Cardy formula

In physics, the Cardy formula gives the entropy of a (1+1)-dimensional conformal field theory (CFT). 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 c is the central charge, L 0 = E R is the product of the total energy and radius of the system, and the shift of c / 24 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 (n+1)-dimensional CFTs, it is also called Cardy-Verlinde formula.

Consider a 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 E_c is the Casimir effect, E total energy. The above reduced formula gives the maximal entropy

when E_c=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 (meaning n>1) CFTs is dependent on exactly marginal couplings, it is believed that a Cardy formula for the entropy is not achievable when n>1. However, for supersymmetric CFTs, a twisted version of the partition function, called "the superconformal index" (related to the Witten index) is shown by Di Pietro and Komargodski to exhibit Cardy-like behavior when n=3 or 5.