Universal terms for the entanglement entropy in 2+1 dimensions

We show that the entanglement entropy and alpha entropies corresponding to spatial polygonal sets in 2 + 1 dimensions contain a term which scales logarithmically with the cutoff. Its coefficient is a universal quantity consisting in a sum of contributions from the individual vertices. For a free scalar field this contribution is given by the trace anomaly in a three-dimensional space with conical singularities located on the boundary of a plane angular sector. We find its analytic expression as a function of the angle. This is given in terms of the solution of a set of non-linear ordinary differential equations. For general free fields, we also find the small-angle limit of the logarithmic coefficient, which is related to the two-dimensional entropic c-functions. The calculation involves a reduction to a two-dimensional problem, and as a byproduct, we obtain the trace of the Green function for a massive scalar field in a sphere where boundary conditions are specified on a segment of a great circle. This also gives the exact expression for the entropies for a scalar field in a two-dimensional de Sitter space. (C) 2007 Elsevier B.V. All rights reserved.