臺灣博碩士論文加值系統

我們首先回顧了Kerr黑洞在極端以及近極端的情況下與共形場論之間的對應關係。根據在共形場與在黑洞視界表面上計算出的熵是相等的，我們可以相信極端Kerr黑洞與共形場之間的確存在對應關係。更進一步藉由無質量的純量場在Kerr黑洞背景之中的散射，計算出在Kerr背景下純量場的吸收截面，與在邊界上的共形場中的兩點函數是一致的，所以Kerr黑洞在近極端的情況下與共形場之間也存在有對應關係。特別的，當無質量的純量場在頻率很低的情況下，其徑向波方程的解空間本身就含有SL(2,R)×SL(2,R)的共形對稱。我們利用這種特性研究Kerr-Newman黑洞，發現有兩種不一樣的共形場的對應關係，我們稱之為J圖像以及Q圖像。最後我們將原先在黑洞背景下計算的兩點函數推廣至三點函數，其結果與共形場中的三點函數相符。

We review the conjecture of Kerr/CFT correspondence in extremal and near-extremal
limit. By the matching of the CFT (Cardy formula) and black hole (Bekenstein-
Hawking formula) entropies, the conjecture of extreme Kerr/CFT correspondence is
confirmed. From the scattering of scalar field near the superradiance region in kerr
geometry, the matching of the absorption cross section and the finite-temperature
two-point function of dual CFT operator also implies the validity of near-extreme
Kerr/CFT correspondence. Moreover, for generic Kerr black holes, the SL(2, R) ×
SL(2, R) conformal symmetry can be obtained in the solution space of the radial wave
equation at low frequency. The black hole/CFT can be extended to the Kerr-Newman
(KN) black holes in which there are two individual pictures of dual CFT description,
called J- and Q-pictures. Finally we discuss the generalization the computation of two-
point function to the three-point function by assuming that there is a cubic interaction
in the bulk geometry. The three-point functions in both J- and Q-pictures of KN
geometry have a divergent term in the bulk integration near the boundary region. It
is consistent with the CFT finite-temperature three-point function.

1 Introduction
1
2 Kerr/CFT: central charges and temperatures
6
2.1 The NHEK geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Asymptotic symmetry group . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Central charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Kerr/CFT: superradiance scattering
14
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Near-NHEK geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Superradiant modes . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Macroscopic greybody factor . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.3 Absorption probability . . . . . . . . . . . . . . . . . . . . . . 20
3.3.4 Near and far factors . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Microscopic greybody factor . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 Conformal dimensions . . . . . . . . . . . . . . . . . . . . . . 21
3.4.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Real-time correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Kerr/CFT: hidden conformal symmetry
26
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Massless scalar wave equation . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Near region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
i4.4 Conformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 CFT interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.5.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 KN/CFTs: twofold hidden conformal symmetries
33
5.1 The Kerr-Newman black hole . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Charged scalar field in the KN background . . . . . . . . . . . . . . . 34
5.3 Angular momentum J-picture . . . . . . . . . . . . . . . . . . . . . . 34
5.3.1 Hidden Conformal Symmetry . . . . . . . . . . . . . . . . . . 34
5.3.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.3 Real-time correlator . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.4 Return to Kerr/CFT . . . . . . . . . . . . . . . . . . . . . . . 39
5.4 Charge Q-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4.1 Hidden Conformal Symmetry . . . . . . . . . . . . . . . . . . 39
5.4.2 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4.3 Real-time correlator . . . . . . . . . . . . . . . . . . . . . . . 42
5.4.4 Return to RN/CFT . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Three-point function
46
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Two- and three-point functions in near-NHEK geometry . . . . . . . 48
6.2.1 Two point function . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2.2 Three-point function . . . . . . . . . . . . . . . . . . . . . . . 50
6.2.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 The three-point function for hidden conformal symmetry of Kerr/CFT 57
6.4 Three-point function in KN/CFTs . . . . . . . . . . . . . . . . . . . . 60
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.4.2 J-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.4.3 Q-picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7 Conclusion
67
A Review of conformal field theory
72
A.1 Comformal group and algebra . . . . . . . . . . . . . . . . . . . . . . 72
A.2 Two-dimensional CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 73
ii
A.3 Conformal generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.4 Primary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.5 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
A.6 The Virasoro algebras in CFT . . . . . . . . . . . . . . . . . . . . . . 76
B Review of Anti de Sitter space
81
B.1 de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.2 Anti de Sitter (AdS) space . . . . . . . . . . . . . . . . . . . . . . . . 82
C Euclidean correlators in AdS/CFT
85
D Lorentzian correlators in AdS/CFT
90
E Symmetry and Casimir operator of AdS3 in KN/CFTs
94

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