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string structure

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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 g^\hat g of the classifying map g:XSpin(n)g : X \to \mathcal{B} Spin(n) through the third nontrivial step String(n)String(n)\mathcal{B}String(n) \to \mathcal{B}String(n) in the Whitehead tower of BO(n)BO(n) to a String group-principal bundle:

String(n) g^ X g Spin(n) \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 XX be an nn-dimensional topological manifold.

Its tangent bundle is canonically associated to a O(n)O(n)-principal bundle, which is in turn classified by a continuous function

XBO(n) X \to B O(n)

from XX to the classifying space of the orthogonal group O(n)O(n).

  • A String structure on XX is the choice of a lift of this map a few steps through the Whitehead tower of BO(n)BO(n).

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

This means the following:

  • there is a canonical map w 1:BO(n)B 2w_1 : B O(n) \to B\mathbb{Z}_2 from the classifying space of O(n)O(n) to that of 2=/2\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z} that represents the generator of the cohomology H 1(BO(n), 2)H^1(B O(n), \mathbb{Z}_2). The classifying space of the group SO(n)SO(n) is the homotopy pullback

    BSO(n) * BO(n) w 1 𝔹 2 \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)B O(n) with the one object groupoid whose space of morphisms is O(n)O(n) and similarly for B 2 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)O(n) to the identity and orientation reversing elements to the nontrivial element in 2\mathbb{Z}_2.

    • an orientation on XX is a choice of lift of the structure group through BSO(n)BO(n)B SO(n) \to B O(n)
      BSO(n) orientation X BO(n). \array{ && B SO(n) \\ & {}^{orientation}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B O(n) } \,.
  • there is a canonical map w 2:BSO(n)B 2 2w_2 : B SO(n) \to B^2 \mathbb{Z}_2 representing the generator of H 2(BSO(n), 2)H^2(B SO(n), \mathbb{Z}_2). The classifying space of the group Spin(n)Spin(n) is the homotopy pullback

    BSpin(n) * BSO(n) w 2 𝔹 2 2 \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 XX is a choice of lift of the structure group through BSpin(n)BSO(n)B Spin(n) \to B SO(n)
      BSpin(n) spinstructure X BSO(n). \array{ && B Spin(n) \\ & {}^{spin structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B SO(n) } \,.
  • there is a canonical map BSpin(n)B 3U(1)B Spin(n) \to B^3 U(1) The classifying space of the group String(n)String(n) is the homotopy pullback

    BString(n) * BSpin(n) 12p 1 𝔹 3U(1) \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 XX is a choice of lift of the structure group through BString(n)BSpin(n)B String(n) \to B Spin(n)
      BString(n) stringstructure X BSpin(n). \array{ && B String(n) \\ & {}^{string structure}\nearrow& \downarrow \\ X &\stackrel{}{\to}& B Spin(n) } \,.
  • there is a canonical map BString(n)B 7U(1)B String(n) \to B^7 U(1) The classifying space of the group Fivebrane(n)Fivebrane(n) is the homotopy pullback

    BFivebrane(n) * BString(n) 16p 2 𝔹 7U(1) \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 XX is a choice of lift of the structure group through BFivebrane(n)BString(n)B Fivebrane(n) \to B String(n)
      BFivebrane(n) fivebranestructure X BString(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 H=\mathbf{H} = ETop∞Grpd or Smooth∞Grpd. Write XX for a topological manifold or smooth manifold of dimension nn, respectively.

Write String(n)String(n) for the string 2-group, a 1-truncated ∞-group object in H\mathbf{H}.

Definition

The 2-groupoid of (topological or smooth) string structures on XX is the hom-space of cocycles XBString(n)X \to \mathbf{B}String(n), or equivalently that of (topological or smooth) String(n)String(n)-principal 2-bundles:

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

Write 12p 1:BSpin(n)B 3U(1)\frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin(n) \to \mathbf{B}^3 U(1) in H\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 XX is the homotopy fiber of 12p 1 X\frac{1}{2}\mathbf{p}_1^X: the (∞,1)-pullback

String(X) * H(X,BSpin(n)) 12p 1 H(X,B 3U(1)). \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 BStringBSpinp 112B 3U(1)\mathbf{B} String \to \mathbf{B}Spin \stackrel{\frac{1}{2}} \mathbf{p_1}{\to} \mathbf{B}^3 U(1). The hom-functor H(X,)\mathbf{H}(X,-) preserves every (∞,1)-limit, hence preserves this fiber sequence.

Definition

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

Twisted and differential string structures

(…)

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

12p 1:Top(X,Spin)Top(X, 4) \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 4c : X \to \mathcal{B}^4 \mathbb{Z} may be thought of as that of twisted string structures.

The Pontryagin class 12p 1\frac{1}{2}p_1 refines to the smooth first fractional Pontryagin class 12p 1:BSpinB 3U(1)\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 SS are as follows

  • if [12p 1(S)]0[\frac{1}{2}\mathbf{p}_1(S)] \neq 0 it is empty: String S(X)String_S(X) \simeq \emptyset;

  • if [12p 1(S)]=0[\frac{1}{2}\mathbf{p}_1(S)] = 0 it is String S(X)H(X,B 2U(1))String_S(X) \simeq \mathbf{H}(X, \mathbf{B}^2 U(1)).

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

    π 0String S(X)H 3(X,). \pi_0 String_S(X) \simeq H^3(X, \mathbb{Z}) \,.
Proof

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

String S(X) String(X) * * S H(X,BSpin(n)) 12p 1 H(X,B 3U(1)). \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 3U(1))\mathbf{H}(X, \mathbf{B}^3 U(1)). Since H(X,)\mathbf{H}(X,-) commutes with forming loop space objects (see fiber sequence for details) we have

String S(X)ΩH(X,B 3U(1))H(X,B 2U(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

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 XX be a manifolds with spin structure S:XBSpinS : X \to \mathbf{B}Spin. Write PXP \to X for the corresponding spin group-principal bundle.

Then a string structure lifting SS is a cohomology class H 3(P,)H^3(P,\mathbb{Z}) such that the restriction of the class to any fiber Spin(n)\simeq Spin(n) is a generator of H 3(Spin(n),mathbZ)H^3(Spin(n), \mathb{Z}) \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

String P^ * Spin P B 2U(1) * * X BString(n) BSpin(n) \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 (,1)(\infty,1)-pullbacks. The map PB 2U(1)P \to B^2 U(1) appears by decomposing the homotopy pullback of the point along XBSpin(n)X \to B Spin(n) into a homotopy pullback first along BString(n)BSpin(n)B String(n) \to B Spin(n) and then along XBString(n)X \to B String(n) using the given String structure. This is the cocycle for a BU(1)\mathbf{B}U(1)-principal 2-bundle on the total space PP of the SpinSpin-principal bundle: a bundle gerbe.

The rest of the diagram is constructed in order to prove the following:

  • The class in H 3(P,)H^3(P, \mathbb{Z}) of this bundle gerbe, represented by PB 2U(1)P \to B^2 U(1) has the property that restricted to the fibers of the Spin(n)Spin(n)-principal bundle PP it becomes the generating class in H 3(Spin(n),)H^3(Spin(n), \mathbb{Z}).

Examples

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
\vdots
\downarrow
fivebrane 6-groupBFivebrane\mathbf{B}Fivebrane fivebrane structuresecond fractional Pontryagin class
\downarrow
string 2-groupBString16p 2B 7U(1)\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)string structurefirst fractional Pontryagin class
\downarrow
spin groupBSpin12p 1B 3U(1)\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)spin structuresecond Stiefel-Whitney class
\downarrow
special orthogonal groupBSOw 2B 2 2\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2orientation structurefirst Stiefel-Whitney class
\downarrow
orthogonal groupBOw 1B 2\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2orthogonal structure/vielbein/Riemannian metric
\downarrow
general linear groupBGL\mathbf{B}GLsmooth 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

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 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

  • 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

Discussion of the moduli stack of twisted differential string structures is in

Revised on January 14, 2014 14:14:23 by Urs Schreiber (89.204.153.51)