nLab vielbein

Idea

A vielbein or solder form on a manifold $X$ is a linear identification of the tangent bundle with a vector bundle with explicit (pseudo-)orthogonal structure.

Any such choice encodes a (pseudo-)Riemannian metric on $X$.

Definition

There are different equivalent perspectives on the notion of vielbein that are closely related:

• In terms of Cartan geometry

• In terms of orthogonal structure

In terms of Cartan geometry

Let $X$ be a smooth manifold of dimension $d$. For definiteness we assume here that $X$ is oriented, but this is not necessary.

A Lie algebra valued 1-form

$$(E,\Omega) : T X \to \mathfrak{iso}(d)$$

with values in the Poincaré Lie algebra encodes a pseudo-Riemannian metric on $X$ (if non-degenerate, at least). In this context the component

$$E : T X \to \mathbb{R}^d$$

of the connection 1-form is called the vielbein . It encodes the metric by

$$g = \langle E \otimes E\rangle \in Sym^2_{C^\infty(X)} \Gamma(T^* X) \,,$$

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.

In terms of orthogonal structure

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.

The class of the tangent bundle

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

$$\phi_j\circ \phi_i^{-1} : U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n$$

form the Jacobian matrix of smooth functions

$$((\lambda_{i j})^{\mu}{}_{\nu}) \coloneqq \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu) \right] : U_i \cap U_j \to GL_n$$

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

$$(\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu} = (\lambda_{i k})^\mu{}_{\nu} \,,$$

hence the equation

$$\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$$

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 $GL(n)$ ):

$$[(\lambda_{i j})] \in H^1_{smooth}(X, GL_n) \,.$$

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

  $$C(\{U_i\}) = (\coprod_{i, j} U_i \cap U_j \stackrel{\to}{\to} \coprod_i U_i)$$

• any Lie group $G$ induces its delooping Lie groupoid

  $$\mathbf{B}G = \left( G \stackrel{\to}{\to} * \right) \,.$$

The above situation is neatly encoded in the existence of a diagram of Lie groupoids of the form

$$\array{ C(\{U_i\}) &\stackrel{\lambda}{\to}& \mathbf{B} GL(n). \\ {}^{\mathllap{\simeq}}\downarrow \\ X } \,,$$

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

$$T X : X \to \mathbf{B} GL(n) \;\; \in \mathbf{H} \,.$$

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.

Reduction of the structure group

Consider the defining inclusion of the orthogonal group into the general linear group

$$O(n) \hookrightarrow GL(n) \,.$$

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

$$\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n) \,.$$

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.

$$\array{ X &&\stackrel{h}{\to}&& \mathbf{B} O(n) \\ & {}_{\mathllap{\lambda}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\mathrlap{\mathbf{c}}} \\ && \mathbf{B}GL(n) } \,.$$

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

  $$((h_{i j}){}^a{}_b) : U_i \cap U_j \to O(n)$$

  such that

  $$h_{i j} \cdot h_{j k} = h_{i k} \,;$$

• the homotopy $E$ is on each chart a function

$$E_i = ((E_i)^a{}_\mu) : U_i \to GL(n)$$

• 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

  $$(E_i)^a{}_{\mu} (\lambda_{i j})^{\mu}{}_\nu = (h_{i j})^a{}_b (E_j]^b{}_\nu$$

  holds. This exhibits the naturality diagram of $E$:

  $$\array{ * &\stackrel{E_i}{\to}& * \\ {}^{\mathllap{h_{i j}}}\downarrow && \downarrow^{\mathrlap{\lambda_{i j}}} \\ * &\stackrel{E_j}{\to}& * }$$

Such a lift $(h,E)$ is an orthogonal structure on $T X$. The component $E$ is called the corresponding vielbein. It exhibits an isomorphism

$$E : T X \stackrel{\simeq}{\to} V$$

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 $O(n)$.

Moduli space of orthogonal structures: twisted cohomology

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

$$\array{ GL(n)/O(n) &\to& \mathbf{B} O(n) \\ && \downarrow \\ && \mathbf{B} GL(n) }$$

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

$$\mathbf{H}(X, O(n)\backslash GL(n)) = C^\infty(X, O(n)\backslash GL(n)) \,.$$

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

$$\array{ O(n)\backslash GL(n) &\to& (O(n)\backslash GL(n)) // GL(n) \\ && \downarrow \\ && \mathbf{B}GL(n) } \,,$$

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

$$\array{ T X \times_{GL(n)} (O(n)\backslash GL(n)) &\to& O(n)\backslash GL(n) // GL(n) \\ \downarrow && \downarrow \\ X &\stackrel{T X}{\to}& \mathbf{B}GL(n) }$$

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.

Moduli stack of orthogonal structures

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 $\mathbf{c} : \mathbf{B}O \to \mathbf{B}GL(n)$

$$O Struc_{TX}(X) \simeq \mathbf{H}_{/\mathbf{B}GL(n)}(T X, \mathbf{c}) \,.$$

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 $\mathbf{H}$.

$$O \mathbf{Struc}_{T X}(X) = [X, \mathbf{B}O]_{\mathbf{B}GL(n)}$$

Differential refinement: Spin connection

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

$$\mathbf{c}_{conn} : \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn} \,.$$

The homotopy fiber of this differential refinement is still the same as before

$$\array{ GL(n)/ O(n) &\to& \mathbf{B} O(n)_{conn} \\ && \downarrow \\ && \mathbf{B} GL(n)_{conn} } \,,$$

so that the moduli space of “differential vielbein fields” is the same as that of plain vielbein fields.

Consider an affine connection

$$\nabla_{T X} : X \to \mathbf{B}GL(n)$$

hence a $GL(n)$-principal connection which locally on out atas is given by the Christoffel symbols

$$\Gamma_i = ((\Gamma_i)_\mu{}{}^{\alpha}{}_\beta) \in \Omega^1(U_i, \mathfrak{gl}(n)) \,.$$

A lift $(\nabla_V, E)$ in

$$\array{ X &&\stackrel{\nabla_{V}}{\to}&& \mathbf{B}O_{conn} \\ & {}_{\mathllap{\nabla_{T X}}}\searrow &\swArrow_{E^{-1}}& \swarrow_{\mathbf{c}_{conn}} \\ && \mathbf{B}GL(n)_{conn} }$$

is in components a “spin connection”

$$\omega_\mu = E d E^{-1} + E \Gamma_\mu E^{-1}$$

$$\omega_\mu{}^a{}_b = E^a{}_\nu \partial_\mu E^\nu{}_b + E^a{}_\nu \Gamma_\mu{}^\nu{}_\lambda E^\lambda{}_b \,.$$

This is the standard formula for the relation between the Christoffel symbols and the spin connection in terms of the vielbein.

Generalized and exceptional vielbein fields

The above discussion seamlessly generalizes to many other related cases. For instance

1. For the coefficient bundle

   $$\array{ O(n)\backslash O(n,n) /O(n) &\to& \mathbf{B} (O(n) \times O(n)) \\ && \downarrow \\ && \mathbf{B}O(n,n) }$$

   one gets the generalized vielbein of type II geometry;

2. for the coefficient bundle

   $$\array{ H_n\backslash E_n &\to& \mathbf{B} H_n \\ && \downarrow \\ && \mathbf{B} E_n }$$

   coming from the inclusion of the maximal compact subgroup of an exceptional Lie group one gets generalized vielbein fields for exceptional generalized geometry;

3. for the coefficient bundle

   $$\array{ && \mathbf{B} E_8 \\ && \downarrow^{\mathbf{a}} \\ && \mathbf{B}^3 U(1) }$$

   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.

In terms of basic forms on the frame bundle

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

$$\tau_{b} \in \Omega^1(Fr(X), \mathbb{R}^n)$$

defined on a tangent vector $v\in \Gamma_{f \in Fr(X)}$ by

$$\tau_b(v) \coloneqq f^{-1}(d \pi(v)) \,,$$

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

$$\tau = \sigma^\ast i^\ast \tau_b \,.$$

(exposition of this in the wider context of integrability of G-structures includes Lott 90, p. 4).

Related concepts

• reduction of structure groups, orthogonal structure

• integrability of G-structures – Examples – Orthogonal structure

• super-vielbein

• generalized vielbein, exceptional generalized geometry

See also at field (physics) the section on Ordinary gravity.

References

Original discussion in Élie Cartan‘s reformulation of Riemannian geometry (Cartan geometry):

• Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Chapter 1 of: Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]

Lecture notes:

• Zuoqin Wang, The method of moving frames, Lecture 11 in: Riemannian geometry (2024) [webpage, pdf]

Review in the context of first-order formulation of gravity:

• Pietro Fré, §3.2.5, §5.2.1 in: Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [doi:10.1007/978-94-007-5361-7]

Discussion in the general context of G-structures includes

• John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)