Gauge supergravity in D = 2 + 2

Abstract

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(R 2 Γ), where R is the OSp(1|4) curvature supermatrix two-form, and Γ a constant supermatrix containing γ 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) ⊕ 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.

Supergravity in the Group‐Geometric Framework: A Primer

“…Two explicit constructions are given in detail: the d = 5 Chern-Simons supergravity action [52,53], and the d = 4 Mac Dowell-Mansouri action [19]. For other odd-dimensional CS supergravity actions we refer to [54,18,55], while evendimensional d = 10 + 2 and d = 2 + 2 gauge supergravity actions have been constructed in [56] and [57] respectively.…”