Rainbow Valley of Colored (Anti) de Sitter Gravity in Three Dimensions

Abstract

We propose a theory of three-dimensional (anti) de Sitter gravity carrying Chan-Paton color charges. We define the theory by Chern-Simons formulation with the gauge algebra $(\mathfrak{gl}_{2}\oplus \mathfrak{gl}_{2})\otimes \mathfrak{u}(N)$, obtaining a color-decorated version of interacting spin-one and spin-two fields. We also describe the theory in metric formulation and show that, among $N^{2}$ massless spin-two fields, only the singlet one plays the role of metric graviton whereas the rest behave as \emph{colored spinning matter} that strongly interacts at large $N$. Remarkably, these \emph{colored spinning matter} acts as Higgs field and generates a non-trivial potential of staircase shape. At each extremum labelled by $k = 0, \ldots, [\frac{N-1}2]$, the $\mathfrak{u}(N)$ color gauge symmetry is spontaneously broken down to $\mathfrak{u}(N-k)\oplus \mathfrak{u}(k)$ and provides different (A)dS backgrounds with the cosmological constants $\big(\frac{N}{N-2k}\big)^{2}\,Ł$\,. When this symmetry breaking takes place, the spin-two Goldstone modes combine with (or are eaten by) the spin-one gauge fields to become partially-massless spin-two fields. We discuss various aspects of this theory and highlight physical implications.