nLab
generalized vielbein

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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 nGL_n to its maximal compact subgroup, the orthogonal group.

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

Definition

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

i:BHBG 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
G/H BH i BG \array{ G/H &\to& \mathbf{B}H \\ && \downarrow^{\mathrlap{i}} \\ && \mathbf{B}G }

exhibiting the coset G/HG/H as the homotopy fiber of ii;

  1. under geometric realization ii becomes an equivalence

    |i|:|BH|=BHBG=|BG| {\vert i\vert} : {\vert \mathbf{B} H\vert} = B H \simeq B G = {\vert \mathbf{B}G\vert}

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

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

The moduli space of ii-generalized vielbeing relative gg is the twisted cohomology

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

Properties

  • Locally on XX the moduli space of generalized vielbeins is the coset G/HG/H.

Examples

  • The ordinary notion of vielbein is obtained for

    GL n/O(n) BO(n) BGL n. \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)×O(n)O(n,n)O(n)\times O(n) \to O(n,n) of the generalized tangent bundle

    O(n)\O(n,n)/O(n) B(O(n)×O(n)) BO(n,n). \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 nE nH_n \to E_n for E nE_n an exceptional Lie group.

References

Section Fields at

Revised on January 17, 2015 10:14:41 by Urs Schreiber (195.113.30.252)