generalized vielbein


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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.


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) \,.


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


  • 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.


Section Fields at

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