∞-Lie theory

# Contents

## Idea

The supergravity Lie 6-algebra is a super L-∞ algebra such that ∞-connections with values in it encode

This is such that the field strengths and Bianchi identities of these fields are governed by certain fermionic super L-∞ algebraic cocycles as suitable for 11-dimensional supergravity.

## Definition

###### Proposition

The supergravity Lie 3-algebra $\mathfrak{sugra}_3(10,1)$ carries an L-∞ algebra cocycle $\mu_7 \in CE(\mathfrak{sugra}_3(10,1))$ of degree 7, given in the standard generators $\{e^a\}$ (vielbein), $\{\omega^{a b}\}$ (spin connection) $\{\psi^\alpha\}$ (gravitino) and $\{c_3\}$ (supergravity C-field) by

$\mu_7 \coloneqq \frac{i}{2}\bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5} + 15 \mu_4 \wedge c_3 \,,$

where

$\mu_4 = \frac{i}{2}\bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}$

is the 4-cocycle which defines $\mathfrak{sugra}_3(10,1)$ as an extension of $\mathbb{R}^{10,1\vert \mathbf{32}}$m and where $c_3$ is the generator that cancels the class of this cocycle, $d_{CE} c_3 \propto \mu_4$.

This appears in (DAuria-Fre, page 18) and Castellani-DAuria-Fre, III.8.3.

###### Proof

One computes

\begin{aligned} d_{CE} \mu_7 = & - \frac{5}{4} \bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_4} \wedge \bar \psi \wedge \Gamma_b \psi \\ & - i 15 \wedge \Gamma^{a b} e_a \wedge \bar \psi \wedge \Gamma_b \psi \wedge c_3 \\ & + \frac{15}{4} \bar \psi \wedge \Gamma_{a b} \psi \wedge e^a \wedge e^b \wedge \bar \psi \wedge \Gamma_{c d} \psi \wedge e^c \wedge e^d \end{aligned} \,.

This expression vanishes due to the Fierz identities

$\bar \psi \wedge \Gamma^{a_1 \cdots a_4 b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 3 \bar \psi \wedge \Gamma^{[a_1 a_2} \psi \wedge \bar \psi \wedge \Gamma^{a_3 a_4 ]} \psi$

and

$\bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 0 \,.$
###### Remark

Hence if we write

$g_4 \coloneqq \mu_4 = \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}$

and

$g_7 \coloneqq \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}$

then

$d g_7 \propto g_4 \wedge g_4 \,.$

This is the structure of the equations of motion of the field strength $G_4$ of the supergravity C-field and its Hodge dual $G_7 = \ast G_4$ in 11-dimensional supergravity.

###### Definition

The supergravity Lie 6-algebra $\mathfrak{sugra}_{7}(10,1)$ is the super Lie 7-algebra that is the $b^6 \mathbb{R}$-extension of $\mathfrak{sugra}_3(10,1)$ classified by the cocycle $\mu_7$ from def. 1.

$b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \,.$
###### Remark

This means that the Chevalley-Eilenberg algebra $CE(\mathfrak{sugra}_6)$ is generated from

• $\{e^a\}$ (vielbein) in degree $(1,even)$

• $\{\omega^{a b}\}$ (spin connection) in degree $(1,even)$;

• $\{\psi^\alpha\}$ (gravitino) in degree $(1,odd)$

• $\{c_3\}$ (supergravity C-field) in degree $(3,even)$

• $\{c_6\}$ (magnetic dual supergravity C-field) in degree $(6,even)$

with differential defined by

$d_{CE} : \omega^{a b} \mapsto \omega^{a c} \wedge \omega_c{}^b$
$d_{CE} : e^a = -\omega^{a b} e_b - \frac{1}{2}i \bar \psi \wedge \Gamma^a \psi$
$d_{CE} : \psi \mapsto - \frac{1}{4}\omega^{a b} \Gamma^{a b}$
$d_{CE} : c_3 \mapsto \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$
$d_{CE} \colon c_6 \mapsto - \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5} - \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3 \,.$

This appears as (Castellani-DAuria-Fre, (III.8.18)).

###### Remark

According to (Castellani-DAuria-Fre, comment below (III.8.18)): “no further extension is possible”.

## Relation to $D = 11$ supergravity

The supergravity Lie 6-algebra is something like the gauge $L_\infty$-algebra of 11-dimensional supergravity, in the sense discussed at D'Auria-Fre formulation of supergravity .

###### Definition

Write $W(\mathfrak{sugra}_6(10,1))$ for the Weil algebra of the supergravity Lie 6-algebra.

Write $g_4$ and $g_7$ for the shifted generators of the Weil algebra corresponding to $c_3$ and $c_6$, respectively.

Define a modified Weil algebra $\tilde W(\mathfrak{sugra}_6(10,1))$ by declaring it to have the same generators and differential as before, except that the differential for $c_6$ is modified to

$d_{\tilde W} c_6 := d_{W} c_6 + 15 g_4 \wedge c_3$

and hence the differential of $g_7$ is accordingly modified in the unique way that ensures $d_{\tilde W}^2 = 0$ (yielding the modified Bianchi identity for $g_7$).

This ansatz appears as (CastellaniDAuriaFre, (III.8.24)).

Note that this amounts simply to a redefinition of curvatures

$\tilde g_7 := g_7 + 15 g_4 \wedge c_3 \,.$
###### Claim

A field configuration of 11-dimensional supergravity is given by L-∞ algebra valued differential forms with values in $\mathfrak{sugra}_6$. Among all of these the solutions to the equations of motion (the points in the covariant phase space) can be characterized as follows:

A field configuration

$\Omega^\bullet(X) \leftarrow \tilde W(\mathfrak{sugra}_6) : \Phi$

solves the equations of motion precisely if

1. all curvatures sit in the ideal of differential forms spanned by the 1-form fields $E^a$ (vielbein) and $\Psi$ (gravitino);

more precisely if we have

• $\tau = 0$

(super-torsion);

• $G_4 = (G_4)_{a_1, \cdots a_4} E^{a_1} \wedge \cdots E^{a_4}$

• $G_7 = (G_7)_{a_1, \cdots a_7} E^{a_1} \wedge \cdots E^{a_7}$

(dual field strength)

• $\rho = \rho_{a b} E^a \wedge E^b + H_a \Psi \wedge E^a$

(Dirac operator applied to gravitino)

• $R^{a b} = R^{a b}_{c d} E^c \wedge E^d + \bar \Theta^{a b}{}_c \Psi \wedge E^c + \bar \Psi \wedge K^{a b} \Psi$

2. such that the coefficients of terms containing $\Psi$s are polynomials in the coefficients of the terms containing no $\Psi$s. (“rheonomy”).

This is the content of (CastellaniDAuriaFre, section III.8.5).

In particular this implies that on-shell the 4- and 7-field strength are indeed dual of each other

$G_7 \propto \star G_4 \,.$

This is the content of (CastellaniDAuriaFre, equation (III.8.52)).

supergravity Lie 6-algebra $\to$ supergravity Lie 3-algebra $\to$ super Poincaré Lie algebra

## References

The supergravity Lie 6-algebra appears first on page 18 of

and is recalled in section 4 of

A textbook discussion is in section III.8.3 of

The same is being recalled for instance in section 3 of

Then it is rediscovered around equation (8.8) in

which gives a detailed and comprehensive discussion.

A discussion in the context of smooth super ∞-groupoids is in

in the last section of

Revised on June 2, 2015 05:36:53 by Urs Schreiber (50.207.161.2)