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 ω^ab, 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 ω^ab and the Weyl projection of ψ for OSp(1|2), and the antiselfdual part of ω^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.