nLab supergravity Lie 6-algebra

Idea

The supergravity Lie 6-algebra (D’Auria & Fré 1982, p. 18) is (with hindsight, following FSS15) a super L-∞ algebra such that $L_\infty$-algebra valued differential forms (local ∞-connections) with values in it encode field histories of

1. the vielbein field with a spin connection,

   hence the first order formulation of gravity for a graviton field

   in 10+1 dimensions;

2. the gravitino field;

3. the supergravity C-field

   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

Our normalization conventions entirely follow Castellani, D’Auria & Fré 1991, §III.8.3, but we denote the super-vielbein 1-forms by $\big\{e^a\big\}_{a=0}^{10}$ instead of by $\big\{V^a\big\}_{a=0}^{10}$.

Proposition

The supergravity Lie 3-algebra $\mathfrak{sugra}_3(10,1)$ carries an L-∞ algebra cocycle $\mu_7 \in CE\big(\mathfrak{sugra}_3(10,1)\big)$ 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

$$\mu_4 \;=\; \frac{1}{2} \overline{\psi} \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2}$$

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

In other words, we have

$$\begin{array}{l} \mathrm{d} \, \mu_4 \;=\; 0 \\ \mathrm{d} \, \mu_7 \;=\; 15\, \mu_4 \wedge \mu_4 \mathrlap{\,.} \end{array}$$

which has the same structure as 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 (to which it is related below)

This appears (in the dual language of Chevalley-Eilenberg algebras) in DAuria & FrFré 1982, page 18 and Castellani, D’Auria & Fré 1991, §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

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

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. .

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

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

$$\begin{array}{ccl} \omega^{a b} &\mapsto& \omega^{a c} \wedge \omega_c{}^b \\ e^a &\mapsto& -\omega^{a b} e_b - \frac{1}{2}i \bar \psi \wedge \Gamma^a \psi \\ \psi &\mapsto& - \frac{1}{4}\omega^{a b} \Gamma^{a b} \\ c_3 &\mapsto& \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b \\ c_6 &\mapsto& - \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5} - \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3 \,. \end{array}$$

This appears in Castellani, D’Auria & Fré, (III.8.18).

Remark

According to Castellani, D’Auria & Fré 1991, 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-Fré formulation of supergravity .

Definition

Write $W\big(\mathfrak{sugra}_6(10,1)\big)$ 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 an adjusted 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

$$d_{\tilde W} \, c_6 \coloneqq d_{W} \, c_6 + 15 \, g_4 \wedge c_3$$

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 in Castellani, D’Auria & Fré 1991 (III.8.24).

Note that this amounts simply to a redefinition of curvatures

$$\tilde g_7 \;\coloneqq\; g_7 + 15 g_4 \wedge c_3 \,.$$

Proposition

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

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

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)

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

     (Dirac operator applied to gravitino)

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

     (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, in paraphrase, the content of Castellani, D’Auria & Fré, section III.8.5.

Remark

In particular, the Bianchi identity for the super-form enhancement of the supergravity C-field flux density is (by III.8.23j, 34, 35, 36 and using hupf):

(1)$$\begin{array}{l} \mathrm{d} \big( \underset{ G_7^{super} }{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} \,+\, \mathrm{i} \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \, e^{a_1} \wedge \cdots \wedge e^{a_5} } } \big) \\ \;=\; -15 \big( \underset{G_4^{super}}{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} + \tfrac{1}{2} \overline{\psi}\Gamma_{a_1 a_2}\psi \, e^{a_1} \wedge e^{a_2} } } \big) \wedge \big( \underset{G_4^{super}}{ \underbrace{ F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4} + \tfrac{1}{2} \overline{\psi}\Gamma_{a_1 a_2}\psi \, e^{a_1} \wedge e^{a_2} } } \big) \mathrlap{\,} \end{array}$$

and its rheonomic solution implies for the “actual” flux densities $F_4 \coloneqq F_{a_1 \cdots a_4} e^{a_1} \wedge \cdots \wedge e^{a_4}\vert_{\psi=0}$ and $G_4 \coloneqq G_{a_1 \cdots a_7} e^{a_1} \wedge \cdots e^{a_7} \vert_{\psi = 0}$ that

1. the CJS higher Maxwell equation holds

   $$\mathrm{d}\, G_7 \;=\; - 15 \, F_4 \wedge F_4$$

   (III.8.53)

2. the Hodge duality holds

   $$G_7 \;=\; \star F_4$$

   (III.8.52)

Related concepts

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

• 4d supergravity Lie 2-algebra

• type II supergravity Lie 2-algebra

• M-theory super Lie algebra

• adjusted Weil algebra

References

The supergravity Lie 6-algebra originates in:

• Riccardo D'Auria, Pietro Fré, p. 18 of: Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5, errata]

Brief review is in:

• Leonardo Castellani, Pietro Fré, F. Giani, Krzysztof Pilch, Peter van Nieuwenhuizen, §4 of: Gauging of $d = 11$ supergravity?, Annals of Physics 146 1 (1983) 35-77 [spire:11998, doi:10.1016/0003-4916(83)90052-0]

• Pietro Fré, Pietro Antonio Grassi, §3 of: Free Differential Algebras, Rheonomy, and Pure Spinors [arXiv:0801.3076, inspire:777785]

• Pietro Fré, §6.4.1 in: Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer (2013) [doi:10.1007/978-94-007-5443-0]

A monograph account is in:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, vol 2, III.8.3 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, epdf, ch III.8: pdf]

It was (apparently) rediscovered in:

• C. Chrysso‌malakos, José de Azcárraga, José M. Izquierdo, C. Pérez Bueno, around equation (8.8) in: The geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000) 293-330 [arXiv:hep-th/9904137, doi:10.1016/S0550-3213(99)00512-X]

where a detailed discussion is given.

These authors all consider the Chevalley-Eilenberg algebra (“FDA”) of the actual super $L_\infty$-algebra. The latter and its relation to smooth super ∞-groupoids is considered in:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, Int. J. Geom. Meth. Modern Physics 12 02 (2015) 1550018 [arXiv:1308.5264, doi:10.1142/S0219887815500188]

• Urs Schreiber, last section of: differential cohomology in a cohesive topos