Cardy formula

In physics, the Cardy formula gives the entropy of a two-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 formula_2 is the central charge, formula_3 is the product of the total energy and radius of the system, and the shift of formula_4 is related to the Casimir effect. These data emerge from the Virasoro algebra of this CFT.
In 2000, E. Verlinde extended this to certain strongly-coupled (n+1)-dimensional CFTs. The resulting Cardy-Verlinde formula was obtained by studying a radiation-dominated universe with the Friedmann–Lemaître–Robertson–Walker metric
where R is the radius of a n-dimensional sphere at time t. The radiation is represented by a (n+1)-dimensional CFT. The entropy of that CFT is then given by the formula
where Ec is the Casimir effect, and E the 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 (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.