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 $g : X \to \mathcal{B} Spin(n)$ through the third nontrivial step $\mathcal{B}String(n) \to \mathcal{B}Spin(n)$ in the Whitehead tower of $B O(n)$ 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 $B O(n)$ 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 $B O(n)$ 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 $B S O(n)$ 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:

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

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 :

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

• 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 $S : X \to \mathbf{B}Spin$. 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 $H^3(Spin(n), \mathbb{Z}) \simeq \mathbb{Z}$.

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

Proposition

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

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

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

Related entries

• orientation

• spin structure, twisted spin structure

• p1-structure, Atiyah 2-framing

• string structure

  differential string structure

  stringor bundle

• fivebrane structure, differential fivebrane structure

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	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)

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

• T.P. Killingback?, World-sheet anomalies and loop geometry Nuclear Physics B Volume 288, 1987, Pages 578-588 doi:10.1016/0550-3213(87)90229-X

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, In: Landweber, P.S. (eds) Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, N.J., Sep 1986). Lecture Notes in Mathematics 1326 (1988) Springer, Berlin, Heidelberg. doi:10.1007/BFb0078045

• Edward Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525–536 doi:10.1007/BF01208956

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)

A precise formulation of Killingback’s original argument in differential K-theory appeared in

• Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle, Commun. Math. Phys. 307 (2011) : 675-712 [arXiv:0909.0846, doi:10.1007/s00220-011-1348-0]

A review of that is in

• Konrad Waldorf, Geometric string structure and supersymmetric sigma-models (pdf)

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

• Oberwolfach Workshop, June 2009 – Friday, June 12

Discussion of the moduli stack of twisted differential string structures:

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics 315 1 (2012) 169-213 [arXiv:0910.4001, doi:10.1007/s00220-012-1510-3]

and its appearance on M5-branes under Hypothesis H:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies twisted String structure on M5-branes, J. of Mathematical Physics 62 (2021) 042301 [arXiv:2002.11093, doi:10.1063/5.0037786]

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

• David Roberts, Explicit string bundles, talk at Workshop on Higher Gauge Theory and Higher Quantization 2014, arXiv:2203.04544, (pdf).

Discussion for indefinite (Lorentzian) signature is in

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

More discussion of relation to spin structures on loop spaces is in

• Alessandra Capotosti, From String structures to Spin structures on loop spaces, Ph.D. thesis, Università degli Studi Roma Tre, Rome, April 2016

• Konrad Waldorf, String structures and loop spaces, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.12998]

A study of (flat) string structures encoded in the bicategory of flat 2-group bundles on an oriented surface via weak representations of the fundamental group is in

• Daniel Berwick-Evans, Emily Cliff, Laura Murray, Apurva Nakade, Emma Phillips, String structures, 2-group bundles, and a categorification of the Freed-Quinn line bundle (arXiv:2110.07571)