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Entanglement entropy across a deformed sphere

I study the entanglement entropy (EE) across a deformed sphere in conformal field theories (CFTs). I show that the sphere (locally) minimizes the universal term in EE among all shapes. In arXiv:1407.7249 it was derived that the sphere is a local extremum, by showing that the contribution linear in the deformation parameter is absent. In this paper I demonstrate that the quadratic contribution is positive and is controlled by the coefficient of the stress tensor two point function, $C_T$. Such a minimization result contextualizes the fruitful relation between the EE of a sphere and the number of degrees of freedom in field theory. I work with CFTs with gravitational duals, where all higher curvature couplings are turned on. These couplings parametrize conformal structures in stress tensor $n$-point functions, hence I show the result for infinitely many CFT examples.