Universal formula for the density of states with continuous symmetry

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 (dim R)2 exp [–c2(R)/(bTd–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 (dimR)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.