nLab 11d SuGra from super C-field flux quantization -- section

11d-SuGra from Super C-Field Flux Quantization

11d-SuGra from Super C-Field Flux Quantization

We discuss (Thm. below, following GSS24) 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 UXU \hookrightarrow X of the locally supersymmetric form

    (2)G 4 s=14!(G 4) a 1a 4e a 1e a 412(ψ¯Γ a 1a 2ψ)e a 1e a 2 G 7 s=17!(G 7) a 1a 7e a 1e a 715!(ψ¯Γ a 1a 5ψ)e a 1e 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 (and apparently nowhere else), and 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 \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):



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 1,D1|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,D1)Iso( 1,D1|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 UXU \hookrightarrow X

      ((e a) a=0 D=1,(ψ α) α=1 N)Ω dR 1(U; 1,D1|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 xXx \in \overset{\rightsquigarrow}{X} the induced map on tangent spaces is an isomorphism

      (e,ψ) x:T xX 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 aω a be b=(ψ¯Γ aψ), \mathrm{d} \, e^a - \omega^a{}_b \, e^b \;=\; \big( \overline{\psi} \,\Gamma^a\, \psi \big) \,,

    where Γ (): 1,D1End (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} \,.


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

  1. its super-torsion is:

    T ade aω 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

    ρdψ+14ω abΓ abψ, \rho \;\coloneqq\; \mathrm{d}\, \psi + \tfrac{1}{4} \omega_{a b}\Gamma^{a b}\psi \,,
  3. its curvature is

    R a bdω a bω a cω c b. R^{a}{}_b \;\coloneqq\; \mathrm{d}\, \omega^{a}{}_b \,-\, \omega^a{}_c \, \omega^c{}_b \,.


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

(4)dR a b = ω a aR a bR a bω 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}


(role of the gravitational Bianchi identities)
Notice that the equations (4) are not conditions but identities satisfied by any super-spacetime (in the sense of Def. , hence even such that T a=0T^a = 0.) But conversely this means that when constructing a super-spacetime (say subject to further contraints, such as Bianchi identities for flux densities), the equations (4) are a necessary condition to be satisfied by any candidate super-vielbein-field, and as such they may play the role of equations of motion for the super-gravitational field, as we will see.

Write now 32Rep (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.


(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.


(Normalization conventions)
Our choice of prefactors and normalization follows CDF91 except for the following changes:

  • our Clifford generators absorb a factor of i \mathrm{i} : Γ a=iΓ a DF\;\;\;\Gamma_a \;=\; \mathrm{i}\, \Gamma_a^{^{DF}}

  • our gravitinos absorb a factor of 2\sqrt{2}: ψ=2ψ DF\;\;\;\psi \;=\; \sqrt{2}\psi^{^{DF}}

  • our 4-flux density absorbs a combinatorial factor of 1/21/2: G 4=12R \;\;\;G_4 = \tfrac{1}{2} R^{\Box}

  • our 7-flux density absorbs a combinatoiral factor of 1/5!1/5!: G 7=15!R \;\;\;G_7 = \tfrac{1}{5!} R^{\otimes}


  • The first rescaling reflects that Γ DF\Gamma^{{}^{\mathrm{DF}}} is not actually a Majorana representation of Pin +(1,10)\mathrm{Pin}^+(1,10), but iΓ DF\mathrm{i}\Gamma^{{}^{\mathrm{DF}}} is.

    This rescaling removes all occurrences of imaginary units in the Bianchi identities, as it should be for algebra over the real numbers with real fermion representations.

  • The second rescaling has the effect that de a=(ψ¯Γ aψ)+\mathrm{d} e^a = \big(\overline{\psi} \Gamma^a \psi\big) + \cdots instead of de a=12(ψ¯Γ aψ)+\mathrm{d}\, e^a = \tfrac{1}{2} \big(\overline{\psi} \Gamma^a \psi\big) + \cdots.


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 1a 4)e ae a 1e a 4=0 ρ=ρ abe ae b+(1613!(G 4) ab 1b 2b 3Γ ab 1b 2b 3+11214!(G 4) b 1b 4Γ ab 1b 4)ψe a (14!ψ α α(G 4) a 1a 4+(ψ¯Γ a 1a 2ρ a 3a 4))e a 1e 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 \,+\, \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 CDF91, (III.8.44-49 & 60b).

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 1a 4e a 1e a 412(ψ¯Γ a 1a 2ψ)e a 1e a 2)=0 {( a(G 4) a 1a 4)e ae a 1e a 4=0 (14!ψ α( α(G 4) a 1a 4)+(ψ¯Γ a 1a 2ρ a 3a 4))e a 1e 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 1a 4(G_4)_{a_1 \cdots a_4}, hence it must be the latter. But then we may also need a component Γ a 1a 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 1b 4Γ ab 1b 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 1b 4(ψ¯Γ a 1a 2Γ a 3b 1b 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 1b 3a 3(ψ¯Γ a 1a 2 b 1b 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.


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 2a 8)e a 1e a 8=12(14!(G 4) a 1a 414!(G 4) a 5a 8)e a 1e a 8 (G 7) a 1a 7=14!ϵ a 1a bb 1b 4(G 4) b 1b 4 (17!ψ α α(G 7) a 1a 7ψ α+25!(ψ¯Γ a 1a 5ρ a 6a 7))e a 1e 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).

The components of the Bianchi identity are

dG 4 s=0 {d(17!(G 7) a 1a 7e a 1e a 715!(ψ¯Γ a 1a 5ψ)e a 1e a 5) =12(14!(G 4) a 1a 4e a 1e a 412(ψ¯Γ a 1a 2ψ))(14!(G 4) a 1a 4e a 1e a 412(ψ¯Γ a 1a 2ψ)) {( a 117!(G 7) a 2a 8=1214!(G 4) a 1a 414!(G 4) a 5a 8)e a 1e a 8 (17!ψ α α(G 7) a 1a 7+25!(ψ¯Γ a 1a 5ρ a 6a 7))e a 1e a 7=0 16!(G 7) a 1a 6b(ψ¯Γ bψ)e a 1e a 6 +21215!14!(G 4) b 1b 4(ψ¯Γ a 1a 5Γ ab 1b 4ψ)e ae a 1e a 5 2615!13!(G 4) ab 1b 2b 3(ψ¯Γ a 1a 5Γ b 1b 2b 3ψ)e ae a 1e a 5 (12(ψ¯Γ a 1a 2ψ)e a 1e a 2)14!(G 4) b 1b 4e b 1e b 4=0,}(G 7) a 1a 6b=14!ϵ a 1a 6bb 1b 4(G 4) b 1b 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}


(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 1a 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 2a 5(ψ¯Γ aa 1ψ)e ae a 1e 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 3a 6 b 1b 2ψ)e a 1e 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 1a 5ψ)(ψ¯Γ a 1)e a 2e 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 1a 6b14!(G 4) b 1b 4ϵ b 1b 4a 1a 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.


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 these source terms:

dG 4 s=0,dG 7 s=12G 4 sG 4 2 {R bm am12δ b aR mn mn=const(G 4) ac 1c 3(G 4) bc 1c 3+constδ b a(G 4) c 1c 4(G 4) c 1c 4(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} \;=\; const\, (G_4)^a{c_1 \cdots c_3} (G_4)_{b c_1 \cdots c_3} + const\, \delta^a_b \, (G_4)^{c_1 \cdots c_4} (G_4)_{c_1 \cdots c_4} \;\;\;\; ({\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}

This is indicated in CDF91, (III.8.54-60). It follows by solving the super-gravitational Bianchi identities (4), using from the previous lemmas only the form of the gravitino field strength (5).

In conlcusion, the above lemmas give Thm. .

Last revised on April 10, 2024 at 17:10:33. See the history of this page for a list of all contributions to it.