11d-SuGra from Super C-Field Flux Quantization
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 -dimensional super-torsion-free super spacetime with super vielbein (the graviton/gravitino-fields) — follow from just the requirement that the duality-symmetric super-C-field flux densities :
-
satisfy their Bianchi identities
(1)
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are on any super-chart of the locally supersymmetric form
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 enters the EoMs as an independent field, whose Hodge duality to is imposed by the Bianchi identity for , 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 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:
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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;
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disregard the gauge potentials and , 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 -Whitehead -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 we here mean:
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a supermanifold
-
which admits an open cover by super-Minkowski supermanifolds ,
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equipped with a super Cartan connection with respect to the canonical subgroup inclusion of the spin group into the super Poincaré group, namely:
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equipped with a super-vielbein , hence on each super-chart
such that at every point the induced map on tangent spaces is an isomorphism
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and with a spin-connection (…),
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such that the super-torsion vanishes, in that on each chart:
(3)
where is a representation of , hence
Definition
(the gravitational field strength)
Given a super-spacetime (Def. ), we say that (super chart-wise):
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its super-torsion is:
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its gravitino field strength is
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its curvature is
Lemma
(super-gravitational Bianchi identities)
By exterior calculus the gravitational field strength tensors (Def. ) satisfy the following identities:
(4)
Write now for the unique non-trivial irreducible real -representation.
Theorem
(11d SuGra EoM from super-flux Bianchi identity) Given
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(super-gravity field:) an -dimensional super-spacetime (Def. ),
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(super-C-field flux densities:) as in (2)
then the super-flux Bianchi identity (1) (the super-higher Maxwell equation for the C-field)
is equivalent to the joint solution by 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 (1) is equivalent to
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the closure of the ordinary 4-flux density
-
the following dependence of on
shown in any super-chart:
(5)
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 in the super-vielbein basis is of the form
where the last term is taken to vanish.l (…).
Therefore, the Bianchi identity has the following components,
(6)
where we used that the quartic spinorial component vanishes identically, due to a Fierz identity (here):
To solve the second line in (6) for (this is CDF91 (III.8.43-49)) we expand in the Clifford algebra (according to this Prop.), observing that for to be a linear combination of the the matrix needs to have a -summand or a -summand. The former does not admit a Spin-equivariant linear combination with coefficients , hence it must be the latter. But then we may also need a component in order to absorb the skew-symmetric product in . Hence must be of this form:
(7)
With this, we compute:
(8)
Here the multiplicities of the nonvanishing Clifford-contractions arise via this Lemma:
and all remaining contractions vanish inside the spinor pairing by this lemma.
Now using (8) in (6) yields:
as claimed.
Lemma
Given the Bianchi identity for (5), then the Bianchi identity for (1) is equivalent to
-
the Bianchi identity for the ordinary flux density
-
its Hodge duality to
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another condition on the gravitino field strength
(9)
This is essentially
CDF91, (III.8.50-53).
Proof
The components of the Bianchi identity are
where:
(i) in the quadratic spinorial component we inserted the expression for from (5), then contracted -factors using again this Lemma, and finally observed that of the three spinorial quadratic forms (see there) the coefficients of and of vanish identically, by a remarkable cancellation of combinatorial prefactors:
-
(check)
-
(check)
(ii) the quartic spinorial component holds identitically, due to the Fierz identity here:
Therefore the only spinorial component of the Bianchi identity which is not automatically satisfied is (with , see there) the vanishing of
which is manifestly the claimed Hodge duality relation.
Lemma
Given the Bianchi identities for (5) and (9), the supergravity fields satisfy their Einstein equations with source the energy momentum tensor of the C-field:
(10)
Cf. e.g.
CDF91, (III.8.54-60); full details are given in
GSS24, Lem. 3.8.
In conclusion, the above lemmas give Thm. .