the d3-d7 model of ads/cft with flavour (with special...

The D3-D7 Model of AdS/CFT with Flavour(with Special Feature)

René Meyer

Max-Planck-Institute for PhysicsMunich, Germany

November 13, 2008

Lot’s of known stuff (Review: hep-th/0711.4467 ) +JHEP 0712:091,2007 (with J. Erdmenger & J. P. Shock)

René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 1 / 43

Outline

1 Introduction to AdS/CFT

2 The D3-D7 model: Adding Flavour to AdS/CFT

3 Electric and Magnetic Field BackgroundsMagnetic FieldElectric Field

4 Summary


Introduction to AdS/CFT

Outline




4 Summary



Motivation

New tools for strongly coupled gauge theories?

QCD in the infrared is strongly coupled (Conformal window?)[Deur, Korsch et. al: hep-ph/0509113]

0.050.060.070.080.090.1

0.2

0.3

0.4

0.50.60.70.80.9

1

2

10-1

1Q (GeV)

α s(Q

)/π

αs,g1/π world data

αs,τ/π OPAL

Burkert-Ioffe

pQCD evol. eq.

αs,g1/π JLab

αs,F3/π

GDH constraint

QGP produced at strong coupling (T >≈ ΛQCD)



Our new tool: AdS/CFT Holography



Our new tool: AdS/CFT Holography

The Original Correspondence (weakest form)

IIB Supergravity on AdS5 × S5 with (R/`s)4 = 2λ 1

ds2AdS5×S5 = R2

(dxµ2 + du2

u2 + dΩ25

)⇔

large Nc limit of N = 4 SU(Nc) Super Yang-Mills with λ = g2YMNc

1 Strong-Weak Coupling Duality: GN ∝ g2s = g4

YM2 Gubser-Klebanov-Polyakov-Witten relation:

〈ei∫

d4xJOO〉 = eiSIIB,onshell[JO]

3 Operator-Field Dictionary:

φm2=∆(∆−4) ' u4−∆JO + u∆〈O〉



A short look at string theory

Basic Objects:Open and Closed Strings, D-Branes

Strings have length: `sStrings have tension: Ts = 1

2πα′ = 12π`2

s

“Minimal Surface” action principle: Nambu-Goto Action

SNG =1

2π`2s

∫dτdσ

√−det P[G]ab

P[G]ab = Gµν∂Xµ

∂σa∂X ν

∂σb

[Becker, Becker, Schwarz]




D-Branes: Dirichlet boundaryconditions for the string (e.g.X 1(σ, τ) = 0 for σ = 0, π)

1st Quantized Strings:Open String: “Gauge Theory”

Aµ ,ΦI ,Fermions

Closed String: “Gravity”

Gµν , Bµν , Φ ,Fermions [hep-ph/0701201]





D-Branes Curve Spacetime:

Open-Closed String Duality





D-Branes Curve Spacetime:

“Graviton” Absorption/Emission





Dp-Branes Have Tension:

Tp =1

(2π)p`(p+1)s gs

gs . . . string coupling

“Graviton” Absorption/Emission





Dp-Branes Have Tension:

Tp =1

(2π)p`(p+1)s gs

gs . . . string coupling String Splitting/Merging[“D-Branes”, C. V. Johnson]





Dp-Branes are “Minimal Hypersurfaces”: (α′ = `2s)

SDp = SDBI + SCS

SDBI = −Tp

∫d(p+1)ξ

√−det (P[G + B]ab + 2πα′Fab)

SCS = +Tpgs∑

p

∫topform

P[C(p+1) ∧ eB] ∧ e2πα′F





Dp-Branes carry Gauge Theories:e.g. flat D3-Brane in flat space in the limit α′ << 1

SDBI,p=3 = − 1(2π)3α′2gs

∫d4ξ

√−det (P[G]ab + 2πα′Fab)

≈ −T3

∫d4ξ

√−det P[η]− 1

8πgs

∫d4ξ

√−det P[η]FabF ab

= −T3Vol(R4)− 18πgs

∫d4ξFabF ab



N = 4 d = 4 Super-Yang-Mills-Theory

More precisely: The gauge theory on a stack of Nc D3-branes is theunique maximally supersymmetric gauge theory in 4D,N = 4 Super-Yang-Mills Theory (SYM)

SUSYs: 16 Poincaré + 16 Conformal = 642 real supercharges = 1

2 BPS

Fields: N = 4 Vector Multiplet in 4D = (Aµ, λαA, φi) = (V ,ΦI)

L = tr

− 1

2g2YM

F 2µν +

ΘI

8π2 Fµν Fµν − i4∑

A=1

λAσµDµλA −6∑

i=1

DµφiDµφi

+g2

YM2

∑i,j

[φi , φj ]2 + gYM

∑A,B,i

(CAB

i λA[φi , λB] + h.c.)

Global Symmetries: SO(4,2)× SU(4)R (β(gYM) = 0 !)



D3-Branes in IIB Supergravity

Have seen:1 D3-Branes carry SU(Nc) N = 4 SYM theory2 D3-Branes curve space in a supersymmetric way

D3-Branes are solitonic 12 BPS solutions to IIB Supergravity [Horowitz, Strominger

Nucl.Phys.B360(1991)]

ds2 = H(r)−12 dxµ2 + H(r)

12 (dr2 + r2dΩ2

5)

C4 = H(r)−1dt ∧ dx ∧ dy ∧ dz , H(r) = 1 +R4

r4 ,R4

α′2= 4πgsNc

Asymptotically flatKilling Horizon: gtt = 0 = H(r)−

12 at r = 0

Near horizon geometry: AdS5 × S5 (w. Nc units of 5-form flux on S5)

ds2AdS5×S5 =

r2

R2 dxµ2 +R2

r2 dr2 + R2dΩ25

Q: Connection between these two descriptions of D3-Branes, i.e.between open and closed string physics?



Maldacena’s Decoupling Argument [Juan Maldacena ’98]

1 Field Theory Perspective: D3-Branes in flat space,GN = g2

s `8s = κ2 << 1 (backreaction ∼ GN )

Seff = SIIB︸︷︷︸1

2κ2

∫ √|g|R+...

+ SD3︸︷︷︸∼ 1

κ

+ Sint︸︷︷︸∼κ#>0

→ Low Energy Limit `s → 0, gs, Nc , . . . fixed :

Free IIB SUGRA & N = 4 SYM Theory

2 Soliton Perspective: Observer at Infinity sees (ω << 1/`s)

1 Low-Energy Bulk Modes σD3 ∼ ω3R8 ≈ 0→ Decoupling of Near-Horizon Region→ Free IIB SUGRA

2 Redshifted modes of arbitrary (!) energy from near the horizon→ IIB STRING THEORY on AdS5 × S5



AdS/CFT Forte, Mezzo & Piano

1 SU(Nc) N = 4 SYM ⇔IIB String Theory on AdS5 × S5 (Ncunits of five-form flux,R4

AdS5/S5 = 4πα′2gsNc = 2α′2g2YMNc )

’t Hooft Limit: Nc →∞, λ = g2YMNc fix

2 Planar SU(Nc =∞)N = 4 SYM

⇔Semiclassical Strings on AdS5 × S5

2πgs = g2YM → 0

Strong Coupling Limit: R4

`4s

= 2λ >> 1

3

Planar SU(Nc =∞)N = 4 SYM atstrong ’t Hooft coupling

⇔ IIB Supergravity on AdS5 × S5



The AdS/CFT Correspondence

The Original Correspondence (weakest form)

IIB Supergravity on AdS5 × S5 with (R/`s)4 = 2λ 1

ds2AdS5×S5 = R2

(dxµ2 + du2

u2 + dΩ25

)⇔

large Nc limit of N = 4 SU(Nc) Super Yang-Mills with λ = g2YMNc

1 Strong-Weak Coupling Duality: GN ∝ g2s = g4

YM2 Gubser-Klebanov-Polyakov-Witten relation:

〈ei∫

d4xJOO〉 = eiSIIB,onshell[JO]

3 Operator-Field Dictionary:

φm2=∆(∆−4) ' u4−∆JO + u∆〈O〉



The AdS/CFT Correspondence

closed string sectorAdS x S geometry

open string sectorlarge N D3stack at bottomof throat

55


The D3-D7 model: Adding Flavour to AdS/CFT

Outline




4 Summary



Adding Flavour to AdS/CFT [Karch & Katz ’02]

N = 4 Vector Multiplet (V ,ΦI), I = 1,2,3: All Adjoint

How to introduce “Quarks” = fundamental degrees of freedom ?→ Open String Sector in Gravity Dual → Probe Branes

Nc D3-Nf D7 Intersection:

L

N D3s

D7s

7−7

3−3

3−7

7−3

D3:(x0, x1, x2, x3)D7:(x0, x1, x2, x3, y4, y5, y6, y7)

N = 2 Supersymmetric

Field Content:3-3: N = 4 vector multiplet7-3: Nc-Nf chiral multiplet QI

3-7: Nc-Nf chiral multiplet QI7-7: U(Nf ) DBI theory

“Quark” masses: mq = LTs = L2πα′




How to A) Not spoil Maldacenas argument?B) Decouple 7-7 U(Nf ) theory from the field theory?

’t Hooft Limit = Probe Limit Nc →∞, Nf = fix

1 Field TheoryDecoupling limit still holds: T7 ∼ 1

(gs`8s)

= gsκ2

U(Nf ) becomes global : λ8 = g2YM,8Nf ∼ gs`

4sNc

NfNc∼ λ`4

sNfNc→ 0

→ Quenched Approximation!2 Gravity Side:

VNewton = GNNf T7 ∼ Nf gs ∼ λNfNc→ 0

→ Neglect D7 backreaction on D3 soliton background




Extended Correspondence: Nf D7 Branes in AdS5 × S5

4d N = 4 SU(N) SuperYang-Mills theory

coupled to4d Nf N = 2 hypermultiplets

in theQuenched Approximation

λ>>1↔

type IIB SUGRA on AdS5 × S5

+Dirac-Born-Infeld & Chern-Simons

theory on D7with

Neglected Backreaction

89

0123

4567

D3N

4R

AdS5

open/closed string duality

7−7

AdS5brane

flavour open/open string duality

conventional

3−7quarks

3−3

SYM

N probe D7f



D3/D7-Model: Field TheoryN = 4 Glue & Nf N = 2 Quarks

L = =[τ

∫d2θd2θ

(tr(ΦieV Φie−V ) + Q†

I eV QI + QI†e−V QI

)+

+τ

∫d2θ

(tr(WαWα) + tr(εijkΦiΦjΦk ) + QI(mq + Φ3)QI

) ],

Φ1 = φ1 + iφ2 + . . . ,Φ2 = φ3 + iφ4 + . . . ,Φ3 = φ5 + iφ6 + . . . ;

τ =θ

2π+ i

4πg2

ym

1 Field Content:N = 4 “Glue” Vector Multiplet: V , Φi , i = 1,2,3N = 2 “Quark” Hypermultiplet: QI , QI

2 Conformal in the Nc →∞ limit: β(λ) = Ncβ(g2ym) ∝ λ2 Nf

Nc→ 0

3 Global Symmetries (mq = 0, VEVs=0):

SO(4,2)× SU(2)Φ × SU(2)R × U(1)R × U(Nf )



D3/D7-Model: Field TheoryN = 4 Glue & Nf N = 2 Quarks

L = =[τ

∫d2θd2θ

(tr(ΦieV Φie−V ) + Q†

I eV QI + QI†e−V QI

)+

+τ

∫d2θ

(tr(WαWα) + tr(εijkΦiΦjΦk ) + QI(mq + Φ3)QI

) ],

Φ1 = φ1 + iφ2 + . . . ,Φ2 = φ3 + iφ4 + . . . ,Φ3 = φ5 + iφ6 + . . . ;

τ =θ

2π+ i

4πg2

ym

components spin SU(2)Φ × SU(2)R U(1)R ∆ U(Nf ) U(1)fΦ1,Φ2 Φ1,Φ2,Φ3,Φ4 0 ( 1

2 ,12 ) 0 1 1 0

λ1, λ212 ( 1

2 , 0) −1 32 1 0

Φ3, Wα Φ5,Φ6 0 (0, 0) +2 1 1 0λ3, λ4

12 (0, 1

2 ) +1 32 1 0

vµ 1 (0, 0) 0 1 1 0Q, Q qm = (q, ¯q) 0 (0, 1

2 ) 0 1 Nf +1ψi = (ψ, ψ†) 1

2 (0, 0) ∓1 32 Nf +1

→ U(1)R acts like a chiral symmetry: . . .mq(iψψ + h.c.) . . .



Supersymmetric Embeddings (Nf = 1)Rewrite AdS5 × S5: r2 = (y4)2 + (y5)2 + (y6)2 + (y7)2 + (z8)2 + (z9)2 = ρ2 + L2

ds2AdS5×S5 =

r2

R2dxµ2 +

R2

r2dr2 + R2dΩ2

5

=ρ2 + L2

R2dxµ2 +

R2

ρ2 + L2

dρ2 + ρ2dΩ23︸︷︷︸

d~y2

+ dL2 + L2dΦ2︸︷︷︸d~z2

Match Geometric Symmetries :

SO(4,2) SU(2)Φ SU(2)R U(1)R U(Nf )Iso(AdS5) SU(2)L,~y SU(2)R,~y U(1)~z U(Nf )

Embedding: ξα = (xµ, ρ,S3), L = L(ρ), Φ = 0

Constant L: LDBI ∝ ρ3√

1 + L′2 ⇒(

ρ3L′√1+L′2

)′

= 0 ⇒ L = 2πα′mq

N = 2 SUSY?: ∂Sonshell∂mq

= 0 (or check κ-symmetry)



Supersymmetric Embeddings (Nf = 1)

Dual Operator: Φ3 = φ+ . . .

Omq = i(ψψ + ψ†ψ†) + mq(qq† + q†q) +√

2(qφq† + q†φq + h.c.)︸︷︷︸∂ 1

2 (|Fq |2+|Fq |2)

∂mq

Why no VEV in AdS5 × S5? A Priori:(ρ3L′√1 + L′2

)′= 0 ⇒ L(ρ)

ρ→∞' 2πα′mq +

(2πα′)3〈Omq 〉ρ2

Reason 1: Om is an F-Term∫

d2θQ(mq + Φ3)QReason 2: Embeddings not well-behaved

ρ3L′√1+L′2

= c1 ⇒ L′ = c1√ρ6−c2

1

ρ→c1+→ ∞ , unless c1 = 0 !



Flavour at Finite Temperature

Flavour Physics at Finite Temperature

AdS-Schwarzschild× S5 (Black brane) , T = THawking ∝ rs

Picture: [hep-th/0611099]

1 Embedding: L(ρ)ρ→∞∼ 2πα′mq + (2πα′)3

ρ2 〈Omq 〉2 No “Spontaneous CSB”: 〈Omq 〉(mq = 0) = 03 Fluctuations: Mesons → Meson Melting Transition



D7 in AdS-Schwarzschild

-1 1 2 3 4 5 6Ρ

-2

-1

1

2L



1st order Phase Transition in AdS-Schwarzschild

0.5 1 1.5 2 2.5 3Quark mass

-0.4

-0.3

-0.2

-0.1

-Quark condensate

1.32 1.33 1.34 1.35 1.36Quark mass

-0.24

-0.23

-0.22

-0.21

-Quark condensate

F (mq,T ) = −SD3,onshell(mq,T )⇒ −〈Omq 〉 =∂F (mq,T )

∂mq


Electric and Magnetic Field Backgrounds

Outline




4 Summary


Electric and Magnetic Field Backgrounds Magnetic Field

Electric/Magnetic B-Field [see also 0709.1547, 0709.1554 (hep-th)]

Ansatz for the Kalb-Ramond Field

Bel = Bdt ∧ dx , Bmag = Bdy ∧ dz

dB = 0 ⇒ No deformation of AdS-Schwarzschild

−T7

g2

√−det (P[G + B] + 2πα′F ) +

∑p

P[Cp ∧ eB] ∧ e2πα′F

Affects Brane (Flavour) physics: D7 embeddings ,thermodynamics, phase transitions , meson spectra ...Effect: Background for U(1)F gauge field, mimics constantU(1) ∈ U(Nc) field strength



Magnetic Bµν

Magnetic Field

Bmag = Bdy ∧ dz

Zero Temperature: [Filev etal.hep-th/0701001]

CSB, Goldstone Boson with M ∝ √mq for small mq, ZeemansplittingFinite Temperature:

Small B: Meson Melting TransitionNo molten phase & CSB above a critical magnetic field strength(GMOR)Phase diagramSpectrum of Pseudoscalar Mesons



Magnetic Finite Temperature Embeddings

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B=0

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B=5

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B=10

2 4 6 8Ρ

-2

-1

1

2

L@ΡD

B=17

χSB : Bcrit ≈ 16 , B ∝ BT 2



Phase Diagram m = 2mq/(√λT ) , T = m−1

Melted phase

Mesonic phase

0 5 10 15B

0.0

0.2

0.4

0.6

0.8

1.0

1.2

m

Melted phase

Mesonic phase

0 20 40 60 80 100T

0

5

10

15

B

Meson Melting Transition below Bcrit

No molten phase and spontaneous CSB above Bcrit

Magnetic KR-Field acts repells the D7s from the origin



Φ Meson Spectrum (Upper Branch, M = M√

λ√πmq

)

0 1 2 3 4 5m

10

20

30

40

M

B

=0

0 2 4 6 8m

10

20

30

40

50

60

M

B

=5

0 2 4 6m

10

20

30

40

50

60

M

B

=10

0 2 4 6 8m

10

20

30

40

50

60

M

B

=17



Goldstone Boson (B = 16)

0.01 0.02 0.03 0.04 0.05m

0.2

0.4

0.6

0.8

1.0

M

M = 5√

m


Electric and Magnetic Field Backgrounds Electric Field

Electric Bµν

Bel = Bdt ∧ dx

Problem: Zero Locus of DBI action√−det P[G + B] = 0 for ρ2

IR + L(ρIR)2 = BR2

2 + 12

√4r4

s + B2R4

Zero Locus

L

rho



Electric Bµν

Bel = Bdt ∧ dx

Solution: U(1)f gauge field Ax(ρ), At(ρ) [hep-th:0705.3890]

∂a

(δLD7δ∂aAb

)= 0 ⇒ Two Conserved Quantities

At(ρ) ' µ− Dρ2 ↔ finite baryon number density 〈Jt〉 = D

Ax(ρ) ' Bρ2 ↔ baryon number current in x-direction 〈Jx〉 = B

⇒ A′x(ρ) = f ′(D,B, ρ) , A′t(ρ) = g′(D,B, ρ) ⇒ Legendre transform ⇒

Require SD7[L(ρ),D,B] to be well-defined :

B = B(ρIR,T ,B,D)

⇒ Well-defined EOMs for L(ρ)!



Electric Embeddings at T = 0: No CSB, Phase Transition

-1 1 2 3 4 5 6Ρ

-1.0

-0.5

0.5

1.0

1.5

2.0

L

→ Dissociation of Mesons?→ Conical singularities?



Condensate vs. Mass

1 2 3 4m

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

c



Condensate vs. Mass: Phase Transition

1.3160 1.3165 1.3170 1.3175 1.3180 1.3185 1.3190m

-0.225

-0.224

-0.223

-0.222

-0.221

-0.220

c

Area∝ F



Φ (l=0) Meson Spectrum at T = 0: ∆M < 0

1 2 3 4 5m

10

20

30

40

50

M

Incoming Wave Boundary Conditions → Dissociation of Mesons!



Electric Bµν: Finite Temperature

What to expect at Finite Temperature?Meson melting enhanced by dissociationAny nonzero electric field will decrease the melting temperatureNo SCSBFinite T: One or two transitions?

Open QuestionsFate of the conically singular solutions? They are either

physical → What creates the singularity?unphysical → What happens in that mass range?

Energy of the system is time-dependent →Is this still equilibrium physics?Thermodynamics in the presence of external currents?


Summary

Outline




4 Summary


Summary

Summary: Electric/Magnetic Background Fields

Magnetic BackgroundInduced SCSBMagnetic Field Stabilizes MesonsMesons don’t melt for large enough B (SCSB)Zeeman splitting

Electric BackgroundNo SCSBDissociation (& Meson Melting)Stark shift


Summary

Backup Slides


Summary

Condensate vs. Mass

1 2 3 4 5 6

-0.30-0.25-0.20-0.15-0.10-0.05

B

=0

1 2 3 4 5 6

-0.30-0.25-0.20-0.15-0.10-0.05

B

=0.5

1 2 3 4 5 6

-0.3

-0.2

-0.1

0.1

0.2

B

=1

1 2 3 4 5 6

12345

B

=5

1 2 3 4 5 6

468

10

B

=10

1 2 3 4 5 602468

1012

B

=12

1 2 3 4 5 6

5

10

15

B

=15

1 2 3 4 5 60

10

15

20

B

=17

-2 2 4 60

1015202530

B

=20

-4 -2 2 4 60

20

30

40

50

B

=30

-4 -2 2 4 6

406080

100120140

B

=60

-10 10 20 30

2000400060008000

10 000

B

=1000


the d3-d7 model of ads/cft with flavour (with special...

Documents