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Breakdown in Super Yang-Mills and Cascade of Gregory-Laflamme Transitions

This article discusses the phase structure of bosonic Yang-Mills on a torus, its relation to supergravity at high temperature, and the cascade of Gregory-Laflamme transitions. It explores the breakdown of U(1) symmetry and the implications in SYM theory and AdS/CFT correspondence.

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Breakdown in Super Yang-Mills and Cascade of Gregory-Laflamme Transitions

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

  2. Outline • Phase structure of bosonic YM on torus • SYM on torus • Supergravity High Temrerature Hint for phase str. AdS/CFT (gauge/gravity) 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. Lμ 3d pure bosonic U(N) YM on torus (1) Global U(1)3 symmetry Spatial Wilson loop

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

  6. We may expect that in YM on p-torus with m adjoint scalars U(1)p breaks down one-by-one. YM on torus with adjoint scalars • If we take L3→0 first, then we obtain 2d YM on torus with 1 adjoint scalars. <W>=0 for all directions All <W> nonzero 1 nonzero <W> L 0 L(2) L(1) YM on T4 has the same pattern.

  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 Tp

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

  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 S1 is unstable if S1 is large.

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

  14. Simulation result of Narayanan et.al. suggests a cascade of Gregory-Laflamme transitions: localized Smeared D0’s on T2 on S1 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 S8-p. • Dp-brane picture: • D0-brane picture:

  18. Comparison of free energies (1) • Compare free energies for Dp-brane with the same temperature TH . • Exact metric for Dp in Tn is not known. Approximate compact directions transvers to brane by noncompact ones.

  19. Comparison of free energies (2) 0-brane 1-brane 2-brane 3-brane 0 2.40 2.67 2.87

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

  22. Schwarzschild case

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

  24. tC(1) 1.28 1.17 tC(2) tC(3) 1.04 Critical temp. for R7×T3

  25. ? t<tC(3) t>tC(3) 0-brane 1-brane 2-brane 3-brane t=tC(3) t=tGL(3)

  26. tC(1) tGL(1) 1.28 1.30 1.17 1.20 tC(2) tGL(2) tC(3) tGL(3) 1.04 1.08 Critical temp. for R7×T3

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

  28. Resolution of a puzzle • “3-brane in R7×T3cannot 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. Small ⇒temporal KK decouple Especially, all fermions decouple. Condition for fermion decoupling(2) • If spatial KK modes decouple first, then…

  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…

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