This is based on this paper, http://arxiv.org/abs/hep-th/0212138

For a CFT on a $S^d$ spacetime (of radius R) it seems to be claimed that the central charge is given by, $ c = \langle \int_{S^d_R} d^dx \sqrt{g} T_\mu ^\mu \rangle $

Their equation 23 (on page 6) seems to indicate that if $W = -log Z$ is the free energy of the theory then it further follows that, $c = \frac{1}{d}R \frac{\partial }{\partial R} W$ (...I believe that the derivative is being evaluated at the value of the radius of the sphere..)

Just below equation 26 it is claimed that, "...the central charge can be read off from the coefficient of log R in an expansion of W[R]..."

I would like to know the proof/derivation of three methods that have been spelt out to calculate the central charge of a CFT.

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user user6818