Strings and Quantum Gravity








RESEARCH > (A)dS/CFT Correspondence and Cosmology > Entropy of conformal field theories

The Bekenstein bound [32] for the ratio of the entropy S to the total energy E of a closed physical system that fits in a sphere in three spatial dimensions reads

\frac{S}{2\pi RE} \leq 1,

where R denotes the radius of the sphere. Despite many efforts, the microscopic origin of the bound remains elusive. A recent interesting development is Verlinde's observation [33] that CFTs possessing AdS duals satisfy a version of the Bekenstein bound. The observation of Verlinde is that for strongly coupled CFTs with AdS duals the entropy is given by a generalized Cardy formula

S=\frac{2\pi R}{D-1}\sqrt{E_C(2E-E_C)}

where EC is the sub-extensive part of the energy (Casimir energy). To show this, one employs the results for the entropy and total energy of the corresponding D-dimensional CFT that fits into a (D-1)-dimensional sphere at finite temperature [34]. These are obtained by virtue of holography from the corresponding thermodynamical quantities of a (D+1)-dimensional Schwarzschild AdS black hole. One obtains the bound

\frac{S}{2\pi R E}\leq \frac{1}{D-1}\,.

In view of the above developments, a natural question arising is whether there exists a microscopic derivation of Verlinde's formula within the thermodynamics of CFTs. This question could be checked in the context of CFTs whose microscopic thermodynamics is well understood, such as free CFTs on ${\mathbb{R}} \times S^{D-1}$. The relevant calculations for dimensions D = 4, 6 were recently undertaken by Kutasov and Larsen [35]. They computed the high temperature limits of various partition functions on $S^1 \times S^{D-1}$, from which all thermodynamical quantities follow. It was then shown that the Verlinde CFT bound is violated for free CFTs.

Dr. Siopsis and his collaborators, A. Petkou, D. Klemm and D. Zanon, performed a further analysis of the results in [35] for free CFTs in dimensions D = 4, 6 [36]. They found that for the specific cases of ${\cal N}=4$ U(N) SYM theory in D = 4 and the (2,0) tensor multiplet in D=6, the ratio of the entropy to the total energy is bounded from above, however the corresponding bounds are less stringent than the Verlinde bound. They showed that general bounds for the ratio of the entropy to the total energy in D-dimensional CFTs arise naturally under the requirement of monotonicity properties with respect to temperature changes of a generalized C-function. This generalized C-function is related to the sub-extensive part of the total energy. Although bounds for the ratio of the entropy to the total energy seem to arise quite generically in CFTs, their exact values depend on the details of the underlying CFT, e.g., it seems that the bounds become more stringent as one goes from weak to strong coupling.

Dr. Siopsis and collaborators then applied their results to the case of strongly coupled CFTs with AdS duals. They showed that the Verlinde formula remains valid also in the case of strongly coupled CFTs in a rotating Einstein universe. They pointed out an intriguing resemblance of the formulas of (D+1)-dimensional AdS black hole thermodynamics to corresponding formulas in the thermodynamics of two-dimensional CFTs. Particularly interesting is the fact that the entropy of the black hole resembles the C-function of a two-dimensional system. Motivated by this, they suggested a simple scaling form for the free energy of a D-dimensional CFT in a space with finite extent at finite temperature. Requiring then that the entropy of such a theory be given by a generalization of the two-dimensional entropy, leads to a simple differential equation whose solution yields a finite-size correlation length that turns out to coincide with the horizon distance of (D+1)-dimensional AdS black holes.

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