nLab
ninebrane structure

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

String theory

\cdots \to ninebrane 10-group \to Fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group.

Contents

Idea

On a space with Fivebrane structure a ninebrane structure is a trivialization of (some fractional multiple of) the 3rd Pontryagin class. This is supposed to be related to parts of the quantum anomaly cancellation of the M9-brane.

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

Last revised on February 4, 2015 at 20:52:36. See the history of this page for a list of all contributions to it.