nLab ninebrane structure

Redirected from "ninebrane structures".
Contents

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.