# Some questions about calculation central charge in a CFT in $d$ spacetime dimensions

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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
"Method" $3$ is an obvious consequence of Method $2$ ($c = \frac{1}{d}R \frac{\partial }{\partial R} W$). If you get $W = a Log R +b$, you will have $c=\frac{a}{d}$

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user Trimok
@Trimok Isn't your argument just one-way? What if it were $W = aR^2 + bR$? Then $c = (R/d)(2aR + b)$. I don't get what you are saying. Firstly one can't have a term like "log(R)" in $W$ since $R$ has dimensions and one can't take log of something with dimensions. So even if such a log term arises it will come compensated with some scale to make the argument dimensionless. So it can still be something like, $W = alog(\Lambda R) + bR^2 + cR$ say and then this method-3 will not work!. And also why should such a "log(R)" term be natural?

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user user6818
@Trimok One way the things can match up is if in the method-3 there is an implicit limit of $R \rightarrow 0$..but why should that be?

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user user6818
If you look at expressions $(36), (40), (48),(50)$, you see that $V_1$ and $V_2$ ($W_f \sim V_1+V_2$), respect the expression $(a \log R+b)$.

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user Trimok
@Trimok But aren't these 3 methods quoted there a general statement about all CFTs? Or are they also claiming that this $a log (\Lambda R) + b$ is the general form for all $Ws$ for CFTs? [...you might have a look at this related question that I had asked - physics.stackexchange.com/questions/73612/… ... ]

This post imported from StackExchange Physics at 2014-03-07 13:48 (UCT), posted by SE-user user6818
@Trimok Also I would wonder how many of these statements are special about $S^d$..wonder what happens to these statements if one is on say $\mathbb{H}^{d-1} \times S^1$ - a natural space for the CFT to live in from the point of view of entanglement entropy.
Although it does not concern the sphere, and it is in the special context of string theory ($d=2$), there is an interesting paragraph in David Tong course, p $82-89$, Chapters $4.1,4.4.1,4.4.2,4.4.3$, about the relations between the central charge $c$, $<T^\mu_\mu>$, and the partition function $Z$.
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