cohomology

# 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 $\stackrel{^}{g}$ of the classifying map $g:X\to ℬ\mathrm{Spin}\left(n\right)$ through the third nontrivial step $ℬ\mathrm{String}\left(n\right)\to ℬ\mathrm{String}\left(n\right)$ in the Whitehead tower of $\mathrm{BO}\left(n\right)$ to a String group-principal bundle:

$\begin{array}{ccc}& & ℬ\mathrm{String}\left(n\right)\\ & {}^{\stackrel{^}{g}}↗& ↓\\ X& \stackrel{g}{\to }& ℬ\mathrm{Spin}\left(n\right)\end{array}$\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 $O\left(n\right)$-principal bundle, which is in turn classified by a continuous function

$X\to BO\left(n\right)$X \to B O(n)

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

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

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

This means the following:

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

$\begin{array}{ccc}B\mathrm{SO}\left(n\right)& \to & *\\ ↓& & ↓\\ BO\left(n\right)& \stackrel{{w}_{1}}{\to }& 𝔹{ℤ}_{2}\end{array}$\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 $BO\left(n\right)$ with the one object groupoid whose space of morphisms is $O\left(n\right)$ and similarly for $B{ℤ}_{2}$. Then the map in question is the one induced from the group homomorphism that sends orientation preserving elements in $O\left(n\right)$ to the identity and orientation reversing elements to the nontrivial element in ${ℤ}_{2}$.

• an orientation on $X$ is a choice of lift of the structure group through $B\mathrm{SO}\left(n\right)\to BO\left(n\right)$

$\begin{array}{ccc}& & B\mathrm{SO}\left(n\right)\\ & {}^{\mathrm{orientation}}↗& ↓\\ X& \stackrel{}{\to }& BO\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && B SO(n) \\ & {}^{orientation}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B O(n) } \,.
• there is a canonical map ${w}_{2}:B\mathrm{SO}\left(n\right)\to {B}^{2}{ℤ}_{2}$ representing the generator of ${H}^{2}\left(B\mathrm{SO}\left(n\right),{ℤ}_{2}\right)$. The classifying space of the group $\mathrm{Spin}\left(n\right)$ is the homotopy pullback

$\begin{array}{ccc}B\mathrm{Spin}\left(n\right)& \to & *\\ ↓& & ↓\\ B\mathrm{SO}\left(n\right)& \stackrel{{w}_{2}}{\to }& {𝔹}^{2}{ℤ}_{2}\end{array}$\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 $B\mathrm{Spin}\left(n\right)\to B\mathrm{SO}\left(n\right)$

$\begin{array}{ccc}& & B\mathrm{Spin}\left(n\right)\\ & {}^{\mathrm{spin}\mathrm{structure}}↗& ↓\\ X& \stackrel{}{\to }& B\mathrm{SO}\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && B Spin(n) \\ & {}^{spin structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B SO(n) } \,.
• there is a canonical map $B\mathrm{Spin}\left(n\right)\to {B}^{3}U\left(1\right)$ The classifying space of the group $\mathrm{String}\left(n\right)$ is the homotopy pullback

$\begin{array}{ccc}B\mathrm{String}\left(n\right)& \to & *\\ ↓& & ↓\\ B\mathrm{Spin}\left(n\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& {𝔹}^{3}U\left(1\right)\end{array}$\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 $B\mathrm{String}\left(n\right)\to B\mathrm{Spin}\left(n\right)$

$\begin{array}{ccc}& & B\mathrm{String}\left(n\right)\\ & {}^{\mathrm{string}\mathrm{structure}}↗& ↓\\ X& \stackrel{}{\to }& B\mathrm{Spin}\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && B String(n) \\ & {}^{string structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B Spin(n) } \,.
• there is a canonical map $B\mathrm{String}\left(n\right)\to {B}^{7}U\left(1\right)$ The classifying space of the group $\mathrm{Fivebrane}\left(n\right)$ is the homotopy pullback

$\begin{array}{ccc}B\mathrm{Fivebrane}\left(n\right)& \to & *\\ ↓& & ↓\\ B\mathrm{String}\left(n\right)& \stackrel{\frac{1}{6}{p}_{2}}{\to }& {𝔹}^{7}U\left(1\right)\end{array}$\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 $B\mathrm{Fivebrane}\left(n\right)\to B\mathrm{String}\left(n\right)$

$\begin{array}{ccc}& & B\mathrm{Fivebrane}\left(n\right)\\ & {}^{\mathrm{fivebrane}\mathrm{structure}}↗& ↓\\ X& \stackrel{}{\to }& B\mathrm{String}\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $H=$ ETop∞Grpd or Smooth∞Grpd. Write $X$ for a topological manifold or smooth manifold of dimension $n$, respectively.

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

###### Definition

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

$\mathrm{String}\left(X\right):=\mathrm{String}\left(n\right)\mathrm{Bund}\left(X\right)\simeq X\left(X,B\mathrm{String}\right)\phantom{\rule{thinmathspace}{0ex}}.$String(X) := String(n) Bund(X) \simeq \mathbf{X}(X,\mathbf{B}String) \,.

Write $\frac{1}{2}{p}_{1}:B\mathrm{Spin}\left(n\right)\to {B}^{3}U\left(1\right)$ in $H$ 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 $\frac{1}{2}{p}_{1}^{X}$: the (∞,1)-pullback

$\begin{array}{ccc}\mathrm{String}\left(X\right)& \to & *\\ ↓& & ↓\\ H\left(X,B\mathrm{Spin}\left(n\right)\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& H\left(X,{B}^{3}U\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $B\mathrm{String}\to B\mathrm{Spin}\stackrel{\frac{1}{2}}{{p}_{1}}\to {B}^{3}U\left(1\right)$. The hom-functor $H\left(X,-\right)$ preserves every (∞,1)-limit, hence preserves this fiber sequence.

###### Definition

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

### Twisted and differential string structures

(…)

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

$\frac{1}{2}{p}_{1}:\mathrm{Top}\left(X,ℬ\mathrm{Spin}\right)\to \mathrm{Top}\left(X,{ℬ}^{4}ℤ\right)$\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 $c:X\to {ℬ}^{4}ℤ$ 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 $\frac{1}{2}{p}_{1}:B\mathrm{Spin}\to {B}^{3}U\left(1\right)$. 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 $\left[\frac{1}{2}{p}_{1}\left(S\right)\right]\ne 0$ it is empty: ${\mathrm{String}}_{S}\left(X\right)\simeq \varnothing$;

• if $\left[\frac{1}{2}{p}_{1}\left(S\right)\right]=0$ it is ${\mathrm{String}}_{S}\left(X\right)\simeq H\left(X,{B}^{2}U\left(1\right)\right)$.

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

${\pi }_{0}{\mathrm{String}}_{S}\left(X\right)\simeq {H}^{3}\left(X,ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_0 String_S(X) \simeq H^3(X, \mathbb{Z}) \,.
###### Proof

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

$\begin{array}{ccccc}{\mathrm{String}}_{S}\left(X\right)& \to & \mathrm{String}\left(X\right)& \to & *\\ ↓& & ↓& & ↓\\ *& \stackrel{S}{\to }& H\left(X,B\mathrm{Spin}\left(n\right)\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& H\left(X,{B}^{3}U\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $H\left(X,{B}^{3}U\left(1\right)\right)$. Since $H\left(X,-\right)$ commutes with forming loop space objects (see fiber sequence for details) we have

${\mathrm{String}}_{S}\left(X\right)\simeq \Omega H\left(X,{B}^{3}U\left(1\right)\right)\simeq H\left(X,{B}^{2}U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $S:X\to B\mathrm{Spin}$. Write $P\to X$ for the corresponding spin group-principal bundle.

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

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

$\begin{array}{ccccc}\mathrm{String}& \to & \stackrel{^}{P}& \to & *\\ ↓& & ↓& & ↓\\ \mathrm{Spin}& \to & P& \to & {B}^{2}U\left(1\right)& \to & *\\ ↓& & ↓& & ↓& & ↓\\ *& \to & X& \to & B\mathrm{String}\left(n\right)& \to & B\mathrm{Spin}\left(n\right)\end{array}$\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 $\left(\infty ,1\right)$-pullbacks. The map $P\to {B}^{2}U\left(1\right)$ appears by decomposing the homotopy pullback of the point along $X\to B\mathrm{Spin}\left(n\right)$ into a homotopy pullback first along $B\mathrm{String}\left(n\right)\to B\mathrm{Spin}\left(n\right)$ and then along $X\to B\mathrm{String}\left(n\right)$ using the given String structure. This is the cocycle for a $BU\left(1\right)$-principal 2-bundle on the total space $P$ of the $\mathrm{Spin}$-principal bundle: a bundle gerbe.

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

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

## Examples

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$⋮$
$↓$
fivebrane 6-group$B\mathrm{Fivebrane}$fivebrane structuresecond fractional Pontryagin class
$↓$
string 2-group$B\mathrm{String}\stackrel{\frac{1}{6}{p}_{2}}{\to }{B}^{7}U\left(1\right)$string structurefirst fractional Pontryagin class
$↓$
spin group$B\mathrm{Spin}\stackrel{\frac{1}{2}{p}_{1}}{\to }{B}^{3}U\left(1\right)$spin structuresecond Stiefel-Whitney class
$↓$
special orthogonal group$B\mathrm{SO}\stackrel{{w}_{2}}{\to }{B}^{2}{ℤ}_{2}$orientation structurefirst Stiefel-Whitney class
$↓$
orthogonal group$BO\stackrel{{w}_{1}}{\to }B{ℤ}_{2}$orthogonal structure/vielbein/Riemannian metric
$↓$
general linear group$B\mathrm{GL}$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)

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

Revised on January 10, 2013 16:29:16 by Urs Schreiber (89.204.153.52)