Higher spin <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>AdS</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> gravity and Tits-Satake diagrams
R. Sammani, Youssra Boujakhrout, El Hassan Saidi, R. Ahl Laamara, L.B. Drissi
Abstract
We investigate higher spin ${\mathrm{AdS}}_{3}$ gravity with real split forms of complex ${\mathrm{A}}_{N}\text{ }{\mathrm{B}}_{N}$, ${\mathrm{C}}_{N}$ and ${\mathrm{D}}_{N}$ Lie algebras. This is done by linking $SO(1,2)$ spin multiplets with splitted root systems using Tits-Satake diagrams of real forms. Unlike $SL(N,R)$, we show that the orthogonal families have two different higher spin (HS) spectrums; vectorial and spinorial. We find amongst others that the spinorial spectrum has an isolated spin ${\mathfrak{j}}_{\mathcal{N}}$ given by $\mathcal{N}(\mathcal{N}+1)/2$ for $SO(\mathcal{N},1+\mathcal{N})$ and $\mathcal{N}(\mathcal{N}\ensuremath{-}1)/2$ for $SO(\mathcal{N},\mathcal{N})$. We implement these results into the computation of the HS partition functions in these gravity theories and identify the individual contributions of the higher spin fields; valuable to manoeuver the HS--Ba\~nados-Teitelboim-Zanelli black hole partition function.