Non-trivial extension of the (1+2)-Poincaré algebra and conformal invariance on the boundary of ${\mathrm{AdS}}_3$

Abstract

Using recent results on strings on AdS 3 × N d , where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2)dimensional-Poincaré algebra obtained by Rausch de Traubenberg and Slupinsky, refs[1]and [29] . We show by explicit computation that this new extension is a special kind of fractional supersymmetric algebra which may be derived from the deformation of the conformal structure living on the boundary of AdS 3 . The two so(1, 2) Lorentz modules of spin ± 1 k used in building of the generalisation of the (1+2) Poincaré algebra are re-interpreted in our analysis as highest weight representations of the left and right Virasoro symmetries on the boundary of AdS 3 . We also complete known results on 2d-fractional supersymmetry by using spectral flow of affine Kac-Moody and superconformal symmetries. Finally we make preliminary comments on the trick of introducing Fth-roots of g-modules to generalise the so(1, 2) result to higher rank Lie algebras g.

NC Effective Gauge Model for Multilayer FQH States

“…where B ex = ε 3µν ∂ µ A ex ν and using eq(4.6), one gets relation the density ρ 0 of electrons and quantum fluxes, namely ρ 0 = ν Bex 2π . Quantum mechanically, there are different field theoretical methods to approach the quantum states of this system, either by using techniques of non relativistic quantum mechanics [36], methods of conformal field theory especially for the study of droplets and edge excitations [31,32] or again by using the CS effective field model [33] describing the limit N → ∞ of electrons where now the Hall system is viewed as a liquid of particles. In this case, the Chern Simons theory in the (2 + 1) dimensional space modeling the FQH Laughlin state of filling fraction ν = 1 k is obtained by interpreting the ρ 0 density as…”

Finite-dimensional Lie algebras of order F

“…Fractional supersymmetry (FSUSY) [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] is among the possible extensions of supersymmetry which have been studied in the literature. Basically, in such extensions, the generators of the Poincaré algebra are obtained as F −fold (F ∈ N ⋆ ) symmetric products of more fundamental generators.…”

“…It is generally accepted that because of the theorems of Coleman & Mandula [4] and Haag, Lopuszanski & Sohnius [5], one cannot go beyond supersymmetry. However, if one weakens the hypotheses of these two theorems, one can imagine symmetries which go beyond supersymmetry [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], the idea being that then the generators of the Poincaré algebra can be obtained as an appropriate product of more than two fundamental additional symmetries.…”