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:[1]

${\displaystyle S=2\pi {\sqrt {{\tfrac {c}{6}}{\bigl (}L_{0}-{\tfrac {c}{24}}{\bigr )}}},}$

Here ${\displaystyle c}$ is the central charge, ${\displaystyle L_{0}=ER}$ is the product of the total energy and radius of the system, and the shift of ${\displaystyle c/24}$ is related to the Casimir effect. These data emerge from the Virasoro algebra of this CFT. The proof of the above formula relies on modular invariance of a Euclidean CFT on the torus.

The Cardy formula is usually understood as counting the number of states of energy ${\displaystyle \Delta =L_{0}+{\bar {L}}_{0}}$ of a CFT quantized on a circle. To be precise, the microcanonical entropy (that is to say, the logarithm of the number of states in a shell of width ${\displaystyle \delta \lesssim 1}$) is given by

${\displaystyle S_{\delta }(\Delta )=2\pi {\sqrt {\frac {c\Delta }{3}}}+O(\ln \Delta )}$

in the limit ${\displaystyle \Delta \to \infty }$. This formula can be turned into a rigorous bound.[2]

In 2000, E. Verlinde extended this to certain strongly-coupled (n+1)-dimensional CFTs.[3] The resulting Cardy–Verlinde formula was obtained by studying a radiation-dominated universe with the Friedmann–Lemaître–Robertson–Walker metric

${\displaystyle ds^{2}=-dt^{2}+R^{2}(t)\Omega _{n}^{2}}$

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

${\displaystyle S={\frac {2\pi R}{n}}{\sqrt {E_{c}(2E-E_{c})}},}$

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

${\displaystyle S\leq S_{max}={\frac {2\pi RE}{n}},}$

when E_c=E, which is the Bekenstein bound. The Cardy–Verlinde formula was later shown by Kutasov and Larsen[4] 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.