Canonical deformation of $N=2$ ${\mathrm{AdS}}_4$ supergravity

It is well known that one can define a consistent theory of extended, $N=2$ anti-de Sitter (AdS) supergravity (SUGRA) in $D=4$. Besides the standard gravitational part (including a negative cosmological constant), this theory involves a single $U(1)$ gauge field and a pair of Majorana vector spinors that can be combined to form a pair of Dirac vector spinors (charged spin-$3/2$ gravitini). The action for $N=2$ ${\mathrm{AdS}}_4$ SUGRA is invariant under diffeomorphisms, $SO(1,3)\times U(1)$ gauge transformations, and under local complex supersymmetry. We present a geometric action that involves two “inhomogeneous” parts: an orthosymplectic $\mathrm{OSp}(4|2)$ gauge-invariant action quadratic in the gauge field strength and a supplementary term invariant under the purely bosonic $SO(2,3)\times U(1)\sim Sp(4)\times SO(2)$ sector of $\mathrm{OSp}(4|2)$, which needs to be added for consistency. This action reduces to $N=2$ ${\mathrm{AdS}}_4$ SUGRA after suitable gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical ($\theta \text{−}\text{constant}$) deformation is performed by using the Seiberg-Witten approach to noncommutative (NC) gauge field theory with Moyal star product. The NC-deformed action is expanded in powers of the deformation parameter ${\theta }^{\mu \nu }$, up to the first order. We show that $N=2$ ${\mathrm{AdS}}_4$ SUGRA has nonvanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action term. For comparison, simple $N=1$ Poinacaré SUGRA can be obtained in the same manner from an $\mathrm{OSp}(4|1)$ gauge-invariant action (without introducing additional terms). The first nonvanishing NC correction is quadratic in the deformation parameter ${\theta }^{\mu \nu }$ and therefore exceedingly difficult to calculate. Under Wigner-Inönü contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincaré superalgebra, and it is not clear whether this relation holds after canonical NC deformation. We present the linear NC correction to $N=2$ ${\mathrm{AdS}}_4$ SUGRA explicitly and discuss its low-energy limit and what remains of it after Wigner-Inönü contraction.