The entanglement spectrum of a bipartite quantum system is given by the distribution of eigenvalues of the modular Hamiltonian.

In this talk, I compute the entanglement spectrum in the vacuum state for a subregion of a d-dimensional conformal field theory (CFT) admitting a holographic dual.

In the case of a spherical (or planar) entangling surface, I recover known results in two dimensions, including the Cardy formula in the high energy regime.

In higher dimensions (d>2), I analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of

CFT spectra in the literature.

In the case of a spherical (or planar) entangling surface, I recover known results in two dimensions, including the Cardy formula in the high energy regime.

In higher dimensions (d>2), I analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of

CFT spectra in the literature.

I also numerically investigate the spectrum at energy levels far above the modular ground state energy.

Then, I extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when d=2,3, and a generalization of the Cardy formula at high energies in generic dimension d.

I consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases, I find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation.

This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.

Then, I extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when d=2,3, and a generalization of the Cardy formula at high energies in generic dimension d.

I consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases, I find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation.

This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.