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 (dim ⁢𝑅)2²exp⁡[−𝑐₂⁡(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 2109.03838 that the probability is proportional to (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.

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