group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A vielbein or solder form on a manifold $X$ is a linear identification of a tangent bundle with a vector bundle with explicit orthogonal structure.
Any such choice encodes a Riemannian metric on $X$.
There are different equivalent perspectives on the notion of vielbein that are closely related:
Let $X$ be a smooth manifold of dimension $d$. For definitness we assume here that $X$ is oriented, but this is not necessary.
with values in the Poincaré Lie algebra encodes a pseudo-Riemannian metric on $X$ (if non-degenerate, at least). In this context the component
of the connection 1-form is called the vielbein . It encodes the metric by
where $\langle -,-\rangle : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ is the canonical bilinear form.
In other words, Given an $(SO(d) \hookrightarrow ISO(d))$-Cartan connection on $X$, the vielbein is the isomorphism in the definition of Cartan connection.
For $d=4$ this is the vierbein , for $d = 3$ the dreibein , etc.
This terminology is used notably in the context of the first-order formulation of gravity.
We discuss here how a choice of vielbein on a manifold is equivalently the reduction of the structure group of the tangent bundle from the general linear group $GL(n)$ to its maximal compact subgroup, the orthogonal group.
The following also introduces the description of this in terms of smooth twisted? cohomology. While of course this is not necessary to understand vielbeins, it does give a very natural conceptual description with the advantage that it seamlessly generalizes to notions of generalized vielbein fields and generally to twisted differential c-structures.
For completeness, we first review how the tangent bundle of a smooth manifold is naturally incarnated as a cocycle in $GL(n)$-valued Cech cohomology and how this in turn is naturally formulated in terms of Lie groupoids/smooth moduli stacks. The reader familiar with these basics should skip to the next section.
Let $X$ be a smooth manifold of dimension $n$.
By definition this means that there is an atlas $\{\phi_i^{-1} : \mathbb{R}^n \simeq U_i \hookrightarrow X\}$ of coordinate charts. On each overlap $U_i \cap U_j$ of two coordinate charts the partial derivatives of the corresponding coordinate transformations
form the Jacobian matrix of smooth functions
with values in invertible matrices, hence in the general linear group $GL(n)$. By construction (by the chain rule), these functions satisfy on triple overlaps of coordinate charts the matrix product equations
hence the equation
in the group $C^\infty(U_i \cap U_j, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlap.
This is the cocycle condition for a smooth Cech cocycle in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the sheaf of smooth functions with values in $GL(n)$ ):
It is useful to formulate this statement in the language of Lie groupoids/differentiable stacks.
$X$ itself is trivially a Lie groupoid $(X \stackrel{\to}{\to} X)$;
from the atlas $\{U_i \to X\}$ we get the corresponding Cech groupoid
any Lie group $G$ induces its delooping Lie groupoid
The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form
where
the left morphism is stalk-wise (around small enough neighbourhoods of each point) an equivalence of groupoids;
the horizontal functor has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.
We want to think of such a diagram as being directly a morphism of smooth groupoids
This is true in the (2,1)-category $\mathbf{H}$ in which stalkwise equivalences $W \subset Mor(PSh(SmthMfd, Grpd))$ have been formally inverted to become homotopy equivalences.
Since all real vector bundles on $X$ are encoded by such morphisms, as are their gauge transformations, we say that $\mathbf{B} GL(n)$ is the moduli stack for real vector bundles.
Of course there is a “smaller” Lie groupoid that also classifies real vector bundles. Passing to this “smaller” Lie groupoid is what the choice of vielbein accomplishes, to which we now turn.
Consider the defining inclusion of the orthogonal group into the general linear group
We may understand this inclusion geometrically in terms of the canonical metric on $\mathbb{R}^n$, but we may also understand it purely Lie theoretically as the the inclusion of the maximal compact subgroup of $GL(n)$. This makes manifest that the inclusion is trivial at the level of homotopy theory (it is a homotopy equivalence) and hence only encodes geometric information.
The inclusion induces a corresponding morphism of moduli stacks
A choice of orthogonal structure on $T X$ a G-structure for $G = O(n)$, hence is a factorization of the above $GL(n)$-valued cocycle through $\mathbf{c}$, up to a smooth homotopy.
This consists of two pieces of data
the morphism $h$ is a $O(n)$-valued 1-cocycle – a collection of orthogonal transition functions – hence on each overlap of coordinate patches a smooth function
such that
the homotopy $E$ is on each chart a function
such that on each double overlap it intertwines the transition functions $\lambda$ of the tangent bundle with the new orthogonal transition functions, meaning that the equation
holds. This exhibits the naturality diagram of $E$:
Such a lift $(h,E)$ is an orthogonal structure on $T X$. The component $E$ is called the corresponding vielbein. It exhibits an isomorphism
between a vector bundle $V \to X$ with structure group explicitly being the orthogonal group and the tangent bundle, hence it exhibits the reduction of the structure group of $T X$ from $GL(n)$ to $O(n)$.
In order to understand the space of choices of vielbein fields on a given tangent bundle, hence the moduli space or moduli stack of orthogonal structures/Riemannian metrics on $X$, it is useful to first consider the homotopy fiber of the morphism $\mathbf{c} : \mathbf{B}O(n) \to \mathbf{B}GL(n)$. One finds that this is the coset $O(n) \backslash GL(n)$. We may think of the fiber sequence
as being a bundle in $\mathbf{H}$ over the moduli stack $\mathbf{B}GL(n)$ with typical fiber $GL(n)/O(n)$. It is the smooth associated bundle to the smooth universal GL(n)-bundle induced by the canonical action of $GL(n)$ on $O(n)\backslash GL(n)$.
This means that if the tangent bundle $T X$ is trivializable, then the coset space $O(n)\backslash GL(n)$ is the moduli space for vielbein fields on $T X$, in that the space of these is
However, if $T X$ is not trivial, then this is true only locally: there is then an atlas $\{U_i \to X\}$ such that restricted to each $U_i$ the moduli space of vielbein fields is $C^\infty(U_i, GL(n)/ O(n))$, but globally these now glue together in a non-trivial way as encoded by the tangent bundle: we may say that
the tangent bundle twists the functions $X \to GL(n)/O(n)$. If – as we may – we think of an ordinary such function as a cocycle in degree-0 cohomology, then this means that a vielbein is a cocycle in $T X$-_twisted cohomology_ relative to the twisting coefficient bundle $\mathbf{c}$.
We can make this more manifest by writing equivalently
where now on the right we have inserted the fibration resolution of the morphism $\mathbf{c}$ as provided by the factorization lemma: this is the morphism out of the action groupoid of the action of $GL(n)$ on $O(n)\backslash GL(n)$.
The pullback
give the non-linear $T X$-associated bundle whose space of sections is the “twisted $O(n)\backslash GL(n)$-0-cohomology”, hence the space of inequivalent vielbein fields.
The above says that the space of vielbein fields is the cohomology of $T X$ in the slice (2,1)-topos $\mathbf{H}_{/\mathbf{B}GL(n)}$ with coefficients in $\mathbf{c} : \mathbf{B}O \to \mathbf{B}GL(n)$
But also this space of choices of vielbein fields has a smooth structure, it is itself a smooth moduli stack. This is obtained by forming the internal hom in the slice over $\mathbf{B}GL(n)$ of the locally cartesian closed (2,1)-category $\mathbf{H}$.
We may further lift this discussion to differential cohomology to get genuine differential $T X$-twisted $\mathbf{c}$-structures.
Write $\mathbf{B}G_{conn}$ for groupoid of Lie-algebra valued forms. As an object of $\mathbf{H} =$ SmoothGrpd this the moduli stack of $G$-connections.
The morphism $\mathbf{c}$ has an evident differential refinement
The homotopy fiber of this differential refinement is still the same as before
so that the moduli space of “differential vielbein fields” is the same as that of plain vielbein fields.
Consider an affine connection
hence a $GL(n)$-principal connection which locally on out atas is given by the Christoffel symbols
A lift $(\nabla_V, E)$ in
is in components a “spin connection”
This is the standard formula for the relation between the Christoffel symbols and the spin connection in terms of the vielbein.
The above discussion seamlessly generalizes to many other related cases. For instance
For the coefficient bundle
one gets the generalized vielbein of type II geometry;
for the coefficient bundle
coming from the inclusion of the maximal compact subgroup of an exceptional Lie group one gets generalized vielbein fields for exceptional generalized geometry;
for the coefficient bundle
coming from the second smooth universal Chern class of E8 one gets part of the geometry of the supergravity C-field
and so on. More examples are discussed for instance at twisted smooth cohomology in string theory.
A G-structure on $X$ for $G = O(n)$ the orthogonal group is equivalently an $O(n)$-principal subbundle of the frame bundle $\pi \colon Fr(X)\to X$.
This frame bundle carries a universal “basic” $\mathbb{R}^n$-valued differential form
defined on a tangent vector $v\in \Gamma_{f \in Fr(X)}$ by
where $d\pi \colon T Fr(X)\to T X$ is the differential of the bundle projection $\pi$ and $f$ is the given frame regarded as a linear isomorphism $f\colon \mathbb{R}^n \stackrel{\simeq}{\longrightarrow} T_x X$.
Then given an orthogonal structure in the form of an $O(n)$-subbundle $i \colon Fr_O(X) \hookrightarrow Fr(X)$ and given finally a local section $\sigma$ of $Fr_O(X)$, then the vielbein field with respect to that local trivialization is the pullback form
(exposition of this in the wider context of integrability of G-structures includes Lott 90, p. 4).
See also at field (physics) the section on Ordinary gravity.
Discussion in the general context of G-structures includes