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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
asked Oct 27, 2013 in Theoretical Physics by user6818 (955 points) [ no revision ]
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"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
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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
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

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