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

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Summary

INTRODUCTION

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.

CLASSICAL ORTHOSYMPLECTIC SUGRA
OSpð4j1Þ SUGRA
NC DEFORMATION
CONCLUSIONS
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