1. Baryon Chemical Potential in AdS/CFT Shin Nakamura 中村 真 Hanyang Univ. and CQUeST （韓国・漢陽大学 ) Ref. S.N.-Seo-Sin-Yogendran, hep-th/0611021 ( Kobayashi-Mateos-Matsuura-Myers, hep-th/0611099)

2. Purpose of this talk I would like to present an overview of AdS/CFT. (Incomplete, but “intuitive” hopefully.) I will report the present status on construction of finite-density AdS/CFT. (What we know and what we do not know.)

3. Motivation Hadron physics is very interesting research area both theoretically and experimentally. RHIC, LHC Nuetron (quark) stars We encounter strongly coupled systems. We need theoretical frameworks which enable us to analyze strongly coupled QCD. Effective theories, Lattice QCD,… AdS/CFT

4. (Original, weak version) Classical Supergravity on 4dim. Large-Nc SU( Nc ) N=4 Super Yang-Mills at the large ‘t Hooft coupling conjecture = Maldacena ‘97 Strongly interacting quantum YM !! 10 dim.

5. What is AdS/CFT? Analogy: Euclidean theory B A 2 solutions: A: Ф=0 “trivial” vacuum B: Ф= Ф B “non-trivial” vacuum -m 2 m2m2

6. Physics around the “non-trivial” vacuum 2 equivalent methods: dynamical = 1. Perturbation theory around the “non-trivial” vacuum. source term 2. Perturbation theory around the “trivial” vacuum (with source). dynamical

7. Propagator around the non-trivial vacuum method 1: (around non-trivial) method 2: (around trivial) +++….. = consistency (Comment after the seminar: we have to understand more about this.)

8. What we have learned Same physics can be described in two different ways: 1. non-trivial vacuum, without source 2. trivial vacuum, with source Re-summation of infinitely many diagrams The source carries non-perturbative information Single Feynmann diagram =

9. Let us do the same thing in string theory Type IIB Superstring Theory Low energy: 10d type IIB supergravity Many different vacua. Two of them: 1. A curved spacetime: black 3-brane solution 2. Flat spacetime Asymptotically flat Extremal black hole “Source for closed strings”: D3-brane Theory of closed strings (perturbatively) 3+1 dim. hypersurface, gauge theory on it Defined in 10d spacetime “non-trivial” “trivial”

10. ? = U(Nc) 3+1 dim N=4 Super YM theory at low energy on the D3-branes Superstring theory around black 3-brane geometry Superstring theory around flat geometry asymptotically flat + source (Nc D3-brane) Black hole (3+1 dim. object) The near horizon limit : We do not want here. SU(N c )

11. AdS/CFT (Original, weak version) Classical Supergravity on 4dim. Large-Nc SU( Nc ) N=4 Super Yang-Mills at the large ‘t Hooft coupling conjecture = Maldacena ‘97 Strongly interacting quantum YM !! 10 dim.

12. What we have learned Same physics can be described in two different ways: 1. non-trivial vacuum, without source 2. trivial vacuum, with source Re-summation of infinitely many diagrams The source carries non-perturbative information Single Feynmann diagram =

13. Construction of gauge/gravity duality 1.Construct a D-brane configuration on which the gague theory you want is realized. 2.Find the supergravity solution which corresponds to the D-brane configuration. (Here, we have a curved spacetie, but no D- brane.) 3.Take near-horizon limit to make the unwanted modes (like gravity in the YM side) decoupled. 4.Take appropriate limits to make the supergravity approximation valid, if necessary.

14. Introduction of quark/antiquarks D3-brane 3+1 dim. AdS 5 string The quark-antiquark pair is a single string coming from the boundary of AdS. The end of the string is a quark or antiquark.

15. N c D3 N f D7 mqmq quark flavor brane Introduction of dynamical quarks gravity dual AdS 5 N f D7 meson AdS 5 + flavor branes

16. AdS/CFT and statistical mechanics AdS/CFT : a useful tool for analysis of strongly coupled YM theories. Finite temperature Finite baryon-number density (chemical potential) Established Yet to be completed We need to describe systems with finite temperature and finite density.

17. AdS/CFT at finite temperature Classical Supergravity on AdS-BH×S 5 4dim. Large-Nc strongly coupled SU( Nc ) N=4 SYM at finite temperature (in the deconfinement phase). conjecture = Witten ‘98 Hawking temp.

18. Phase transitions Transition of bulk geometry at the same β(=1/T). Thermal AdSAdS-BH “confinement” phase“de-confinement” phase Hawking-Page transition Transition related to quark condensate Transition of flavor-brane configuration, on a common branch of bulk geometry

19. gravity dual AdS-BH D7 horizon Minkowski branch Black-hole branch 1 st order T<Tc Tc<T N c D3 N f D7 mqmq quark flavor brane Phase transition related to quarks

20. Brane configurations Minkowski branch Black-hole branch BH y0y0 y ρ yHyH D7 y0y0

21. How to introduce finite density (or chemical potential)? Kim-Sin-Zahed, 2006/8 Horigome-Tanii, 2006/8 S.N.-Seo-Sin-Yogendran, 2006/11 Kobayashi-Mateos-Matsuura-Myers- Thomson, 2006/11

22. The system we consider: D3-D7 system YM theory: N=2 large-N c SYM with quarks Flavor branes: N f D7-branes Flavor symmetry: U(N f ) Quarks are massive (in general): m q Probe approximation (N c >>N f ) Free energy ～ Flavor-brane action No back reaction to the bulk gometry from the flavor branes. ( ～ quenched approx.)

23. AdS/CFT at finite R-charge chemical potential R-symmetry: SO(6) on the S 5 R-charge: angular momentum on the S 5 electric charge of the BH From the AdS 5 point of view 10 dim. Electric potential A 0 at the boundary is interpreted as a chemical potential Chamblin-Emparan-Johnson-Myers,1999 Cvetic-Gubser,1999

24. First law in charged black hole Mass Hawking temperature Entropy from the area of the horizon Electromagnetic potential Charge plays as a chemical potential

25. How about finite baryon-number density? We need flavor branes （ D8,D7,….) U(1) B symmetry: Local (gauge) symmetry on the flavor branes U(1) B charge:“electric charge” for the U(1) gauge field on the flavor brane A 0 on the flavor brane at the boudary U(1) B chemical potential?? Kim-Sin-Zahed,2006/8; Horigome-Tanii,2006/8 D4-D8-D8 case

26. How about gauge invariance? We should use A “physical” ? meaning: a work necessary to bring a single quark charge from the boundary to ρ min against the electric field. S.N.-Seo-Sin-Yogendran,2006/11 ρ E D7 ρ boundary Kobayashi-Mateos-Matsuura- Myers-Thomson,2006/11 AdS-BH

27. More standard AdS/CFT language U(1) part of the U(N f ) gauge symmetry: A μ A μ couples the U(1) B current (density): the boundary value of A 0 corresponds to the source for the U(1) B number density op. μ (N c D3-N f D7 case)

28. Thermodynamics as classical electromagnetism DBI action of the flavor D7-branes with F ρ0 : Gauss-law constraint: “electric charge” density A function of A 0 ’: grand potential in the grand canonical ensemble. =Ω quark number density ρ-derivative

29. Legendre transformation “Hamiltonian” is interpreted as the Helmholtz free energy in the canonical ensemble.

30. A problem Kobayashi-Mateos-Matsuura-Myers (KMMM) claims: “the Minkowski branch is unphysical.” Our (S.N.-Seo-Sin-Yogendran) treatment: with the Minkowski branch. (Analysis: canonical ensemble in both papers)

31. KMMM’s claim AdS-BH D7 horizon Minkowski branch Black-hole branch 1 st order Gauss-law constraint: charged source F1 D7 falls into the BH and no Minkowski branch. 1 st order in canonical ensemble E E

32. However, However, if we use only the black-hole branch, we have another serious problem. (S.N.-Seo-Sin-Yogendran, to appear) In the grand canonical ensemble, KMMM has only high-temperature region. (Full temperature region cannot be covered within their framework.)

33. Brane configurations Minkowski branch (y 0 / y H >1) Black-hole branch (y 0 / y H <1) BH y0y0 y ρ yHyH D7 y0y0

34. If black-hole branch only, No flavor brane! μ=const. BH branchMinkowski branch No low-temp. region in the theory?? Q=const. y 0 /y H 1/T

35. Conclusion Basic ideas of AdS/CFT have been reviewed in this talk. Attempts to introduce U(1) B -chemical potential have been started last year. The KMMMT’s claim looks reasonable, but we found that their proposal produces another serious problem. AdS/CFT with U(1) B -chemical potential is still under construction.