松尾 善典 Based on YM-Tsukioka-Yoo [arXiv: ] YM-Nishioka [arXiv: ]

 Kerr/CFT 対応において Left mover は Extremal でのエントロピーを、 Right mover は Non-extremal の補正を与える。  Hidden conformal symmetry の解析から、 Central charge は c L = c R = 12J となると予想される。  しかし、 Near horizon limit における Asymptotic symmetry を用 いた解析では c L = 12J, c R = 0 となる。  そこで、新しい Near horizon limit を導入する。  この新しい Near horizon limit のもとで Asymptotic symmetry を 用いて Central charge を計算すると c L = c R = 12J となる。  このとき Left mover と Right mover それぞれの Leading order での エントロピーへの寄与は、 Bekenstein-Hawking エントロピーの示 唆する値と一致する。

≤≳ ∲ ∽ ⊡ ⊢ ⊽ ∲ ∨ ≤≴ ⊡ ≡ ≳≩≮ ∲ ⊵≤⋁ ∩ ∲ ∫ ≳≩≮ ∲ ⊵ ⊽ ∲ ⊣ ∨ ≲ ∲ ∫ ≡ ∲ ∩ ≤⋁ ⊡ ≡≤≴ ⊤ ∲ ∫ ⊽ ∲ ⊢ ≤≲ ∲ ∫ ⊽ ∲ ≤⊵ ∲ ⊢∽ ≲ ∲ ⊡ ∲ ≍≲ ∫ ≡ ∲ ∻⊽ ∲ ∽ ≲ ∲ ∫ ≡ ∲ ≣≯≳ ∲ ⊵ ≍ ≁≄≍ ∽ ≍ ≇ ≎ ∻≊ ∽ ≡≍ ≇ ≎ ≔ ≈ ∽ ≲ ∫ ⊡ ≲ ⊡ ∴ ⊼≍≲ ∫ ∻≓ ∽ ∲ ⊼≍≲ ∫ ≇ ≎ Metric of Kerr black hole is given by where 2 parameters are related to the ADM mass and angular momentum as Temperature and entropy are given by

⊲ ∡ ∰ ≴ ∽∲ ⊲ ⊡ ∱ ≡ ≞ ≴∻≲ ∽ ≡ ∨∱∫ ⊲ ≞ ≲ ∩ ∻⋁ ∽ ≞ ⋁ ∫ ≴ ∲ ≡ ≤≳ ∲ ∽ ⊡ ∨≞ ≲ ∲ ⊡ ≞ ≲ ∲ ≈ ∩ ≦ ∰ ∨ ⊵ ∩ ≤ ≞ ≴ ∲ ∫ ≦ ⋁ ∨ ⊵ ∩∨ ≤ ≞ ⋁ ∫≞ ≲≤ ≞ ≴ ∩ ∲ ∫ ≦ ∰ ∨ ⊵ ∩ ≤ ≞ ≲ ∲ ≞ ≲ ∲ ⊡ ≞ ≲ ∲ ≈ ∫ ≦ ∰ ∨ ⊵ ∩ ≤⊵ ∲ ≍ ∽ ≡ ⊵ ∱∫ ⊲ ∲ ≞ ≲ ∲ ≈ ∲ ⊶ ≦ ∰ ∨ ⊵ ∩∽ ≡ ∲ ∨∱∫≣≯≳ ∲ ⊵ ∩ ∻≦ ⋁ ∨ ⊵ ∩∽ ∴ ≡ ∲ ≳≩≮ ∲ ⊵ ∱∫≣≯≳ ∲ ⊵ We define near horizon coordinates as We consider near-extremal case And take the limit of Then, the metric becomes where

We first introduce a boundary condition In this case ⋂ ⊹⊺ ∽ ≲ ⊡ ≮ ⊹⊺ ≨ ⊹⊺ ∽ ≏ ∨ ⋂ ⊹⊺ ∩ Next, we introduce perturbations of same order Then, if the metric satisfies the following condition: ⊱ ⊻ ≧ ⊹⊺ ∽∤ ⊻ ≧ ⊹⊺ ∽ ≏ ∨ ⋂ ⊹⊺ ∩ ≧ ⊹⊺ ∽⊹ ≧ ⊹⊺ ∫ ≨ ⊹⊺ where The geometry is asymptotically symmetric, namely, ⊹ ≧ ⊹⊺ ∫ ≏ ∨ ⋂ ⊹⊺ ∩ ∡ ⊹ ≧ ⊹⊺ ∫ ≏ ∨ ⋂ ⊹⊺ ∩ ≏ ∨ ⋂ ⊹⊺ ∩

≛ ⊻ ≮ ∻⊻ ≭ ≝∽ ⊡ ≩ ∨ ≮ ⊡ ≭ ∩ ⊻ ≮∫≭ For left movers, the boundary condition is given by [Guica-Hartman-Song-Strominger,’08] Then, the asymptotic symmetry is given by This symmetry forms the Virasoro algebra where ⊲ ⊻ ≮ ∨ ≞ ⋁ ∩∽ ≥ ≩≮ ≞ ⋁ ⊻ ∽ ⊳ ⊲ ⊻ ∨ ≞ ⋁ ∩∫ ≏ ∨ ≲ ⊡ ∲ ∩ ⊴ ≀ ≞ ⋁ ∫ ⊳ ⊡ ≞ ≲⊲ ∰ ⊻ ∨ ≞ ⋁ ∩∫ ≏ ∨ ≲ ∰ ∩ ⊴ ≀ ≞≲ ∫ ⊳ ≃ ∫ ≏ ∨ ≲ ⊡ ∳ ∩ ⊴ ≀ ≞ ≴ ∫ ≏ ∨≞ ≲ ⊡ ∳ ∩ ≀ ⊵

We impose the following boundary condition ≛ ⊻ ≮ ∻⊻ ≭ ≝∽∨ ≮ ⊡ ≭ ∩ ⊻ ≮∫≭ Then, the asymptotic symmetry is given by ⊻ ∽ ⊳ ⊲ ⊻ ∨ ≞ ≴ ∩∫ ⊲ ∰∰ ⊻ ∨ ≞ ≴ ∩ ∲≞ ≲ ∲ ⊴ ≀ ≞ ≴ ∫ ⊳ ⊡ ≞ ≲⊲ ∰ ⊻ ∨ ≞ ≴ ∩∫ ⊲ ∰∰∰ ⊻ ∨ ≞ ≴ ∩ ∲≞ ≲ ⊴ ≀ ≞≲ ∫ ⊳ ≃ ⊡ ⊲ ∰∰ ⊻ ∨ ≞ ≴ ∩ ≞ ≲ ⊴ ≀ ≞ ⋁ ∫ ≏ ∨≞ ≲ ⊡ ∳ ∩ This symmetry forms the Virasoro algebra where ⊲ ⊻ ≮ ∨ ≞ ≴ ∩∽ ≞ ≴ ≮∫∱

≑ ⊻ ≛ ≨ ≝∽ ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≛ ≨∻ ⊹ ≧ ≝ ≾ ≫ ⊹⊺ ⊻ ≛ ≨∻ ⊹ ≧ ≝∽ ∱ ∲ ≨ ⊻ ⊹ ≄ ⊺ ≨ ⊡ ⊻ ⊹ ≄ ⊸ ≨ ⊸⊺ ∫ ⊡ ≄ ⊹ ≨ ⊺⊸ ⊢ ⊻ ⊸ ∫ ∱ ∲ ≨≄ ⊹ ⊻ ⊺ ⊡ ≨ ⊹⊸ ≄ ⊸ ⊻ ⊺ ∫ ∱ ∲ ≨ ⊹⊸ ∨ ≄ ⊺ ⊻ ⊸ ∫ ≄ ⊸ ⊻ ⊺ ∩ ⊡ ∨ ⊹ ∤ ⊺ ∩ ≩ ≫ ⊻ ≛ ≨∻ ⊹ ≧ ≝∽ ≾ ≫ ⊹⊺ ⊻ ≛ ≨∻ ⊹ ≧ ≝ ⊡ ≤ ∲ ≸ ⊢ ⊹⊺ ⊱ ⊳ ≑ ⊻ ∽ ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≛∤ ⊳ ⊹ ≧∻ ⊹ ≧ ≝∫ ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≛∤ ⊳ ≨∻ ⊹ ≧ ≝ ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≮ ≛∤ ⊻ ≭ ⊹ ≧∻ ⊹ ≧ ≝∽ ⊡ ≩⊱ ≮∫≭∻∰ ≮ ∳ ≣ ∱∲ ∫ ≏ ∨ ≮ ∩ Asymptotic Charge is defined as [Barnich-Brandt-Compere] where We consider transform of the charge itself The central charge can be read off from the first term

∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≮ ≛∤ ⊻ ≭ ⊹ ≧∻ ⊹ ≧ ≝∽ ⊡ ≩⊱ ≮∫≭∻∰ ∨ ≮ ∳ ∫∲ ≮ ∩ ≡ ∲ ≇ ≎ ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≫ ⊻ ≮ ≛∤ ⊻ ≭ ⊹ ≧∻ ⊹ ≧ ≝∽∰ ≣ ≌ ∽ ∱∲ ≡ ∲ ≇ ≎ ⊻ ∱∲ ≊∻≣ ≒ ∽∰ For left movers, we obtain And for right movers, Then, the central charges become

≔ ≌ ∽ ∱ ∲ ⊼ ∻≔ ≒ ∽ ≞ ≲ ≈ ∲ ⊼ Frolov-Thorne temperature is defined as ≥≸≰ ⊷ ⊡ ∡ ≔ ≈ ∫ ⊭ ≈ ≔ ≈ ≭ ⊸ ∽≥≸≰ ⊷ ⊡ ≮ ≌ ≔ ≌ ⊡ ≮ ≒ ≔ ≒ ⊸ In this case, we obtain By using the Cardy formula, ≓ ∽ ⊼ ∲ ∳ ≣ ≌ ≔ ≌ ∫ ⊼ ∲ ∳ ≣ ≒ ≔ ≒ ∽ ∲ ⊼≡ ∲ ≇ ≎ For left mover, the Cardy formula reproduce the entropy of the extremal Kerr black hole. For right mover, we obtain c R = 0, and does not contribute to the entropy.

Quasi-local charge is defined in a similar fashion to the GKPW ⊰ ⊹⊺ : Induced metric ≔ ⊹⊺ ∽ ∲ ≰ ⊡ ⊰ ⊱≓ ≧≲≡≶ ⊱⊰ ⊹⊺ The quasi-local charge is defined by We first define the surface energy-momentum tensor For Einstein gravity, it can be written as ≔ ⊹⊺ ∽ ∱ ∸ ⊼≇ ≎ ∨ ≋ ⊹⊺ ⊡ ⊰ ⊹⊺ ≋ ∩ ≋ ⊹⊺ : extrinsic curvature ⊿ ⊹⊺ ∽ ≔ ⊹⊺ ⊡ ≔ ≣≴ ⊹⊺ We regularize the surface energy-momentum tensor as ≵ ⊹ ⊻ ⊹ ⊾ ⊹⊺ ≑ ⊻ ∽ ≚ ≤ ∲ ≸ ≰ ⊡ ⊾≵ ⊹ ⊿ ⊹⊺ ⊻ ⊺ : timelike unit normal : Killing vector : Induced metric on timeslice at boundary

⊱≑ ⊻ ∽ ≡ ∲ ≇ ≎ ⊤ ⊲ ∰∰∰ ⊻ ∨ ≴ ∩ The central charge can be read off from the anomaly where we put the boundary at ≲ ∽ ⊤. Then, the central charge is ≣ ≒ ∽ ∱∲ ≡ ∲ ≇ ≎ ⊤ ⊹ ≌ ∰ ∽ ≍ ∽ ≡ ∲ ∲ ≇ ≎ ⊤ ∨∲ ⊼≔ ∩ ∲ For finite temperature, we obtain Then, the Cardy formula gives ≓ ∽∲ ⊼ ≲ ≣ ≒ ⊹ ≌ ∰ ∶ ∽ ∨∲ ⊼ ∩ ∲ ≡ ∲ ≔ ≇ ≎ ⊤

 The Cardy formula gives the non-extremal correction of the entropy, if we identify.  If is kept finite, the geometry is approximated by near horizon geometry in near horizon region.  The boundary of the near horizon geometry should be taken around. Therefore, we identify. For near-extremal case, the entropy is ≓ ∽ ∲ ⊼≡ ∲ ≇ ≎ ∨∱∫ ⊲ ≞ ≲ ≈ ∫ ⊢⊢⊢ ∩ By using the Frolov-Thorne temperature, ≓ ∽ ∲ ⊼≡ ∲ ≞ ≲ ≈ ≇ ≎ ⊤ ⊤∽∱ ∽⊲ ⊲ ≲ ⊡ ≲ ∫ ⊿ ≡ ≞ ≲ ∮ ⊲ ⊡ ∱ ⊤∽∱ ∽⊲

EOM for radial part of Scalar in Kerr background For small ∡, this equation can be approximated as [Castro-Maloney-Strominger ’10] We consider the scalar field in Kerr background. ≀ ⊹ ⊡ ≰ ⊡ ≧≧ ⊹⊺ ≀ ⊺ ⊩∨ ≴∻≲∻⋁∻⊵ ∩ ⊢ ∽∰ ∺ Then, the scalar field can be factorized as ⊩∨ ≴∻≲∻⋁∻⊵ ∩∽ ≥ ⊡ ≩∡≴∫≩≭⋁ ≒ ∨ ≲ ∩ ≓ ∨ ⊵ ∩ ⊷ ≀ ≲ ⊢≀ ≲ ∫ ∨∲≍≲ ∫ ∡ ⊡ ≡≭∩ ∲ ∨≲ ⊡ ≲ ∫ ∩∨≲ ∫ ⊡ ≲ ⊡ ∩ ⊡ ∨∲≍≲ ⊡ ∡ ⊡ ≡≭∩ ∲ ∨≲ ⊡ ≲ ⊡ ∩∨≲ ∫ ⊡ ≲ ⊡ ∩ ⊸ ≒∨≲∩∽≋≒∨≲∩ ⊷ ≀ ≲ ⊢≀ ≲ ∫ ∨∲≍≲ ∫ ∡ ⊡ ≡≭∩ ∲ ∨≲ ⊡ ≲ ∫ ∩∨≲ ∫ ⊡ ≲ ⊡ ∩ ⊡ ∨∲≍≲ ⊡ ∡ ⊡ ≡≭∩ ∲ ∨≲ ⊡ ≲ ⊡ ∩∨≲ ∫ ⊡ ≲ ⊡ ∩ ∫∨≲ ∲ ∫∲≍≲∫∴≍ ∲ ∩∡ ∲ ⊸ ≒∨≲∩∽≋≒∨≲∩

We define conformal coordinates as ≷ ∫ ∽ ≲ ≲ ⊡ ≲ ∫ ≲ ⊡ ≲ ⊡ ≥ ∲⊼≔ ≒ ⋁ ∻≷ ⊡ ∽ ≲ ≲ ⊡ ≲ ∫ ≲ ⊡ ≲ ⊡ ≥ ∲⊼≔ ≌ ⋁ ⊡ ≴ ∲≍ ∻ ≹ ∽ ≲ ≲ ∫ ⊡ ≲ ⊡ ≲ ⊡ ≲ ⊡ ≥ ⊼∨≔ ≒ ∫≔ ≌ ∩⋁ ⊡ ≴ ∴≍ ∻ where ≔ ≌ ∽ ≲ ∫ ∫ ≲ ⊡ ∴ ⊼≡ ∻≔ ≒ ∽ ≲ ∫ ⊡ ≲ ⊡ ∴ ⊼≡ Then, the laplacian becomes that on AdS 3. ≀ ≲ ⊢ ≀ ≲ ∫ ≩ ∨∲ ≍≲ ∫ ≀ ≴ ∫ ≡≀ ⋁ ∩ ∲ ∨ ≲ ⊡ ≲ ∫ ∩∨ ≲ ∫ ⊡ ≲ ⊡ ∩ ⊡ ≩ ∨∲ ≍≲ ⊡ ≀ ≴ ∫ ≡≀ ⋁ ∩ ∲ ∨ ≲ ⊡ ≲ ⊡ ∩∨ ≲ ∫ ⊡ ≲ ⊡ ∩ ∽ ∱ ∴ ≹ ∳ ≀ ≹ ∱ ≹ ≀ ≹ ∫ ≹ ∲ ≀ ∫ ≀ ⊡

We define the “light-cone” coordinates as We define the “light-cone” coordinates as Then, the “laplacian” for radial part becomes ≸ ∫ ∽⋁∻≸ ⊡ ∽⋁ ⊡ ≡ ∲≍ ∲ ≴ In the Kerr background, ⋁ has a periodicity ⋁ ⊻ ⋁ ∫∲ ⊼ The approximated background is not equivalent to the AdS3, but its quotient. BTZ black hole ≀ ≲ ⊢ ≀ ≲ ∫ ∱ ⊢ ⊷ ∴ ≡ ∲ ≲ ∫ ∫ ≲ ⊡ ⊵ ≲ ⊡ ≲ ∫ ∫ ≲ ⊡ ∲ ⊶ ≀ ∫ ≀ ⊡ ⊡ ≡ ∲ ≀ ∲ ∫ ⊡ ≡ ∲ ∨ ≲ ∫ ⊡ ≲ ⊡ ∩ ∲ ∨ ≲ ∫ ∫ ≲ ⊡ ∩ ∲ ≀ ∲ ⊡ ⊸

≤≳ ∲ ∽ ⊡ ∨⊽ ∲ ⊡ ≲ ∲ ∫ ∩∨⊽ ∲ ⊡ ≲ ∲ ⊡ ∩ ⊽ ∲ ≤⊿ ∲ ∫ ≬ ∲ ⊽ ∲ ≤⊽ ∲ ∨⊽ ∲ ⊡ ≲ ∲ ∫ ∩∨⊽ ∲ ⊡ ≲ ∲ ⊡ ∩ ∫⊽ ∲ ⊵ ≤∧ ⊡ ≲ ∫ ≲ ⊡ ⊽ ∲ ≤⊿ ⊶ ∲ ≸ ⊧ ∽∧ ⊧ ⊿∻⊽ ∲ ∽∨≲ ∫ ∫≲ ⊡ ∩≲ ⊡ ≲ ∫ ≲ ⊡ ≔ ≌ ∽ ≲ ∫ ∫≲ ⊡ ∲⊼≬ ∻≔ ≒ ∽ ≲ ∫ ⊡ ≲ ⊡ ∲⊼≬ The metric of the BTZ black hole can be written as The Frolov-Thorne temperatures are given by By introducing the following coordinates The laplacian in the BTZ background is expressed as Therefore, the approximated laplacian on Kerr geometry equals to that in BTZ if we identify ≬∽∲≡ ∴ ≬ ∲ ≀ ≲ ⊢ ≀ ≲ ∫ ∱ ⊢ ⊷ ∴ ≲ ∫ ∫ ≲ ⊡ ⊵ ≲ ⊡ ≲ ∫ ∫ ≲ ⊡ ∲ ⊶ ≀ ∫ ≀ ⊡ ⊡ ≀ ∲ ∫ ⊡ ∨ ≲ ∫ ⊡ ≲ ⊡ ∩ ∲ ∨ ≲ ∫ ∫ ≲ ⊡ ∩ ∲ ≀ ∲ ⊡ ⊸

≤≳ ∲ ∽ ⊡ ∨≞ ≲ ∲ ⊡ ≞ ≲ ∲ ≈ ∩ ≦ ∰ ∨ ⊵ ∩∨ ≤≸ ∫ ∩ ∲ ∫ ≦ ⋁ ∨ ⊵ ∩∨ ≤≸ ⊡ ∫≞ ≲≤≸ ∫ ∩ ∲ ∫ ≦ ∰ ∨ ⊵ ∩ ≤ ≞ ≲ ∲ ≞ ≲ ∲ ⊡ ≞ ≲ ∲ ≈ ∫ ≦ ∰ ∨ ⊵ ∩ ≤⊵ ∲ ⊻ ∽ ⊳ ⊲ ⊻ ∨ ≸ ∫ ∩∫ ⊲ ∰∰ ⊻ ∨ ≸ ∫ ∩ ∲≞ ≲ ∲ ⊴ ≀ ∫ ∫ ⊳ ⊡ ≞ ≲⊲ ∰ ⊻ ∨ ≸ ∫ ∩∫ ⊲ ∰∰∰ ⊻ ∨ ≸ ∫ ∩ ∲≞ ≲ ⊴ ≀ ≞≲ ∫ ⊳ ≃ ⊡ ⊲ ∰∰ ⊻ ∨ ≸ ∫ ∩ ≞ ≲ ⊴ ≀ ⊡ ∫ ≏ ∨≞ ≲ ⊡ ∳ ∩ In the near horizon limit, the metric becomes This geometry has the following periodicity The asymptotic symmetry for right mover is which should be expanded as which should be expanded as ≸ ∫ ⊻ ≸ ∫ ∫∲ ⊼≮⊲∻≸ ⊡ ⊻ ≸ ⊡ ∫∲ ⊼≮ ⊲ ⊻ ≮ ∨ ≸ ∫ ∩∽ ⊲≥ ≩≮≸ ∫ ∽⊲ We define new near horizon coordinates as ≸ ∫ ∽ ⊲⋁∻≸ ⊡ ∽ ⋁ ⊡ ≡≴ ∲ ≍ ∲ ∻≲ ∽ ≡ ∨∱∫ ⊲ ≞ ≲ ∩

≔ ∫ ∽ ≲ ∫ ⊡ ≲ ⊡ ∴ ⊼≡ ∡ ≞ ≲ ≈ ∲ ⊼ ⊲∻≔ ⊡ ∽ ≲ ∫ ∫ ≲ ⊡ ∴ ⊼≡ ∡ ∱ ∲ ⊼ ≣ ≒ ∽ ∱∲ ≡ ∲ ≇ ≎ ∽ ≣ ≌ Integrating on a time-slice, the following component contributes to the central charge: Then, the central charge becomes The Frolov-Thorne temperatures are given by The entropy can be reproduced by Cardy formula ∱ ∸ ⊼≇ ≎ ≚ ≀⊧ ≾ ≫ ⊡ ≲ ⊻ ≭ ≛∤ ⊻ ≮ ⊹ ≧∻ ⊹ ≧ ≝ ≤≸ ∫ ≤⊵ ∽ ⊱ ≮∫≭∻∰ ≮ ∳ ≡ ∲ ≇ ≎ ≓ ∽ ⊼ ∲ ∳ ≣ ≌ ≔ ≌ ∫ ⊼ ∲ ∳ ≣ ≒ ≔ ≒ ∽ ∲ ⊼≡ ∲ ≇ ≎ ∨∱∫ ⊲ ≞ ≲ ≈ ∩ ⊻ ∲ ⊼≍≲ ∫ ≇ ≎

 We define a new near horizon limit. By using this limit, we obtain the central charge c L = c R = 12J.  This new definition corresponds to a modification of the asymptotic symmetry.  There are higher order corrections from metric and Killing vectors of the asymptotic symmetry.  Left movers gives O (ε 0 ) contributions but right movers gives O (ε ).  To be exact, we have calculated only the leading term for left and right movers, respectively.  They agree with the expected result.  However, the next-to-leading term from the left movers is at the same order to the leading term from right movers.  It is left to be checked that this term vanishes.