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