1. U(1) Breakdown in Super Yang-Mills and Cascade of Gregory-Laflamme Transitions Masanori Hanada (RIKEN) With Tatsuma Nishioka (Kyoto U., D1 ) Weizmann inst. （ Israel ） Oct. ～ arXiv: 0706.0188[hep-th]

2. Outline  Phase structure of bosonic YM on torus  SYM on torus  Supergravity AdS/CFT (gauge/gravity) High Temrerature Hint for phase str. Confirm it in gravity side.

3. 1.Phase structure of bosonic YM on torus (simulation by Narayanan et.al.) 2.SYM at high temperature 3.Relation to gravity side: Cascade of Gregory-Laflamme transitions

4. 3d pure bosonic U(N) YM on torus (1) LμLμ Global U(1) 3 symmetry Spatial Wilson loop

5. All nonzero 3d pure bosonic U(N) YM on torus (2)  When L μ becomes small, becomes nonzero.  If we take all L μ to be the same, L, then… Narayanan-Neuberger-Reynoso, arXiv:0704.2591[hep-lat] L 0 L (1) =0 for all directions 1 nonzero 2 nonzero L (2) L (3)

6. YM on torus with adjoint scalars  If we take L 3 → 0 first, then we obtain 2d YM on torus with 1 adjoint scalars. L 0 L (1) =0 for all directions 1 nonzero L (2) All nonzero YM on T 4 has the same pattern. We may expect that in YM on p -torus with m adjoint scalars U (1) p breaks down one-by-one.

7. 1.Phase structure of bosonic YM on torus 2.SYM at high temperature 3.Relation to gravity side: Cascade of Gregory-Laflamme transitions

8. Bosonic YM as High Temp. limit of SYM  Consider SYM on T p+1 with (9-p) adjoint scalars. (In this talk, p=0,1,2,3.)  Finite temperature → ・ size of temporal circle = β= 1/ T ・ antiperiodic b.c. for fermion Fermions decouple at high temperature Bosonic YM on T p

9. Temporal KK decouple bosonic YM Spatial KK decouple One-by-one breakdown of U(1) At weak coupling and high-temperature, bosonic YM can be used. One-by-one breakdown of U(1) exist also in SYM.

10. 1.Phase structure of bosonic YM on torus 2.SYM at high temperature 3.Relation to gravity side : Cascade of Gregory-Laflamme transitions

11.  Assume (or believe) that one-by-one breakdown of U(1) in SYM persists to strong coupling. AdS/CFT [gauge/gravity] correspondence SYM at strong coupling can be described using supergravity. U(1) breakdown Gregory-Laflamme Susskind, Barbon-Kogan-Rabinovici, Li-Martinec-Sahakian, Aharony-Marsano-Minwalla-Wiseman, Harmark-Obers, …

12. Gregory-Laflamme transition  Black string winding on S 1 is unstable if S 1 is large.

13. Phase of spatial Wilson loop Position of D-brane T-dual Taking T-dual along all directions of torus, we have a system of D0-branes. Then, condensation of spatial Wilson loop Condensation of D0-branes “ Gregory-Laflamme ”

14.  Simulation result of Narayanan et.al. suggests a cascade of Gregory-Laflamme transitions: Smeared D0 ’ s on T 2 on S 1 localized Check it.

15. Metric for Dp-brane, etc. Cf) Itzhaki-Maldacena- Sonnenschein-Yankieloewicz

16. T-dual picture: D0-branes

17. When SUGRA approximation is good? Winding mode along torus and massive tower of string oscillation should be heavier than KK mode along S 8-p.  Dp-brane picture:  D0-brane picture:

18. Comparison of free energies (1)  Compare free energies for D p -brane with the same temperature T H.  Exact metric for D p in T n is not known. Approximate compact directions transvers to brane by noncompact ones.

19. Comparison of free energies (2) 02.402.672.87 0-brane1-brane2-brane3-brane

20. Comparison of free energies (3) t>2.87/λ’ 1/2 3-brane 2-brane ? t<2.87/λ’ 1/2 t=2.87/λ’ 1/2

21. Remarks  Transition takes place where D0-brane picture is valid.  In D0-picture, small t ⇔ large 1/L. internal space large internal space large low dim.object favored. low dim.object favored.

22. Schwarzschild case

23. Schwarzschild-type black brane Schwarzschild BH Torus (flat)

24. t C(1) t C(2) t C(3) 1.28 1.17 1.04 Critical temp. for R 7 × T 3

25. t<t C(3) 3-brane 2-brane 1-brane 0-brane ? t=t GL(3) t=t C(3) t>t C(3)

26. t C(1) t C(2) t C(3) 1.28 1.17 1.04 Critical temp. for R 7 × T 3 t GL(1) t GL(2) t GL(3) 1.30 1.20 1.08

27. Cascade of first order transitions: 3-brane→ 2-brane→ 1-brane→ 0-brane 3 2 1 0 3-brane cannot decay to 1- or 0-brane

28. Resolution of a puzzle  “ 3-brane in R 7 ×T 3 cannot decay to 0-brane. ”  3-brane cannot decay to 0-brane directly, but it can decay as 3-brane→ 2-brane→ 1-brane→ 0-brane ! (Kol-Sorkin, 2004)

29. Summary  Black brane on torus goes through a cascade of Gregory-Laflamme transitions.  This cascade is related to one-by-one breakdown of U(1) in Yang-Mills theory.

30. おまけ

31. Take it small. Condition for fermion decoupling(1)

32.  If spatial KK modes decouple first, then … Condition for fermion decoupling(2) Small ⇒ temporal KK decouple Especially, all fermions decouple.

33. Small → ・ spatial KK decouple ・ U(1) breaks one-by-one (result from bosonic model) When U(1) p ?  If temporal KK modes decouple first, then …