# nLab torsion constraints in supergravity

Contents

### Context

#### Gravity

gravity, supergravity

supersymmetry

## Applications

#### Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

The equations of motion of supergravity typically imply – or are even equivalent to (Candiello-Lechner 93, Howe 97), that the super-torsion of the super-vielbein fields vanishes. At least in some cases these supergravity torsion constraints may naturally be understood as saying that supergravity solutions are (higher) super-Cartan geometry modeled on extended super Minkowski spacetime with its canonical torsion of a G-structure, due to the fact that the left invariant 1-forms on super-Minkowski space are not closed.

### From the canonical torsion of super-Minkowski spacetime

The torsion constraint is naturally understood by regarding supergravity as Cartan geometry for the inclusion of the orthogonal group into a super Poincare group and by noticing that the corresponding local model space, which is super-Minkowski spacetime $\mathbb{R}^{d|N}$, canonically has non-vanishing torsion.

Let $(x^a, \theta^\alpha)$ be the canonical coordinates on the supermanifold $\mathbb{R}^{d|N}$ underlying the super translation group. Then the left-invariant 1-forms are

• $\psi^\alpha = d \theta^\alpha$.

• $e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta$.

Here the extra summand in the equation for $e^a$ (necessary to make it left-invariant) causes it to be non-closed:

\begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,.

Taking the spin connection $(\omega^a{}_b)$ on $\mathbb{R}^{d|N}$ to vanish, as usual, this means that there is non-vanishing torsion:

\begin{aligned} \tau^a & = \mathbf{d} e^a + \omega^a{}_b \wedge e^b \\ & = \mathbf{d} e^a \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned}

Depending on perspective one might say that it is the supertorsion that vanishes (see at super-Minkowski spacetime and at D'Auria-Fre formulation of supergravity for this perspective), or, alternatively, that one is dealing with Cartan geometry/G-structure whose local model space carries non-vanishing torsion, see below.

Notice that the torison-full but left-invariant forms are of course obtained from the torsion-free but non-left-invartiant forms by a $GL(\mathbb{R}^{d|N})$-valued function:

$\left( \array{ e^a \\ \psi^\alpha } \right) = \left( \array{ id & \tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha \\ 0 & id } \right) \left( \array{ \mathbf{d}x^a \\ \mathbf{d}\theta^\alpha } \right)$
$\left( \array{ \mathbf{d}x^a \\ \mathbf{d}\theta^\alpha } \right) = \left( \array{ id & -\tfrac{i}{2}\Gamma^a{}_{\alpha \beta} \theta^\alpha \\ 0 & id } \right) \left( \array{ e^a \\ \psi^\alpha } \right)$

This shows that regarding

$(E^A) \coloneqq (E^a, E^\alpha) \coloneqq (e^a, \Psi^\alpha)$

as a super-vielbein is consistent: this is indeed a homotopy in

$\array{ \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N}) \\ & {}_{\mathllap{\tau_{\mathbb{R}^{d|N}}}}\searrow & \swArrow_{E} & \swarrow_{\mathrlap{O(\mathbb{R}^{d|N})\mathbf{Struc}}} \\ && \mathbf{B}GL(\mathbb{R}^{d|N}) }$

but not the tautological one given by

$\array{ \mathbb{R}^{d|N} &\to& \ast &\to& \mathbf{B}O(\mathbb{R}^{d|N}) \\ & \searrow & \downarrow & \swarrow \\ && \mathbf{B}GL(\mathbb{R}^{d|N}) }$

where the left triangle is that which exhibits the canonical trivialization of the frame bundle of $\mathbb{R}^{d|N}$.

### In terms of torsion-twisted $G$-structure

Given a subgroup $G\hookrightarrow GL(V)$ of the general linear group of a linear model space $V$ (e.g. super-Minkowski spacetime $\mathbb{R}^{d|N}$), then a G-structure is first-order integrable if on the first-order infinitesimal neighbourhoods of any point it is equal to the canonical (trivial) $G$-structure on $V$. Ordinarily the standard torsion on $V$ vanishes, and if so then so does that of any first-order integrable $G$-structure, which is the reason why for these the torsion of a G-structure vanishes.

But in the situation of $V$ being super-Minkowski spacetime as above, the torsion of the local model space does not vanish, and so accordingly neither does that of a first-order integrable $G$-structure in this case.

This perspective on the torsion constraints in supergravity is adopted in (Lott 01), see there around (38) of the original article or section 4 of the review on the arXiv.

## Properties

### Relation to supergravity equations of motion

The supergravity equations of motion typically imply the torsion constraints. See at super p-brane – On curved spacetimes for more.

With enough supersymmetry, the torsion constraints (always together with the Bianchi identities on the superfields, see at D'Auria-Fre formulation of supergravity) may even become equivalent to the supergravity equations of motion. This is so for 11-dimensional supergravity (Candiello-Lechner 93, Howe 97, see Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4) and maybe its maximally supersymmetric KK-compactifications. See at Examples – 11d SuGra.

### Relation to CR-geometry

A close analogy between CR geometry and supergravity superspacetimes (as both being torsion-ful integrable G-structures) is pointed out in (Lott 01 exposition (4.2)).

## Examples

In accord with the above, typically the equations of motion of a supergravity theory constrain the spinorial part of the torsion to have components $(\Gamma^a)_{\alpha \beta}$.

### 11d supergravity

The torsion constraint for 11-dimensional supergravity is discussed for instance in (Bergshoeff-Sezgin-Townsend 87, equation (14)).

Here something special happens:

The articles (Candiello-Lechner 93 (5.6), Howe 97, see Cederwall-Gran-Nilsson-Tsimpis 04, section 2.4) show that imposing the torsion constraint (on some chart) $\mathbf{d} E^a + \omega^{a}{}_b \wedge E^b - \bar \psi \Gamma^a \psi = 0$ as well as $(\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi)_{\theta \theta} = 0$ implies the equations of motion of 11d supergravity.

Moreover, setting $\mathbf{d} \Psi +\tfrac{1}{4}\omega^{a b} \Gamma_{a b}\Psi = 0$ generally (not just the component proportional to the wedge product of two fermionic 1-forms, hence requiring the full supertorsion tensor to be that of super-Minkowski spacetime)) then (Candiello-Lechner 93, (5.8) with (6.5)) this in addition puts the field strength of the supergravity C-field to 0. Hence this implies solutions to the ordinary vacuum Einstein equations in 11d. Such solutions are considered notably in the context of M-theory on G2-manifolds (e.g. Acharya 02, p. 9). See also at M-theory on G2-manifolds – Details – Vacuum solution and torsion constraints.

### 10d Heterotic supergravity

For heterotic supergravity in 10d the equations of motion are equivalent to the condition that

1. the super-torsion of the bosonic part $\{e^a\}$ of the super vielbein is a bosonic form

$\mathcal{D}e^a - \overline{\psi} \Gamma^a \psi = T^a{}_{b c} e^b \wedge e^c$
2. the super-torsion of the odd part $\psi^\alpha$ of the super vielbein is of the form

$\mathcal{D} \psi^\alpha = T^\alpha{}_{b c} e^b \wedge e^c + \overline{\psi}^\beta \Gamma_a{}_{\beta \gamma} \phi^{\alpha \gamma}$

for

$\phi^{\alpha \beta} \propto tr(\chi^\alpha \chi^\beta) - tr(T^\alpha T^\beta)$

proportional to the bispinor formed by tracing the square of the gaugino field $\chi$

3. the curvature 2-form of the gauge field has vanishing bispinorial component:

$F_{\alpha \beta} = 0$

(this is the 10d super Yang-Mills theory sector)

This is due to (Witten 86 (5)+(27)), see also (Atick-Dhar-Ratra 86 (4.1)). These authors do not state explicitly that $\phi^{\alpha \beta} \propto tr(\lambda^\alpha \lambda^\beta) - tr (T T)$. (Among authors using a similar but different parameterization this statement is made explicit in Candiello-Lechner 93 (2.5) with (2.29)). But this follows by taking the differential of the bispinorial part of the 3-form field (which is the cocycle term for the heterotic Green-Schwarz superstring)

$d \left( \overline{\psi} \wedge \Gamma_a \psi \wedge e^a \right) \propto \underset{i}{\sum} \underset{= F^i_{(1,1)}}{ \underbrace{ \left( \overline{\psi} \Gamma_a \chi^i \right) \wedge e^a } } \wedge \underset{ = F^i_{(1,1)}}{ \underbrace{ \left( \overline{\chi^i} \Gamma_b \psi \right) \wedge e^b } } - tr(R R)_{(2,2)}$

where we used the relation (Witten 86 (8)) (recalled for instance in Bonora-Bregola-Lechner-Pasti-Tonin 87 (2.28), Lechner-Tonin 08 (2.13)).

According to (Bonora-Bregola-Lechner-Pasti-Tonin 90) in fact all these constraints follow from just $T^a_{\alpha \beta} \propto \Gamma^a_{\alpha \beta}$.

### General

The formulation of supergravity equations of motion in terms of constraints on the torsion tensor goes back to

A mathematical formulation in terms of torsion-full first-order integrable G-structures on supermanifolds (for low dimensional supergravity theories) is given in

which is followed up in

### For 11d supergravity

Discussion of torsion constrains for 11-dimensional supergravity from the point of view of consistency of the membrane Green-Schwarz action functional is in

The claim that this torsion constraint in 11-dimensional supergravity is already equivalent to all of the equations of motion is due to

concisely reviewed in

For commentary see also (Nilsson 00, section 2) and

also

• Lars Brink, Paul Howe, Eleven-dimensional supergravity on the mass shell in superspace, Phys. Lett. , B91:384–386, 1980

Discussion of possible deformations of the torsion constraint (M-theory corrections) includes

### For 10d heterotic supergravity

Discussion of torsion constraints for heterotic supergravity goes back to (Nilsson 81) and includes

### For 4d supergravity

For d=4 N=1 supergravity the torsion is again constrained to be equal to the left-invariant torsion of super-Minkowski spacetime, see for instance

### For 2d supergravity / superstring worldsheets / super Riemann surfaces

• Suresh Govindarajan, Burt Ovrut, A geometric interpretation for the torsion constrains of $(2,0)$-heterotic worldhseet supergravity, Mod. Phys. Lett. A6(1991), 3341. (pdf)