nLab
Fivebrane structure

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

String theory

Contents

Idea

The notion of Fivebrane structure is the next higher analog of that of spin structure and string structure.

Recall from the discussion there that a string structure on manifold XX with spin structure is a lift g^\hat g of the classifying map g:XBSpin(n)g : X \to B Spin(n) of the tangent bundle associated to a Spin group-principal bundle through the next step in the Whitehead tower of O(n)O(n), called BString(n)B String(n) – the delooping of the String group:

BString(n) g^ X g BSpin(n). \array{ && B String(n) \\ & {\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& B Spin(n) } \,.

The names “Spin” and “String” both derive from the role these structures play in quantum field theory: a spin structure is required on XX for it to serve as a target space for spinning particles (superparticles), while a string structure is required for it to serves as a target for “spinning strings” – superstrings – (see heterotic string theory for more). Topologists just say (said) O(n)2O(n)\langle 2\rangle for Spin(n)Spin(n) and O(n)6O(n)\langle 6\rangle for String(n)String(n), respectively.

They wrote O(n)8O(n)\langle 8\rangle for the next step in the Whitehead tower of O(n)O(n) (note that this is only the next step for n>6n \gt 6; for lower nn there are intermediate steps, as can be seen in the table at orthogonal group).

It was Hisham Sati who first realized that a lift of the tangent bundle TXT X to this highly connected structure group is related to XX serving as a target for “spinning 5-branes” – super-5-branes – in what is called dual heterotic string theory. Following the history of the term String group he gave the topological group O(n)8O(n)\langle 8\rangle the name Fivebrane group: Fivebrane(n)Fivebrane(n).

Accordingly, a Fivebrane structure(n) on a manifold XX with string structure is a lift of g^:XBString(n)\hat g : X \to B String(n) to g^^\hat \hat g

BFivebrane(n) g^^ X g^ BString(n). \array{ && B Fivebrane(n) \\ & {\hat \hat g}\nearrow & \downarrow \\ X &\stackrel{\hat g}{\to}& B String(n) } \,.

The obstruction class to this lift is a fractional multiply of the second Pontrjagin class. Namely the generator of H 8(BString,)H^8(B String, \mathbb{Z}) is 16p 2\frac{1}{6}p_2,

BString 1/6p 2 B 8 6 BSO p 2 B 8. \array{ B String &\stackrel{1/6 p_2}{\to}& B^8 \mathbb{Z} \\ \downarrow && \downarrow^{\mathrlap{\cdot 6}} \\ B SO &\stackrel{p_2}{\to}& B^8 \mathbb{Z} } \,.

(stated in SSS2, then in DHH, also follows from the index theory argument leading to (3.3) in Witten 96).

The Fivebrane group is the loop space object of the corresponding homotopy fiber

BFivebrane * BString 16 B 7U(1) \array{ B Fivebrane &\to& * \\ \downarrow && \downarrow \\ B String &\stackrel{\frac{1}{6}}{\to}& B^7 U(1)& }

and so, by the universal property of the homotopy pullback, String-structures g^\hat g lift to Fivebrane structures precisely if 16p 2(g^)\frac{1}{6}p_2(\hat g) is trivial in cohomology

BFivebrane * g^^ X g^ BString 16p 2 B 7U(1). \array{ && B Fivebrane &\to& * \\ & {}^{\mathllap{\hat \hat g}}\nearrow & \downarrow && \downarrow \\ X &\stackrel{\hat g}{\to}& B String &\stackrel{\frac{1}{6}p_2}{\to}& B^7 U(1) } \,.

In (SSS2) the physical interpretation of this topological lift was established by comparison with known quantum anomaly cancellaton conditions in dual heterotic string theory.

The term “Fivebrane” apparently quickly caught on in the mathematical community, for instance in (DouglasHenriquesHill).

Since gauge theory is not just about principal bundles, but about principal bundles with connection, what matters in physics is not just the topological Spin-, String- and Fivebrane structures, but their refinement to differential nonabelian cohomology. See differential fivebrane structure.

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
\vdots
\downarrow
ninebrane 10-groupBNinebrane\mathbf{B}Ninebrane ninebrane structurethird fractional Pontryagin class
\downarrow
fivebrane 6-groupBFivebrane1np 3B 11U(1)\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)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 notion was introduced in

It is briefly mentioned in

Related structures are also mentioned around p. 9 of

The differential refinement is discussed in

and

Articles that use Fivebrane structures include

  • Boris Botvinnik, Mohammed Labbi, Highly connected manifolds of positive pp-curvature (arXiv:1201.1849)

Revised on November 27, 2014 06:49:48 by David Roberts (129.127.210.201)