Cardy formula


In physics, the Cardy formula gives the entropy of a two-dimensional conformal field theory. In recent years, this formula has been especially useful in the calculation of the entropy of BTZ black holes and in checking the AdS/CFT correspondence and the holographic principle.
In 1986 J. L. Cardy derived the formula:
Here is the central charge, is the product of the total energy and radius of the system, and the shift of is related to the Casimir effect. These data emerge from the Virasoro algebra of this CFT.
Since E. Verlinde extended this formula in 2000 to arbitrary -dimensional CFTs, it is also called Cardy-Verlinde formula. Consider an AdS space with the metric
where R is the radius of a n-dimensional sphere. The dual CFT lives on the boundary of this AdS space. The entropy of the dual CFT can be given by this formula as
where Ec is the Casimir effect, E total energy. The above reduced formula gives the maximal entropy
when Ec=E, which is the Bekenstein bound. The Cardy-Verlinde formula was later shown by Kutasov and Larsen to be invalid for weakly interacting CFTs. In fact, since the entropy of higher dimensional CFTs is dependent on exactly marginal couplings, it is believed that a Cardy formula for the entropy is not achievable when n>1.