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supergravity Lie 6-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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

μ 7:=12ψ¯Γ a 1a 5ψe a 1e a 5132ψ¯Γ a 1a 2ψe a 1e a 2c 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

dμ 7 =54ψ¯Γ a 1a 4bψe a 1e a 4ψ¯Γ bψ i15Γ abe aψ¯Γ bψc 3 +154ψ¯Γ abψe ae bψ¯Γ cdψe ce d. \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

ψ¯Γ a 1a 4bψψ¯Γ bψ=3ψ¯Γ [a 1a 2ψψ¯Γ a 3a 4]ψ \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

ψ¯Γ abψψ¯Γ bψ=0. \bar \psi \wedge \Gamma^{a b} \psi \wedge \bar \psi \wedge \Gamma_b \psi = 0 \,.
Definition

The supergravity Lie 6-algebra 𝔰𝔲𝔤𝔯𝔞 7(10,1)\mathfrak{sugra}_{7}(10,1) is the super Lie 7-algebra that is the b 6b^6 \mathbb{R}-extension of 𝔰𝔲𝔤𝔯𝔞 3(10,1)\mathfrak{sugra}_3(10,1) classified by the cocycle μ 7\mu_7 from def. 1.

b 5𝔰𝔲𝔤𝔯𝔞 6𝔰𝔲𝔤𝔯𝔞 3. b^5 \mathbb{R} \to \mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \,.
Note

This means that the Chevalley-Eilenberg algebra CE(𝔰𝔲𝔤𝔯𝔞 6)CE(\mathfrak{sugra}_6) is generated from

with differential defined by

d CE:ω abω acω c b d_{CE} : \omega^{a b} \mapsto \omega^{a c} \wedge \omega_c{}^b
d CE:e a=ω abe b12iψ¯Γ aψ d_{CE} : e^a = -\omega^{a b} e_b - \frac{1}{2}i \bar \psi \wedge \Gamma^a \psi
d CE:ψ14ω abΓ ab d_{CE} : \psi \mapsto - \frac{1}{4}\omega^{a b} \Gamma^{a b}
d CE:c 312ψ¯Γ abψe ae b d_{CE} : c_3 \mapsto \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
d CE:c 612ψ¯Γ a 1a 5ψe a 1e a 5132ψ¯Γ a 1a 2ψe a 1e a 2c 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=11D = 11 supergravity

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

Definition

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

Write g 4g_4 and g 7g_7 for the shifted generators of the Weil algebra corresponding to c 3c_3 and c 6c_6, respectively.

Define a modified Weil algebra W˜(𝔰𝔲𝔤𝔯𝔞 6(10,1))\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 6c_6 is modified to

d W˜c 6:=d Wc 6+15g 4c 3 d_{\tilde W} c_6 := d_{W} c_6 + 15 g_4 \wedge c_3

and hence the differential of g 7g_7 is accordingly modified in the unique way that ensures d W˜ 2=0d_{\tilde W}^2 = 0 (yielding the modified Bianchi identity for g 7g_7).

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

Note that this amounts simply to a redefinition of curvatures

g˜ 7:=g 7+15g 4c 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 𝔰𝔲𝔤𝔯𝔞 6\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

Ω (X)W˜(𝔰𝔲𝔤𝔯𝔞 6):Φ \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 aE^a (vielbein) and Ψ\Psi (gravitino);

    more precisely if we have

    • τ=0\tau = 0

      (super-torsion);

    • G 4=(G 4) a 1,a 4E a 1E a 4G_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,a 7E a 1E a 7G_7 = (G_7)_{a_1, \cdots a_7} E^{a_1} \wedge \cdots E^{a_7}

      (dual field strength)

    • ρ=ρ abE aE b+H aΨE a\rho = \rho_{a b} E^a \wedge E^b + H_a \Psi \wedge E^a

      (Dirac operator applied to gravitino)

    • R ab=R cd abE cE d+Θ¯ ab cΨE c+Ψ¯K abΨR^{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 Ψ\Psis are polynomials in the coefficients of the terms containing no Ψ\Psis. (“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 7G 4. 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

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 September 5, 2013 07:10:58 by Urs Schreiber (89.204.130.92)