U(1) Breakdown in Super Yang-Mills andCascade of Gregory-Laflamme Transitions

Masanori Hanada (RIKEN)With

Tatsuma Nishioka (Kyoto U., D1 )

Weizmann inst.（ Israel）Oct.～

arXiv: 0706.0188[hep-th]

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.

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

3d pure bosonic U(N) YM on torus (1)

Lμ

Global U(1)3 symmetry

Spatial Wilson loop

All<W> nonzero

3d pure bosonic U(N) YM on torus (2)When Lμ becomes small, <Wμ> beco

mes nonzero. If we take all Lμto be the same, L, the

n…

Narayanan-Neuberger-Reynoso, arXiv:0704.2591[hep-lat]

L0 L(1)

<W>=0 for all directions

1 nonzero <W>

2 nonzero

L(2)L(3)

YM on torus with adjoint scalarsIf we take L3→0 first, then we obtain

2d YM on torus with 1 adjoint scalars.

L0 L(1)

<W>=0 for all directions

1 nonzero <W>

L(2)

All <W> nonzero

YM on T4 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.

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) adjoi

nt 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

Temporal KK decouple　　　 bosonic YMSpatial 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.

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.

AdS/CFT [gauge/gravity] correspondenceSYM 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 transitionBlack string winding on S1 is

unstable if S1 is large.

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”

Simulation result of Narayanan et.al. suggests a cascade of Gregory-Laflamme transitions:

Smeared D0’s on T2

on S1 localized

Check it .

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

T-dual picture: D0-branes

When SUGRA approximation is good?Winding mode along torus and massive tower of str

ing oscillation should be heavier than KK mode along

S8-p. Dp-brane picture:

D0-brane picture:

Comparison of free energies (1)Compare free energies for Dp-brane with th

e 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 2.40 2.67 2.87

0-brane1-brane 2-brane 3-brane

Comparison of free energies (3)

t>2.87/λ’1/2

3-brane2-brane

?

t<2.87/λ’1/2

t=2.87/λ’1/2

RemarksRemarks

Transition takes place where D0-brane Transition takes place where D0-brane picture is valid.picture is valid.

In D0-picture, small In D0-picture, small tt ⇔ large 1/ ⇔ large 1/L. L. internal space large internal space large low dim.object favorelow dim.object favore

d.d.

Schwarzschild case

Schwarzschild-type black brane

Schwarzschild BH

Torus (flat)

tC(1)

tC(2)

tC(3)

1.28

1.17

1.04

Critical temp. for R7×T3

t<tC(3)

3-brane2-brane

1-brane

0-brane

?t=tGL(3)

t=tC(3)

t>tC(3)

tC(1)

tC(2)

tC(3)

1.28

1.17

1.04

Critical temp. for R7×T3

tGL(1)

tGL(2)

tGL(3)

1.30

1.20

1.08

Cascade of first order transitions:3-brane→ 2-brane→ 1-brane→ 0-brane

3

2

10

3-brane cannot decay to 1- or 0-brane

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)

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.

おまけ

Take it small.

Condition for fermion decoupling(1)

If spatial KK modes decouple first, then…

Condition for fermion decoupling(2)

Small ⇒temporal KK decoupleEspecially, all fermions decouple.

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…