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Contents

Idea

N=1N=1 supergravity in d=11d = 11.

for the moment see the respective section at D'Auria-Fre formulation of supergravity

11d-SuGra from Super C-Field Flux Quantization

We discuss (Thm. below, following GSS24, §3) how the equations of motion of D=11 supergravity — on an 11|3211\vert\mathbf{32}-dimensional super-torsion-free super spacetime XX with super vielbein (e,ψ)(e,\psi) (the graviton/gravitino-fields) — follow from just the requirement that the duality-symmetric super-C-field flux densities (G 4 s,G 7 s)∈Ω dR 4(X)×Ω dR 7(X)(G_4^s, G_7^s) \,\in\, \Omega^4_{dR}(X) \times \Omega^7_{dR}(X):

  1. satisfy their Bianchi identities

    (1)dG 4 s=0 dG 7 s=12G 4 sG 4 s \begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \mathrm{d} \, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^s \end{array}

  2. are on any super-chart U↪XU \hookrightarrow X of the locally supersymmetric form

    (2)G 4 s=14!(G 4) a 1⋯a 4e a 1⋯e a 4−12(ψ¯Γ a 1a 2ψ)e a 1e a 2 G 7 s=17!(G 7) a 1⋯a 7e a 1⋯e a 7−15!(ψ¯Γ a 1⋯a 5ψ)e a 1⋯e a 5. \begin{array}{l} G_4^s \;=\; \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} \,-\, \tfrac{1}{2} \big(\overline{\psi}\Gamma_{a_1 a_2} \psi\big) e^{a_1} \, e^{a_2} \\ G_7^s \;=\; \tfrac{1}{7!} (G_7)_{a_1 \cdots a_7} e^{a_1} \cdots e^{a_7} \,-\, \tfrac{1}{5!} \big(\overline{\psi}\Gamma_{a_1 \cdots a_5} \psi\big) e^{a_1} \cdots e^{a_5} \mathrlap{\,.} \end{array}

Up to some mild (but suggestive, see below) re-arrangement, the computation is essentially that indicated in CDF91, §III.8.5 (where some of the easy checks are indicated) which in turn is a mild reformulation of the original claim in Cremmer & Ferrara 1980 and Brink & Howe 1980 (where less details were given). A full proof is laid out in GSS24, §3, whose notation we follow here.

The following may be understood as an exposition of this result, which seems to stand out as the only account that is (i) fully first-order and (ii) duality-symmetric (in that G 7G_7 enters the EoMs as an independent field, whose Hodge duality to G 4G_4 is imposed by the Bianchi identity for G 7 sG_7^s, remarkably).

Notice that the discussion in CDF91, §III.8 amplifies the superspace-rheonomy principle as a constraint that makes the Bianchi identities on (in our paraphrase) a supergravity Lie 6-algebra-valued higher vielbein be equivalent to the equations of motion of D=11 SuGra. But we may observe that the only rheonomic constraint necessary is that (2) on the C-field flux density — and this is the one not strictly given by rules in CDF91, p. 874, cf. around CDF91, (III.8.41) —; while the remaining rheonomy condition on the gravitino field strength ρ\rho is implied (Lem. below), and the all-important torsion constraint (3) (which is also outside the rules of rheonomy constraints, cf. CDF91, (III.8.33)) is naturally regarded as part of the definition of a super-spacetime in the first place (Def. below).

In thus recasting the formulation of the theorem somewhat, we also:

  1. re-define the super-flux densities as above (2), highlighting that it is (only) in this combination that the algebraic form of the expected Bianchi identity (1) extends to superspace;

  2. disregard the gauge potentials C 3C_3 and C 6C_6, whose role in CDF91, §III.8.2-4 is really just to motivate the form of the Bianchi identities equivalent to (1), but whose global nature is more subtle than acknowledged there, while being irrelevant for just the equations of motion.

Indeed, the point is that, in consequence of our second item above, the following formulation shows that one may apply flux quantization of the supergravity C-field on superspace in formally the same way as bosonically (for instance in Cohomotopy as per Hypothesis H, or in any other nonabelian cohomology theory whose classifying space has the ℚ\mathbb{Q}-Whitehead L ∞ L_\infty -algebra of the 4-sphere), and in fact that the ability to do so implies the EoMs of 11d SuGra. Any such choice of flux quantization is then what defines, conversely, the gauge potentials, globally. Moreover, by the fact brought out here, that the super-flux Bianchi identity already implies the full equations of motion, this flux quantization is thereby seen to be compatible with the equations of motion on all of super spacetime.

For the present formulation, we find it suggestive to regard the all-important torsion constraint (3) as part of the definition of the super-gravity field itself (since it ties the auxiliary spin-connection to the super-vielbein field which embodies the actual super-metric structure):

Definition

(super-spacetime)
For

by a super-spacetime of super-dimension D|ND\vert \mathbf{N} we here mean:

  1. a supermanifold

  2. which admits an open cover by super-Minkowski supermanifolds ℝ 1,D−1|N\mathbb{R}^{1,D-1\vert \mathbf{N}},

  3. equipped with a super Cartan connection with respect to the canonical subgroup inclusion Spin(1,D−1)↪Iso(ℝ 1,D−1|N)Spin(1,D-1) \hookrightarrow Iso(\mathbb{R}^{1,D-1\vert\mathbf{N}}) of the spin group into the super Poincaré group, namely:

    1. equipped with a super-vielbein (e,ψ)(e, \psi), hence on each super-chart U↪XU \hookrightarrow X

      ((e a) a=0 D=1,(ψ α) α=1 N)∈Ω dR 1(U;ℝ 1,D−1|N) \big( (e^a)_{a=0}^{D=1} ,\, (\psi^\alpha)_{\alpha=1}^N \big) \;\in\; \Omega^1_{dR}\big( U ;\, \mathbb{R}^{1,D-1\vert \mathbf{N}} \big)

      such that at every point x∈X⇝x \in \overset{\rightsquigarrow}{X} the induced map on tangent spaces is an isomorphism

      (e,ψ) x:T xX⟶∼ℝ 1,10|N. (e,\psi)_x \;\colon\; T_x X \overset{\sim}{\longrightarrow} \mathbb{R}^{1,10\vert \mathbf{N}} \,.

    2. and with a spin-connection ω\omega (…),

  4. such that the super-torsion vanishes, in that on each chart:

    (3)de a−ω a be b=(ψ¯Γ aψ), \mathrm{d} \, e^a - \omega^a{}_b \, e^b \;=\; \big( \overline{\psi} \,\Gamma^a\, \psi \big) \,,

    where Γ (−):ℝ 1,D−1⟶End ℝ(N)\Gamma^{(-)} \,\colon\, \mathbb{R}^{1,D-1} \longrightarrow End_{\mathbb{R}}(\mathbf{N}) is a representation of Pin + ( 1 , 10 ) Pin^+(1,10) , hence

    Γ aΓ b+Γ bΓ a=+2diag(−,+,+,⋯,+) ab. \Gamma_{a} \Gamma_b + \Gamma_{b} \Gamma_a \;=\; + 2\, diag(-, +, +, \cdots, +)_{a b} \,.

Definition

(the gravitational field strength)
Given a super-spacetime (Def. ), we say that (super chart-wise):

  1. its super-torsion is:

    T a≔de a−ω a be b−(ψ¯Γ aψ) T^a \;\coloneqq\; \mathrm{d}\, e^a \,-\, \omega^a{}_b \, e^b \,-\, \big( \overline{\psi}\Gamma^a\psi \big)

  2. its gravitino field strength is

    ρ≔dψ+14ω abΓ abψ, \rho \;\coloneqq\; \mathrm{d}\, \psi + \tfrac{1}{4} \omega_{a b}\Gamma^{a b}\psi \,,

  3. its curvature is

    R a b≔dω a b−ω a cω c b. R^{a}{}_b \;\coloneqq\; \mathrm{d}\, \omega^{a}{}_b \,-\, \omega^a{}_c \, \omega^c{}_b \,.

Lemma

(super-gravitational Bianchi identities)
By exterior calculus the gravitational field strength tensors (Def. ) satisfy the following identities:

(4)dR a b = ω a a′R a′ b−R a b′ω b′ b dT a = −R a be b+2(ψ¯Γ aρ) dρ = 14R abΓ abψ \begin{array}{ccl} \mathrm{d} \, R^{a}{}_b &=& \omega^a{}_{a'} \, R^{a'}{}_b - R^{a}{}_{b'} \, \omega^{b'}{}_{b} \\ \mathrm{d} \, T^a &=& - R^{a}{}_b \ e^b + 2 \big( \overline{\psi} \,\Gamma^a\, \rho \big) \\ \mathrm{d} \, \rho &=& \tfrac{1}{4} R^{a b} \Gamma_{a b} \psi \end{array}

Write now 32∈Rep ℝ(Spin(1,10))\mathbf{32} \in Rep_{\mathbb{R}}\big(Spin(1,10)\big) for the unique non-trivial irreducible real Spin ( 1 , 10 ) Spin(1,10) -representation.

Theorem

(11d SuGra EoM from super-flux Bianchi identity) Given

  1. (super-gravity field:) an 11|3211\vert\mathbf{32}-dimensional super-spacetime XX (Def. ),

  2. (super-C-field flux densities:) (G 4 s,G 7 s)(G^s_4,\, G^s_7) as in (2)

then the super-flux Bianchi identity (1) (the super-higher Maxwell equation for the C-field)

dG 4 s=0 dG 7 s=12G 4 sG 4 s \begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \mathrm{d} \, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^s \end{array}

is equivalent to the joint solution by (e,ψ,ω,G 4 s,G 7 s)\big(e, \psi, \omega, G_4^s,\, G_7^s\big) of the equations of motion of D=11 supergravity.

This is, in some paraphrase, the result of CDF91, §III.8.5, We indicate the proof broken up in the following Lemmas , , and .

In all of the following lemmas one expands the Bianchi identoties in their super-vielbein form components.

Lemma

The Bianchi identity for G 4 sG^s_4 (1) is equivalent to

  1. the closure of the ordinary 4-flux density G 4G_4

  2. the following dependence of ρ\rho on G 4G_4

shown in any super-chart:

(5)dG 4 s=0 ⇔{(∇ a(G 4) a 1⋯a 4)e ae a 1⋯e a 4=0 ρ=ρ abe ae b+(1613!(G 4) ab 1b 2b 3Γ ab 1b 2b 3−11214!(G 4) b 1⋯b 4Γ ab 1⋯b 4)⏟H aψe a (14!ψ α∇ α(G 4) a 1⋯a 4+(ψ¯Γ a 1a 2ρ a 3a 4))e a 1⋯e a 4=0. \begin{array}{l} \mathrm{d}\, G^s_4 \;=\; 0 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a} (G_4)_{a_1 \cdots a_4} \big) e^{a} \, e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \rho \;=\; \rho_{a b} \, e^{a} \, e^b \,+\, \underset{ H_a }{ \underbrace{ \Big( \tfrac{1}{6} \, \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \,\Gamma^{a b_1 b_2 b_3}\, \, - \tfrac{1}{12} \, \tfrac{1}{4!} (G_4)_{b_1 \cdots b_4} \,\Gamma^{a b_1 \cdots b_4}\, \Big) } } \psi \, e^a \\ \Big( \tfrac{1}{4!} \psi^\alpha \nabla_\alpha (G_4)_{a_1 \cdots a_4} \;+\; \big( \overline{\psi} \Gamma_{a_1 a_2} \rho_{a_3 a_4} \big) \Big) e^{a_1} \cdots e^{a_4} \;=\; 0 \,. \end{array} \right. \end{array}

This is essentially the claim in CDF91 (III.8.44-49 & 60b); full proof is given in GSS24, Lem. 3.2.

Proof

The general expansion of ρ\rho in the super-vielbein basis is of the form

ρ:=ρ abe ae b+H aψe a+ψ¯κψ⏟=0, \rho \;:=\; \rho_{a b} \, e^a\, e^b + H_a \psi \, e^a + \underset{ = 0 }{ \underbrace{ \overline{\psi} \,\kappa\, \psi } } \,,

where the last term is taken to vanish.l (…).

Therefore, the Bianchi identity has the following components,

(6)d(14!(G 4) a 1⋯a 4e a 1⋯e a 4−12(ψ¯Γ a 1a 2ψ)e a 1e a 2)=0 ⇔{(∇ a(G 4) a 1⋯a 4)e ae a 1⋯e a 4=0 (14!ψ α(∇ α(G 4) a 1⋯a 4)+(ψ¯Γ a 1a 2ρ a 3a 4))e a 1⋯e a 4=0 13!(G 4) ab 1b 2b 3(ψ¯Γ aψ)e b 1b 2b 3+(ψ¯Γ a 1a 2H bψ)e a 1e a 2e b=0, \begin{array}{l} \mathrm{d} \Big( \, \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \, e^{a_1}\, e^{a_2} \Big) \;=\; 0 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a} (G_4)_{a_1 \cdots a_4} \big) e^{a}\, e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \Big( \tfrac{1}{4!} \psi^\alpha \big( \nabla_\alpha (G_4)_{a_1 \cdots a_4} \big) \;+\; \big( \overline{\psi} \Gamma_{a_1 a_2} \rho_{a_3 a_4} \big) \Big) e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma^a\, \psi \big) \, e^{b_1 b_2 b_3} + \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, H_b \psi \big) e^{a_1} \, e^{a_2} \, e^b \;=\; 0 \,, \end{array} \right. \end{array}

where we used that the quartic spinorial component vanishes identically, due to a Fierz identity (here):

−12(ψ¯Γ a 1a 2ψ)(ψ¯Γ a 1ψ)e a 2=0. - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \big( \overline{\psi} \Gamma^{a_1} \psi \big) e^{a_2} \;=\; 0 \,.

To solve the second line in (6) for H aH_a (this is CDF91 (III.8.43-49)) we expand H aH_a in the Clifford algebra (according to this Prop.), observing that for Γ a 1a 2H a 3\Gamma_{a_1 a_2} H_{a_3} to be a linear combination of the Γ a\Gamma_a the matrix H aH_a needs to have a Γ a 1\Gamma_{a_1}-summand or a Γ a 1a 2a 3\Gamma_{a_1 a_2 a_3}-summand. The former does not admit a Spin-equivariant linear combination with coefficients (G 4) a 1⋯a 4(G_4)_{a_1 \cdots a_4}, hence it must be the latter. But then we may also need a component Γ a 1⋯a 5\Gamma_{a_1 \cdots a_5} in order to absorb the skew-symmetric product in Γ a 1a 2H a\Gamma_{a_1 a_2} H_a. Hence H aH_a must be of this form:

(7)H a=const 113!(G 4) ab 1b 2b 3Γ b 1b 2b 3+const 214!(G 4) b 1⋯b 4Γ ab 1⋯b 4. H_a \;=\; \mathrm{const}_1 \, \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \Gamma^{b_1 b_2 b_3} + \mathrm{const}_2 \, \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \Gamma_{a b_1 \cdots b_4} \,.

With this, we compute:

(8)(ψ¯Γ a 1a 2H a 3ψ)e a 1e a 2e a 3 =const 113!(G 4) a 3b 1b 2b 3(ψ¯Γ a 1a 2Γ b 1b 2b 3ψ)e a 1e a 2e a 3 +const 214!(G 4) b 1⋯b 4(ψ¯Γ a 1a 2Γ a 3b 1⋯b 4ψ)e a 1e a 2e a 3 =1const 113!(G 4) a 3b 1b 2b 3(ψ¯Γ a 1a 2 b 1b 2b 3ψ)e a 1e a 2e a 3 +6const 113!(G 4) b 3a 1a 2a 3(ψ¯Γ b 3ψ)e a 1e a 2e a 3 +8const 214!(G 4) b 1⋯b 3a 3(ψ¯Γ a 1a 2 b 1⋯b 3ψ)e a 1e a 2e a 3. \begin{array}{ll} \big( \overline{\psi} \Gamma_{a_1 a_2} H_{a_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} & =\; \mathrm{const}_1 \, \tfrac{1}{3!} (G_4)_{a_3 b_1 b_2 b_3} \, \big( \overline{\psi} \Gamma_{a_1 a_2} \Gamma^{b_1 b_2 b_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, \mathrm{const}_2 \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_4} \, \big( \overline{\psi} \Gamma_{a_1 a_2} \Gamma_{a_3 b_1 \cdots b_4} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;=\; 1 \, \mathrm{const}_1 \, \tfrac{1}{3!} \, (G_4)_{a_3 b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma_{a_1 a_2}{}^{b_1 b_ 2 b_3}\, \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, 6 \, \mathrm{const}_1 \, \tfrac{1}{3!} \, (G_4)_{b_3 a_1 a_2 a_3} \big( \overline{\psi} \,\Gamma^{b_3}\, \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, 8 \, \mathrm{const}_2 \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_3 a_3} \, \big( \overline{\psi} \Gamma^{a_1 a_2}{}_{b_1 \cdots b_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \,. \end{array}

Here the multiplicities of the nonvanishing Clifford-contractions arise via this Lemma:

1=0!(20)(30) 6=2!(22)(32) 8=1!(21)(41), \begin{array}{l} 1 \;=\; 0! \Big( {2 \atop 0} \Big) \Big( {3 \atop 0} \Big) \\ 6 \;=\; 2! \Big( {2 \atop 2} \Big) \Big( {3 \atop 2} \Big) \\ 8 \;=\; 1! \Big( {2 \atop 1} \Big) \Big( {4 \atop 1} \Big) \,, \end{array}

and all remaining contractions vanish inside the spinor pairing by this lemma.

Now using (8) in (6) yields:

const 1=−1/6, const 2=−4!/3!const 1/8=+1/12, \begin{array}{l} \mathrm{const}_1 = -1/6 \,, \\ \mathrm{const}_2 = - 4!/3! \, \mathrm{const}_1 / 8 = + 1/12 \,, \end{array}

as claimed.

Lemma

Given the Bianchi identity for G 4 sG^s_4 (5), then the Bianchi identity for G 7 sG^s_7 (1) is equivalent to

  1. the Bianchi identity for the ordinary flux density G 7G_7

  2. its Hodge duality to G 4G_4

  3. another condition on the gravitino field strength

(9)dG 7 s=12G 4 sG 4 s ⇔{(∇ a 117!(G 7) a 2⋯a 8)e a 1⋯e a 8=12(14!(G 4) a 1⋯a 414!(G 4) a 5⋯a 8)e a 1⋯e a 8 (G 7) a 1⋯a 7=14!ϵ a 1⋯a bb 1⋯b 4(G 4) b 1⋯b 4 (17!ψ α∇ α(G 7) a 1⋯a 7ψ α+25!(ψ¯Γ a 1⋯a 5ρ a 6a 7))e a 1⋯e a 7=0 \begin{array}{l} \mathrm{d} \, G^s_7 \;=\; \tfrac{1}{2} G^s_4 \, G^s_4 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a_1} \tfrac{1}{7!} (G_7)_{a_2 \cdots a_8} \big) e^{a_1} \cdots e^{a_8} \;=\; \tfrac{1}{2} \big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, \tfrac{1}{4!} (G_4)_{a_5 \cdots a_8} \big) e^{a_1} \cdots e^{a_8} \\ (G_7)_{a_1 \cdots a_7} \;=\; \tfrac{1}{4!} \epsilon_{a_1 \cdots a_b b_1 \cdots b_4} (G_4)^{b_1 \cdots b_4} \\ \Big( \tfrac{1}{7!} \psi^\alpha \nabla_\alpha (G_7)_{a_1 \cdots a_7} \psi^\alpha \;+\; \frac{2}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \rho_{a_6 a_7} \big) \Big) e^{a_1} \cdots e^{a_7} \;=\; 0 \end{array} \right. \end{array}

This is essentially CDF91, (III.8.50-53).

Proof

The components of the Bianchi identity are

dG 4 s=0 ⇒{d(17!(G 7) a 1⋯a 7e a 1⋯e a 7−15!(ψ¯Γ a 1⋯a 5ψ)e a 1⋯e a 5) =12(14!(G 4) a 1⋯a 4e a 1⋯e a 4−12(ψ¯Γ a 1a 2ψ))(14!(G 4) a 1⋯a 4e a 1⋯e a 4−12(ψ¯Γ a 1a 2ψ)) ⇔{(∇ a 117!(G 7) a 2⋯a 8=1214!(G 4) a 1⋯a 414!(G 4) a 5⋯a 8)e a 1⋯e a 8 (17!ψ α∇ α(G 7) a 1⋯a 7+25!(ψ¯Γ a 1⋯a 5ρ a 6a 7))e a 1⋯e a 7=0 16!(G 7) a 1⋯a 6b(ψ¯Γ bψ)e a 1⋯e a 6 +21215!14!(G 4) b 1⋯b 4(ψ¯Γ a 1⋯a 5Γ ab 1⋯b 4ψ)e ae a 1⋯e a 5 −2615!13!(G 4) ab 1b 2b 3(ψ¯Γ a 1⋯a 5Γ b 1b 2b 3ψ)e ae a 1⋯e a 5 −(12(ψ¯Γ a 1a 2ψ)e a 1e a 2)14!(G 4) b 1⋯b 4e b 1⋯e b 4=0,}⇔(G 7) a 1⋯a 6b=14!ϵ a 1⋯a 6bb 1⋯b 4(G 4) b 1⋯b 4 \begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \Rightarrow \left\{ \begin{array}{l} \mathrm{d} \Big( \tfrac{1}{7!} (G_7)_{a_1 \cdots a_7} \, e^{a_1} \cdots e^{a_7} - \tfrac{1}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \big) e^{a_1} \cdots e^{a_5} \Big) \\ \;=\; \tfrac{1}{2} \Big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \Big) \Big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \Big) \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \Big( \nabla_{a_1} \tfrac{1}{7!} (G_7)_{a_2 \cdots a_8} \;=\; \;\tfrac{1}{2}\; \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, \tfrac{1}{4!} (G_4)_{a_5 \cdots a_8} \Big) e^{a_1} \cdots e^{a_8} \\ \Big( \tfrac{1}{7!} \psi^\alpha \nabla_\alpha (G_7)_{a_1 \cdots a_7} + \frac{2}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \rho_{a_6 a_7} \big) \Big) e^{a_1} \cdots e^{a_7} \;=\; 0 \\ \left. \begin{array}{l} \tfrac{1}{6!} (G_7)_{a_1 \cdots a_6 b} \big( \overline{\psi} \,\Gamma^b\, \psi \big) e^{a_1} \cdots e^{a_6} \\ \;\;\;+\, \tfrac{2}{12} \, \tfrac{1}{5!} \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_4} \big( \overline{\psi} \, \Gamma_{a_1 \cdots a_5} \, \Gamma_{a b_1 \cdots b_4}\, \psi \big) e^a \, e^{a_1} \cdots e^{a_5} \\ \;\;-\; \tfrac{2}{6} \tfrac{1}{5!} \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma_{a_1 \cdots a_5}\, \Gamma^{b_1 b_2 b_3} \psi \big) e^{a} \, e^{a_1} \cdots e^{a_5} \\ \;\;\;-\, \Big( \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) e^{a_1} \, e^{a_2} \Big) \tfrac{1}{4!} (G_4)_{b_1 \cdots b_4} \, e^{b_1} \cdots e^{b_4} \;\;=\;\; 0 \,, \end{array} \right\} \Leftrightarrow (G_7)_{a_1 \cdots a_6 b} \;=\; \tfrac{1}{4!} \epsilon_{a_1 \cdots a_6 b b_1 \cdots b_4} (G_4)^{b_1 \cdots b_4} \end{array} \right. \end{array} \right. \end{array}

where:

(i) in the quadratic spinorial component we inserted the expression for ρ\rho from (5), then contracted Γ\Gamma-factors using again this Lemma, and finally observed that of the three spinorial quadratic forms (see there) the coefficients of (ψ¯Γ a 1a 2ψ)\big(\overline{\psi}\Gamma_{a_1 a_2} \psi\big) and of (ψ¯Γ a 1⋯a 6ψ)\big(\overline{\psi}\Gamma_{a_1 \cdots a_6} \psi\big) vanish identically, by a remarkable cancellation of combinatorial prefactors:

  • (−21215!14!4!(54)(44)+2615!13!3!(53)(33)−1214!)⏟=0(G 4) a 2⋯a 5(ψ¯Γ aa 1ψ)e ae a 1⋯e a 6\underset{= 0 }{\underbrace{\bigg(- \frac{2}{12} \frac{1}{5!} \frac{1}{4!} 4! \Big( { 5 \atop 4 } \Big) \Big( { 4 \atop 4 } \Big) \;+\; \frac{2}{6} \frac{1}{5!} \frac{1}{3!} 3! \Big( { 5 \atop 3 } \Big) \Big( { 3 \atop 3 } \Big) \;-\; \frac{1}{2} \frac{1}{4!} \bigg) } } \; (G_4)_{a_2 \cdots a_5} \big( \overline{\psi} \,\Gamma_{a a_1}\, \psi \big) e^{a} \, e^{a_1} \cdots e^{a_6} \;\;\; (check)

  • (21215!14!2(52)(42)−2615!13!1(51)(31))⏟=0(G 4) a 1a 2b 1b 2(ψ¯Γ a 3⋯a 6 b 1b 2ψ)e a 1⋯e a 6\underset{ = 0 }{ \underbrace{ \bigg( \frac{2}{12} \frac{1}{5!} \frac{1}{4!} 2 \Big( { 5 \atop 2 } \Big) \Big( { 4 \atop 2 } \Big) \;-\; \frac{2}{6} \frac{1}{5!} \frac{1}{3!} 1 \Big( { 5 \atop 1 } \Big) \Big( { 3 \atop 1 } \Big) \bigg) } } \; (G_4)_{a_1 a_2 b_1 b_2} \big( \overline{\psi} \,\Gamma_{a_3 \cdots a_6}{}^{b_1 b_2}\, \psi \big) e^{a_1} \cdots e^{a_6} \;\;\; (check)

(ii) the quartic spinorial component holds identitically, due to the Fierz identity here:

−14!(ψ¯Γ a 1⋯a 5ψ)(ψ¯Γ a 1)e a 2⋯e a 5=18((ψ¯Γ a 1a 2ψ)e a 1e a 2)((ψ¯Γ a 1a 2ψ)e a 1e a 2). -\tfrac{1}{4!} \big( \overline{\psi} \,\Gamma_{a_1 \cdots a_5}\, \psi \big) \big( \overline{\psi} \Gamma^{a_1} \big) e^{a_2} \cdots e^{a_5} \;=\; \tfrac{1}{8} \Big( \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, \psi \big) e^{a_1} e^{a_2} \Big) \Big( \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, \psi \big) e^{a_1} e^{a_2} \Big) \,.

Therefore the only spinorial component of the Bianchi identity which is not automatically satisfied is (with Γ 012⋯=ϵ 012⋯\Gamma_{0 1 2 \cdots} = \epsilon_{0 1 2 \cdots}, see there) the vanishing of

16!((G 7) a 1⋯a 6b−14!(G 4) b 1⋯b 4ϵ b 1⋯b 4a 1⋯a 6b)(ψ¯Γ bψ), \tfrac{1}{6!} \Big( (G_7)_{a_1 \cdots a_6 b} - \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \epsilon_{b_1 \cdots b_4 a_1 \cdots a_6 b} \Big) \big( \overline{\psi} \,\Gamma^b\, \psi \big) \,,

which is manifestly the claimed Hodge duality relation.

Lemma

Given the Bianchi identities for G 4 sG_4^s (5) and G 7 sG_7^s (9), the supergravity fields satisfy their Einstein equations with source the energy momentum tensor of the C-field:

(10)dG 4 s=0,dG 7 s=12G 4 sG 4 2 ⇒{R bm am−12δ b aR mn mn=−112((G 4) ac 1⋯c 3(G 4) bc 1⋯c 3−18(G 4) c 1⋯c 4(G 4) c 1⋯c 4δ b a(Einstein equation) Γ ba 1a 2ρ a 1a 2=0(Rarita-Schwinger equation) \begin{array}{l} \mathrm{d}\, G_4^s \;=\;0 \,, \;\;\; \mathrm{d}\, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^2 \\ \;\Rightarrow\; \left\{ \begin{array}{l} R^{a m}_{b m} - \tfrac{1}{2} \delta^a_b\, R^{m n}_{m n} \;=\; - \tfrac{1}{12} \Big( \, (G_4)^a{c_1 \cdots c_3} (G_4)_{b c_1 \cdots c_3} - \tfrac{1}{8} (G_4)^{c_1 \cdots c_4} (G_4)_{c_1 \cdots c_4} \delta^a_b \;\;\;\; ({\color{darkblue}\text{Einstein equation}}) \\ \Gamma^{b a_1 a_2} \rho_{a_1 a_2} \;=\; 0 \;\;\;\; ({\color{darkblue}\text{Rarita-Schwinger equation}}) \end{array} \right. \end{array}

Cf. e.g. CDF91, (III.8.54-60); full details are given in GSS24, Lem. 3.8.

In conclusion, the above lemmas give Thm. .

The action functional

(…)

Kinetic terms

(…)

The higher Chern-Simons term

under construction

∫ X(16(C∧G∧G−C∧18(p 2+(12p 1) 2))) \int_X \left( \frac{1}{6} \left( C \wedge G \wedge G - C \wedge \frac{1}{8} \left( p_2 + (\frac{1}{2}p_1)^2 \right) \right) \right)

where p ip_i is the iith Pontryagin class.

λ≔12p 1. \lambda \coloneqq \frac{1}{2}p_1 \,.

Concerning the integrality of the I8-term

I 8≔148(p 2+(λ) 2) I_8 \coloneqq \frac{1}{48}(p_2 + (\lambda)^2)

on a spin manifold XX. (Witten96, p.9)

First, the index of a Dirac operator on XX is

I=11440(7(12p 1) 2−p 2)∈ℤ. I = \frac{1}{1440}(7 (\frac{1}{2}p_1)^2 - p_2) \in \mathbb{Z} \,.

Notice that 1440=6×8×301440 = 6 \times 8 \times 30. So

p 2−(12p 2) 2=6((12p 1) 2−30×8I) p_2 - (\frac{1}{2}p_2)^2 = 6 ( (\frac{1}{2}p_1)^2 - 30 \times 8 I)

is divisible by 6.

Assume that (12p 1)(\frac{1}{2}p_1) is further divisible by 2 (see the relevant discussion at M5-brane).

(12p 1)=2x. (\frac{1}{2}p_1) = 2 x \,.

Then the above becomes

p 2−(12p 2) 2=24(x 2−30×2I) p_2 - (\frac{1}{2}p_2)^2 = 24 ( x^2 - 30 \times 2 I)

and hence then p 2+(12p 1) 2p_2 + (\frac{1}{2}p_1)^2 is divisible at least by 24.

But moreover, on a Spin manifold the first fractional Pontryagin class 12p 1\frac{1}{2}p_1 is the Wu class ν 4\nu_4 (see there). By definition this means that

x 2=x(12p 1)mod2 x^2 = x (\frac{1}{2}p_1) \; mod \; 2

and so when (12p 1) 2(\frac{1}{2}p_1)^2 is further divisible by 2 we have that p 2−(12p 1) 2p_2 - (\frac{1}{2}p_1)^2 is divisible by 48. Hence I 8I_8 is integral.

Higher curvature corrections

Possible higher curvature corrections to 11-dimensional supergravity are discussed in the references listed below.

The first correction is an R 4R^4-term at order ℓ P 3\ell^3_{P} (11d Planck length). In Tsimpis 04 it is shown that part of this is a topological term (total derivative) which relates to the shifted C-field flux quantization.

For effects of higher curvature corrections in a Starobinsky model of cosmic inflation see there.

The hidden deformation

There is in fact a hidden 1-parameter deformation of the Lagrangian of 11d sugra. Mathematically this was maybe first noticed in (D’Auria-Fre 82) around equation (4.25). This shows that there is a topological term which may be expressed as

∝∫ X 11G 4∧G 7 \propto \; \textstyle{\int}_{X_11} G_4 \wedge G_7

where G 4G_4 is the curvature 3-form of the supergravity C-field and G 7G_7 that of the magnetically dual C6-field. However, (D’Auria-Fre 82) consider only topologically trivial (trivial instanton sector) configurations of the supergravity C-field, and since on them this term is a total derivative, the authors “drop” it.

The term then re-appears in the literatur in (Bandos-Berkovits-Sorokin 97, equation (4.13)). And it seems that this is the same term later also redicovered around equation (4.2) in (Tsimpis 04).

(hm, check)

BPS states

The basic BPS states of 11d SuGra are

(e.g. EHKNT 07)

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D(-2)-brane\,\,
D0-brane\,\,BFSS matrix model
D2-brane\,\,\,
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
D8-brane\,\,
(D=2n+1)(D = 2n+1)type IIB\,\,
D(-1)-brane\,\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
D5-brane\,\,\,
D7-brane\,\,\,
D9-brane\,\,\,
(p,q)-string\,\,\,
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
A-brane\,
B-brane\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
M-wave
topological M2-branetopological M-theoryC3-field on G₂-manifold
topological M5-brane\,C6-field on G₂-manifold
S-brane
SM2-brane,
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

References

General

That there is a maximal dimension d=11d = 11 in which supergravity may exist was found in

The theory was then actually constructed (as a Lagrangian field theory) in

The claim of the derivation of D=11D=11 supergravity in supergeometry, by solving the torsion constraint and Bianchi identities on super spacetime supermanifolds (“superspace”) is due to

and in the mild variation (using a manifestly duality-symmetric super-C-field flux density) due to

A proof of this claim is laid out in

using further lemmas and then heavy computer algebra checks (here).

With focus on the Kaluza-Klein compactification to 4d anti de Sitter spacetime:

The history as of the 1990s, with an eye towards the development to M-theory:

The description of 11d supergravity in terms of the D'Auria-Fré-Regge formulation of supergravity originates in

of which a textbook account is in

reviewed again in

The topological deformation (almost) noticed in equation (4.25) of D’Auria-Fre 82 later reappears in (4.13) of

and around (4.2) of Tsimpis 04

The D'Auria-Fré formulation is a first-order formulation of D=11D=11 supergravity; for more on this see:

More recent textbook accounts include

Discussion of the equivalence of the 11d SuGra equations of motion with the supergravity torsion constraints is in

following

Much computational detail is displayed in

In terms of pure spinors:

Duality-symmetric formulation

Formulation of the equations of motion of D=11 supergravity in superspace on fields including a flux density G 7G_7 a priori independent of the flux density G 4G_4 of the supergravity C-field:

Discussion of Lagrangian densities for D=11 supergravity with an a priori independent dual C-field field and introduction of the “duality-symmetric” terminology:

Discussion in the context of shifted C-field flux quantization:

Supergravity C-Field gauge algebra

Identifying the super-graded gauge algebra of the C-field in D=11 supergravity (with non-trivial super Lie bracket [v 3,v 3]=−v 6[v_3, v_3] = -v_6):

Identification as an L ∞ L_\infty -algebra (a dg-Lie algebra, in this case):

  • Hisham Sati, (4.9) in: Geometric and topological structures related to M-branes, in Superstrings, Geometry, Topology, and C *C^\ast-algebras, Proc. Symp. Pure Math. 81 (2010) 181-236 [ams:pspum/081, arXiv:1001.5020]

and identificatoin with the rational Whitehead L ∞ L_\infty -algebra (the rational Quillen model) of the 4-sphere (cf. Hypothesis H):

Classical solutions and BPS states

Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the branes and duality paradigm and the AdS/CFT correspondence.

One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund-Rubin compactification with geometry AdS 4×S 7AdS_4 \times S^7 and 4-form flux proportional to the volume form on AdS 4AdS_4.

  • Peter Freund, Mark Rubin, Dynamics of Dimensional Reduction Phys.Lett. B97 (1980) 233-235 (inSpire)

The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon geometry of coincident M2-branes.

Shortly after the original Freund-Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the S 7S^7; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in AdS 4AdS_4 and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the SO(8) symmetry of the sphere to an SO(7)SO(7) subgroup.

  • Francois Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity Phys.Lett. B119 (1982) 339 (inSPIRE)

Some of the above is taken from this TP.SE thread.

See also

Classification of symmetric solutions:

Discussion of black branes and BPS states includes

Discussion of black hole horizons includes

See also

  • Teng Fei, Bin Guo, Duong H. Phong, A Geometric Construction of Solutions to 11D Supergravity (arXiv:1805.07506)

Resolution of scalar field-dressed Schwarzschild black holes in D=11 supergravity:

Higher curvature corrections to D=11D=11 supergravity

Discussion of higher curvature corrections to 11-dimensional supergravity (i.e. in M-theory):

Via 11d superspace cohomology

Via 11d superspace-cohomology (or mostly):

Via superparticle scattering in 11d

Via analysis of would-be superparticle scattering amplitudes on D=11 supergravity backgrounds:

Via exceptional geometry

Via geodesic motion on the coset space of the U-duality Kac-Moody group E 10 E_{10} by its “maximal compact” subgroup K(E 10)K(E_{10}):

Via lifting 10d stringy corrections

From lifting alpha'-corrections in type IIA string theory through the duality between M-theory and type IIA string theory:

Via type IIB supergravity:

Via the ABJM M2-brane model

From the ABJM model for the M2-brane:

In terms of D=4 supergravity:

See also

See also

Discussion in view of the Starobinsky model of cosmic inflation is in

and in view of de Sitter spacetime vacua:

Scattering amplitudes and Effective action

Computation of Feynman amplitudes/scattering amplitudes and effective action in 11d supergravity:

Truncations and compactifications

Kaluza-Klein compactifications of supergravity and its consistent truncations:

Discussion of Freund-Rubin compactifications:

Topology and anomaly cancellation

Discussion of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles. (For more see at M5-brane – anomaly cancellation).

Description by exceptional generalized geometry

Review of U-duality and exceptional generalized geometry in KK-compactification of D=11 supergravity:

Last revised on February 12, 2025 at 18:27:04. See the history of this page for a list of all contributions to it.