It is familiar that Einstein gravity may be formulated in terms of a vielbein field together with a “spin connection”, subject to the constraint that the torsion vanishes. There is a little industry trying to suggestively rewrite Einstein’s field equations in this “first-order formalism” and speculating about how this might shed light on the deeper nature of gravity and possibly its quantization.

But maybe the most spectacular observation is due to Candiello and Lechner ’93 & Howe ’97 and remains little known: In the special case of 11-dimensional super-gravity, they showed that the torsion constraint alone already implies the field equations.

Even better: while constraining just the bosonic component of the super-torsion tensor to vanish implies the 11d super-gravity equations of motion, constraining the entire super-torsion tensor to vanish furthermore constrains the gravitino field strength to vanish and hence implies the purely bosonic Einstein equations in 11d. In effect the super-partners then serve as auxiliary fields that allow to encode the ordinary Einstein’s equations of motion purely as a torsion constraint.

I now explain the statement of this result in a little bit more detail.

First-order gravity

People like to say that gravity is the result of locally gauging Lorentz symmetry. A slick way to make precise what this means is to say that a spacetime is a ##G/H##-Cartan geometry, where ##G## is the symmetry group of the local model of spacetime and ##H \subset G## is the subgroup that fixes a given point.

In a component-independent way, this simply means that spacetime locally looks like the coset ##G/H## and that a field configuration of gravity is encoded in a choice of “soldering” the tangent bundle to a vector bundle whose gauge group is ##H## — mathematically this is a choice of ##H##-structure.

A physics account of this that does it right and still sticks to the component notation that you are used to from the physics textbooks is

chapter I.4 Poincaré gravity

of the excellent textbook

L. Castellani. R. D’Auria, P.Fré: Supergravity and Superstrings – A Geometric Perspective, World Scientific (1991)

The original is

For instance take the local model spacetime to be Minkowski spacetime ##\mathbb{R}^{d-1,1}##. Then

##G = \mathrm{Iso}(\mathbb{R}^{d-,1})## is the Poincaré group, ##H = O(d-1,1)## is the Lorentz group, we recover Minkowski spacetime as the coset ##\mathbb{R}^{d-1,1} \simeq \mathrm{Iso}(\mathbb{R}^{d-1,1}) / O(d-1,1) ## a choice of ##H##-structure is precisely a choice of vielbein, hence of metric; in components, the vielbein is locally a choice of 1-forms ##E^a = E^a_\mu d x ^\mu## with ##a \in \{0, \cdots, d-1\}##, and the metric that it enoces is ## ds^2 = E^a E^b \eta_{a b}##.

This is often called the “first-order formulation of gravity”.

Torsion

On our local model spacetime, the Minkowski spacetime ##\mathbb{R}^{d-1,1}##, there is a canonical vielbein field: this is because we have canonical coordinates ##x^\mu = x^a## and hence may set

$$ E^a = d x^a \,.$$

This canonical Minkowski vielbein field has the obvious but important property that it is translation invariant, expressed by the equation

$$ d E^a = 0 \,.$$

To have a spacetime be “locally modeled” on ##G/H = \mathbb{R}^{d-1,1}##, we should not only ask that at each point of spacetime (at each “event”) the tangent space looks like ##\mathbb{R}^{d-1,1}##, but that it does so also at least in a tiny vicinity of that point (mathematically: in a first-order infinitesimal neighbourhood of that point). This condition is essentially the principle of equivalence.

That spacetime looks like the model spacetime ##G / H = \mathbb{R}^{d-1,1}## around the first-order infinitesimal neighbourhood of each point must mean that the above infinitesimal translation invariance of the vielbein field on ##\mathbb{R}^{d-1,1}## is shared by the vielbein field ##E^a## on spacetime, in that the covariant derivative

$$ \tau^a :=

abla E^a $$

vanishes:

$$ \tau^a = 0$$

Here ##\tau## is called the torsion of the ##H##structure, or just torsion for short.

This should be intuitively clear. The precise mathematical statement (and its generalization to higher order torsions) is due to the remarkable article

A more concise exposition of the key idea is in

section 3 Torsion

of

Now, in stating the above torsion constraint, we replaced the exterior derive ##d## (sufficient to express infinitesimal translation invariance on Minkowski spacetime, all whose tangent spaces are canonically identified with itself), with the covariant derivative ##

abla##, which takes care of the fact that ##E^a## is subject to a kind of gauge transformations by the Lorentz group ##H = O(d-1,1)##:

generally we should not require that the change ##d E^a## under infinitesimal translation is strictly zero, only that it is so up to a contribution from a Lorenz gauge transformation.

To choose such gauge transformation means to pick, locally, a 1-form ##\Omega## with values in the Lie algebra of ##H = O(d-1,1)##. In components this is a collection of 1-forms ##\Omega^{a b}## skew symmetric in the indices ##a## and ##b##. In terms of any such choice, then the corresponding torsion is

$$ \tau^a = d E^a + \Omega^{a}{}_b \wedge E^b $$

Here it is important to notice that, while the vielbein ##E^a## carries genuine geometric/physics information (the metric, the field of gravity), the choice of ##\Omega## is auxiliary. As we change the choice of ##\Omega## by ##\Delta \Omega##, the torsion changes by

$$\Delta \tau^a = \Delta \Omega^a{}_b E^b = \Delta \Omega_{c}{}^a{}_b E^b \wedge E^c$$.

Since this is pure gauge, the intrinsic torsion is ##\tau^a## modulo shifts of this form, and it is this intrinsic torsion which should vanish.

One may prove that the case at hand with ##H = O(d-1,1)## the Lorentz group, then the intrinsic torsion does vanish, which in turn means that given any vielbein field ##E^a## then there always exists a choice of ##\Omega##, and in fact a unique choice in this case, such that

$$ \tau^a = d E^a + \Omega^a{}_b \wedge E^b = 0\,.$$

This ##\Omega^{a}{}_b## is of course the (“spin connection” associated to) the Levi-Civita connection that is associated to the metric ##d s^2 = E^a E^b \eta_{a b}##.

Einstein equations

So far this defines a torsion-free spacetime geometry. In general however, this of course need not yet satisfy the equations of motion of gravity. In order to phrase these, one defines the Riemann curvature as the 2-form

$$ R^{a}{}_b := d \Omega^a{}_b + \tfrac{1}{2} \Omega^a{}_c \wedge \Omega^c{}_b .$$

with components

$$ R^{a}{}_b = R_{\mu

u}{}^a{}_b \, d x^\mu \wedge d x^

u $$.

The extra Einstein equations that are to be enforced to ensure that ##E^a## is actually a physical field of gravity is the equation

$$

E^{

u’ b} R_{\mu

u’}{}^{a}{}_b

– \tfrac{1}{2} R_{\mu’

u’}{}^{d e} E^{\mu’}{}_d E^{

u’}{}_e

= 0

$$

(displayed for the “vacuum” case, with no other fields sourcing gravity than gravity itself).

Super-Gravity

But we could make other choices for the spacetime symmetry group ##G##, for instance we could choose the (anti-)de Sitter group. See chapter I.4.4 of Supergravity and Superstrings – A Geometric Perspective for more on the resulting “(anti-)de Sitter gravity”.

Last time we saw that Deligne’s theorem on tensor categories says that the most general class of spacetime symmtry groups ##G## that we may choose from, in order to get a sensible collection of fundamental particle species on spacetime, are algebraic super-symmetry group.

If ##N## is a real spin representation of ##\mathrm{Spin}(d-1,1)##, then there is

a canonical super-symmetry group extension of the Poincaré group, called the super Poincaré group denoted ##G = \mathrm{Iso}(\mathbb{R}^{d-1,1\vert N})##. (Now ##N## is called the “number of super-symmetries”). Taking ##H = \mathrm{Spin}(d-1,1)## the spin group cover of the Lorentz group, this defines super Minkowski spacetime (Minkowski superspace) as the quotient ## \mathbb{R}^{d-1,1 \vert N} \simeq \mathrm{Iso}(\mathbb{R}^{d-1,1\vert N}) / \mathrm{Spin}(d-1,1) ##.

A super-Cartan geometry now is a super-spacetime locally modeled on this superspace, and an ##H##-structure on this is a choice of super-vielbein, encoding a field configuration of supergravity (graviton and gravitino). A super-vielbein is a super-1-form with local bosonic components

$$

E^a = E^a_\mu \, d x^\mu

$$

as before, together with fermionic components

$$ \Psi^\alpha = \Psi^\alpha_

u \, d x^\mu\,,$$

where the spinor index ##\alpha## ranges over a basis of that chosen spin representation ##N##. Since ##\Psi## is a 1-form with values in spinors, it is of spin 3/2, the “Rarita-Schwinger field” or “gravitino”.

Super-Torsion

In direct analogy to the previous case, there is a canonical super-vielbein on super-Minkowski spacetime ##\mathbb{R}^{d-1,1\vert N}##. First of all, the latter has canonical bosonic coordinates ##\{x^a\}## and fermionic coordinates ##\{\theta^\alpha\}##. We might hence be tempted to define

$$ E^a = d x^a \,\;\;\;\; \mathrm{and} \;\;\;\;\; \Psi^\alpha = d \theta^\alpha \;\;\;\; (\mathrm{wrong})$$

This however is wrong, in that the ##E^a## defined this way is not super-translation invariant. The correct expression is

$$ E^a = d x^a + \overline{\theta} \Gamma^a d \theta \;\;\;\; \mathrm{and} \;\;\;\;\; \Psi^\alpha = d \theta ^\alpha\,,$$

where we are using contraction of a spinor index with a charge-conjugated spinor index via a Gamma-matrix.

To see that this expression, with the correction term for ##E^a## included, is in fact super-translation invariant, notice that generally the (left- or right-) invariant 1-forms on a Lie group are those 1-forms ##\{\mu^r\}## which satisfy the Maurer-Cartan equation ##d \mu^r +\tfrac{1}{2}f_{s,t}{}^r \mu^s \wedge \mu^t = 0##, where ##f_{s,t}{}^r## are the structure constants of the given Lie algebra. Precisely the same holds true also for super-Lie groups, just with a sign thrown in whener two odd-graded elements change position.

Now, the only non-trivial bracket of the super-translation Lie algebra ##\mathbb{R}^{d-1,1\vert N}## is the famous one that pairs two spinorial transformations ##Q## to an ordinary translation ##P_a##:

$$ \{Q_1, Q_2\} = (\overline{Q_1}\Gamma^a Q_2) P_a \,.$$

W immediately see from the above formula for the corrected ##E^a## on ##\mathbb{R}^{d-1,1\vert N}## that

$$ d E^a = – \overline{\Psi} \wedge \Gamma^a \Psi \,$$

and hence this is precisely the dual Maurer-Cartan incarnation of the previous fundamental superbracket. Hence ##E^a = d x^a + \overline{\theta} \Gamma^a d\theta## is super-translation invariant (while ##d x^a## alone is not).

This basic fact of super-symmetry has the following profound implication: as opposed to ordinary bosonic Minkowski spacetime, its extension to super-Minkowski spacetime has a built-in non-vanishing torsion, with components

$$\tau_{\alpha \beta}^{a} = \Gamma^a_{\alpha \beta}{}$$

given by the Gamma-matrices.

While there is hence non-trivial torsion in super-Minkowski spacetime if we decompose it into bosonic and fermionic components, it may be more natural to regard it as a unified structure and regard the combination

$$ \tau_{\mathrm{super}} = d E^a + \overline{\Psi} \wedge \Gamma^a \Psi$$

as the super-torsion. This does vanish.

Therefore we may now generalize all of the previous story of gravity encoded in Cartan geometry to super-Cartan geometry. Hence we are asking for a super-spacetime locally modeled on ##\mathbb{R}^{d-1,1\vert N}## and equipped with a super-vielbein, locally given by bosonic 1-forms ##E^a## and fermionic 1-forms ##\Psi^\alpha##.

To enforce the condition that first-order infinitesimally around every point the super-spacetime looks like super-Minkowski spacetime (the “super-equivalence principle”) we pick a spin connection ##\Omega## as before, define the super-torsion 2-form with bosonic components

$$ \tau^a_{super} = d E^a + \Omega^a{}_b \wedge E^b + \overline{\Psi} \wedge \Gamma \Psi $$

and fermionic components

$$ \tau^\alpha_{\mathrm{super}} = d \Psi^\alpha + \tfrac{1}{4}\Omega_{a b} \Gamma^{a b} \Psi$$

and then we constrain this to be zero.

Now a miracle happens.

11-dimensional Supergravity

Choose ##d = 11## and choose ##N = 1 = \mathbf{32}## the Weyl representation of ##Spin(10,1)##. The resulting super-Minkowski spacetime is the 11-dimensional ##N = 1## super-Minkowski spacetime ##\mathbb{R}^{10,1\vert \mathbf{32}}##.

Now impose the condition that

$$\tau^a_{\mathrm{super}} = 0\,.$$

The remarkable result of

(see around equation (5.6)) with a little strengthening due to

P. Howe, Weyl Superspace, Physics Letters B Volume 415, Issue 2, 11 December 1997, Pages 149–155 (arXiv:hep-th/9707184)

(see the first two pages)

says that imposing the constraint ##\tau^a_{\mathrm{super}} = 0## — saying that the bosonic part of the super-torsion of the vielbein field vanishes — is already equivalent to the equations of motion of 11-dimensional supergravity.

(For a convenient listing of these equations and all the torsion and Bianchi identities that go into them, see pages 908-910 of Supergravity and Superstrings – A Geometric Perspective).

No further super-Einstein equations have to be imposed by hand. They are already implied (and do imply) the bosonic super-torsion constraint.

Bosonic 11d Gravity

Moreover, the remaining fermionic component ##\tau^\alpha_{\mathrm{super}}## of the super-torsion is identified with the covariant derivative of the Rarita-Schwinger field, hence with the “gravitino field strength”.

$$ \tau^\alpha_{\mathrm{super}} = \mathcal{D} \Psi^\alpha \,.$$

Hence setting also this to zero, hence constraining the entire super-torsion to vanish, further restricts from general solutions to 11-dimensional supergravity to purely bosonic solutions of 11-dimensional gravity.

Namely, since the supergravity equations of motion (obtained via the above from ##\tau^a_{\mathrm{super}} = 0##) imply that the components ##(\tau^\alpha_{\mathrm{super}})_{\mu \beta}## are given by the field strength of the supergravity C-field — the “4-form flux” — this also implies that this 4-flux vanishes in these bosonic solutions.

What remains is an ordinary field of gravity (in 11 d) subject to the ordinary bosonic vacuum Einstein equations (in 11d).

We obtained all this from imposing nothing but the super-torsion constraint, hence from asking nothing but for a super-spacetime that in each first-order infinitesimal neighbourhood of any point looks like ##11d## ##N = 1## super-Minkowski spacetime ##\mathbb{R}^{10,1\vert \mathbf{32}}##.

Outlook

Incidentally, it is precisely these bosonic and flux-less solutions of 11-dimensional supergravity which give the semi-realistic Kaluza-Klein compactifications on ##G_2##-manifolds down to 3+1 dimensions.

This is the beginning of another story, which I may tell another time.