Introduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Phase Transitions of Charged Kerr-AdS Black Holesfrom Large N Gauge Theories
Tatsuma Nishioka(Kyoto Univ.)

withK. Murata, N. Tanahashi, H. Yumisaki (Kyoto U)
arXiv:0806.2314

SummaryIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Introduction
AdS/CFT Correspondence relates (Maldacena ’98)
Type IIB SUGRA on AdS5 × S5
N = 4 Super Yang-Mills (CFT) on the boundary of AdS5
At finite temperature (Witten ’98)
AdS Schwarzschild black hole
Super Yang-Mills on S1 × S3
Gravity theory is expected to correspond to the stronglycoupled gauge theory

Introduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFTSummary

Hawking-Page transition and AdS/CFTThe AdS/CFT relates (Witten ’98)
The Hawking-Page transition in the gravity theory
The confinement/deconfinement transition in the gauge
theoryThe gauge theory at finite temperaturedefined on S1 × S3
described as the unitary matrix model (Sundborg ’99, Aharony et al. ’03)
exhibits the phase transition in the large N limit even atzero couplingIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Including chemical potentials
What happens when we include chemical potentials?
Including R-symmetry chemical potentials has been
considered (Yamada-Yaffe ’06)
This corresponds to the R-charged black hole
The resulting phase diagram of the gauge theory issimilar to that of dual gravity
T

DeconfinedConfined0
Unstable1µT
MetastableBlackHole
Black Hole 

Unknown

Thermal AdS 


01µ

SummaryIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Our purpose
We treat most general case withthree R-symmetry chemical potentials
two SO(4)-symmetry chemical potentials associated withS3
This is dual to charged Kerr-AdS black hole
the solutions with four degree of freedom are known(Mei-Pope ’07, Cvetic-Lu-Pope ’04, Cvetic-Gibbons-Lu-Pope ’05)

Most general solution has not been yet discovered
We can study the property of such a undiscovered blackhole from the gauge theory sideSummary

Introduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Resulting phase diagram
We found a gap between the confinement/deconfinementtransition line and the boundary of the phase diagram in
the gauge theory
We also found such a gap in dual gravity to study the its
thermodynamics and instability
T

DeconfinedConfined0Unstable


µmax 1µTBlack Hole

Thermal AdS
0µmax 1



Unknown




µIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

N = 4 SYM and chemical potentials
We study the N = 4 SYM on S3Z
IN=4 = −"p1d4 x −g tr (Fµν )2 + (Dµ φm )2 + φ2m + iλ̄A Γµ Dµ λA2#g22A m− [φm , φn ] − gλ̄ Γ [φm , λA ] ,2

where the covariant derivative is defined byDµ = ∇µ + ig[Aµ , · ]This action has
SO(4) symmetry from S3
SO(6) ' SU(4) symmetry from the R-symmetry

SummaryIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Partition function with chemical potentials

SO(4) symmetry has two charges Ĵ1 , Ĵ2
SU(4) symmetry has three charges Q̂1 , Q̂2 , Q̂2The grand canonical partition function is given byZ(β) = Tr[e−β(Ĥ−µa Q̂a −Ωi Ĵi ) ]Notice that the partition function of dual gravity is written ashi
Zgravity = exp −Igraviry= exp [−β(E − µa Qa − Ωi Ji ) + S]

SummaryIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Large N gauge theory and unitary matrix model
Consider the N = 4 SYM on S1 × S3 at zero coupling
ZZN=4 =DAµ Dψi Dφa exp[−IN=4 ], (a = 1, . . . , 6, i = 1, . . . , 4)

The only temporal gauge field A0 has zero mode α
We must maintain Gauss’ law constraint even at zerocoupling
After integrate out the massive modes
ZZN=4 =[dU] exp[−I(U)],U ≡ eiαIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Summary of the gauge theory
We determine the maximal chemical potentials µmax where
the transition line ends at T = 0The gap appears when there are more than four chemicalpotentials
We found the phase diagram is bounded by µa = 1, Ωi = 1,above which some modes of the scalar fields becometachyonicT

DeconfinedConfined0
Unstable

µmax 1µIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Charged Kerr-AdS black hole
The five dimensional charged Kerr-AdS black hole with(J1 , J1 , Q1 , Q2 , Q3 ) charges is defined by the following metricY − f3
f1 − r6 H
r4 H1/3 2ds2 = − 4 2/3 dt2 +dr + r2 H1/3 dΩ23 + 4 2/3 (sin2 θdφ1 + cos2 θdφ2 )2YrHr H
2f2− 4 2/3 dt(sin2 θdφ1 + cos2 θdφ2 )r H
Aa =o2 nsa ca dt + a(ca sb sc − sa cb cc )(sin2 θdφ1 + cos2 θdφ2 ) ,r2 Ha

Xa = Ha−1 H1/3 ,dΩ23 = dθ2 + sin2 θdφ21 + cos2 θdφ22 ,2ms2a,
sa = sinh δa ,ca = cosh δa ,r2

YYXYY Y Y
f1 = r6 H + 2ma2 r2 + 4m2 a2 2ca −sasb −s2a s2b  ,f2 = 2maca −sa r2 + 4m2 asa ,H = H1 H2 H3 ,af3

Ha = 1 +abaabaa<bY YX Y
ca −sasb −s2a s2b  ,
= 2ma2 (1 + r2 ) + 4m2 a2 2a<b

Y = f3a+ r6 H + r4 − 2mr2a

Introduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Phase transition in dual gravity
The action relative to pure AdS space
Igravity = (M − TS − 2ΩJ − µ1 Q1 − µ2 Q2 − µ3 Q3 )/T .
The Hawking-Page transition occurs at Igravity = 0
Below the lines (Igravity > 0): Thermal AdS space is favored
Above the lines (Igravity < 0): Charged Kerr-AdS black holeis favored(a) µ1 ≡ µ, µ2 = µ3 = 0 (b) µ1 = µ2 ≡ µ, µ3 = 0 (c) µ1 = µ2 = µ3 ≡ µIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Instability of charged Kerr-AdS black hole
It has been suggested that this unitarity line in the gauge
theory corresponds to the thermodynamical instabilityline on the phase diagram in the dual gravity theory(Yamada-Yaffe ’06)

We search such a line and obtained the below phase
diagramTBlack Hole

Thermal AdS
0µmax 1
UnknownµSummary

Introduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

Summary

Comparison via AdS/CFT
It is interesting to compare the phase diagrams beteen thegauge theory and dual gravity
We found remarkable agreement in the phase diagrams!

(d) µ1 ≡ µ, µ2 = µ3 = 0(e) µ1 = µ2 ≡ µ, µ3 = 0(f) µ1 = µ2 = µ3 ≡ µIntroduction

Large N gauge theory

Dual Gravity

Comparison via AdS/CFT

SummarySummary
We have constructed the gauge theory on S3 dual to thecharged Kerr-AdS black hole by adding the chemical
potentials associated with SO(4) and R-symmetry
This model shows the confinement/deconfinementtransition which is dual to the Hawking-Page transition in
the gravity theory
The phase diagrams have remarkable similarities in bothsides
Especially we found a new gap between the tansition line
and the boundary of the phase diagram when there are
more than four chemical potentials
We have studied the ratios of the thermodynamicalquantities and showed quantatively that the gauge tehory
describes the dual gravity even near the criticaltemperature