KerrCFT Nonextremal Based on YMTsukiokaYoo ar Xiv 0907

Kerr/CFT対応における Non-extremal補正について 松尾 善典 Based on YM-Tsukioka-Yoo [ar. Xiv: 0907. 0303] YM-Nishioka [ar. Xiv: 1010. 4549]

Kerr/CFT対応におけるNon-extremal補正について � � � 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 = 12 J 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 = 12 J, 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 = 12 J 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。

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

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

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

Cardy formula for right mover 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 ≲ ≣≒≌⊹ ∰ ≓ ∽ ∲⊼ ∶ ∲ ∲ ∨∲⊼∩ ≡ ≔ ∽ ≇≎ ⊤

Non-extremal correction By using the Frolov-Thorne temperature, ∲≈ ∲⊼≡ ≞ ≲ ≓ ∽ ≇≎ ⊤ For near-extremal case, the entropy is ∲ ⊢⊢⊢ ∲⊼≡ ≈ ≓ ∽ ≇≎ ∨∱ ∫ ⊲≞≲ ∫ ∩ � � 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. ∫ be taken The boundary of the near horizon geometry should ⊡∱. Therefore, we identify around. ⊤ ∽ ∱∽⊲ ⊲ � ≲≞ ∮ ⊲ ⊡ ⊿ ≲ ≲ ≡ ⊤ ∽ ∱∽⊲

HCS and BTZ black hole In the Kerr background, ⋁ has a periodicity ⊻ ⋁ ⋁ ∫ ∲⊼ The approximated background is not equivalent to the Ad. S 3, but its quotient. BTZ black hole We define the “light-cone” coordinates as ⊡ ⊡ ≸ ∽ ⋁ ∲≍≡ ∲ ≴ Then, the “laplacian” for radial part becomes ⊷ ⊵ ⊶ ⊸ ⊡ ∲ ∲ ⊡ ⊡ ∫ ∫ ≀≲ ⊢≀≲ ∫ ⊢∱ ≲∫∴≡∫ ≲⊡ ≲ ∫∲ ≲ ≀∫≀⊡ ⊡ ≡∲≀∫∲ ⊡ ≡∲ ∨≲∨≲∫ ∫ ≲≲⊡∩∩∲ ≀⊡∲ ≸∫ ∽ ⋁∻

Conclusion and outlook � � � � We define a new near horizon limit. By using this limit, we obtain the central charge c. L = c. R = 12 J. 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.