Universal Multi-Region Entanglement Spectra in Conformal Field Theory

In any unitary conformal field theory (CFT), we obtain the precise joint spectrum and eigenvalue distribution of modular Hamiltonians for arbitrary nested spherical regions in this letter. We establish that all modular Hamiltonians are affine functions of a single self-adjoint operator, resulting in perfect linear correlations between entanglement spectra, using the thermal mapping to hyperbolic space and conformal covariance. Using a precise multi-boundary replica formula, the joint eigenvalue distribution is supported on a one-dimensional curve parametrised by the hyperbolic thermal spectrum. The findings demonstrate the universal geometric structure that underlies multi-region entanglement in CFTs and are non-perturbative and independent of model.