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 $GL_n$ 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.

Related concepts

• super-vielbein

References

For instance

• Ralph Blumenhagen, Pascal du Bosque, Falk Hassler, Dieter Lüst: section 5.1 of: Generalized Metric Formulation of Double Field Theory on Group Manifolds, J. High Energ. Phys. 2015 56 (2015) [arXiv:1502.02428, doi:10.1007/JHEP08(2015)056]

See also the section Fields at

• geometry of physics