A universal formula for the density of states with continuous symmetry

TL;DR

This paper derives a universal formula for the density of states in conformal field theories with continuous symmetry groups, relating the probability of states in specific representations to their dimensions and a domain wall tension, verified across various theories.

Contribution

It introduces a general formula for state probabilities in CFTs with continuous symmetries, extending previous results and connecting to black hole thermodynamics.

Findings

Formula verified for free field theories

Formula verified for holographic CFTs

Relation between constant b and domain wall tension

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.