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.

See also

• BTZ black hole
• AdS/CFT correspondence
• Holographic principle
• Conformal field theory

References

1. Cardy, John (1986), Operator content of two-dimensional conformal invariant theory, Nucl. Phys. B, vol. 270 186
2. Mukhametzhanov, Baur; Zhiboedov, Alexander (2019). "Modular invariance, tauberian theorems and microcanonical entropy". Journal of High Energy Physics. Springer Science and Business Media LLC. 2019 (10). arXiv: 1904.06359 . doi: 10.1007/jhep10(2019)261 . ISSN   1029-8479.
3. Verlinde, Erik (2000). "On the Holographic Principle in a Radiation Dominated Universe". arXiv: hep-th/0008140 .
4. D. Kutasov and F. Larsen (2000). "Partition Sums and Entropy Bounds in Weakly Coupled CFT". Journal of High Energy Physics. 2001: 001. arXiv: hep-th/0009244 . Bibcode:2001JHEP...01..001K. doi:10.1088/1126-6708/2001/01/001.

• Carlip, Steven (2005), "Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole", Classical and Quantum Gravity, 22: R85–R123, arXiv: gr-qc/0503022 , Bibcode:2005CQGra..22R..85C, doi:10.1088/0264-9381/22/12/R01