# nLab generalized vielbein

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

An ordinary vielbein/orthogonal structure is a reduction of the structure group of the tangent bundle of a smooth manifold from the general linear group ${\mathrm{GL}}_{n}$ to its maximal compact subgroup, the orthogonal group.

Accordingly, whenever we have a reduction of structure groups along the inclusion $H↪G$ of a maximal compact subgroup, we may speak of a generalized vielbein.

## Definition

Let $G$ be a Lie group and let $H↪G$ be the inclusion of a maximal compact subgroup. Write

$i:BH\to BG$i : \mathbf{B}H \to \mathbf{B}G

for the induced morphism of smooth moduli stacks of principal bundles.

Notice that

1. these form a bundle
$\begin{array}{ccc}G/H& \to & BH\\ & & {↓}^{i}\\ & & BG\end{array}$\array{ G/H &\to& \mathbf{B}H \\ && \downarrow^{\mathrlap{i}} \\ && \mathbf{B}G }

exhibiting the coset $G/H$ as the homotopy fiber of $i$;

1. under geometric realization $i$ becomes an equivalence

$\mid i\mid :\mid BH\mid =BH\simeq BG=\mid BG\mid${\vert i\vert} : {\vert \mathbf{B} H\vert} = B H \simeq B G = {\vert \mathbf{B}G\vert}

Then for $X$ a smooth manifold or more generally a smooth infinity-groupoid equiped with a map $g:X\to BG$ an $i$-generalized vielbein is a lift $e$ in

$\begin{array}{ccc}& & BH\\ & {}^{e}↗& {↓}^{i}\\ X& \stackrel{g}{\to }& BG\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && \mathbf{B}H \\ & {}^{\mathllap{e}}\nearrow & \downarrow^{\mathrlap{i}} \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.

The moduli space of $i$-generalized vielbeing relative $g$ is the twisted cohomology

${H}_{/BG}\left(g,i\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}_{/\mathbf{B}G}(g,i) \,.

## Properties

• Locally on $X$ the moduli space of generalized vielbeins is the coset $G/H$.

## Examples

• The ordinary notion of vielbein is obtained for

$\begin{array}{ccc}{\mathrm{GL}}_{n}/O\left(n\right)& \to & BO\left(n\right)\\ & & ↓\\ & & B{\mathrm{GL}}_{n}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ GL_n/O(n) &\to& \mathbf{B}O(n) \\ && \downarrow \\ && \mathbf{B}GL_n } \,.
• in the context of generalized complex geometry one considers generalized vielbeins arising from reduction along $O\left(n\right)×O\left(n\right)\to O\left(n,n\right)$ of the generalized tangent bundle

$\begin{array}{ccc}O\left(n\right)\O\left(n,n\right)/O\left(n\right)& \to & B\left(O\left(n\right)×O\left(n\right)\right)\\ & & ↓\\ & & BO\left(n,n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B}(O(n) \times O(n)) \\ && \downarrow \\ && \mathbf{B}O(n,n) } \,.
• in the context of exceptional generalized geometry one considers vielbeins arising from reduction along ${H}_{n}\to {E}_{n}$ for ${E}_{n}$ an exceptional Lie group.

## References

Section Fields at

Revised on January 10, 2013 17:58:01 by Urs Schreiber (89.204.153.52)