group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry
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 $\mathcal{B}String(n) \to \mathcal{B}String(n)$ in the Whitehead tower of $BO(n)$ to a String group-principal bundle:
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
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
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 $\mathbb{Z}_2$.
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
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
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
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 $\mathbf{H}$.
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:
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).
The 2-groupoid of string structure on $X$ is the homotopy fiber of $\frac{1}{2}\mathbf{p}_1^X$: the (∞,1)-pullback
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.
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:
(…)
The 2-groupoid of string structures is the homotopy fiber of
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.
(…)
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
Apply the pasting law for (∞,1)-pullbacks on the diagram
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
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.
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).
Every string structure in the sense of def. 2 induces a string structure in the sense of def. 3.
Consider the pasting diagram of (∞,1)-pullbacks
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:
string structure, differential string structure
smooth ∞-group | Whitehead tower of smooth moduli ∞-stacks | G-structure/higher spin structure | obstruction |
---|---|---|---|
$\vdots$ | |||
$\downarrow$ | |||
fivebrane 6-group | $\mathbf{B}Fivebrane$ | fivebrane structure | second fractional Pontryagin class |
$\downarrow$ | |||
string 2-group | $\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$ | string structure | first fractional Pontryagin class |
$\downarrow$ | |||
spin group | $\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$ | spin structure | second Stiefel-Whitney class |
$\downarrow$ | |||
special orthogonal group | $\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$ | orientation structure | first 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)
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
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
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 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
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