Entanglement entropy of a Maxwell field on the sphere

Understand

We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=4$ dimensions.

• In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum.
• We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of $l=0$ has been removed.
• This shows the relation $c^M_{\log}=2 (c^S_{\log}-c^{S_{l=0}}_{\log})$ between the logarithmic coefficient in the entropy for a Maxwell field $c^M_{\log}$, the one for a $d=4$ massless scalar $c_{\log}^S$, and the logarithmic coefficient $c^{S_{l=0}}_{\log}$ for a $d=2$ scalar with Dirichlet boundary condition at the origin.