# Cardy formula The high-energy (large $h,\bar h$) spectrum of any 2d CFT is constrained by the Cardy formula: $\overline{\rho(h, \bar{h})}=\rho_{0}(h) \rho_{0}(\bar{h}), \quad \rho_{0}(h) \approx \exp \left[2 \pi \sqrt{\frac{c}{6}\left(h-\frac{c}{24}\right)}\right],$ where the bar means averaging over some window of weights $(h,\bar h)$. This formula is universal in that it is true in any 2d CFT with $c>1$ and does not depend on any details of the theory. In holography ($c\gg1$), it is valid for a much larger regime, $h,\bar h\gtrsim c/12$, i.e., almost all the way down to the black hole threshold, as shown in [[2014#Hartman, Keller, Stoica]], assuming a certain sparseness of the spectrum. ## Refs - original: - [[1986#Cardy]] - higher dimensions: - [[1991#Cardy]] - in general dimensions: - [[Shaghoulian2015]] - much stronger form: - [[2014#Hartman, Keller, Stoica]] - [[2024#Dey, Pal, Qiao]]: proof - large spin: - [[2018#Kusuki]] - [[2019#Kusuki, Miyaji]] - [[2019#Benjamin, Ooguri, Shao, Wang]] - [[2019#Maxfield]] - similar universal formula for OPE coefficients: - [[2019#Collier, Maloney, Maxfield, Tsiares]] ## Comments - the averaging should be interpreted as a microcanonical averaging - according to [[2018#Mukhametzhanov, Zhiboedov]] - the Cardy formula can be obtained by modular transforming the statement that the identity operator has dimensions $h=\bar h=0$ ## BCFT In [[0548 Boundary CFT|BCFT]], the Cardy density for "open-string" states is given by$\rho_{a b}^{(\text{open})}(P) \sim e^{\frac{1}{2}\left(\mathfrak{s}_a+\mathfrak{s}_b\right)} \rho_0(P) \text { as } P \rightarrow \infty,$where $\mathfrak{s}_a$ is the boundary entropy for