Recently it has been shown in 0901.0931 [hep-th] that the approach to extremality for the non-extremal Reissner-Nordstrom black hole is not continuous. The non-extremal RN black hole splits into two spacetimes at the extremality: an extremal black hole and a disconnected $AdS_2\times S^2$ space which has been called the "compactification solution". As a possible resolution for understanding the entropy of extremal RN black hole, it has been speculated that the entropy of the non-extremal black hole may be carried by the latter solution. By uplifting the four dimensional "compactification solution" with electric charge $Q_e$ to a five dimensional solution, we show that this solution is dual to a CFT with central charge $c=6Q_e^3$. The Cardy formula then shows that the microscopic entropy of the CFT is the same as the macroscopic entropy of the "compactification solution".
Extremal black holes have vanishing Hawking temperatures. In this article, we argue that for asymptotically AdS black holes, at extremality, a particular class of correlators in the dual CFT can exhibit exponential, maximally chaotic growth with a non-vanishing temperature. Our approach, at extremality, is two-fold. First, we geometrically investigate the modes that are responsible for chaos. Secondly, we study the dynamics of a probe string to capture chaos in worldsheet correlators. For rotating BTZ at extremality, the corresponding Lyapunov exponent is determined by the left-moving temperat…
Any unitary compact two-dimensional CFT with $c>1$ and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for $\bar{h}>\bar{h}_\text{extr}\sim \frac{c-1}{24}$, of a universal form determined by modular invariance. By including the contribution of light primary operators and multi-twist composites constructed from them in the modular bootstrap, we find that $\bar{h}_\text{extr}$ receives corrections in a large spin expansion, which we compute at finite $c$. The analysis uses a formulation of the modular S-transform as a Fourier transform acting on t…
The extremal functional method determines approximate solutions to the constraints of crossing symmetry, which saturate bounds on the space of unitary CFTs. We show that such solutions are characterized by extremality conditions, which may be used to flow continuously along the boundaries of parameter space. Along the flow there is generically no further need for optimization, which dramatically reduces computational requirements, bringing calculations from the realm of computing clusters to laptops. Conceptually, extremality sheds light on possible ways to bootstrap without positivity, extend…
Kaluza-Klein theory, AdS/CFT correspondence and black hole entropy
通过反演卡鲁扎-克莱因约化得到的提升（近极值）Reissner-Nordstrom黑洞的近视界几何的渐近对称性解释了贝肯斯坦-霍金熵偏离极值的原因。我们指出的事实是，额外的维度使我们能够证明使用Virasoro模式分解沿类时边界的近视界几何，AdS $_2\times$ S $^n$，低维（Reissner-Nordstrom）时空。
We will study the AdS/CFT correspondence in an intermediate region between the strong form of this correspondence (string theory on AdS being dual to a boundary CFT), and the weak form of this correspondence (supergravity on AdS being dual to a boundary CFT). We will go beyond the supergravity approximation in the AdS by using the fact that strings have an extended structure. We will also calculate the CFT dual to such string corrections in the bulk, and demonstrate that they are consistent with the strong form of the AdS/CFT correspondence. So, even though the conformal dimensions of both the…
