Higher spin AdS$_{3}$ gravity and Tits-Satake diagrams

TL;DR

This paper explores higher spin AdS3 gravity theories based on real split forms of classical Lie algebras, revealing distinct spectra and computing partition functions relevant for understanding higher spin black hole thermodynamics.

Contribution

It introduces a novel analysis of higher spin spectra using Tits-Satake diagrams for real forms of Lie algebras, highlighting differences from previous models like SL(N,R).

Findings

Orthogonal families exhibit vectorial and spinorial spectra.

Spinorial spectrum has an isolated spin j_N with specific formulas.

Partition functions for higher spin fields are explicitly computed.

Abstract

We investigate higher spin AdS$_{3}$ gravity with real split forms of complex A$_{N}$ B$_{N}$, C$_{N}$ and 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 j$_{\mathcal{N}}$ given by $\mathcal{N}\left(\mathcal{N}+1\right)/2$ for $SO(\mathcal{N},1+\mathcal{N})$ and $\mathcal{N} \left( \mathcal{N}-1\right) /2$ 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-BTZ black hole partition function