U1 Breakdown in Super YangMills and Cascade of

U(1) Breakdown in Super Yang-Mills and Cascade of Gregory-Laflamme Transitions Masanori Hanada (RIKEN) With Tatsuma Nishioka (Kyoto U. , D 1 ) ar. Xiv: 0706. 0188[hep-th] Weizmann inst. （Israel）Oct. ～

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

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

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

3 d 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… 2 nonzero All<W> nonzero 1 nonzero <W>=0 for all directions L 0 L(3) L(2) L(1) Narayanan-Neuberger-Reynoso, ar. Xiv: 0704. 2591[hep-lat]

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

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

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

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

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

| Assume (or believe) that one-by-one breakdown of U(1) in SYM persists to strong coupling. Ad. S/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, …

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

Phase of spatial Wilson loop T-dual Position of D-brane Taking T-dual along all directions of torus, we have a system of D 0 -branes. Then, condensation of spatial Wilson loop Condensation of D 0 -branes “Gregory-Laflamme”

| Simulation result of Narayanan et. al. suggests a cascade of Gregory-Laflamme transitions: Smeared D 0’s on T 2 on S 1 Check it. localized

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

T-dual picture: D 0 -branes

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: | D 0 -brane picture:

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.

Comparison of free energies (2) 0 -brane 1 -brane 2 -brane 3 -brane 0 2. 40 2. 67 2. 87

Comparison of free energies (3) t>2. 87/λ’ 1/2 3 -brane t<2. 87/λ’ 1/2 2 -brane t=2. 87/λ’ 1/2 ?

Remarks Ø Transition takes place where D 0 -brane picture is valid. Ø In D 0 -picture, small t ⇔ large 1/L. internal space large low dim. object favored.

Schwarzschild case

Schwarzschild-type black brane Schwarzschild BH Torus (flat)

t. C(1) 1. 28 t. C(2) 1. 17 t. C(3) 1. 04 Critical temp. for R 7×T 3

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

t. C(1) 1. 28 t. GL(1) 1. 30 t. C(2) 1. 17 t. GL(2) 1. 20 t. C(3) 1. 04 t. GL(3) 1. 08 Critical temp. for R 7×T 3

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

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

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.