Canonical Deformation of $N=2$ $Ad{S}_4$ SUGRA

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 $N=2$ $Ad{S}_4$ 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 $OSp(4|2)$ 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 $OSp(4|2)$, that needs to be added for consistency. This action reduces to $N=2$ $Ad{S}_4$ 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 $N=2$ $Ad{S}_4$ SUGRA has non-vanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action. For comparison, simple $N=1$ Poinacaré SUGRA can be obtained in the same manner, directly from an $OSp(4|1)$ gauge-invariant action. The first non-vanishing NC correction is quadratic in ${\theta }^{\mu \nu }$ and therefore exceedingly difficult to calculate. Under Wigner-Inönü (WI) contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincaré superalgebra, and it is not clear whether this relation holds after canonical deformation. We present the linear NC correction to $N=2$ $Ad{S}_4$ SUGRA explicitly, discuss its low-energy limit, and what remains of it after WI contraction.

arXiv comment25 pages

References in corpus (38)

• String Theory and Noncommutative Geometry
• Deformation quantization of Poisson manifolds, I
• A Gravity Theory on Noncommutative Spaces
• Noncommutative Geometry and Gravity
• Construction of non-Abelian gauge theories on noncommutative spaces
• New concept of relativistic invariance in NC space-time: twisted Poincaré symmetry and its implications
• Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces
• Emergent Geometry and Gravity from Matrix Models: an Introduction
• Deforming Einstein's Gravity
• MacDowell-Mansouri gravity and Cartan geometry
• Noncommutative SO(n) and Sp(n) Gauge Theories
• Riemann-Einstein Structure from Volume and Gauge Symmetry
• Noncommutative Gravity
• SL(2,C) Gravity with Complex Vierbein and Its Noncommutative Extension
• Explicit Fermi Coordinates and Tidal Dynamics in de Sitter and Goedel Spacetimes
• Noncommutative deformation of four dimensional Einstein gravity
• Supersymmetry of a different kind
• Cosmological solutions of emergent noncommutative gravity
• Noncommutative supergravity in D=3 and D=4
• Gravity and the Structure of Noncommutative Algebras
• Seiberg--Witten Maps to All Orders
• Noncommutative gravity at second order via Seiberg-Witten map
• Noncommutative o*(N) and usp*(2N) algebras and the corresponding gauge field theories
• NC $SO(2,3{)}_{\star }$ gravity: noncommutativity as a source of curvature and torsion
• Local supersymmetry without SUSY partners
• Spherically Symmetric Noncommutative Space: d = 4
• Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies
• Gravity with de Sitter and Unitary Tangent Groups
• Noncommutative SO(2,3) gauge theory and noncommutative gravity
• Fermi coordinates, simultaneity, and expanding space in Robertson-Walker cosmologies
• AdS-inspired noncommutative gravity on the Moyal plane
• OSp(1|4) supergravity and its noncommutative extension
• On some models of geometric noncommutative general relativity
• Who Ordered the Anti-de Sitter Tangent Group?
• Noncommutative gauge fields coupled to noncommutative gravity
• Noncommutative gravity and the relevance of the $\theta$-constant deformation
• Noncommutative Electrodynamics from $SO(2,3{)}_{\star }$ Model of Noncommutative Gravity
• Dirac field and gravity in NC $SO(2,3{)}_{\star }$ model

Cited by in corpus (3)

• Noncommutative $SO(2,3{)}_{\star }$ Gauge Theory of Gravity
• Resonant superalgebras for supergravity
• $\mathcal{N}=2$ resonant superalgebra for supergravity