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 to its maximal compact subgroup, the orthogonal group.
Accordingly, whenever we have a reduction of structure groups along the inclusion of a maximal compact subgroup, we may speak of a generalized vielbein.
Let be a Lie group and let be the inclusion of a maximal compact subgroup. Write
for the induced morphism of smooth moduli stacks of principal bundles.
- these form a bundle
exhibiting the coset as the homotopy fiber of ;
under geometric realization becomes an equivalence
Then for a smooth manifold or more generally a smooth infinity-groupoid equiped with a map an -generalized vielbein is a lift in
The moduli space of -generalized vielbeing relative is the twisted cohomology
- Locally on the moduli space of generalized vielbeins is the coset .
The ordinary notion of vielbein is obtained for
in the context of generalized complex geometry one considers generalized vielbeins arising from reduction along of the generalized tangent bundle
in the context of exceptional generalized geometry one considers vielbeins arising from reduction along for an exceptional Lie group.
Section Fields at
Revised on January 10, 2013 17:58:01
by Urs Schreiber