Non Trivial Extension of the (1+2)-Poincar\'e Algebra and Conformal Invariance on the Boundary of $AdS_3$

TL;DR

This paper explores a novel fractional supersymmetric extension of the (1+2)-Poincaré algebra derived from boundary conformal structures in $AdS_3$, revealing new algebraic relationships and representations.

Contribution

It introduces a fractional supersymmetric algebra extension of the (1+2) Poincaré algebra based on boundary conformal symmetries of $AdS_3$, connecting Lorentz modules to Virasoro representations.

Findings

Identifies the extension as a fractional supersymmetric algebra.

Reinterprets Lorentz modules as Virasoro highest weight representations.

Uses spectral flow to extend 2d-fractional supersymmetry results.

Abstract

Using recent results on string on $AdS_{3}\times N^d$, where N is a d-dimensional compact manifold, we re-examine the derivation of the non trivial extension of the (1+2) dimensional-Poincar\'e 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 $\pm{1\over k}$ used in building of the generalisation of the (1+2) Poincar\'e 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.