# nLab generalized vielbein

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $GL_n$ to its maximal compact subgroup, the orthogonal group.

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

## Definition

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

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

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

Notice that

1. these form a bundle
$\array{ G/H &\to& \mathbf{B}H \\ && \downarrow^{\mathrlap{i}} \\ && \mathbf{B}G }$

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

1. under geometric realization $i$ becomes an equivalence

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

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

$\array{ && \mathbf{B}H \\ & {}^{\mathllap{e}}\nearrow & \downarrow^{\mathrlap{i}} \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.$

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

$\mathbf{H}_{/\mathbf{B}G}(g,i) \,.$

## Properties

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

## Examples

• The ordinary notion of vielbein is obtained for

$\array{ GL_n/O(n) &\to& \mathbf{B}O(n) \\ && \downarrow \\ && \mathbf{B}GL_n } \,.$
• in the context of generalized complex geometry one considers generalized vielbeins arising from reduction along $O(n)\times O(n) \to O(n,n)$ of the generalized tangent bundle

$\array{ O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B}(O(n) \times O(n)) \\ && \downarrow \\ && \mathbf{B}O(n,n) } \,.$
• 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.

Section Fields at