Abstract
It is 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, this theory involves a single $U(1)$ gauge field and a pair of Majorana vector-spinors that can be mixed into a pair of charged spin-$3/2$ gravitini. The action for $N=2$ $AdS_{4}$ SUGRA is invariant under $SO(1,3)\times U(1)$ gauge transformations, and under local SUSY. We present a geometric action that involves two "inhomogeneous" parts: an orthosymplectic $OSp(4\vert 2)$ gauge-invariant action of the Yang-Mills type, and a supplementary action invariant under purely bosonic $SO(2,3)\times U(1)\sim Sp(4)\times SO(2)$ sector of $OSp(4\vert 2)$, that needs to be added for consistency. This action reduces to $N=2$ $AdS_{4}$ SUGRA after gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical deformation is performed by using the Seiberg-Witten approach to noncommutative (NC) gauge field theory with the Moyal product. The NC-deformed action is expanded in powers of the deformation parameter $\theta^{\mu\nu}$ up to the first order. We show that $N=2$ $AdS_{4}$ SUGRA has non-vanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action. For comparison, simple $N=1$ Poinacar\'{e} SUGRA can be obtained in the same manner, directly from an $OSp(4\vert 1)$ gauge-invariant action. The first non-vanishing NC correction is quadratic in $\theta^{\mu\nu}$ and therefore exceedingly difficult to calculate. Under Wigner-In\"{o}n\"{u} (WI) contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincar\'{e} superalgebra, and it is not clear whether this relation holds after canonical deformation. We present the linear NC correction to $N=2$ $AdS_{4}$ SUGRA explicitly, discuss its low-energy limit, and what remains of it after WI contraction.
Highlights
In our quest for the theory of “quantum gravity,” we must be prepared to go beyond some deeply rooted assumptions on which we are accustomed; in particular, at very short distances, we might have to abandon the notion of a continuous space-time and the associated mathematical concept of a smooth manifold that describes it
We propose a geometric way of obtaining N 1⁄4 2 AdS4 SUGRA action and perform its NC deformation
We present a geometric action that consists of two “inhomogeneous” parts: an OSpð4j2Þ gauge-invariant action quadratic in gauge field strength and a supplementary action, invariant under the purely bosonic SOð2; 3Þ × Uð1Þ sector of OSpð4j2Þ, that has to be included in order to obtain complete N 1⁄4 2 AdS4 SUGRA at the classical level; this additional bosonic term produces a nontrivial linear NC correction to N 1⁄4 2 AdS4 SUGRA, after deformation
Summary
In our quest for the theory of “quantum gravity,” we must be prepared to go beyond some deeply rooted assumptions on which we are accustomed; in particular, at very short distances (very high energies), we might have to abandon the notion of a continuous space-time and the associated mathematical concept of a smooth manifold that describes it. The “distortion” from the original, “undeformed” one can be somehow parametrized In physics, this so-called deformation parameter appears as some fundamental constant of nature that measures the deviation from the classical (i.e., undeformed) theory. One obtains NC corrections to classical gravity, invariant under SOð1; 3Þ gauge transformations. We present a geometric action that consists of two “inhomogeneous” parts: an OSpð4j2Þ gauge-invariant action quadratic in gauge field strength and a supplementary action, invariant under the purely bosonic SOð2; 3Þ × Uð1Þ sector of OSpð4j2Þ, that has to be included in order to obtain complete N 1⁄4 2 AdS4 SUGRA at the classical level; this additional bosonic term produces a nontrivial linear NC correction to N 1⁄4 2 AdS4 SUGRA, after deformation.
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