Gauge supergravity in $D=2+2$

We present an action for chiral $N=(1,0)$ supergravity in $2+2$ dimensions. The fields of the theory are organized into an $OSp(1|4)$ connection supermatrix, and are given by the usual vierbein $V^a$, spin connection $\omega^{ab}$, and Majorana gravitino $\psi$. In analogy with a construction used for $D=10+2$ gauge supergravity, the action is given by $\int STr ({\bf R}^2 {\bf \Gamma})$, where ${\bf R}$ is the $OSp(1|4)$ curvature supermatrix two-form, and ${\bf \Gamma}$ a constant supermatrix containing $\gamma_5$. It is similar, but not identical to the MacDowell-Mansouri action for $D=2+2$ supergravity. The constant supermatrix breaks $OSp(1|4)$ gauge invariance to a subalgebra $OSp(1|2) \oplus Sp(2)$, including a Majorana-Weyl supercharge. Thus half of the $OSp(1|4)$ gauge supersymmetry survives. The gauge fields are the selfdual part of $\omega^{ab}$ and the Weyl projection of $\psi$ for $OSp(1|2)$, and the antiselfdual part of $\omega^{ab}$ for $Sp(2)$. Supersymmetry transformations, being part of a gauge superalgebra, close off-shell. The selfduality condition on the spin connection can be consistently imposed, and the resulting"projected"action is $OSp(1|2)$ gauge invariant.


Introduction
In most approaches to supergravity, local supersymmetry appears as the "square root" of diffeomorphisms, and has a natural interpretation as coordinate transformation along Grassmann directions. In this framework supersymmetry is part of a superdiffeomorphism algebra in superspace.
In Chern-Simons supergravities, on the other hand, supersymmetry "lives" on the fiber of a gauge supergroup rather than on a (super) base space. It is part of a gauge superalgebra of transformations leaving the Chern-Simons action invariant, up to boundary terms.
These two conceptually different ways of interpreting supersymmetry are fused together in the group geometric approach (a.k.a. group manifold or rheonomic framework, see for ex. [1]). Recent advances in superintegration theory [2] have shown how this approach interpolates between superspace and component actions.
In this paper we work in the gauge supersymmetry paradigm, that has been explored since long ago [3,4,5] and has allowed the construction of Chern-Simons supergravities in odd dimensions [6,7,8]. Recently it has been used to construct chiral gauge supergravity in D = 10 + 2 dimensions [9]. The twelve dimensional action is written in terms of the OSp(1|64) curvature supermatrix, but is invariant only under its OSp(1|32) ⊕ Sp(32) subalgebra. Supersymmetry is part of this superalgebra: it is generated by a Majorana-Weyl supercharge and closes off-shell. The constructive procedure relies on the existence of Majorana-Weyl fermions, and can in principle be applied in all even dimensions with signatures (s, t) satisying s − t = 0 (mod 8).
Here we apply it for the case s = 2, t = 2 to find an action for D = 2 + 2 chiral supergravity. This action is given by ST r(R ∧ RΓ), where R is the OSp(1|4) curvature supermatrix two-form, and Γ is a constant supermatrix involving γ 5 . Due to the presence of Γ, the action is not invariant under the full OSp(1|4) superalgebra, but only under a subalgebra OSp(1|2) ⊕ Sp(2) that includes a Majorana-Weyl supercharge. Thus chiral (1,0) supersymmetry survives. This is an important difference with the MacDowell-Mansouri action, for which gauge supersymmetry is completely broken 1 Supergravity theories in D = 2 + 2 dimensions have been considered by many authors in the past (for a very partial list of references see [11,12]). They are candidate backgrounds for the N = 2 superstring [13,14,15], and are related after dimensional reduction to integrable models in D = 2 [16,17,18]. The actions were obtained in most cases by supersymmetrizing the self-dual Einstein-Palatini action with (2,2) signature, supersymmetry invariance being of the "base space" type, as in usual supergravity in D = 3 + 1. The version we propose here differs because supersymmetry is chiral (1,0), and part of a gauge superalgebra, entailing automatic off-shell closure without need of auxiliary fields.
The paper is organized as follows. Section 2 recalls the definitions of OSp(1|4) connection and curvature, and their 5 × 5 supermatrix representation. Section 3 deals with the chiral D = 2 + 2 action, its invariances, field transformation laws, the explicit expression of the action in terms of component fields, field equations and selfduality condition. Section 4 contains some conclusions. Gamma matrix conventions in D = 2 + 2 are summarized in the Appendix.

OSp(1|4) connection and curvature
This section and the next one closely parallel the analogous sections for D = 10 + 2 supergravity of ref. [9].

The algebra
The OSp(1|4) superalgebra is given by: where M ab and P a , dual to the one-forms ω ab (spin connection) and V a (vierbein), generate the Sp(4) ≈ SO(3, 2) bosonic subalgebra, and the superchargeQ α is dual to the Majorana gravitino ψ α . Conventions on D = 2 + 2 gamma matrices and charge conjugation C αβ are given in the Appendix.

The 5 × 5 supermatrix representation
The above superalgebra can be realized by the 5 × 5 supermatrices: To verify the anticommutations (2.4), one needs the identity deducible from the Fierz identity (A.4) by factoring out the two spinor Majorana one-forms.

Connection and curvature
The 1-form OSp(1|4)-connection is given by In the 5 × 5 supermatrix representation: The corresponding OSp(1|4) curvature two-form supermatrix is where simple matrix algebra yields 2 : We have also used the Fierz identity for 1-form Majorana spinors in (A.4).

Action
The action is written in terms of the OSp(1|4) curvature two-form R as: where ST r is the supertrace and Γ is the constant matrix: All boldface quantities are 5 × 5 supermatrices.
Finally, ω ab − is inert under supersymmetry, This will be important for the consistency of the selfduality condition ω ab − = 0, see Section 3.6.

The action in terms of component fields
Recalling that ST r(RR) is a topological term, we have: up to boundary terms. Carrying out the supertrace leads to: with R and Σ as defined in Section 2.3, and P − = (1 − γ 5 )/2. After inserting the curvature definitions the action takes the form We have dropped the topological term R ab − R cd − ǫ abcd (sum of Euler and Pontryagin forms), and used the identities ψγ a ψR a = −2ργ a ψV a + total derivative (3.24) and the Bianchi identities consequences of the definitions (2.16) and (3.21) of R a and ρ. This action is similar to a MacDowell-Mansouri action in D = 2 + 2, or also to R 2 -type actions previously considered in the literature concerning self-dual supergravity, but with the important difference that it is invariant under a gauge (chiral) supersymmetry, closing off-shell.

Equations of motion
The variational equations for the action (3.20) read: These equations admit the vacuum solution OSp(1|4) curvatures = 0.

Self-dual D=2+2 supergravity
We can impose the selfduality condition on the spin connection: Recalling that we can implement the selfduality condition (3.32) in the action by discarding the R ab − component of R ab in the first term of (3.20), and the R ab − part of R ab − in the last term. The resulting action is invariant under the transformations of ω ab + , V a , ψ + , ψ − given in Section 3.3, with ε ab − = 0, i.e. under OSp(1|2) transformations whose gauge fields are ω ab + and ψ + . Indeed the condition ω ab − = 0 breaks OSp(1|2) × Sp(2) to its first factor OSp(1|2).
Second order formalism is retrieved by solving the torsion equation of motion (3.31), which for ω ab − = 0 allows to express ω ab + as a function of V a , ψ + and ψ − .

Conclusions
We have presented a D = 2+2 supergravity action, made out of the fields contained in the OSp(1|4) connection. It is invariant only under a subalgebra OSp(1|2)⊕Sp (2) of OSp(1|4). This closely resembles what happens for the Mac Dowell-Mansouri action in D = 3 + 1: there too the supergavity fields are organized in a OSp(1|4) connection, but the action itself is invariant only under the Lorentz subalgebra, whereas in the present paper also (1,0) supersymmetry survives, being part of the invariance subalgebra of the action. A selfdual condition can be imposed on the spin connection, and breaks the OSp(1|2) ⊕ Sp(2) invariance to the first factor OSp(1|2).