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 Va, spin connection ωab, and Majorana gravitino ψ. In analogy with a construction used for D = 10 + 2 gauge supergravity, the action is given by ∫STr(R2Γ), where R is the OSp(1|4) curvature supermatrix two-form, and Γ a constant supermatrix containing γ5. 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) ⊕ 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 ωab and the Weyl projection of ψ for OSp(1|2), and the antiselfdual part of ωab 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.
In this paper we work in the gauge supersymmetry paradigm, that has been explored since long ago [8][9][10][11] and has allowed the construction of Chern-Simons supergravities in odd dimensions [12][13][14][15]. Recently it has been used to construct chiral gauge supergravity in D = 10 + 2 dimensions [16]. 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) JHEP10(2017)062 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 STr(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 [18][19][20]). They are candidate backgrounds for the N = 2 superstring [21][22][23][24][25], and are related after dimensional reduction to integrable models in D = 2 [26][27][28]. 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. [16].

The algebra
The OSp(1|4) superalgebra is given by: It is "restored" in second order formalism, or by modifying the spin connection transformation law, see for ex. [11,17].

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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.

Action
The action is written in terms of the OSp(1|4) curvature two-form R as: where STr is the supertrace and Γ is the constant matrix: All boldface quantities are 5 × 5 supermatrices.
The antiselfdual connection ω ab − is inert under supersymmetry, This will be important for the consistency of the selfduality condition ω ab − = 0, see section 3.6. Finally, the OSp(1|2)⊕Sp(2) transformation laws for the curvatures deduced from (3.3) read:

The action in terms of component fields
Recalling that STr(RR) is a topological term, we have: 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 (3.29) and the Bianchi identities consequences of the definitions (2.16) and (3.26) of R a and ρ. This action is similar to a MacDowell-Mansouri (MDM) 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.
As in the MDM case, we can introduce a length parameter λ by rescaling V → λV , ψ → √ λψ, and dividing the action by λ 2 . Contrary to the MDM case, here the Poincaré limit λ → 0 is singular, due to the term 1 2λ R ab −ψ − γ cd ψ − ǫ abcd originating from the last term in the action (3.25). The Poincaré limit however exists in the "projected" selfdual action, where ω ab − is set to zero (and therefore R ab − = 0), see section 3.6.

Equations of motion
The variational equations for the action (3.25) read: These equations admit the vacuum solution OSp(1|4) curvatures = 0. The torsion equation does not imply R a = 0, as in the MDM case, because of the second term in (3.36). As a consequence, the ψ − equation acquires the term γ a ψ − R a , absent in the MDM case. This term contains a mass parameter λ −1 , due to the second term in the torsion equation (after rescaling this term is multiplied by λ −1 ), and could provide a mass term for ψ − . To settle this issue one needs to solve the torsion equation. On the other hand, we know that in the MDM model the gravitino is massless even if gauge supersymmetry is completely broken. 3 3.6 Self-dual D = 2 + 2 supergravity We can impose the selfduality condition on the spin connection: Recalling that we can implement the selfduality condition (3.37) in the action by discarding the R ab − component of R ab in the first term of (3.25), and the R ab − part of R ab − in the last term. The resulting action (the "projected" 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 (3.36), which for ω ab − = 0 allows to express ω ab + as a function of V a , ψ + and ψ − . To make contact with the selfdual supergravities considered in [18][19][20], one can further impose ρ − = 0, so that ψ − has no dynamics and becomes an auxiliary field. This requirement is consistent with Bianchi identities, and with the field equations and the transformation rules of sections 3.5 and 3.3. As in refs. [18][19][20], it cannot be obtained as a field equation from the projected action.

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

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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).
A D = 2 + 2 γ matrices Clifford algebra. (to prove it, just multiply both sides by γ c or γ cd and take the trace on spinor indices).
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