A universal formula for the density of states with continuous symmetry

TL;DR

This work derives a universal high-temperature formula for the density-of-states of a $d$-dimensional unitary CFT with a compact Lie symmetry $G$, showing that the probability to lie in an irrep $R$ scales as $P_R=( ext{dim}R)^2(rac{4\, extpi}{bT^{d-1}})^{ ext{dim}G/2}\exp[-c_2(R)/(bT^{d-1})+ ext{O}(T^{-(d-1)})]$, with $b$ tied to domain-wall tension. The derivation uses spurion analysis via the twisted partition function $Z(T,g)$ and a character expansion based on the heat equation on $G$, providing a clear bridge to the two-dimensional Yang–Mills structure and connecting to black hole thermodynamics in AdS/CFT. The authors verify the formula in free-field theories and holographic CFTs, computing explicit $a$ and $b$ coefficients in these cases, and they extend the analysis to non-abelian $G$, where the high-$T$ behavior remains governed by $( ext{dim}R)^2$ and $c_2(R)$, while bulk holography reveals that black holes with non-abelian hair can be thermodynamically preferred at finite temperature. Overall, the work unifies finite- and continuous-symmetry cases and highlights deep links between high-temperature density of states, group-theoretic data, and black hole thermodynamics in AdS.

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 $(\operatorname{dim}R)^2\,\exp[-c_2(R)/(b\, T^{d-1})]$. 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 arXiv:2109.03838 that the probability is proportional to $(\operatorname{dim}R)^2$ when $G$ is a finite group. As a by-product of this analysis, we clarify thermodynamical properties of black holes with non-abelian hair in anti-de Sitter space.

Figures (2)

• Figure 5.1: They are plotted at a fixed value of $\Lambda=-1,\,v=1$ and $\sqrt{-\Lambda}\,G_NQ=100$.
• Figure 6.1: $\log\left[ G_N(F_{RN}-F)/r_H\right]$ as a function of ${h'_H\sqrt{v/-\Lambda}}$ when $\Lambda=-1$