PROBE BRANES ON FLAVORED ABJM BACKGROUND Javier Mas Universidad de - - PowerPoint PPT Presentation

PROBE BRANES ON FLAVORED ABJM BACKGROUND

Javier Mas

Universidad de Santiago de Compostela

Heraklion

June 2013

Niko Jokela, J. M., Alfonso V. Ramallo & Dimitrios Zoakos arXiv: 1211.0630 based on Eduardo Conde & Alfonso V. Ramallo 1105.6045

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

PLAN OF THE TALK

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

The ABJM theory

field theory

Chern-Simons-matter theories in 2+1 dimensions

gauge group: U(N)k × U(N)−k

field content (bosonic)

• Two gauge fields Aµ, ˆ

Aµ

• Four complex scalar fields: CI (I = 1, · · · , 4)

bifundamentals (N, ¯ N)

action

S = k CS[A] − k CS[ ˆ A] − k Dµ CI † Dµ CI − Vpot(C)

Vpot(C) → sextic scalar potential

Aharony, Bergman, Jafferis & Maldacena 0806.1218

The ABJM theory

The ABJM model has N = 6 SUSY in 3d

N → rank of the gauge groups

k → CS level (1/k ∼ gauge coupling) ’t Hooft coupling λ ∼ N

k

it is a CFT in 3d with very nice properties

it has two parameters

• partition function and Wilson loops can be obtained from localization
• has many integrability properties (Bethe ansatz, Wilson loop/

amplitude relation, ...)

• connection to FQHE?

it is the 3d analogue of N=4 SYM

Fujita, Li, Ryu & Takayanagi, 0901.0924

Gaiotto&Jafferis 0903.2175

• T. Klose, 1012.3999

Drukker, Mariño & Putrov 1003.3837

The ABJM theory

sugra description in type IIA :

AdS4 × CP3 + fluxes

CP3 = C4/(zi ∼ λzi)

Effective description for N

1 5 << k << N

J → Kahler form of CP3

1 2π

• CP1 F2 = k

F2 = 2k J F4 = 3π √ 2

• kN

1

2 ΩAdS4

eφ = 2L k = 2√π 2N k5 1

4

ds2 = L2ds2

AdS + L2ds2 CP3

L4 = 2π2 N k = 2π2λ

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

D6-branes extended in AdS4 and wrapping RP3 ⊂ CP3

Introduce quarks in the (N, 1) and (1, N) representation

Hohenegger&Kirsch 0903.1730

Gaiotto&Jafferis 0903.2175

Q1 → (N, 1)

Q2 → (1, N)

˜ Q1 → ( ¯ N, 1)

˜ Q2 → (1, ¯ N)

Q†

1 e−V Q1 + Q† 2 e− ˆ V Q2 + antiquarks

coupling to the vector multiplet

V, ˆ V vector supermultiplets for A , ˆ A

coupling to the bifundamentals

˜ Q1 AiBi Q1 , ˜ Q2 Bi Ai Q2

CI = (A1, A2, B†

1, B† 2)

plus quartic terms in Q, ˜ Q’s

Flavors in the ABJM background

Flavors in the ABJM background

ω1 = cos ˆ ψdˆ θ + sin ˆ ψ sin ˆ θd ˆ ϕ ω1 = sin ˆ ψdˆ θ − cos ˆ ψ sin ˆ θd ˆ ϕ ω3 = d ˆ ψ + cos ˆ θd ˆ ϕ where Ai = − ξ2 1 + ξ2 ωi

SU(2) instanton on S4

x1 = sin θ cos ϕ x2 = sin θ sin ϕ x3 = cos θ

Write CP3 as an S2-bundle over S4

( x0, x1, x2, r | {z }

AdS4

, ξ, ˆ θ, ˆ ψ, ˆ ϕ | {z }

S4

, θ, ϕ |{z}

S2

)

ds2 = L2ds2

AdS + L2ds2 CP3

ds2 = L2ds2

AdS4 + L2

 4 (1 + ⇠2)2 ✓ d⇠2 + ⇠2 4

• (!1)2 + (!2)2 + (!3)2◆

+ ⇣ dxi + ✏ijkAjxk⌘2

Flavors in the ABJM background

D6-branes extended in AdS4 and wrapping RP3 ⊂ CP3

S = SDBI + SW Z = −TD6 Z d7ζ e−φ p −det ˆ g7 + TD6 Z ˆ C7

m0

θ(r)

θ(r) = π/2 ⇒ m0 = 0

( x0, x1, x2, r | {z }

AdS4

, ξ, ˆ θ = 0, ˆ ψ, ˆ ϕ = 0 | {z }

S4

, θ = θ(r) , ϕ | {z }

S2

)

the idea is now to smear over positions and orientations

• no delta-function sources

preserve N=1 SUSY

• much simpler (analytic) solutions

−flavor symmetry : U(1)Nf

Sflav = PNf

i=1

⇣ −TD6 R

M(i) d7ζ e−φ √−det ˆ

g7 + TD6 R

M(i) ˆ

C7 ⌘

Backreaction

m0

• E. Conde & A. V. Ramallo 1105.6045

The ABJM flavored background

→ 1 κ2

10

✓ − Z d10xe3φ/4p − det g10|Ω| + Z d10x C7 ∧ Ω ◆

Ω is a charge distribution 3-form

C7 = e−φK is the calibration form

The ABJM flavored background

modified Bianchi identity

dF2 = 2π Ω

The ABJM flavored background

modified Bianchi identity

dF2 = 2π Ω

θ(r) = π/2

solution for massless flavors

The ABJM flavored background

modified Bianchi identity

dF2 = 2π Ω

go to vielbein basis

Ea = (E1, E2)

Si = (S1, S2, S3, S4)

along the base

S4

along the fiber

S2

(dξ, ω1, ω2, ω3, dθ, dϕ) → θ(r) = π/2

solution for massless flavors

The ABJM flavored background

modified Bianchi identity

dF2 = 2π Ω

go to vielbein basis

Ea = (E1, E2)

Si = (S1, S2, S3, S4)

along the base

S4

along the fiber

S2

(dξ, ω1, ω2, ω3, dθ, dϕ) →

Sξ = 2 1 + ξ2 dξ S1 = ξ 1 + ξ2

• sin ϕ ω1 − cos ϕ ω2

S2 = ξ 1 + ξ2

• sin θ ω3 − cos θ
• cos ϕ ω1 + sin ϕ ω2

S3 = ξ 1 + ξ2

• − cos θ ω3 − sin θ
• cos ϕ ω1 + sin ϕ ω2

E1 = dθ + ξS1 E2 = sin dϕ − ξS2

θ(r) = π/2

solution for massless flavors

F2 = k 2 h E1 ∧ E2 − (S4 ∧ S3 + S1 ∧ S2) i

ds2 = L2ds2

AdS + L2 4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 ! F4 = 3k 2 L2 ΩAdS4

The ABJM flavored background

F2 = k 2 h E1 ∧ E2 − (S4 ∧ S3 + S1 ∧ S2) i

ds2 = L2ds2

AdS + L2 4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 ! F4 = 3k 2 L2 ΩAdS4

The ABJM flavored background

η

Flavor backreaction η = 1 + 3 4 Nf k

F2 = k 2 h E1 ∧ E2 − (S4 ∧ S3 + S1 ∧ S2) i

ds2 = L2ds2

AdS + L2 4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 ! F4 = 3k 2 L2 ΩAdS4

The ABJM flavored background

η

Flavor backreaction η = 1 + 3 4 Nf k

q 1 b2

F2 = k 2 h E1 ∧ E2 − (S4 ∧ S3 + S1 ∧ S2) i

ds2 = L2ds2

AdS + L2 4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 ! F4 = 3k 2 L2 ΩAdS4

The ABJM flavored background

η

Flavor backreaction η = 1 + 3 4 Nf k

q 1 b2

q = 3 + 9 4 Nf k − 2 s 1 + 3 4 Nf k + ✓3 4 ◆4 ✓Nf k ◆2

   1 + 3

8 Nf k − ...

q → 5

3

( Nf

k → ∞)

b = 4 + 39

16 Nf k −

r 1 + 3

4 Nf k +

⇣

9 16 Nf k

⌘2 3 + 3

2 Nf k

   1 + 3

16 Nf k − ...

q → 5

4

( Nf

k → ∞)

F2 = k 2 h E1 ∧ E2 − (S4 ∧ S3 + S1 ∧ S2) i

ds2 = L2ds2

AdS + L2 4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 ! F4 = 3k 2 L2 ΩAdS4

The ABJM flavored background

η

Flavor backreaction η = 1 + 3 4 Nf k L2 = π √ 2λ σ

where is related to the quark-antiquark potential screening

σ

q 1 b2

q = 3 + 9 4 Nf k − 2 s 1 + 3 4 Nf k + ✓3 4 ◆4 ✓Nf k ◆2

   1 + 3

8 Nf k − ...

q → 5

3

( Nf

k → ∞)

b = 4 + 39

16 Nf k −

r 1 + 3

4 Nf k +

⇣

9 16 Nf k

⌘2 3 + 3

2 Nf k

   1 + 3

16 Nf k − ...

q → 5

4

( Nf

k → ∞)

q Σ b 5 10 15 20 25 30 Ε

• 2

4

q

b

σ

Nf k

The ABJM flavored background

Vq¯

q = −Q

l ; → Q = 4 √ 2π3 Γ(1/4)4 √ λ σ

potential screening

σ = 1 4 q3/2(η + q)2(2 − q)1/2 (q + ηq − η)5/2

8 < : 1 − 3

8 Nf k − ...

→ q

k Nf

( Nf

k → ∞)

q Σ b 5 10 15 20 25 30 Ε

• 2

4

q

b

σ

Nf k

The ABJM flavored background

Vq¯

q = −Q

l ; → Q = 4 √ 2π3 Γ(1/4)4 √ λ σ

potential screening dilaton shifts

eφ = 4√π ✓2N k5 ◆1/4 (2 − q)5/4 (η + q)[q(q + ηq − η)]1/4

σ = 1 4 q3/2(η + q)2(2 − q)1/2 (q + ηq − η)5/2

8 < : 1 − 3

8 Nf k − ...

→ q

k Nf

( Nf

k → ∞)

q Σ b 5 10 15 20 25 30 Ε

• 2

4

q

b

σ

Nf k

The ABJM flavored background

Vq¯

q = −Q

l ; → Q = 4 √ 2π3 Γ(1/4)4 √ λ σ

potential screening dilaton shifts

eφ = 4√π ✓2N k5 ◆1/4 (2 − q)5/4 (η + q)[q(q + ηq − η)]1/4

regime of validity

N 1/5 ⌧ Nf ⌧ N

σ = 1 4 q3/2(η + q)2(2 − q)1/2 (q + ηq − η)5/2

8 < : 1 − 3

8 Nf k − ...

→ q

k Nf

( Nf

k → ∞)

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

Probe Branes

D6-branes extended in AdS4 and wrapping RP3 ⊂ CP3

new cartesian-like coordinates

R

ρ

( x0, x1, x2, r | {z }

AdS4

, ξ, ˆ θ = 0, ˆ ψ, ˆ ϕ = 0 | {z }

S4

, θ(r) , ϕ | {z }

S2

)

L2 b2(ρ2 + R2)

• dρ2 + dR2

u = rb

R = u cos θ ; ρ = u sin θ

L2 ✓dr2 r2 + dθ2 b2 ◆

profile

R = R(ρ)

θ(r)

DBI+WZ action embedding

Probe Branes

S = SDBI + SW Z = TD6 ✓ − Z d7ζe−φp − det ˆ g7 + Z d7ζ ˆ C7 ◆

R(ρ)

∼ Z ρ dρ ρ(ρ2 + R2)

3 2b 1(

p 1 + R02 − 1)

R ∼ m + c r3−2b φ ∼ φ0 r∆−3 + O r∆

∆ = 3 − b

compare with

φ0 is the source of O ∆ → dimension of O asymptotic behavior

ρ → 0 ⇒ ∂ρ ⇣ ρ3/b∂ρR ⌘ = 0

Probe Branes

dim( ¯ ψψ) = 3 − b

In our case O ∼ ¯ ψψ

2 − 3 16 Nf k + 63 512 ✓Nf k ◆2 + ... , → 7 4 ✓Nf k → ∞ ◆

anomalous dimension

Probe Branes

dim( ¯ ψψ) = 3 − b

In our case O ∼ ¯ ψψ

2 − 3 16 Nf k + 63 512 ✓Nf k ◆2 + ... , → 7 4 ✓Nf k → ∞ ◆

anomalous dimension R = constant

SUSY solution

Depends on the gauge for C7!!

C7 → C7 + dΛ6 generates boundary conterterms C7 = e−φK → SUSY scheme

• n shell action

is automatically finite

S = 0

ρ → ∞ ⇒ S ∼ ρ2−3/b

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

The ABJM flavored thermal background

replace AdS by Schwarzschild-AdS

T = 3 rh 4π

entropy

ds2 = L2 ✓ −r2h(r)dt2 + dr2 r2h(r) + r2(dx2

1 + dx2 2)

◆ + L2 1 b2 q

4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 !

h(r) = ✓ 1 − r3

h

r3 ◆

blackening factor

sback = 2π κ2

10

A8 V2 = 1 3 √ 2 ✓4π 3 ◆2 √ k N 3/2 ξ ✓Nf k ◆ T 2

The ABJM flavored thermal background

replace AdS by Schwarzschild-AdS

T = 3 rh 4π

entropy

ds2 = L2 ✓ −r2h(r)dt2 + dr2 r2h(r) + r2(dx2

1 + dx2 2)

◆ + L2 1 b2 q

4

X

i=1

(Si)2 +

2

X

a=1

(Ea)2 !

h(r) = ✓ 1 − r3

h

r3 ◆

blackening factor

sback = 2π κ2

10

A8 V2 = 1 3 √ 2 ✓4π 3 ◆2 √ k N 3/2 ξ ✓Nf k ◆ T 2

free energy

Fback = EADM − Tsback = − 1

9 √ 2 ✓4π 3 ◆2 √ k N 3/2 ξ ✓Nf k ◆ T 3

EADM = − 1 κ2

10

p |Gtt| Z

t0,r∞

p det G8 (KT − K0)

energy density

unflavored term ∼ N

3 2

field theory match by Drukker et al. (1007.3837) !

The ABJM flavored thermal background

ξ Nf k

• ≡ 1

16 q

5 2 (η + q)4

(2 − q)

1 2 (q + ηq − η) 7 2

8 > > > < > > > : = 1 + 3 4 Nf k − 9 64 ✓Nf k ◆2 + . . . Nf → 0 ∼ 1.389 r Nf k Nf → ∞

Ξ q Σ b 5 10 15 20 25 30 Ε

• 2

4 6 8

Nf k

ξ ✓Nf k ◆ = 1 16 q5/2(η + q)4 √2 − q(q + ηq − η)7/2

Comparison with 3-Sasakian

(U(Nf), N = 3 flavors)

Gaiotto&Jafferis 0903.2175

Couso-Santamaria et al. 1011.6281

The ABJM flavored background

Localized solution in 11d for coincident massless flavors

AdS4 × M7 with M7 a hyperkahler 3-Sasakian manifold N = 3 with U(Nf) flavor symmetry

ξ3S ✓Nf k ◆ = 1 + Nf

k

q 1 + Nf

2k

10 20 30 40 Nf k 2 4 6 8 Ξ Nf k

TriSasakian Smeared

ξ ✓Nf k ◆ = 1 16 q5/2(η + q)4 √2 − q(q + ηq − η)7/2

Comparison with 3-Sasakian

(U(Nf), N = 3 flavors)

Gaiotto&Jafferis 0903.2175

Couso-Santamaria et al. 1011.6281

The ABJM flavored background

Localized solution in 11d for coincident massless flavors

AdS4 × M7 with M7 a hyperkahler 3-Sasakian manifold N = 3 with U(Nf) flavor symmetry

ξ3S ✓Nf k ◆ = 1 + Nf

k

q 1 + Nf

2k

• The ABJM theory
• The flavored ABJM background
• Probes on the flavored ABJM background
• The flavored thermal ABJM background
• Probes on the flavored thermal ABJM background
• Conclusions

the embeddings are governed by the DBI+WZ action

S = SDBI + SW Z = TD6 ✓ − Z d7ζe−φp − det ˆ g7 + Z d7ζ ˆ C7 ◆

Probes on the ABJM flavored thermal background

SDBI = N Z d3x Z ∞

rmin

dr r2 r3

h

sin θ r 1 + r2h b2 ˙ θ2

the embeddings are governed by the DBI+WZ action

S = SDBI + SW Z = TD6 ✓ − Z d7ζe−φp − det ˆ g7 + Z d7ζ ˆ C7 ◆

Probes on the ABJM flavored thermal background

N = 2 √ 2π2 27 N 3/2√ k T 3 ζ ✓Nf k ◆ SDBI = N Z d3x Z ∞

rmin

dr r2 r3

h

sin θ r 1 + r2h b2 ˙ θ2

the embeddings are governed by the DBI+WZ action

S = SDBI + SW Z = TD6 ✓ − Z d7ζe−φp − det ˆ g7 + Z d7ζ ˆ C7 ◆

Probes on the ABJM flavored thermal background

ζ ✓Nf k ◆ = 1 32 √2 − q(η + q)4q5/2 (q + ηq − η)9/2

→ 8 < : 1 − 3

8 Nf k + . . . (Nf → 0)

q

k Nf + . . .

(Nf → ∞)

N = 2 √ 2π2 27 N 3/2√ k T 3 ζ ✓Nf k ◆ SDBI = N Z d3x Z ∞

rmin

dr r2 r3

h

sin θ r 1 + r2h b2 ˙ θ2

SW Z = TD6 Z d7ζ ˆ C7 = TD6 Z d7ζ ⇣ e−φ ˆ K + δC7 ⌘

e−φ ˆ K = L7q b3 e−φd3x ∧ r3 b sin θ cos θdθ + r2 sin2 θdr

• ∧ Ξ3

must be improved to get a consistent thermodynamics

C7

represent the improvement term as follows

δC7 = L7q b3 e−φd3x ∧ h L1(θ)dθ + L2(r)dr i ∧ Ξ3 ⇒ d δC7 = 0

the angular part of must vanish at the horizon

C7

L1(θ) = −r3

h

b sin θ cos θ

Probes on the ABJM flavored thermal background

Jensen 1006.3066

define (a zero point energy)

Z drL2(r) = ∆0

the total action is now

∂L ∂ ˙ θ

• r=rh

= 0

and satisfies that the canonical momentum vanishes at the horizon in order to fix let us compare the free energy (density) of the probe and the background

∆0

F = T SE

F = − S

• d3x

Probes on the ABJM flavored thermal background

S = N Z d3x " − 4b r3

h

Z dr r2 sin θ r 1 + r2h(r) b2 ˙ θ2 − sin θ − rh(r) b cos θ ˙ θ ! + ∆0 #

N = 2 √ 2π2 27 N 3/2√ k T 3 ζ ✓Nf k ◆

a) infinite mass limit ⇒ decoupling Consistency check:

Probes on the ABJM flavored thermal background

∆0 = 1

lim

m→∞ F = N (1 − ∆0) = 0

a) infinite mass limit ⇒ decoupling b) zero mas limit ⇒ add to backreaction

Fback ✓Nf k ◆ + Fprobe(m = 0) = Fback ✓Nf + 1 k ◆ = Fback ✓Nf k ◆ + 1 k F 0

back + ...

hence Consistency check:

Probes on the ABJM flavored thermal background

∆0 = 1

lim

m→∞ F = N (1 − ∆0) = 0

a) infinite mass limit ⇒ decoupling b) zero mas limit ⇒ add to backreaction

Fback ✓Nf k ◆ + Fprobe(m = 0) = Fback ✓Nf + 1 k ◆ = Fback ✓Nf k ◆ + 1 k F 0

back + ...

hence Consistency check:

Probes on the ABJM flavored thermal background

∆0 = 1

lim

m→∞ F = N (1 − ∆0) = 0

1 √ 2 ✓4π 9 ◆2 N 3/2 √ k T 3 3 4ζ ✓Nf k ◆ ∆0 = 1 √ 2 ✓4π 9 ◆2 N 3/2√ k T 3 1 k ξ0 ✓Nf k ◆

a) infinite mass limit ⇒ decoupling b) zero mas limit ⇒ add to backreaction

Fback ✓Nf k ◆ + Fprobe(m = 0) = Fback ✓Nf + 1 k ◆ = Fback ✓Nf k ◆ + 1 k F 0

back + ...

hence Consistency check:

Probes on the ABJM flavored thermal background

∆0 = 1

lim

m→∞ F = N (1 − ∆0) = 0

1 √ 2 ✓4π 9 ◆2 N 3/2 √ k T 3 3 4ζ ✓Nf k ◆ ∆0 = 1 √ 2 ✓4π 9 ◆2 N 3/2√ k T 3 1 k ξ0 ✓Nf k ◆

a) infinite mass limit ⇒ decoupling b) zero mas limit ⇒ add to backreaction

Fback ✓Nf k ◆ + Fprobe(m = 0) = Fback ✓Nf + 1 k ◆ = Fback ✓Nf k ◆ + 1 k F 0

back + ...

hence Consistency check:

Probes on the ABJM flavored thermal background

∆0 = 1

lim

m→∞ F = N (1 − ∆0) = 0

1 √ 2 ✓4π 9 ◆2 N 3/2 √ k T 3 3 4ζ ✓Nf k ◆ ∆0 = 1 √ 2 ✓4π 9 ◆2 N 3/2√ k T 3 1 k ξ0 ✓Nf k ◆

∆0 = 1

ξ ✓Nf k ◆ = 1 16 q5/2(η + q)4 √2 − q(q + ηq − η)7/2

ζ ✓Nf k ◆ = 1 64 √2 − q(η + q)4q5/2 (q + ηq − η)9/2

3 4ζ = ξ0

the same mechanism for the entropy yields stotal = sback + s ≈ 1 3 4π 3 2 N 2 √ 2λ ξ Nf + 1 k

• T 2 ,

(m → 0) hence, massless probe entropy ≈ increase in area of the horizon Now that the action is completely fixed we may derive the correct equations of motion and the solutions, as well as the thermodynamics

Probes on the ABJM flavored thermal background

S = N Z d3x " − 4b r3

h

Z dr r2 sin θ r 1 + r2h(r) b2 ˙ θ2 − sin θ − rh(r) b cos θ ˙ θ ! + 1 #

1 2 3 4 Ρ 0.0 0.5 1.0 1.5 2.0 2.5 R

embeddings

R = u cos θ ; ρ = u sin θ

Black hole embeddings

Probes on the ABJM flavored thermal background

isotropic coordinates

dr2 r2h(r) + dθ2 b2 = 1 u2b2

• du2 + u2dθ2

u = 2 4 ✓ r rh ◆ 3

2

+ ✓ r rh ◆3 − 1 ! 1

2 3

5

2b 3

Minkowski embeddings R(ρ) ∼    R0 + ... (ρ → 0) m + c ρ3/b−2 + ... (ρ → ∞) χ(u) = cos θ(u) ∼    χh + ... (u → uh) m u + c u3/b−1 + ... (u → ∞)

χh

R0

Om = −2

2 3 π

9 (3 − 2b) b q σ N T 2 c

Probes on the ABJM flavored thermal background

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 m 1.5 1.0 0.5 0.0 c

Numerical c = c(m)

Mq = 21/3π 3 √ 2λ σ T m1/b

mass and condensate

D7 massive probes: condensate

0, 0.2 , 0.4

✏h =

0.5 1.0 1.5 2.0 m 0.05 0.10 0.15 ΨΨ

0.89 0.90 0.91 0.92 0.93 m Mq

1 2

Λ T 0.01 0.02 0.03 0.04 0.05 0.06 0.07 ΨΨ

meson melting phase transition temperature increases

Probes on the ABJM flavored thermal background

√ λ Tc Mq ∼ 1 m1/b

c

σ ∼ p Nf

5 10 15 20 25 Ε

• 0.0

0.5 1.0 1.5 2.0 Λ Tc mq

√ λ Tc Mq

Nf k

Probes on the ABJM flavored thermal background

self-similar behavior c and m oscillate

critical embedding

1.8 2.0 2.2 2.4 2.6 2.8 m 1.5 1.0 0.5 c

χh

R0

χh

R0

Mateos, Myers & Thomson hep-th/0701132

Probes on the ABJM flavored thermal background

self-similar behavior c and m oscillate

critical embedding

1.8 2.0 2.2 2.4 2.6 2.8 m 1.5 1.0 0.5 c

m∗

m∗

χh

R0

χh

R0

Mateos, Myers & Thomson hep-th/0701132

Probes on the ABJM flavored thermal background

2.5 2.0 1.5 1.0 0.5 0.5 1.0 7 4 Π logR01 2 1 1 2 cc R0 132 2.5 2.0 1.5 1.0 0.5 0.5 1.0 7 4 Π logR01 2 1 1 2 mm R0 132

1.5 1.0 0.5 7 8 Π log1Χh 3 2 1 1 2 3 cc 1 Χh34

1.5 1.0 0.5 7 8 Π log1Χh 3 2 1 1 2 3 mm 1 Χh34

free energy density

0.5 1.0 1.5 2.0 2.5 3.0 1 m 1.0 0.8 0.6 0.4 0.2 0.0 F

• Probes on the ABJM flavored thermal background

= N ⇣ G(m) − 1 ⌘

F = SE R d3x = N Z d3x " 4b r3

h

Z ∞

rmin(m)

dr r2 sin θ r 1 + r2h(r) b2 ˙ θ2 − sin θ − rh(r) b cos θ ˙ θ ! − 1 # = 2 √ 2π2 27 N √ λ T 3 ! ζ ✓Nf k ◆ ⇣ G(m) − 1 ⌘ Nf k = 0 Nf k = 10

0.3 0.4 0.5 0.6 0.7 0.8 1 m 0.2 0.1 0.0 0.1 0.2 F

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T s

• entropy

s = −∂F ∂T = − F N ∂N ∂T − N ∂ ∂T ✓ F N ◆ = − 3 T F N − m T (3 − 2b)c

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 E

• internal energy

E = −2F − N(3 − 2b) c m

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 m 0.0 0.5 1.0 1.5 2.0 2.5 3.0 E

• internal energy

E = −2F − N(3 − 2b) c m

0.0 0.2 0.4 0.6 0.8 Λ T mq 0.25 0.20 0.15 0.10 0.05 ∆vs2 Λ N

speed of sound

v2

s = ∂P

∂E = s cv = 1 2 + δv2

s

Conclusions

• the flavored ABJM theory dual is a conformal field theory
• the thermal deformation is analytic and fully under control.
• we have added massive probe flavors to the theory and examined

the thermodynamics

• the scheme dependence can be fixed by demanding a compatibility
• f the UV and IR behavior of the probe brane
• The flavors introduce cuantitative shifts but no cualitative change in

the picture. For example rises like p

Nf

Tc

Further work

• add chemical potential to the probe brane and study transport

properties (conductivity etc.)

• constructing the flavored thermal ABJM theory with chemical

potential (dilaton stops being constant, and H3 enters the game)

• adding B field, could study magnetic catalysis
• smearing massive flavors in ABJM at zero T
• ....

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