nLab
string structure

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

$$\begin{array}{ccc}& & ℬ\mathrm{String}(n)\\ & {}^{\hat{g}}\nearrow & \downarrow \\ X& \stackrel{g}{\to }& ℬ\mathrm{Spin}(n)\end{array}$$\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.

Definition

Let $X$ be an $n$-dimensional manifold. Its tangent bundle is canonically associated to a $O(n)$-principal bundle, which is in turn classified by a map

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

from $X$ to the classifying space of the 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 $\mathrm{BO}(n)$.

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

This means the following:

• there is a canonical map $w_1:BO(n)\to B{\mathbb{Z}}_2$ 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 $H^1(BO(n),{\mathbb{Z}}_2)$. The classifying space of the group $\mathrm{SO}(n)$ is the homotopy pullback

  $$\begin{array}{ccc}B\mathrm{SO}(n)& \to & *\\ \downarrow & & \downarrow \\ BO(n)& \stackrel{w_1}{\to }& 𝔹{\mathbb{Z}}_2\end{array}$$\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 $BO(n)$ 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 $B\mathrm{SO}(n)\to BO(n)$

    $$\begin{array}{ccc}& & B\mathrm{SO}(n)\\ & {}^{\mathrm{orientation}}\nearrow & \downarrow \\ X& \stackrel{}{\to }& BO(n)\end{array}.$$\array{ && B SO(n) \\ & {}^{orientation}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B O(n) } \,.

• there is a canonical map $w_2:B\mathrm{SO}(n)\to B^2{\mathbb{Z}}_2$ representing the generator of $H^2(B\mathrm{SO}(n),{\mathbb{Z}}_2)$. The classifying space of the group $\mathrm{Spin}(n)$ is the homotopy pullback

  $$\begin{array}{ccc}B\mathrm{Spin}(n)& \to & *\\ \downarrow & & \downarrow \\ B\mathrm{SO}(n)& \stackrel{w_2}{\to }& {𝔹}^2{\mathbb{Z}}_2\end{array}$$\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 $B\mathrm{Spin}(n)\to B\mathrm{SO}(n)$

    $$\begin{array}{ccc}& & B\mathrm{Spin}(n)\\ & {}^{\mathrm{spin}\mathrm{structure}}\nearrow & \downarrow \\ X& \stackrel{}{\to }& B\mathrm{SO}(n)\end{array}.$$\array{ && B Spin(n) \\ & {}^{spin structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B SO(n) } \,.

• there is a canonical map $B\mathrm{Spin}(n)\to B^{3}U(1)$ The classifying space of the group $\mathrm{String}(n)$ is the homotopy pullback

  $$\begin{array}{ccc}B\mathrm{String}(n)& \to & *\\ \downarrow & & \downarrow \\ B\mathrm{Spin}(n)& \stackrel{\frac{1}{2}{p}_1}{\to }& {𝔹}^{3}U(1)\end{array}$$\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 $B\mathrm{String}(n)\to B\mathrm{Spin}(n)$

    $$\begin{array}{ccc}& & B\mathrm{String}(n)\\ & {}^{\mathrm{string}\mathrm{structure}}\nearrow & \downarrow \\ X& \stackrel{}{\to }& B\mathrm{Spin}(n)\end{array}.$$\array{ && B String(n) \\ & {}^{string structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B Spin(n) } \,.

• there is a canonical map $B\mathrm{String}(n)\to B^{7}U(1)$ The classifying space of the group $\mathrm{Fivebrane}(n)$ is the homotopy pullback

  $$\begin{array}{ccc}B\mathrm{Fivebrane}(n)& \to & *\\ \downarrow & & \downarrow \\ B\mathrm{String}(n)& \stackrel{\frac{1}{6}{p}_2}{\to }& {𝔹}^{7}U(1)\end{array}$$\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 $B\mathrm{Fivebrane}(n)\to B\mathrm{String}(n)$

    $$\begin{array}{ccc}& & B\mathrm{Fivebrane}(n)\\ & {}^{\mathrm{fivebrane}\mathrm{structure}}\nearrow & \downarrow \\ X& \stackrel{}{\to }& B\mathrm{String}(n)\end{array}.$$\array{ && B Fivebrane(n) \\ & {}^{fivebrane structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B String(n) } \,.

Description in terms of classes on the total space

One can reformulate an

• orientation-

• Spin-

• String-

• Fivebrane-

structure in terms of the existence of a certain class on the total space of the given bundle.

We write this out for the case of string structures, all other cases work entirely analogously.

Let $X$ be an oriented Spin manifold and let $P\to X$ be the corresponding $\mathrm{Spin}(n)$-bundle. Notice that this, like any principal bundle (see generalized universal bundle, principal bundle and principal infinity-bundle) is the homotopy pullback

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \stackrel{}{\to }& B\mathrm{Spin}(n)\end{array}$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\stackrel{}{\to}& B Spin(n) }

of the point along the classifying map from $X$ to the classifying space $B\mathrm{Spin}(n)$.

In other words, there is a fibration sequence

$$\cdots \to (\Omega B\mathrm{Spin}=\mathrm{Spin}(n))\to P\to X\to B\mathrm{Spin}(n).$$\cdots \to (\Omega B Spin = Spin(n)) \to P \to X \to B Spin(n) \,.

If now $X$ does admit a String structure, i.e. a decomposition of $X\to B\mathrm{Spin}(n)$ into a map $X\to B\mathrm{String}(n)\to B\mathrm{Spin}(n)$ then we obtain the following diagram, where each square is a homotopy pullback

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

The map $P\to B^{2}U(1)$ appears by decomposing the homotopy pullback of the point along $X\to B\mathrm{Spin}(n)$ into a homotopy pullback first along $B\mathrm{String}(n)\to B\mathrm{Spin}(n)$ and then along $X\to B\mathrm{String}(n)$ using the given String structure. This is the cocycle for a $BU(1)$-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 $H^3(P,\mathbb{Z})$ of this bundle gerbe, represented by $P\to B^{2}U(1)$ has the property that restricted to the fibers of the $\mathrm{Spin}(n)$-principal bundle $P$ it becomes the generating class in $H^3(\mathrm{Spin}(n),\mathbb{Z})$.

Proof.

Here $\hat{P}\to X$ denotes the $\mathrm{String}(n)$-principal bundle classified by $X\to B\mathrm{String}(n)$.

This uses

• the fact that pasting compositites of homotopy pullbacks are again homtopy pullbacks.

• the fact that the homotopy pullback of the point to itself produces the loop space object, e.g.

  $$\begin{array}{ccc}\Omega B\mathrm{Spin}(n)\simeq \mathrm{Spin}(n)& \to & *\\ \downarrow & & \downarrow \\ *& \to & B\mathrm{Spin}\end{array}$$\array{ \Omega B Spin(n) \simeq Spin(n) &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& B Spin }

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

• Witten

  • blog entry on String(n)

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

• Stolz, Teichner, What is an elliptic object.

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)

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

• Oberwolfach Workshop, June 2009 -- Friday, June 12