∞-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 ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{3}\left(10,1\right)$ carries an L-∞ algebra cocycle ${\mu }_{7}\in \mathrm{CE}\left({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{3}\left(10,1\right)\right)$ of degree 7, given in the standard generators $\left\{{e}^{a}\right\}$ (vielbein), $\left\{{\omega }^{ab}\right\}$ (spin connection) $\left\{{\psi }^{\alpha }\right\}$ (gravitino) and $\left\{{c}_{3}\right\}$ (supergravity C-field) by

${\mu }_{7}:=-\frac{1}{2}\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{5}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{5}}-\frac{13}{2}\overline{\psi }{\Gamma }^{{a}_{1}{a}_{2}}\psi \wedge {e}_{{a}_{1}}\wedge {e}_{{a}_{2}}\wedge {c}_{3}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{rl}d{\mu }_{7}& =-\frac{5}{4}\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{4}b}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{4}}\wedge \overline{\psi }\wedge {\Gamma }_{b}\psi \\ & -i15\wedge {\Gamma }^{ab}{e}_{a}\wedge \overline{\psi }\wedge {\Gamma }_{b}\psi \wedge {c}_{3}\\ & +\frac{15}{4}\overline{\psi }\wedge {\Gamma }_{ab}\psi \wedge {e}^{a}\wedge {e}^{b}\wedge \overline{\psi }\wedge {\Gamma }_{cd}\psi \wedge {e}^{c}\wedge {e}^{d}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{4}b}\psi \wedge \overline{\psi }\wedge {\Gamma }_{b}\psi =3\overline{\psi }\wedge {\Gamma }^{\left[{a}_{1}{a}_{2}}\psi \wedge \overline{\psi }\wedge {\Gamma }^{{a}_{3}{a}_{4}\right]}\psi$\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

$\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge \overline{\psi }\wedge {\Gamma }_{b}\psi =0\phantom{\rule{thinmathspace}{0ex}}.$\bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 0 \,.
###### Definition

The supergravity Lie 6-algebra ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{7}\left(10,1\right)$ is the super Lie 7-algebra that is the ${b}^{6}ℝ$-extension of ${\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{3}\left(10,1\right)$ classified by the cocycle ${\mu }_{7}$ from def. 1.

${b}^{5}ℝ\to {\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{6}\to {\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{3}\phantom{\rule{thinmathspace}{0ex}}.$b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \,.
###### Note

This means that the Chevalley-Eilenberg algebra $\mathrm{CE}\left({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{6}\right)$ is generated from

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

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

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

• $\left\{{c}_{3}\right\}$ (supergravity C-field) in degree $\left(3,\mathrm{even}\right)$

• $\left\{{c}_{6}\right\}$ (magnetic dual supergravity C-field) in degree $\left(6,\mathrm{even}\right)$

with differential defined by

${d}_{\mathrm{CE}}:{\omega }^{ab}↦{\omega }^{ac}\wedge {\omega }_{c}{}^{b}$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
${d}_{\mathrm{CE}}:\psi ↦-\frac{1}{4}{\omega }^{ab}{\Gamma }^{ab}$d_{CE} : \psi \mapsto - \frac{1}{4}\omega^{a b} \Gamma^{a b}
${d}_{\mathrm{CE}}:{c}_{3}↦\frac{1}{2}\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge {e}_{a}\wedge {e}_{b}$d_{CE} : c_3 \mapsto \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
${d}_{\mathrm{CE}}:{c}_{6}↦-\frac{1}{2}\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{5}}\psi \wedge {e}_{{a}_{1}}\wedge \cdots \wedge {e}_{{a}_{5}}-\frac{13}{2}\overline{\psi }{\Gamma }^{{a}_{1}{a}_{2}}\psi \wedge {e}_{{a}_{1}}\wedge {e}_{{a}_{2}}\wedge {c}_{3}\phantom{\rule{thinmathspace}{0ex}}.$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 ${L}_{\infty }$-algebra of 11-dimensional supergravity, in the sense discussed at D'Auria-Fre formulation of supergravity .

###### Definition

Write $W\left({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{6}\left(10,1\right)\right)$ 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 $\stackrel{˜}{W}\left({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{6}\left(10,1\right)\right)$ by declaring it to have the same generators and differential as before, except that the differential for ${c}_{6}$ is modified to

${d}_{\stackrel{˜}{W}}{c}_{6}:={d}_{W}{c}_{6}+15{g}_{4}\wedge {c}_{3}$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 ${d}_{\stackrel{˜}{W}}^{2}=0$ (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

${\stackrel{˜}{g}}_{7}:={g}_{7}+15{g}_{4}\wedge {c}_{3}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\Omega }^{•}\left(X\right)←\stackrel{˜}{W}\left({\mathrm{𝔰𝔲𝔤𝔯𝔞}}_{6}\right):\Phi$\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);

• ${G}_{4}=\left({G}_{4}{\right)}_{{a}_{1},\cdots {a}_{4}}{E}^{{a}_{1}}\wedge \cdots {E}^{{a}_{4}}$

• ${G}_{7}=\left({G}_{7}{\right)}_{{a}_{1},\cdots {a}_{7}}{E}^{{a}_{1}}\wedge \cdots {E}^{{a}_{7}}$

(dual field strength)

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

(Dirac operator applied to gravitino)

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

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

${G}_{7}\propto \star {G}_{4}\phantom{\rule{thinmathspace}{0ex}}.$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

A textbook discussion is in section III.8.3 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 the last section of

Revised on May 21, 2013 20:44:56 by Urs Schreiber (89.204.153.30)