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:^[1]

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

Here $c$ is the central charge, $L_{0}=ER$ 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,^[2] it is also called Cardy-Verlinde formula.

Consider a AdS space with the metric

ds^{2}=-dt^{2}+R^{2}\Omega _{n}^{2}

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

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

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

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^[3] 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^[4] to exhibit Cardy-like behavior when n=3 or 5.

References

↑ Cardy, John (1986), Operator content of two-dimensional conformal invariant theory, Nucl. Phys. B, 270 186
↑ Verlinde, Erik (2000). "On the Holographic Principle in a Radiation Dominated Universe". arXiv:hep-th/0008140.
↑ D. Kutasov and F. Larsen (2000). "Partition Sums and Entropy Bounds in Weakly Coupled CFT". arXiv:hep-th/0009244.
↑ L. Di Pietro, Z. Komargodski (2014). "Cardy Formulae for SUSY Theories in d=4 and d=6". arXiv:1407.6061.

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

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Cardy formula

See also

References