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

(1)

The D3-D7 Model of AdS/CFT with Flavour

(with Special Feature)

René Meyer

Max-Planck-Institute for Physics Munich, Germany

November 13, 2008

Lot’s of known stuff (Review: hep-th/0711.4467 ) +

JHEP 0712:091,2007 (with J. Erdmenger & J. P. Shock)

(2)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

René Meyer (MPI Munich) Flavour AdS/CFT November 13, 2008 2 / 43

(3)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(4)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(5)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(6)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(7)

Introduction to AdS/CFT

Motivation

New tools for strongly coupled gauge theories?

QCD in the infrared is strongly coupled (Conformal window?)

[Deur, Korsch et. al: hep-ph/0509113]

0.05 0.06 0.07 0.08 0.090.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 2

10^-1 1

Q (GeV) αs(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 >≈ Λ _QCD )

(8)

Our new tool: AdS/CFT Holography

(9)

Our new tool: AdS/CFT Holography

The Original Correspondence (weakest form)

IIB Supergravity on AdS ₅ × S ⁵ with (R/` _s ) ⁴ = 2λ 1

ds _AdS ²

5

×S

⁵

= R ²

dx ^µ2 + du ² u ² + dΩ ² ₅

⇔

large N _c limit of N = 4 SU(N _c ) Super Yang-Mills with λ = g ² _YM N _c

1

Strong-Weak Coupling Duality: G _N ∝ g _s ² = g _YM ⁴

2

Gubser-Klebanov-Polyakov-Witten relation:

he ⁱ ^R ^d

⁴

^xJ

^O

^O i = e ^iS

IIB,onshell

[J

O

]

3

Operator-Field Dictionary:

φ _m

2

=∆(∆−4) ' u ^4−∆ J O + u ^∆ hOi

(10)

A short look at string theory

Basic Objects:

Open and Closed Strings, D- Branes

Strings have length: ` s

Strings have tension: T

s

=

_2πα¹0

=

¹

2π`²_s

“Minimal Surface” action principle: Nambu-Goto Action

S

NG

= 1 2π`

²_s

Z

dτdσ p

− det P[G]

ab

P[G]

ab

= G

µν

∂X

^µ

∂σ

^a

∂X

^ν

∂σ

^b

[Becker, Becker, Schwarz]

(11)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

1 ^st Quantized Strings:

Open String: “Gauge Theory”

A

µ

, Φ

^I

, Fermions Closed String: “Gravity”

G

µν

, B

µν

, Φ , Fermions

[hep-ph/0701201]

(12)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

D-Branes Curve Spacetime:

Open-Closed String Duality

(13)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

D-Branes Curve Spacetime:

“Graviton” Absorption/Emission

(14)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

Dp-Branes Have Tension:

T

p

= 1 (2π)

^p

`

^(p+1)_s

g

s

g

_s

. . . string coupling

“Graviton” Absorption/Emission

(15)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

Dp-Branes Have Tension:

T

_p

= 1 (2π)

^p

`

^(p+1)_s

g

_s

g

_s

. . . string coupling String Splitting/Merging

[“D-Branes”, C. V. Johnson]

(16)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

Dp-Branes are “Minimal Hypersurfaces”: (α ⁰ = ` ² _s )

S _Dp = S _DBI + S _CS

S _DBI = −T _p Z

d ^(p+1) ξ p

− det (P[G + B] _ab + 2πα ⁰ F _ab )

S _CS = +T _p g _s X

p

Z

topform

P[C ^(p+1) ∧ e ^B ] ∧ e ^2πα

⁰

^F

(17)

A short look at string theory

D-Branes: Dirichlet boundary conditions for the string (e.g.

X ¹ (σ, τ ) = 0 for σ = 0, π)

Dp-Branes carry Gauge Theories:

e.g. flat D3-Brane in flat space in the limit α ⁰ << 1

S

_DBI,p=3

= − 1

(2π)

³

α

⁰²

g

s

Z d

⁴

ξ p

− det (P[G]

_ab

+ 2πα

⁰

F

_ab

)

≈ −T

₃

Z

d

⁴

ξ p

− det P[η] − 1 8πg

_s

Z d

⁴

ξ p

− det P[η]F

ab

F

^ab

= −T

₃

Vol( R

⁴

)− 1 8πg

s

Z

d

⁴

ξF

_ab

F

^ab

(18)

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

More precisely: The gauge theory on a stack of N

_c

D3-branes is the unique maximally supersymmetric gauge theory in 4D,

N = 4 Super-Yang-Mills Theory (SYM)

SUSYs: 16 Poincaré + 16 Conformal =

⁶⁴₂

real supercharges =

¹₂

BPS Fields: N = 4 Vector Multiplet in 4D = (A

µ

, λ

_αA

, φ

ⁱ

) = (V , Φ

_I

)

L = tr (

− 1

2g

_YM²

F

_µν²

+ Θ

_I

8π

²

F

µν

F ˜

^µν

− i

4

X

A=1

¯ λ

^A

σ ¯

^µ

D

µ

λ

_A

−

6

X

i=1

D

µ

φ

ⁱ

D

^µ

φ

ⁱ

+ g

_YM²

2 X

i,j

[φ

ⁱ

, φ

^j

]

²

+ g

YM

X

A,B,i

C

_i^AB

λ

_A

[φ

ⁱ

, λ

_B

] + h.c .







Global Symmetries: SO(4, 2) × SU(4)

R

(β(g

YM

) = 0 !)

(19)

D3-Branes in IIB Supergravity

Have seen:

1

D3-Branes carry SU(N

_c

) N = 4 SYM theory

2

D3-Branes curve space in a supersymmetric way

D3-Branes are solitonic

¹₂

BPS solutions to IIB Supergravity

[Horowitz, Strominger Nucl.Phys.B360(1991)]

ds

²

= H(r )

⁻¹²

dx

^µ2

+ H(r)

¹²

(dr

²

+ r

²

dΩ

²₅

)

C

₄

= H(r )

⁻¹

dt ∧ dx ∧ dy ∧ dz , H(r ) = 1 + R

⁴

r

⁴

, R

⁴

α

⁰²

= 4πg

_s

N

_c

Asymptotically flat

Killing Horizon: g

tt

= 0 = H(r)

⁻¹²

at r = 0

Near horizon geometry: AdS

₅

× S

⁵

(w. N

_c

units of 5-form flux on S

⁵

) ds

_AdS² ₅_×S5

= r

²

R

²

dx

^µ2

+ R

²

r

²

dr

²

+ R

²

dΩ

²₅

Q: Connection between these two descriptions of D3-Branes, i.e.

between open and closed string physics?

(20)

Maldacena’s Decoupling Argument

[Juan Maldacena ’98]

1

Field Theory Perspective: D3-Branes in flat space, G

_N

= g

_s²

`

⁸_s

= κ

²

<< 1 (backreaction ∼ G

_N

)

S

_eff

= S

_IIB

|{z}

1 2κ2

R

√

|g|R+...

+ S

_D3

|{z}

∼_κ¹

+ S

_int

|{z}

∼κ^#>0

→ Low Energy Limit `

_s

→ 0, g

s

, N

c

, . . . fixed :

Free IIB SUGRA & N = 4 SYM Theory

2

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

s

)

1

Low-Energy Bulk Modes σ

_D3

∼ ω

³

R

⁸

≈ 0

→ Decoupling of Near-Horizon Region→ Free IIB SUGRA

2

Redshifted modes of arbitrary (!) energy from near the horizon

→ IIB STRING THEORY on AdS

5

× S

⁵

(21)

AdS/CFT Forte, Mezzo & Piano

1

SU(N

_c

) N = 4 SYM ⇔

IIB String Theory on AdS

₅

× S

⁵

(N

_c

units of five-form flux,

R

⁴_AdS

5/S⁵

= 4πα

⁰²

g

s

N

c

= 2α

⁰²

g

_YM²

N

c

)

’t Hooft Limit: N

_c

→ ∞, λ = g

²_YM

N

_c

fix

2

Planar SU(N

c

= ∞)

N = 4 SYM ⇔ Semiclassical Strings on AdS

₅

× S

⁵

2πg

_s

= g

_YM²

→ 0

Strong Coupling Limit:

^R_`4⁴ s

= 2λ >> 1

3

Planar SU(N

c

= ∞) N = 4 SYM at

strong ’t Hooft coupling

⇔ IIB Supergravity on AdS

₅

× S

⁵

(22)

The AdS/CFT Correspondence

The Original Correspondence (weakest form)

IIB Supergravity on AdS ₅ × S ⁵ with (R/` _s ) ⁴ = 2λ 1

ds _AdS ²

5

×S

⁵

= R ²

dx ^µ2 + du ² u ² + dΩ ² ₅

⇔

large N _c limit of N = 4 SU(N _c ) Super Yang-Mills with λ = g ² _YM N _c

1

Strong-Weak Coupling Duality: G _N ∝ g _s ² = g _YM ⁴

2

Gubser-Klebanov-Polyakov-Witten relation:

he ⁱ ^R ^d

⁴

^xJ

^O

^O i = e ^iS

IIB,onshell

[J

O

]

3

Operator-Field Dictionary:

φ _m

2

=∆(∆−4) ' u ^4−∆ J O + u ^∆ hOi

(23)

The AdS/CFT Correspondence

closed string sector AdS x S geometry

open string sector large N D3

stack at bottom of throat

5 5

(24)

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(25)

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

Adding Flavour to AdS/CFT

[Karch & Katz ’02]

N = 4 Vector Multiplet (V , Φ

_I

), I = 1, 2, 3: All Adjoint

How to introduce “Quarks” = fundamental degrees of freedom ?

→ Open String Sector in Gravity Dual → Probe Branes N

c

D3-N

f

D7 Intersection:

0000 1111

00 11

L

N D3s D7s

7−7

3−3 3−7

7−3

D3:(x

⁰

, x

¹

, x

²

, x

³

)

D7:(x

⁰

, x

¹

, x

²

, x

³

, y

⁴

, y

⁵

, y

⁶

, y

⁷

) N = 2 Supersymmetric

Field Content:

3-3: N = 4 vector multiplet 7-3: N

c

-N

_f

chiral multiplet Q

^I

3-7: N ¯

c

- N ¯

_f

chiral multiplet Q ˜

I

7-7: U(N

_f

) DBI theory

“Quark” masses: m

q

= LT

s

=

_2πα^L0

(26)

Adding Flavour to AdS/CFT

How to A) Not spoil Maldacenas argument?

B) Decouple 7-7 U(N _f ) theory from the field theory?

’t Hooft Limit = Probe Limit N c → ∞, N _f = fix

1

Field Theory

Decoupling limit still holds: T

7

∼

¹

(gs`⁸_s)

=

_κ^g^s₂

U(N

_f

) becomes global : λ

₈

= g

_YM,8²

N

_f

∼ g

_s

`

⁴_s

N

_c^N_N^f

c

∼ λ`

⁴_s_N^N^f

c

→ 0

→ Quenched Approximation!

2

Gravity Side:

V _Newton = G _N N _f T ₇ ∼ N _f g _s ∼ λ _N ^N

^f

c

→ 0

→ Neglect D7 backreaction on D3 soliton background

(27)

Adding Flavour to AdS/CFT

Extended Correspondence: N

f

D7 Branes in AdS

5

× S

⁵

4d N = 4 SU(N) Super

Yang-Mills theory coupled to

4d N

_f

N = 2 hypermultiplets in the

Quenched Approximation

λ>>1

↔

type IIB SUGRA on AdS

5

× S

⁵

+

Dirac-Born-Infeld & Chern-Simons theory on D7

with

Neglected Backreaction

000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000

111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111

0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000

1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111

89 0123

4567

D3 N

R

4

AdS

₅

open/closed string duality

7−7

AdS

₅

brane

flavour open/open

string duality conventional

3−7

quarks

3−3

SYM

N probe D7

_f

(28)

D3/D7-Model: Field Theory

N = 4 Glue & N

_f

N = 2 Quarks

L = = h τ

Z

d

²

θd

²

θ ¯

tr( ¯ Φ

_i

e

^V

Φ

_i

e

^−V

) + Q

^†_I

e

^V

Q

^I

+ ˜ Q

^I†

e

^−V

Q ˜

_I

+ +τ

Z d

²

θ

tr(W

^α

W

α

) + tr(

_ijk

Φ

_i

Φ

_j

Φ

_k

) + ˜ Q

I

(m

q

+ Φ

₃

)Q

^I

i , Φ

₁

= φ

₁

+ i φ

₂

+ . . . , Φ

₂

= φ

₃

+ i φ

₄

+ . . . , Φ

₃

= φ

₅

+ iφ

₆

+ . . . ;

τ = θ

2π + i 4π g

_ym²

1

Field Content:

N = 4 “Glue” Vector Multiplet: V , Φ

_i

, i = 1, 2, 3 N = 2 “Quark” Hypermultiplet: Q

^I

, Q ˜

_I

2

Conformal in the N

_c

→ ∞ limit: β (λ) = N

_c

β(g

_ym²

) ∝ λ

²_N^N^f

c

→ 0

3

Global Symmetries (m

_q

= 0, VEVs=0):

SO(4, 2) × SU(2)

Φ

× SU(2)

R

× U(1)

R

× U(N

_f

)

(29)

D3/D7-Model: Field Theory

N = 4 Glue & N

_f

N = 2 Quarks

L = = h τ

Z

d

²

θd

²

θ ¯

tr( ¯ Φ

_i

e

^V

Φ

_i

e

^−V

) + Q

^†_I

e

^V

Q

^I

+ ˜ Q

^I†

e

^−V

Q ˜

I

+ +τ

Z d

²

θ

tr(W

^α

W

α

) + tr(

_ijk

Φ

_i

Φ

_j

Φ

_k

) + ˜ Q

_I

(m

_q

+ Φ

₃

)Q

^I

i , Φ

₁

= φ

₁

+ i φ

₂

+ . . . , Φ

₂

= φ

₃

+ i φ

₄

+ . . . , Φ

₃

= φ

₅

+ iφ

6

+ . . . ;

τ = θ

2π + i 4π g

_ym²

components spin SU(2)_Φ×SU(2)R U(1)R ∆ U(N_f) U(1)_f Φ₁,Φ₂ Φ₁,Φ₂,Φ₃,Φ₄ 0 (¹₂,¹₂) 0 1 1 0

λ₁, λ₂ ¹₂ (¹₂,0) −1 ³₂ 1 0

Φ₃,Wα Φ₅,Φ₆ 0 (0,0) +2 1 1 0

λ₃, λ₄ ¹₂ (0,¹₂) +1 ³₂ 1 0

v_µ 1 (0,0) 0 1 1 0

Q,Q˜ q^m= (q,¯˜q) 0 (0,¹₂) 0 1 N_f +1

ψ_i= (ψ,ψ˜^†) ¹₂ (0,0) ∓1 ³₂ N_f +1

→ U(1)

R

acts like a chiral symmetry: . . . m

q

(i ψψ ˜ + h.c.) . . .

(30)

Supersymmetric Embeddings (N _f = 1)

Rewrite AdS

₅

× S

⁵

:

^r²^{= (y}⁴)²+ (y⁵)²+ (y⁶)²+ (y⁷)²+ (z⁸)²+ (z⁹)²=ρ²+L² ds²_AdS

5×S5 = r²

R²dx^µ2+R²

r²dr²+R²dΩ²₅

= ρ²+L²

R² dx^µ2+ R² ρ²+L²







dρ²+ρ²dΩ²₃

| {z } d~y2

+dL²+L²dΦ²

| {z } d~z2







Match Geometric Symmetries :

SO(4, 2) SU(2)

Φ

SU(2)

R

U(1)

R

U(N

_f

) Iso(AdS

₅

) SU(2)

_L,~_y

SU(2)

_R,~_y

U(1)

~z

U(N

_f

) Embedding: ξ

^α

= (x

^µ

, ρ, S

³

), L = L(ρ), Φ = 0

Constant L: L

_DBI

∝ ρ

³

√

1 + L

⁰²

⇒

ρ³L⁰

√

1+L⁰²

0

= 0 ⇒ L = 2πα

⁰

m

q

N = 2 SUSY?:

^∂S_∂m^onshell

q

= 0 (or check κ-symmetry)

(31)

Supersymmetric Embeddings (N _f = 1)

Dual Operator: Φ ₃ = φ + . . .

O _m

_q

= i( ˜ ψψ + ψ ^† ψ ˜ ^† ) + m _q (˜ q q ˜ ^† + q ^† q) +

√

2(˜ qφ˜ q ^† + q ^† φq + h.c.)

| {z }

∂1

2(|Fq|2+|F˜q|2)

∂mq

Why no VEV in AdS ₅ × S ⁵ ? A Priori:

ρ ³ L ⁰

√ 1 + L ⁰²

⁰

= 0 ⇒ L(ρ) ^ρ→∞ ' 2πα ⁰ m _q + (2πα ⁰ ) ³ hO _m

_q

i ρ ² Reason 1: O _m is an F-Term R

d ² θ Q(m ˜ _q + Φ 3 )Q Reason 2: Embeddings not well-behaved

ρ

³

L

⁰

√

1+L

⁰²

= c ₁ ⇒ L ⁰ = √ ^c

¹

ρ

⁶

−c

²₁

ρ→c

1+

→ ∞ , unless c ₁ = 0 !

(32)

Flavour at Finite Temperature

Flavour Physics at Finite Temperature

AdS-Schwarzschild × S

⁵

(Black brane) , T = T

_Hawking

∝ r

_s

Picture: [hep-th/0611099]

1

Embedding: L(ρ) ^ρ→∞ ∼ 2πα ⁰ m q + ^(2πα _ρ

2⁰

⁾

³

hO _m

_q

i

2

No “Spontaneous CSB”: hO _m

_q

i(m _q = 0) = 0

3

Fluctuations: Mesons → Meson Melting Transition

(33)

D7 in AdS-Schwarzschild

-1 1 2 3 4 5 6

Ρ

-2

-1

1

2 L

(34)

1 ^st order Phase Transition in AdS-Schwarzschild

0.5 1 1.5 2 2.5 3 Quark mass

-0.4 -

0.3

-0.2 -

0.1

-Quark condensate

1.32 1.33 1.34 1.35 1.36 Quark mass

-0.24

-

0.23

-0.22 -0.21 -Quark condensate

F (m _q , T ) = −S _D3,onshell (m _q , T ) ⇒ −hO _m

_q

i = ∂F (m _q , T )

∂m q

(35)

Electric and Magnetic Field Backgrounds

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(36)

Electric/Magnetic B-Field

[see also 0709.1547, 0709.1554 (hep-th)]

Ansatz for the Kalb-Ramond Field

B _el = Bdt ∧ dx , B mag = Bdy ∧ dz

dB = 0 ⇒ No deformation of AdS-Schwarzschild

− T ₇ g ₂

p −det (P[G + B] + 2πα ⁰ F ) + X

p

P[C _p ∧ e ^B ] ∧ e ^2πα

⁰

^F

Affects Brane (Flavour) physics: D7 embeddings , thermodynamics, phase transitions , meson spectra ...

Effect: Background for U(1) _F gauge field, mimics constant U(1) ∈ U(N c ) field strength

(37)

Electric and Magnetic Field Backgrounds Magnetic Field

Magnetic B µν

Magnetic Field

B _mag = Bdy ∧ dz

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

CSB, Goldstone Boson with M ∝ √

m _q for small m _q , Zeeman splitting

Finite Temperature:

Small B: Meson Melting Transition

No molten phase & CSB above a critical magnetic field strength (GMOR)

Phase diagram

Spectrum of Pseudoscalar Mesons

(38)

Magnetic Finite Temperature Embeddings

2 4 6 8

Ρ

-2 -1

1 2 L@ΡD

B

=0

2 4 6 8

Ρ

-2 -1

1 2 L@ΡD

B

=5

2 4 6 8

Ρ

-2 -1

1 2 L@ΡD

B

=10

2 4 6 8

Ρ

-2 -1

1 2 L@ΡD

B

=17

χSB : B ˜ _crit ≈ 16 , B ˜ ∝ B T ²

(39)

Phase Diagram

m^˜=2mq/(√

λT),T˜= ˜m⁻¹

Melted phase

Mesonic phase

0 5 10 15

B

0.0 0.2 0.4 0.6 0.8 1.0 1.2

m

Melted phase Mesonic phase

0 20 40 60 80 100

T

0 5 10 15

B

Meson Melting Transition below B ˜ _crit

No molten phase and spontaneous CSB above B ˜ _crit

Magnetic KR-Field acts repells the D7s from the origin

(40)

Φ Meson Spectrum (Upper Branch,

M^˜= ^M

√

√ λ πmq

)

0 1 2 3 4 5

m 10

20 30 40 M

B

=0

0 2 4 6 8

m 10

20 30 40 50 60 M

B

=5

0 2 4 6

m 10

20 30 40 50 60 M

B

=10

0 2 4 6 8

m 10

20 30 40 50 60 M

B

=17

(41)

Goldstone Boson ( B ˜ = 16)

0.01 0.02 0.03 0.04 0.05

m

0.2 0.4 0.6 0.8 1.0

M

M ˜ = 5

√ m ˜

(42)

Electric B µν

B _el = Bdt ∧ dx Problem: Zero Locus of DBI action

p − det P[G + B] = 0 for ρ ² _IR + L(ρ _IR ) ² = ^BR ₂

²

+ ¹ ₂ p

4r _s ⁴ + B ² R ⁴

Zero Locus

L

rho

(43)

Electric and Magnetic Field Backgrounds Electric Field

Electric B µν

B _el = Bdt ∧ dx

Solution: U(1) _f gauge field A _x (ρ), A _t (ρ) [hep-th:0705.3890]

∂ _a _δL

D7

δ∂

a

A

b

= 0 ⇒ Two Conserved Quantities A _t (ρ) ' µ − _ρ ^D

₂

↔ finite baryon number density hJ _t i = D A x (ρ) ' ^B

ρ

²

↔ baryon number current in x-direction hJ _x i = B

⇒ A ⁰ _x (ρ) = f ⁰ (D, B, ρ) , A ⁰ _t (ρ) = g ⁰ (D, B, ρ) ⇒ Legendre transform ⇒

Require S ˜ _D7 [L(ρ), D, B] to be well-defined :

B = B(ρ _IR , T , B, D)

⇒ Well-defined EOMs for L(ρ)!

(44)

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

-1 1 2 3 4 5 6

Ρ

-1.0 -0.5 0.5 1.0 1.5 2.0 L

→ Dissociation of Mesons?

→ Conical singularities?

(45)

Condensate vs. Mass

1 2 3 4

m

-0.30 -

0.25

-0.20 -0.15 -0.10 -0.05

0.00

c

(46)

Condensate vs. Mass: Phase Transition

1.3160 1.3165 1.3170 1.3175 1.3180 1.3185 1.3190

m -0.225

-0.224 -0.223 -0.222 -0.221 -0.220 c

Area∝ F

(47)

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

1 2 3 4 5 m

10 20 30 40 50 M

Incoming Wave Boundary Conditions → Dissociation of Mesons!

(48)

Electric B µν : Finite Temperature

What to expect at Finite Temperature?

Meson melting enhanced by dissociation

Any nonzero electric field will decrease the melting temperature No SCSB

Finite T: One or two transitions?

Open Questions

Fate 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?

(49)

Summary

Outline

1 Introduction to AdS/CFT

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

3 Electric and Magnetic Field Backgrounds Magnetic Field

Electric Field

4 Summary

(50)

Summary: Electric/Magnetic Background Fields

Magnetic Background Induced SCSB

Magnetic Field Stabilizes Mesons

Mesons don’t melt for large enough B (SCSB) Zeeman splitting

Electric Background No SCSB

Dissociation (& Meson Melting) Stark shift

(51)

Summary

Backup Slides

(52)

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

1 2 3 4 5

B

=5

1 2 3 4 5 6

4 6 8 10

B

=10

1 2 3 4 5 6

0 2 4 6 8 10 12

B

=12

1 2 3 4 5 6

5 10 15

B

=15

1 2 3 4 5 6

0 10 15 20

B

=17

-2 0 2 4 6

10 15 20 25 30

B

=20

-4 -2 0 2 4 6

20 30 40 50 B

=30

-4 -2 2 4 6

40 60 80 100 120 140 B

=60

-10 10 20 30

2000 4000 6000 8000 10 000

B

=1000

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