nLab
generalized vielbein

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 $H\hookrightarrow G$ 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

$$i:BH\to BG$$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

$$\begin{array}{ccc}G/H& \to & BH\\ & & {\downarrow }^i\\ & & BG\end{array}$$\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

$$\begin{array}{ccc}& & BH\\ & {}^e\nearrow & {\downarrow }^i\\ X& \stackrel{g}{\to }& BG\end{array}.$$\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

$$H_{/BG}(g,i).$$\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

  $$\begin{array}{ccc}{\mathrm{GL}}_n/O(n)& \to & BO(n)\\ & & \downarrow \\ & & B{\mathrm{GL}}_n\end{array}.$$\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

  $$\begin{array}{ccc}O(n)\backslash O(n,n)/O(n)& \to & B(O(n)\times O(n))\\ & & \downarrow \\ & & BO(n,n)\end{array}.$$\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

• geometry of physics