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研究生： 唐富一
Fu-Yi Tang
論文名稱： 克爾-紐曼/共形場中的三點關聯函數
Three-point correlators in Kerr-Newman/CFTs
指導教授： 陳江梅
Chiang-Mei Chen
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
畢業學年度： 99
語文別： 英文
論文頁數： 95
中文關鍵詞： 三點關聯函數 、AdS/CFT對偶 、黑洞全像原理 、黑洞
外文關鍵詞： Holographic principle, black hole, AdS/CFT correspondence, Three-point correlators
相關次數： 點閱：5  下載：0 
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    • 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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