}(X, \mathbf{B} Spin(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1)) } \,.$

The outer diagram defines the loop space object of $\mathbf{H}(X, \mathbf{B}^3 U(1))$. Since $\mathbf{H}(X,-)$ commutes with forming loop space objects (see fiber sequence for details) we have

$String_S(X) \simeq \Omega \mathbf{H}(X, \mathbf{B}^3 U(1)) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1)) \,.$

### String structures by gerbes on a bundle

One can reformulate an

structure in terms of the existence of a certain class in abelian cohomolgy on the total space of the given principal bundle. This decomposition is a special case of th general Whitehead principle of nonabelian cohomology.

###### Definition

Let $X$ be a manifolds with spin structure $S : X \to \mathbf{B}Spin$. Write $P \to X$ for the corresponding spin group-principal bundle.

Then a string structure lifting $S$ is a cohomology class $H^3(P,\mathbb{Z})$ such that the restriction of the class to any fiber $\simeq Spin(n)$ is a generator of $H^3(Spin(n), \mathb{Z}) \simeq \mathbb{Z}$.

This kind of definition appears in (Redden, def. 6.4.2).

###### Proposition

Every string structure in the sense of def. 2 induces a string structure in the sense of def. 3.

###### Proof

Consider the pasting diagram of (∞,1)-pullbacks

$\array{ String &\to& \hat P &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ Spin &\to& P &\to& B^2 U(1) &\to& {*} \\ \downarrow && \downarrow && \downarrow && \downarrow \\ {*} &\to& X &\to& B String(n) &\to& B Spin(n) }$

This uses repeatedly the pasting law for $(\infty,1)$-pullbacks. The map $P \to B^2 U(1)$ appears by decomposing the homotopy pullback of the point along $X \to B Spin(n)$ into a homotopy pullback first along $B String(n) \to B Spin(n)$ and then along $X \to B String(n)$ using the given String structure. This is the cocycle for a $\mathbf{B}U(1)$-principal 2-bundle on the total space $P$ of the $Spin$-principal bundle: a bundle gerbe.

The rest of the diagram is constructed in order to prove the following:

• The class in $H^3(P, \mathbb{Z})$ of this bundle gerbe, represented by $P \to B^2 U(1)$ has the property that restricted to the fibers of the $Spin(n)$-principal bundle $P$ it becomes the generating class in $H^3(Spin(n), \mathbb{Z})$.

## Examples

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$\vdots$
$\downarrow$
ninebrane 10-group$\mathbf{B}Ninebrane$ninebrane structurethird fractional Pontryagin class
$\downarrow$
fivebrane 6-group$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$fivebrane structuresecond fractional Pontryagin class
$\downarrow$
string 2-group$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$string structurefirst fractional Pontryagin class
$\downarrow$
spin group$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$spin structuresecond Stiefel-Whitney class
$\downarrow$
special orthogonal group$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$orientation structurefirst Stiefel-Whitney class
$\downarrow$
orthogonal group$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$orthogonal structure/vielbein/Riemannian metric
$\downarrow$
general linear group$\mathbf{B}GL$smooth manifold

(all hooks are homotopy fiber sequences)

## References

The relevance of String structures (like that of Spin structures half a century before) was recognized in the physics of spinning strings, therefore the name.

The article

• Killingback, World-sheet anomalies and loop geometry Nuclear Physics B Volume 288, 1987, Pages 578-588

was (it seems) the first to derive the Green-Schwarz anomaly cancellation condition of the effective background theory as the quantum anomaly cancellation condition for the worldsheet theory of the heterotic string’s sigma-model by direct generalization of the way the condition of a spin structure may be deduced from anomaly cancellation for the superparticle.

String stuctures had at that time been discussed in terms of their transgressions to loop spaces

• Edward Witten, The Index of the Dirac Operator in Loop Space Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986.
• Edward Witten, Elliptic Genera and Quantum Field Theory Commun.Math.Phys.109:525,1987

later it was reformulated in terms of the classes down on base space just mentioned in

The relation between the two pictures is analyzed for instance in

• A. Asada, Characteristic classes of loop group bundles and generalized string classes , Differential geometry and its applications (Eger, 1989), 33–66, Colloq. Math. Soc. János Bolyai, 56, North-Holland, Amsterdam, 1992. (pdf)

A precise formulation of Killingbacks original argument in differential K-theory appeared in

A review of that is in

The definition of string structures by degree-3 classes on the total space of the spin bundle is used in

• Corbett Redden, Canonical metric connections associated to string structures PhD Thesis, (2006)(pdf)

For discussion of String-structures using 3-classes on total spaces see for instance the work by Corbett Redden and Konrad Waldorf described at

Discussion of the moduli stack of twisted differential string structures is in

An explicit cocycle construction of the essentially unique string 2-group-principal 2-bundle lift of the tangent bundle of the 5-sphere is given in

Discussion for indefinite (Lorentzian) signature is in

• Hyung Bo Shim, Indefinite string structure, thesis 2013 (web)

Revised on March 9, 2015 22:52:49 by Nikolaj K (62.158.131.248)