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.

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

$$X\to BO(n)$$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 $\mathrm{BO}(n)$.

• 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) } \,.

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 $\mathrm{String}(n)$ 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 $X\to B\mathrm{String}(n)$, or equivalently that of (topological or smooth) $\mathrm{String}(n)$-principal 2-bundles:

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

Write $\frac{1}{2}{p}_1:B\mathrm{Spin}(n)\to B^{3}U(1)$ 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

$$\begin{array}{ccc}\mathrm{String}(X)& \to & *\\ \downarrow & & \downarrow \\ H(X,B\mathrm{Spin}(n))& \stackrel{\frac{1}{2}{p}_1}{\to }& H(X,B^{3}U(1))\end{array}.$$\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 $B\mathrm{String}\to B\mathrm{Spin}\stackrel{\frac{1}{2}}{p_1}\to B^{3}U(1)$. The hom-functor $H(X,-)$ preserves every (∞,1)-limit, hence preserves this fiber sequence.

Definition

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

Twisted and differential string structures

(…)

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

$$\frac{1}{2}{p}_1:\mathrm{Top}(X,ℬ\mathrm{Spin})\to \mathrm{Top}(X,{ℬ}^4\mathbb{Z})$$\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 $c:X\to {ℬ}^4\mathbb{Z}$ 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 $\frac{1}{2}{p}_1:B\mathrm{Spin}\to B^{3}U(1)$. 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 $[\frac{1}{2}{p}_1(S)]\ne 0$ it is empty: ${\mathrm{String}}_S(X)\simeq \varnothing$;

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

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

  $${\pi }_0{\mathrm{String}}_S(X)\simeq H^3(X,\mathbb{Z}).$$\pi_0 String_S(X) \simeq H^3(X, \mathbb{Z}) \,.

Proof

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

$$\begin{array}{ccccc}{\mathrm{String}}_S(X)& \to & \mathrm{String}(X)& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \stackrel{S}{\to }& H(X,B\mathrm{Spin}(n))& \stackrel{\frac{1}{2}{p}_1}{\to }& H(X,B^{3}U(1))\end{array}.$$\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 $H(X,B^{3}U(1))$. Since $H(X,-)$ commutes with forming loop space objects (see fiber sequence for details) we have

$${\mathrm{String}}_S(X)\simeq \Omega H(X,B^{3}U(1))\simeq H(X,B^{2}U(1)).$$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

• orientation-

• Spin-

• String-

• Fivebrane-

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 $H^3(P,\mathbb{Z})$ such that the restriction of the class to any fiber $\simeq \mathrm{Spin}(n)$ is a generator of $H^3(\mathrm{Spin}(n),mathbZ)\simeq \mathbb{Z}$.

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

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

This uses repeatedly the pasting law for $(\mathrm{\infty },1)$-pullbacks. 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 $H^3(\mathrm{Spin}(n),\mathbb{Z})$.

Examples

• On a 3-dimensional oriented manifold string structures are equivalently 2-framings.

Related entries

• orientation

• spin structure, twisted spin structure

• 2-framing

• string structure, differential string structure

• fivebrane structure, differential fivebrane structure

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$⋮$
	$\downarrow$
fivebrane 6-group	$B\mathrm{Fivebrane}$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$B\mathrm{String}\stackrel{\frac{1}{6}{p}_2}{\to }{B}^{7}U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$B\mathrm{Spin}\stackrel{\frac{1}{2}{p}_1}{\to }{B}^{3}U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$B\mathrm{SO}\stackrel{w_2}{\to }{B}^2{\mathbb{Z}}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$BO\stackrel{w_1}{\to }B{\mathbb{Z}}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
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

• Stefan Stolz, Peter 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)

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

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