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

Question

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

Comments

"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

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@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

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@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

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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

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@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

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In the large $R$ limit (see text between equations $26$ and $27$), $fR^{d-2\Delta} \to \infty$ , that is $fg_l \to \infty$, we may replace, in equation $25$, $\log (1+fg_l)$ by $\log (fg_l)$, and, in equation $24$, we have $g_l \sim R^{d-2\Delta}$, so we naturally get $\log R$ terms in the large $R$ limit.

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

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@Trimok hmm..that is interesting..so they are using this $f$ coupling in the Lagrangian to set a scale for what is "large" and "small" R and arguing that in the large R limit W is always logarithmic in R...is that so? So they are making an asymptotic statement that for CFTs on any dimensional sphere in the limit of large radius W will always asymptotically scale as log R...right?[...but this argument also means that in the limit of small radius or no double-trace deformation the free energy of the CFT vanishes..how does one understand that?..]

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

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@Trimok It would be great to know of your views on the other points too.. :)

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

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@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.

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

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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 $Z$.

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