Gauge supergravity in $D=2+2$

TL;DR

This paper formulates a chiral supergravity action in 2+2 dimensions using an $OSp(1|4)$ connection, breaking gauge symmetry to a subalgebra and maintaining off-shell supersymmetry, with a focus on selfdual spin connections.

Contribution

It introduces a novel gauge supergravity action in 2+2 dimensions based on an $OSp(1|4)$ supermatrix, with partial gauge symmetry breaking and off-shell supersymmetry.

Findings

The action is similar to but distinct from the MacDowell-Mansouri formulation.

Half of the original gauge supersymmetry remains after symmetry breaking.

Selfduality of the spin connection can be imposed consistently, preserving gauge invariance.

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 $\omega^{ab}$, 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.