The boundary F-theorem for free fields
  The boundary free energy, as defined by Gaiotto, is further analysed for free scalars on a hemisphere and shown to be the same as the N-D determinant that earlier occurred in a treatment of GJMS operators. It is also shown to be identical, up to spin degeneracy, to the free energy for a spin-half field on the hemisphere boundary. This is also true if the hemisphere is replaced by a lune. The calculations are carried out in arbitrary dimensions. 
J.S.Dowker
hep-th