Canonical deformation of N=2 AdS4 supergravity

Abstract

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 AdS4 SUGRA is invariant under diffeomorphisms, SO(1,3)×U(1) gauge transformations, and under local complex supersymmetry. We present a geometric action that involves two "inhomogeneous" parts: an orthosymplectic OSp(4|2) gauge-invariant action quadratic in the gauge field strength and a supplementary term invariant under the purely bosonic SO(2,3)×U(1)∼Sp(4)×SO(2) sector of OSp(4|2), which needs to be added for consistency. This action reduces to N=2 AdS4 SUGRA after suitable gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical (θ-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 θμν, up to the first order. We show that N=2 AdS4 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 OSp(4|1) gauge-invariant action (without introducing additional terms). The first nonvanishing NC correction is quadratic in the deformation parameter θμν 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 AdS4 SUGRA explicitly and discuss its low-energy limit and what remains of it after Wigner-Inönü contraction.