A 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 $(\dim R{)}^2\exp [-c_2(R)/(b{T}^{d-1})]$. 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 $(\dim R{)}^2$ 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.

21 pages + references, 2 figures, 1 table

References in corpus (4)

• Cardy Formulae for SUSY Theories in d=4 and d=6
• A universal formula for the density of states in theories with finite-group symmetry
• Proof of the universal density of charged states in QFT
• EFT Asymptotics: the Growth of Operator Degeneracy

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• Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes
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