A locally supersymmetric $SO(10,2)$ invariant action for $D=12$ supergravity

We present an action for $N=1$ supergravity in $10+2$ dimensions, containing the gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with $n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino $\psi$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$ respectively. The action is not gauge invariant under the full $OSp(1|64)$ superalgebra, but only under a subalgebra ${\tilde F}$ (containing the $F$ algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$ and the Weyl projected Majorana gravitino ${1 \over 2} (1+\Gamma_{13}) \psi$. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply $\int STr ({\bf R}^6 {\bf \Gamma})$ where ${\bf R}$ is the $OSp(1|64)$ curvature supermatrix two-form, and ${\bf \Gamma}$ is a constant supermatrix involving $\Gamma_{13}$ and breaking $OSp(1|64)$ to its ${\tilde F}$ subalgebra. The action includes the usual Einstein-Hilbert term.


Introduction
Supergravity theories in dimensions greater than D = 11 are believed to be inconsistent, since their reduction to D = 4 would produce more than N = 8 supersymmetries, involving multiplets with spin ≥ 2, and it is known that coupling of gravity with a finite number of higher spins is problematic.
On the other hand a twelve dimensional theory with signature (10,2) avoids this difficulty, since fermions can be both Majorana and Weyl in D = 10+2, with 32 real components, and therefore giving rise to at most eight supercharges when reduced to D = 4. This fact has encouraged over the years various attempts and proposals ([1] - [13]) for a twelve-dimensional field theory of supergravity.
A D = 10 + 2 structure emerges also from string/brane theory, and has been named F -theory [14]. The OSp(1|32) superalgebra, a natural choice for the gauge algebra of a D = 10 + 2 supergravity, is also called F algebra [13].
Here we follow the "gauge supergravity" road, inspired by Chern-Simons supergravities [18], and adapted to even dimensions via a higher curvature generalization of the Mac Dowell-Mansouri (MDM) action [21]. In the MDM framework (see also [23]) the fields of N = 1, D = 4 supergravity are seen as parts of the OSp(1|4) superconnection, as originally proposed in [22], and enter the MDM action only via the OSp(1|4) curvature. The MDM action however is not invariant under OSp(1|4) gauge transformations: indeed, besides the OSp(1|4) curvature, it also contains a constant matrix involving γ 5 , that breaks OSp(1|4) to its Lorentz subalgebra. Since all supertranslations are broken, one may wonder how could this action be locally supersymmetric. In fact one finds that it is supersymmetric in second order formalism (i.e. expressing the spin connection in terms of the vielbein and the gravitino via the supertorsion constraint). But also remaining in first order formalism supersymmetry can be restored by modifying the supersymmetry transformation law of the spin connection. The MDM action, after dispensing with topological terms, becomes the action of (anti)de Sitter N = 1 supergravity in D = 4. Note that supersymmetry here is not gauge supersymmetry, since it is not part of a superalgebra of transformations. There is no guarantee that the algebra of the "restored supersymmetry" closes off-shell, and indeed for this one needs auxiliary fields.
This being the state of affairs in D = 4, can we do something similar in D = 12 dimensions ? The present paper provides an affirmative answer. The superalgebra OSp(1|4) is replaced by OSp(1|64), the corresponding "would be" gauge fields being one-forms B (n) with n=1,2,5,6,9,10 antisymmetric Lorentz indices and a Majorana gravitino ψ. The vielbein and the spin connection are identified with B (1) and B (2) respectively. These one-forms are organized into an OSp(1|64) connection, in an explicit 65 × 65 dimensional supermatrix representation. The D = 12 action is constructed using exclusively the OSp(1|64) supermatrix curvature, and a constant matrix involving Γ 13 . This constant matrix ensures that the action is not topological (similarly to the MDM action) and breaks OSp(1|64) to a subalgebrã F = OSp(1|32) ⊕ Sp(32), under which the action is invariant 1 . Here part of the supersymmetry of OSp(1|64) survives, in contrast to the D = 4 case. Supersymmetry is then a gauge symmetry, and closes off-shell. Twelve dimensional Lorentz symmetry SO(10, 2) is also part of theF gauge symmetry, so that the action is SO(10, 2) invariant.
Finally, let us emphasize that by "D = 12 supergravity" we really mean a field theory in 10 + 2 dimensions, described by a geometrical action (the integral of a 12-form Lagrangian), containing the usual Einstein-Hilbert term and invariant under a gauge algebra that includes supersymmetry. This is much in the spirit of Chern-Simons supergravities in odd dimensions [24]. The implications of the field equations for the correct counting of degrees of freedom are still to be investigated.
The plan of the paper is as follows. Section 2 is a revisitation of the Mac Dowell-Mansouri action and its symmetries. Section 3 establishes notations for the OSp(1|64) superalgebra, its connection and curvature components. In Section 4 the D = 12, N = 1 supergravity action is presented, and the proof of its invariance underF gauge transformations is provided. Field transformation rules are given, and field equations are briefly discussed. Section 5 contains some conclusions and outlook.

Mac Dowell-Mansouri action revisited
This Section is just a review, and is included here because of its similarity to the D = 12 construction. Most of it is taken from ref. [25].
The Mac Dowell-Mansouri action [21] is a R 2 -type reformulation of (anti)de Sitter supergravity in D = 4. It is based on the supergroup OSp(1|4), and the fields V a (vierbein), ω ab (spin connection) and ψ (gravitino 2 ) are 1-forms contained in the OSp(1|4) connection Ω, in a 5 × 5 supermatrix representation: The corresponding OSp(1|4) curvature supermatrix is and straightforward matrix algebra yields 3 : We have also used the Fierz identity for 1-form Majorana spinors: (to prove it, just multiply both sides by γ c or γ cd and take the trace on spinor indices). The Mac Dowell-Mansouri action can be written in terms of the OSp(1|4) curvature R as: where ST r is the supertrace and G, Γ are the following constant matrices: All boldface quantities are 5 × 5 supermatrices. Carrying out the supertrace, and then the spinor trace, leads to the familiar expression of the MacDowell-Mansouri action: After inserting the curvature definitions the action takes the form We have dropped the topological term R ab R cd ǫ abcd (Euler form), and used the gravitino Bianchi identity and the gamma matrix identity 2γ ab γ 5 = iǫ abcd γ cd to recognize that 1 2 R abψ γ cd ψǫ abcd − 4iργ 5 ρ is a total derivative. The action (2.12) describes N = 1, D = 4 anti-De Sitter supergravity, the last term being the supersymmetric cosmological term. After rescaling the vielbein and the gravitino as V a → λV a , ψ → √ λψ and dividing the action by λ 2 , the usual (Minkowski) N = 1, D = 4 supergravity is retrieved by taking the limit λ → 0. This corresponds to the Inonü-Wigner contraction of OSp(1|4) to the superPoincaré group.

Invariances
As is well known, the action (2.9), although a bilinear in the OSp(1|4) curvature, is not invariant under the OSp(1|4) gauge transformations: where ǫ is the OSp(1|4) gauge parameter: In fact it is not a Yang-Mills action (involving the exterior product of R with its Hodge dual), nor a topological action ST r(RR): the constant supermatrices G and Γ ruin the OSp(1|4) gauge invariance, and break it to its Lorentz subgroup. Indeed the gauge variation of the action (2.9) vanishes when ǫ commutes with Γ (and therefore with G), and this happens only when ǫ in (2.16) has ε a = ǫ = 0, so that only Lorentz rotations leave the action invariant.
Specializing the gauge parameter ǫ to describe supersymmetry variations (i.e. only ǫ = 0 in (2.16)), eq. (2.17) yields the supersymmetry variation of the Mac Dowell-Mansouri action: with R defined in (2.3). This variation is proportional to the torsion R a , since only R a γ a in R has a nonzero commutator with γ 5 . Therefore in second-order formalism, i.e. using the torsion constraint R a = 0 to express ω ab in terms of V a and ψ, the action is indeed supersymmetric. Another way to recover supersymmetry is by modifying the supersymmetry variation of the spin connection, see for ex. [23]. In both cases supersymmetry is not part of a gauge superalgebra: off-shell closure of the supersymmetry transformations is not automatic, and indeed necessitates the introduction of auxiliary fields.
We shall now jump to 12 dimensions, and write a geometrical R 6 -type action that resembles the R 2 -type Mac Dowell-Mansouri action of D = 4. The action will be invariant under theF subalgebra of OSp(1|64). Contrary to the D = 4 case, N = 1 supersymmetry (with a Majorana-Weyl supercharge) survives as part of thẽ F algebra, and closes off-shell.

Connection and curvature
The 1-form OSp(1|64)-connection is given by In the 65 × 65 supermatrix representation: The corresponding curvature supermatrix 2-form is The bosonic curvature components R (n) are easily determined: where we used the Fierz identity for 1-form Majorana spinors in (A.10). For example for R (2) we find: where ST r is the supertrace and Γ is the constant matrix
Note: the analogous action in four dimensions STr(R 2 Γ), where Γ is as in (4.2) with Γ 13 substituted by γ 5 , is not invariant under (chiral) gauge supersymmetry since the supersymmetry parameter cannot be Majorana-Weyl in D = 3 + 1.
Thus we see that theF gauge fields Bn, ψ + transform with theF covariant derivative of the gauge parameters, whereas the "matter fields" Bṅ, ψ − transform homogeneously. These last include the vielbein V = B (1) . Note also that gauge and matter fields do not mix, separating into a gauge and a matter multiplet underF transformations.

Einstein-Hilbert term
The Lagrangian 12-form ST r(R 6 Γ) contains the usual Einstein-Hilbert term R a 1 a 2 ∧ V a 3 ∧ · · · ∧ V a 12 ε a 1 −a 12 (4.12) where R is the Lorentz curvature Indeed R 6 Γ includes a term (in the upper left corner of the supermatrix): Tracing on spinor indices produces the EH term (4.12), since T r[Γ a 1 a 2 Γ a 3 a 4 · · · Γ a 11 a 12 Γ 13 ] is proportional to ε a 1 −a 12 .

Conclusions
The D = 12 supergravity action we have presented here is made out of the fields contained in the OSp(1|64) connection, but is invariant only under a subalgebraF of OSp(1|64). This closely resembles what happens for the Mac Dowell-Mansouri action in D = 4, where the supergavity fields are organized in a OSp(1|4) connection, but the action itself is invariant only under the Lorentz subalgebra. In the D = 12 case, however, supersymmetry is part of the invariance subalgebra of the action, since a Majorana-Weyl supercharge is included inF . This supercharge can give rise to at most N = 8 supersymmetries once the action is reduced to four dimensions. We could try to export the constructive procedure adopted in this paper to the case of D = 9 + 1 supergravity. The relevant "starting" superalgebra is OSp(1|32), generated by Z (n) with n= (1,2,5,6,9,10) antisymmetric D = 10 Lorentz indices, and a Majorana charge. The same argument of Section 4 would lead to an action ST r(R 5 Γ) where Γ involves Γ 11 . This action however is invariant only under the bosonic subalgebra of OSp(1|32) generated by the Z (n) with n=(2,6,10), and not under the supersymmetry generated by a Weyl-projected Majorana charge. Indeed, contrary to the D = 10 + 2 case, Γ 11 ǫ = ǫ impliesǭΓ 11 = −ǭ (note the minus sign), so that the analogue of the commutator (4.6) does not vanish for supersymmetry variations.