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Abstract: Hellerman et al. (arXiv:1505.01537) have shown that in a generic CFT the spectrum of operators carrying a large U(1) charge can be analyzed semiclassically in an expansion in inverse powers of the charge. The key is the operator state correpondence by which such operators are associated with a finite density
superfluid phase for the theory quantized on the cylinder. The dynamics is dominated by the corresponding Goldstone hydrodynamic mode and the derivative expansion coincides with the inverse charge expansion. I illustrate and hopefully clarify this situation by considering simple quantum mechanical analoques. I then systematize the approach by employing the coset construction for non-linearly realized space-time symmetries. Focussing on CFT3 I illustrate the case of higher rank and non-abelian groups and the computation of higher point functions. Three point function coefficients turn out to satisfy universal scaling laws and correlations as the charge and spin are varied.
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