引力的全息性，AdS/CFT和其它 蔡 荣 根 中国科学院理论物理研究所 (中国科大，2011.12.9)

* 引力是自然界四种基本相互作用力之一。 （其它三种：电磁作用，强相互作用和弱相互作用） * 从牛顿的万有引力定律到爱因斯坦的广义相对论 问题： （1）广义相对论成立从亚毫米到太阳系尺度 （2）广义相对论是经典理论，引力的量子性质如何？

引力理论新进展: (E. Witten, 1995, J. Polchinski, 1995) 引力的全息性质:引力的本质 超弦理论:第二次革命 (E. Witten, 1995, J. Polchinski, 1995) 引力的全息性质:引力的本质 ( ‘t Hooft, 1993, L. Susskind, 1994) 膜世界图象:一种新的世界观 (L. Randall and R. Sundrum, 1999) 现代宇宙学：精确宇宙学时代 （since COBE, 1992)

今天要讨论的主题是： 黑洞热力学，引力的全息性 和它的应用

Einstein’s Equations (1915): {Geometry matter (energy-momentum)}

第一个黑洞解：Schwarzschild 黑洞 真空的爱因斯坦场方程的精确解， 描写一个天体的引力场。当这一 天体的半径<2GM,它就是一个黑洞。 (K. Schwarzschild,1873-1916)

黑洞的无毛定理（no-hair theorem): Schwarzschild solution 2) Reissner-Nordstrom solution 3) Kerr-Newman solution (W. Israel) (Israel, 1967,1968, Muller zum Hagen et al, 1973,Robinson 1977,….)

Kerr-Newman 黑洞 M, J, Q 无毛定理（No Hair Theorem ） 视界

黑洞的经典性质： 1960s（1974年以前），黑洞的黑暗时期！ (D. Sciama,1926-1999) (J. Wheeler, 1911-2008) (Y.Zel’dovich,1914-1987) (D. Sciama,1926-1999)

(S. Hawking, 1942-present) (R. Pensore, 1931-present)

(J.M. Bardeen,B. Carter, S. Hawking, CMP,1973) 经典黑洞的性质：黑洞力学四定律 The 0th law k =const. The 1st law d M=k dA/8πG + Ω d J+Φ dQ The 2nd law d A >0 The 3rd law k ->0 （k, 表面引力，类似于引力加速度） (J.M. Bardeen,B. Carter, S. Hawking, CMP,1973)

Wheeler 问： 假如某个热力学体系掉入黑洞，将导致什么？ 热力学第二定律将违背吗？ J. Bekenstein（1973）：黑洞有热力学熵！ S ~ A, 视界面积

Hawking 不以为然, 大力反对Bekenstein 的观点！ 可是考虑了黑洞周围的量子力学后， Hawking （1974，1975）发现黑洞不黑，有热辐射！ For Schwarzschild black hole,

黑洞热力学四定律： The 0th law T=Const. on the horizon The 1st law d M= T d S + Ω d J+Φ d Q The 2nd law d (SBH +Smatter)>=0 The 3rd law T->0

Hawking radiation and Information Loss Paradox S. Hawking, PRD, 1976; a bet established in 1997

事情的另一方面是关于黑洞熵的统计自由度！ （String Theory, 1996) (A. Strminger) (C. Vafa)

In Loop Quantum Gravity (A.Ashtekar,J.Baez,A. Corichi and K. Krannov, PRL,1998)

Black hole is a window to quantum gravity Thermodynamics of black hole： （S.Hawking, 1974, J. Bekenstein, 1973)

Entropy in a system with surface area A: S<A/4G Holography of Gravity Entropy in a system with surface area A: S<A/4G             (‘t Hooft) (L. Susskind) The world is a hologram？

AdS/CFT correspondence (J. Maldacena, 1997) IIB superstring theory on AdS5 x S5 N=4 SYM Theory “Real conceptual change in our thinking about Gravity.” (E. Witten, Science 285 (1999) 512)

AdS/CFT 对偶性字典 : 这里 有二个解释: 在引力这边: 这些对应在AdS bulk中传播的场的边界值。 在AdS边界上场论这一边: 这里 有二个解释: 在引力这边: 这些对应在AdS bulk中传播的场的边界值。 在AdS边界上场论这一边: 这些场相应于边界共形场论许多算符耦合的外源。

超引力和D-brane 10维超引力: IIA: D0, D2, D4, D6, D8 IIB: Ｄ(-1), D1, D3, D5, D7, D9 11维超引力: 　　　Ｍ２　 Ｍ５ AdS5/CFT4 AdS7/CFT6 AdS4/CFT3

One has two decoupled theories: Consider N D3-branes Closed string 3+1 Open string 6 Take low energy limit: One has two decoupled theories: free IIB supergravity in bulk and N=4 SYM on brane

Another viewpoint of D-branes: massive charged objects The energy E_p of an object as measured by an observer at constant r and the energy E measured by an observer at infinity Take low energy limit two kinds of low energy excitations: infinity and near branes

YM side: Supergravity side: Strong Weak

quantum field theory d-spacetime dimensions operator Ο (quantum field theory) quantum gravitational theory (d+1)-spacetime dimenions dynamical field φ (bulk) (0909.3553, S. Hartnoll)

（1）整体对称性: AdS_5 X S^5 ~N=4 SYM （2）关联函数的计算 AdS/CFT, 证据如何？ （1）整体对称性: AdS_5 X S^5 ~N=4 SYM （2）关联函数的计算 AdS/CFT Conjecture： 理论价值？ 应用价值？ 强耦合系统 QCD 高温超导 ……

如何从AdS/CFT来理解低能QCD的性质？ (1) top-down (从上到下） (2) bottom-up （从下往上） 1) AdS/QCD QCD 相图： 如何从AdS/CFT来理解低能QCD的性质？ (1) top-down (从上到下） (2) bottom-up （从下往上）

¾ factor Finite temperature: (E. Witten, 1998) Take an example: AdS_5 black hole: where ¾ factor

Effective action to $\alpha’^3$: where (S. Gubser et al, hep-th/9805156)

deconfinement phase transition Hawking-Page phase transition and deconfinement phase transition One horizon is located at f(r)=0. The inverse temperature The maximum value

HP transition C > 0 C < 0 If r_+ < l Small BHs If r_+ > l Large BHs HP transition C > 0 The thermal radiation in AdS: C < 0

QCD 禁闭/退 紧闭相变 x_0=323 Mev In the soft-wall model T_c=191 Mev

(1) up-down i) Witten’s QCD model (1998) 高温相：

如何加味场(favor)? ii) Karch-Katz Model (2002)

介子谱： 有限温度相变： T

(iii) Sakai-Sugimoto Model (2004) hep-th/0604161

RHIC’s heavy Ion Collision PRL98, 172301(2007), nucl-ex/0611018 PRL99, 172301(2007), nucl-ex/0706.1522 AdS/CFT: Kovtun, Son and Starinet, PRL (05) (Brigante et al, PRL 2008) （Cai et al, 2008, 2009, 2010)

(2) bottom-up: 全息 AdS/QCD 模型 i) hard-wall model (J. Erlich et al, PRL (2005))

Model A: z_m=1/323Mev m_q=2.29 Mev sigma=(327Mev)^3 均方差：15% 9% 模型只有三个参数： z_m, m_q, sigma Model A: z_m=1/323Mev m_q=2.29 Mev sigma=(327Mev)^3 均方差：15% 9% Model B: z_m=1/346Mev m_q=2.3Mev sigma=(308Mev)^3

ii) Soft-wall model ( A. Karch et al, 2006) rho meson

Holographic superconductor models (全息超导） 2) AdS/CMT Holographic superconductor models (全息超导） 超导现象： 电阻消失 (H. Onnes, 1911) Meissner effect (1933) 1950, Landau-Ginzburg theory 1957, BCS theory: interactions with phonons 1980’s: 高温超导， 2000’s: 铁基超导

如何实现一个全息超导模型？ CFT AdS/CFT 引力 整体对称性 阿贝尔规范场 标量算符 标量场 温度 黑洞 相变 高温/无毛；低温/有毛

Building a holographic superconductor S. Hartnoll, C.P. Herzog and G. Horowitz, arXiv: 0803.3295 PRL 101, 031601 (2008) 高温相（无毛黑洞):

考虑 m^2L^2=-2，此时它相应于一个共形耦合的标量 In the probe limit and A_t= Phi 在大 r的边界上: 在对偶场论中标量算符 O_i的凝聚

电导率 麦克斯韦尔方程： (零动量，有时间依赖性 exp [-i w t]） 视界上: 入射波 边界上: AdS/CFT 流 源 电导率:

电导率的频率依赖性

一个普适的能隙（gap): ~ 10% BCS theory: 3.5 K. Gomes et al, Nature 447, 569 (2007)

Analytical studies on holographic superconductors in GB gravity H.F. Li, R.G. Cai and H.Q. Zhang, JHEP 04 (2011) 028 (R.G Cai, PRD,2002） S-wave & p-wave:

Analytical studies on holographic Insulator/superconductor phase transition R.G. Cai, H.F. Li and H.Q. Zhang, PRD83:126007 (2011) Ads soliton: Operator dimension: 3/2 5/2 Similarly it works for the p-wave case

Flavor Superconductivity & Superfluidity, 超导的磁效应（五维磁黑洞），Meissner 效应，vortex 全息超导现象中哪些是普适的？哪些方面没有普适性 全息超导基态的引力构型？具有什么对称性？ 临界行为？（临界指数，热力学性质，输运系数…) 非相对论性？ 高温超导的配对机制（从上到下的模型） Flavor Superconductivity & Superfluidity, M. Kaminski, arXiv: 1002.4886 d-wave 超导？ h) 其它方面： 量子(分数）霍尔效应，奇异金属，拓扑绝缘体 Kosterlitz-Thouless 相变，费米液体，非费米液 体, Josephson Junction 等。

3) Holographic optics and negative refractive index A. Amariti et al., arXiv: 1006.5714 The free propagation of light in curved spacetime The propagation of light inside a material in flat space Metamaterials: cloaking devices, perfect lenses, photonic black holes Some with negative refractive index (left-handed materials) [V.G. Veselago, Sov. Phys. Usp 10, 509 (1968)] [ R.A. Shelby et al., Science 292, 77 (2001)]

For isotropic media: Transverse dispersion relation

Negative Refractive index: <0

Holographic model:

4）引力理论和流体力学的关系 引力与热力学和流体力学一样，具有相当的普适性 研究这两者的关系 利用非平衡流体力学来研究动力学时空的性质 Black holes: The Membrane Paradigm, 1986 Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence M. Rangamani, Class. Quant. Grav. 26: 224003, 2009 研究这两者的关系 利用非平衡流体力学来研究动力学时空的性质 动力学黑洞的热力学性质，应用于QGP 利用引力理论来研究流体力学中的问题如激波，湍流 ……

AdS/CFT 和七个千禧年问题 Clay Mathematics Institute (http://www.claymath.org/millennium) The seven Millennium Prize Problems (US$7 million): Birch and Swinnerton-Dyer Conjecture 2) Hodge Conjecture 3) Navier-Stokes equations 4) P vs NP 5) Poincare Conjecture 6) Riemann Hypothesis 7) Yang-Mills theory

Navier-Stokes Equation: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.

Yang-Mills and Mass Gap: The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

谢谢！