Renyi Entropy and Geometry

Entanglement entropy in even dimensional conformal field theories (CFTs) contains well– known universal terms arising from the conformal anomaly. Rényi entropies are natural gen-eralizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the Rényi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four–dimensional Rényi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide ev-idence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four–dimensional free massless Rényi entropies across cylindrical entangling surfaces to corresponding three–dimensional massive Rényi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional Rényi entropy, including the massless renormalized Rényi entropy and calculable contributions to the perimeter law