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 smooth 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 was originally recognized in the physics of spinning strings (super string theory) in higher dimensional generalization of the theory (from half a century earlier) of spin structures for spinning particles, therefore the name string structure.

The coinage “string structure” is due to p. 585 (shown on the right) of:

• Timothy Killingback: World-sheet anomalies and loop geometry, Nuclear Physics B 288 578-588 (1987) [doi:10.1016/0550-3213(87)90229-X, inSpire:237516, pdf]

where this structure is understood as a de-transgression (both the naming and this insight are often incorrectly attributed to various mathematicians) of the required “spin structure on loop space”, due to

• Edward Witten: The Index of the Dirac Operator in Loop Space, in Elliptic Curves and Modular Forms in Algebraic Topology (Princeton, N.J., Sep 1986), Lecture Notes in Mathematics 1326, Springer (1988) [doi:10.1007/BFb0078045]

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

which arises as an anomaly cancellation condition on the worldsheet of the superstring‘s (specifically the heterotic string’s) sigma-model, analogous to how the condition of spin structure may be understood (Witten 1985) as an anomaly cancellation on the worldline of spinning particles such as superparticles.

See also:

• R. Coquereaux, K. Pilch: String structures on loop bundles, Commun. Math. Phys. 120 (1989) 353–378 (1989) [doi:10.1007/BF01225503]

The understanding of Killingback’s string structure equivalently as the condition to lifting the structure group of the tangent bundle from a spin group to a string group (or string 2-group) was introduced or at least popularized by:

• Stephan Stolz, Peter Teichner: What is an elliptic object?, in: Topology, geometry and quantum field theory, London Math. Soc. LNS 308, Cambridge Univ. Press (2004) 247-343 [pdf, pdf]

The relation between the two pictures is further 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 mathematically rigorous formulation of Killingback 1987‘s original argument, now using differential K-theory, is given by

• 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]

in turn reviewed 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 (also already mentioned by Killingback 1987) is used in

• Corbett Redden, Canonical metric connections associated to string structures PhD Thesis (2006) [pdf]

• Redden & Waldorf 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)