Universal formula for the density of states with continuous symmetry

Abstract

We consider a $d$-dimensional unitary conformal field theory with a compact Lie group global symmetry $G$ and show that, at high temperature $T$ and on a compact Cauchy surface, the probability of a randomly chosen state being in an irreducible unitary representation $R$ of $G$ is proportional to $(\mathrm{dim}\text{ }R{)}^{2}\mathrm{exp}[\ensuremath{-}{c}_{2}(R)/(b{T}^{d\ensuremath{-}1})]$. We use the spurion analysis to derive this formula and relate the constant $b$ to a domain wall tension. We also verify it for free field theories and holographic conformal field theories and compute $b$ in these cases. This generalizes the result in 2109.03838 that the probability is proportional to $(\mathrm{dim}R{)}^{2}$ when $G$ is a finite group. As a byproduct of this analysis, we clarify thermodynamical properties of black holes with non-Abelian hair in anti--de Sitter space.