nLab
supergravity Lie 6-algebra

Idea

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

• a vielbein field and a spin connection, hence the first order formulation of gravity for a graviton field in 11-dimensions;

• a gravitino field;

• the supergravity C-field in degree 3 and its magnetic dual.

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 ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_3(10,1)$ carries an L-∞ algebra cocycle ${\mu }_7\in \mathrm{CE}({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_3(10,1))$ of degree 7, given in the standard generators $\{e^a\}$ (vielbein), $\{{\omega }^{ab}\}$ (spin connection) $\{{\psi }^{\alpha }\}$ (gravitino) and $\{c_3\}$ (supergravity C-field) by

$${\mu }_7:=-\frac{1}{2}\overline{\psi }\wedge {\Gamma }^{a_1\cdots a_5}\psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_5}-\frac{13}{2}\overline{\psi }{\Gamma }^{a_1{a}_2}\psi \wedge e_{a_1}\wedge e_{a_2}\wedge c_3.$$\mu_7 := - \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 in (DAuriaFre, page 18) and CastellaniDAuriaFre, III.8.3.

Proof

One computes

$$\begin{array}{rl}d{\mu }_7& =-\frac{5}{4}\overline{\psi }\wedge {\Gamma }^{a_1\cdots a_{4}b}\psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_4}\wedge \overline{\psi }\wedge {\Gamma }_b\psi \\ & -i15\wedge {\Gamma }^{ab}{e}_a\wedge \overline{\psi }\wedge {\Gamma }_b\psi \wedge c_3\\ & +\frac{15}{4}\overline{\psi }\wedge {\Gamma }_{ab}\psi \wedge e^a\wedge e^b\wedge \overline{\psi }\wedge {\Gamma }_{cd}\psi \wedge e^c\wedge e^d\end{array}.$$\begin{aligned} d \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

$$\overline{\psi }\wedge {\Gamma }^{a_1\cdots a_{4}b}\psi \wedge \overline{\psi }\wedge {\Gamma }_b\psi =3\overline{\psi }\wedge {\Gamma }^{[a_1{a}_2}\psi \wedge \overline{\psi }\wedge {\Gamma }^{a_3{a}_4]}\psi$$\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

$$\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge \overline{\psi }\wedge {\Gamma }_b\psi =0.$$\bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 0 \,.

Definition

The supergravity Lie 6-algebra ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_7(10,1)$ is the super Lie 7-algebra that is the $b^6ℝ$-extension of ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_3(10,1)$ classified by the cocycle ${\mu }_7$ from def. 1.

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

Note

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

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

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

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

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

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

with differential defined by

$$d_{\mathrm{CE}}:{\omega }^{ab}\mapsto {\omega }^{ac}\wedge {\omega }_c{}^b$$d_{CE} : \omega^{a b} \mapsto \omega^{a c} \wedge \omega_c{}^b

$$d_{\mathrm{CE}}:e^a=-{\omega }^{ab}{e}_b-\frac{1}{2}i\overline{\psi }\wedge {\Gamma }^a\psi$$d_{CE} : e^a = -\omega^{a b} e_b - \frac{1}{2}i \bar \psi \wedge \Gamma^a \psi

$$d_{\mathrm{CE}}:\psi \mapsto -\frac{1}{4}{\omega }^{ab}{\Gamma }^{ab}$$d_{CE} : \psi \mapsto - \frac{1}{4}\omega^{a b} \Gamma^{a b}

$$d_{\mathrm{CE}}:c_3\mapsto \frac{1}{2}\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge e_a\wedge e_b$$d_{CE} : c_3 \mapsto \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b

$$d_{\mathrm{CE}}:c_6\mapsto -\frac{1}{2}\overline{\psi }\wedge {\Gamma }^{a_1\cdots a_5}\psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_5}-\frac{13}{2}\overline{\psi }{\Gamma }^{a_1{a}_2}\psi \wedge e_{a_1}\wedge e_{a_2}\wedge c_3.$$d_{CE} : 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 (CastellaniDAuriaFre, (III.8.18)).

Note

According to (CastellaniDAuriaFre, 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 $L_{\mathrm{\infty }}$-algebra of 11-dimensional supergravity, in the sense discussed at D'Auria-Fre formulation of supergravity .

Definition

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

$$d_{\tilde{W}}{c}_6:=d_W{c}_6+15g_4\wedge c_3$$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

$${\tilde{g}}_7:=g_7+15g_4\wedge c_3.$$\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 ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_6$. 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

$${\Omega }^{•}(X)\leftarrow \tilde{W}({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_6):\Phi$$\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);

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

     (field strength of supergravity C-field)

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

     (dual field strength)

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

     (Dirac operator applied to gravitino)

   • $R^{ab}=R_{cd}^{ab}{E}^c\wedge E^d+{\overline{\Theta }}^{ab}{}_c\Psi \wedge E^c+\overline{\Psi }\wedge K^{ab}\Psi$

     (Riemann tensor: field strength of gravity)

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

$$G_7\propto \star G_4.$$G_7 \propto \star G_4 \,.

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

Related concepts

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

• type II supergravity Lie 2-algebra

References

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

• R. D’Auria, P. Fré Geometric Supergravity in D=11 and its hidden supergroup

A textbook discussion is in section III.8.3 of

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre,

  Supergravity and Superstrings - A Geometric Perspective

Then it is rediscovered around equation (8.8) in

• C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces (arXiv:hep-th/9904137)

which gives a detailed and comprehensive discussion.

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

• Urs Schreiber, differential cohomology in a cohesive topos