  Renyi entropies S_q are useful measures of quantum entanglement; they can be
calculated from traces of the reduced density matrix raised to power q, with
q>=0. For (d+1)-dimensional conformal field theories, the Renyi entropies
across S^{d-1} may be extracted from the thermal partition functions of these
theories on either (d+1)-dimensional de Sitter space or R x H^d, where H^d is
the d-dimensional hyperbolic space. These thermal partition functions can in
turn be expressed as path integrals on branched coverings of the
(d+1)-dimensional sphere and S^1 x H^d, respectively. We calculate the Renyi
entropies of free massless scalars and fermions in d=2, and show how using
zeta-function regularization one finds agreement between the calculations on
the branched coverings of S^3 and on S^1 x H^2. Analogous calculations for
massive free fields provide monotonic interpolating functions between the Renyi
entropies at the Gaussian and the trivial fixed points. Finally, we discuss
similar Renyi entropy calculations in d>2.
