  In three dimensions, the pure Maxwell theory with compact U(1) gauge group is
dual to a free compact scalar, and flows from the Maxwell theory with
non-compact gauge group in the ultraviolet to a non-compact free massless
scalar theory in the infrared. We compute the vacuum disk entanglement entropy
all along this flow, and show that the renormalized entropy F(r) decreases
monotonically with the radius r as predicted by the F-theorem, interpolating
between a logarithmic growth for small r (matching the behavior of the S^3 free
energy) and a constant at large r (equal to the free energy of the conformal
scalar). The calculation is carried out by the replica trick, employing the
scalar formulation of the theory. The Renyi entropies for n>1 are given by a
sum over winding sectors, leading to a Riemann-Siegel theta function. The
extrapolation to n=1, to obtain the von Neumann entropy, is done by analytic
continuation in the large- and small-r limits and by a numerical extrapolation
method at intermediate values. We also compute the leading contribution to the
renormalized entanglement entropy of the compact free scalar in higher
dimensions. Finally, we point out some interesting features of the reduced
density matrix for the compact scalar, and its relation to that for the
non-compact theory.
