• The Sakai-Sugimoto model

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Andreas Schmitt

Institut f¨ur Theoretische Physik Technische Universit¨at Wien 1040 Vienna, Austria

QCD at finite temperature and density from holography

• The Sakai-Sugimoto model

• T -µ phase diagram (and comparison to large-N _c QCD)

• Effects of magnetic fields:

“chiral spirals”, “chiral magnetic effect”

A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009); JHEP 1001, 026 (2010)

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• The gauge/gravity duality (page 1/2): basic idea

J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998)

For a pedagogical review, see S. S. Gubser, A. Karch, Ann. Rev. Nucl. Part. Sci. 59, 145 (2009)

string theory

(in higher dimensions) ⇔ gauge theory

(on boundary)

original “AdS/CFT correspondence”:

string theory on AdS ₅ × S ⁵ ⇔ N = 4 SU (N _c ) SYM theory on R ^3,1 R ⁴

ℓ ⁴ _s = g _YM ² N _c ≡ λ

R curvature radius; ℓs string length

ℓ _s ≪ R

supergravity limit ( easy! )

⇔

λ ≫ 1

strong coupling limit

( difficult! )

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• The gauge/gravity duality (page 2/2): overview

• QCD

– compare with N = 4 SYM

compute plasma properties ( → heavy-ion collisions)

∗

^viscosity G. Policastro, D. T. Son, A. O. Starinets, PRL 87, 081601 (2001)

∗

jet quenching H. Liu, K. Rajagopal, U. A. Wiedemann, PRL 97, 182301 (2006)

∗

expanding plasma R. A. Janik, R. B. Peschanski, PRD 73, 045013 (2006)

– find gravity dual of QCD

∗

add flavor to AdS/CFT A. Karch, E. Katz, JHEP 0206, 043 (2002)

∗

Sakai-Sugimoto model T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005)

∗

bottom-up approach Erlich, Katz, Son, Stephanov, PRL 95, 261602 (2005)

• Other applications: cold atoms (unitary Fermi gas),

D. T. Son, PRD 78, 046003 (2008); K. Balasubramanian, J. McGreevy, PRL 101, 061601 (2008)

(high-T _c ) superconductivity, etc ...

S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, PRL 101, 031601 (2008)

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• The Sakai-Sugimoto model in two steps

1. Background geometry with D4-branes

E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)

M. Kruczenski, D. Mateos, R. C. Myers, D. J. Winters, JHEP 0405, 041 (2004)

2. Add flavor D8-branes

T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005)

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• The Sakai-Sugimoto model:

background geometry (page 1/3)

E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998)

M. Kruczenski, D. Mateos, R. C. Myers, D. J. Winters, JHEP 0405, 041 (2004)

N _c D4-branes compactified on circle x ₄ ≡ x ₄ + 2π/M _KK

N D4−branes

c

• 4-4 strings → adjoint scalars & fermions, gauge fields

• periodic x ₄ → break SUSY by giving mass

∼ M _KK to scalars & fermions

⇒ SU (N _c ) gauge theory

λ = g ₅ ² N _c 2π/M _KK

λ ≪ 1 λ ≫ 1 dual to large-N _c QCD √

x (at energies ≪ M _KK )

supergravity works x √

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• Background geometry (page 2/3): two solutions Confined phase

ds²_conf = u R

3/2

[dτ² +dx² +f(u)dx²₄] +

R

u

3/2

du²

f(u) + u²dΩ²₄

u u= 8

u= u_KK 1/M_KK x₄

1/(2 T)π τ

MKK = 3 2

u^1/2_KK

R³^/² f(u) ≡ 1− u³_KK u³

Wick rotated regular geometry

Deconfined phase

ds²_deconf = u R

3/2

[f˜(u)dτ² +δ_ijdx² + dx²₄] +

R

u

³/2

du²

f˜(u) +u²dΩ²₄

u u= 8

u= u 1/M_KK x₄

T

1/(2 T)

τ π

T = 3 4π

u¹_T^/²

R^3/2 f˜(u) ≡ 1 − u³_T u³

Wick rotated black brane

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• Background geometry (page 3/3):

deconfinement phase transition

x₄ x₄ x₄

deconfined confined

temperature

τ τ

τ

T _c = M _KK

2π

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• Add flavor (page 1/2)

T. Sakai, S. Sugimoto, Prog. Theor. Phys. 113, 843 (2005)

• add N _f D8- and D8-branes, separated in x ₄

0 1 2 3 4 5 6 7 8 9

D4 x x x x x

D8/D8 x x x x x x x x x

D8 D8

D4

L

x x

x₄

0−3 5−9

• 4-8, 4-¯8 strings

→ fundamental, massless chiral fermions

under U (N _f ) _L × U (N _f ) _R

⇒ quarks & gluons

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• Add flavor (page 2/2): Chiral symmetry breaking

• background geometry unchanged if N _f ≪ N _c (“probe branes”)

→ “quenched” approximation

• gauge symmetry on the branes → global symmetry at u = ∞

SU(N )

_{f R}

SU(N )

f _L

L x

₄

D8

D8 L x

₄

SU(N )

_{f L+R}

u

• chiral symmetry breaking

SU (N _f ) _L × SU (N _f ) _R → SU (N _f ) _L+R

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• Conjectured QCD phase diagram in T -µ plane

“real” QCD

N. Cabibbo, G. Parisi, PLB 59, 67 (1975)

large-N _c QCD

L. McLerran, R. D. Pisarski, NPA 796, 83 (2007) L. McLerran, K. Redlich, C. Sasaki,

NPA 824, 86 (2009)

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• More (conjectured) details

liq

T

µ

gas

QGP

CFL

nuclear superfluid

heavy ion collider

neutron star

non−CFL hadronic

M. G. Alford, A. Schmitt, K. Rajagopal, T. Sch¨afer, RMP 80, 1455 (2008)

Physical systems:

heavy-ion collisions (intermediate T , small µ)

neutron stars

(small T , intermediate µ)

• Challenging (strongly coupled!) regions at intermediate T , µ

• Try large-N _c (Sakai-Sugimoto) approach

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• T -µ phase diagram (page 1/4)

• chemical potential introduced through gauge fields on D8 branes

• confinement ⇔ chiral symmetry breaking

Tc

T

µ

deconfined

χ S restored

confined χ S broken

• M _KK = 949 MeV (fit to ρ mass) ⇒ T _c ≃ 150 MeV

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• T -µ phase diagram (page 2/4)

• deconfined, chirally broken phase for L < 0.3 π/M _KK

O. Aharony, J. Sonnenschein, S. Yankielowicz, Annals Phys. 322, 1420 (2007) N. Horigome, Y. Tanii, JHEP 0701, 072 (2007)

deconfined S broken χ

T

µ

Tc

deconfined χS restored

confined χ S broken

L

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• T -µ phase diagram (page 3/4)

• “NJL limit” L ≪ π/M _KK

confined χS broken

deconfined S broken χ

T

µ

Tc

χ S restored deconfined

• differences to “QCD limit”: crystalline phases vs. “chiral spiral”?

D. Nickel, PRD 80, 074025 (2009)

T. Kojo, Y. Hidaka, L. McLerran, R. D. Pisarski, arXiv:0912.3800 [hep-ph]

F. Preis, A. Rebhan, A. Schmitt, work in progress

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• T -µ phase diagram (page 4/4)

Holographic nuclear matter:

T

µ

Tc

deconfined χS restored

n = 0

_B

source 4−brane

n = 0

_B

• baryons introduced as instantons

H. Hata, T. Sakai, S. Sugimoto, S. Yamato, Prog. Theor. Phys. 117, 1157 (2007)

• homogeneous baryon number from 4-branes wrapped on S ⁴

O. Bergman, G. Lifschytz, M. Lippert, JHEP 0711, 056 (2007)

• same diagram from large-N _c QCD, “quarkyonic matter”

L. McLerran, R. D. Pisarski, Nucl. Phys. A 796, 83 (2007)

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• Supercurrents in chirally broken phase (page 1/2)

A. Rebhan, A. Schmitt, S.A. Stricker, JHEP 0905, 084 (2009)

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

→ anisotropic (or even crystalline) “pairing”

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

• solution to the EOMs on D8-branes with

A ₁ (z = ±∞ ) = − x ₂ B:

B=0

Μ

ÑΠ⁰

-40 -20 0 20 40

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

zu_KK

gaugefields

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• Supercurrents in chirally broken phase (page 1/2)

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

→ anisotropic (or even crystalline) “pairing”

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

• solution to the EOMs on D8-branes with

A ₁ (z = ±∞ ) = − x ₂ B:

B=0.5

Μ

ÑΠ⁰

-40 -20 0 20 40

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

zu_KK

gaugefields

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• Supercurrents in chirally broken phase (page 1/2)

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

→ anisotropic (or even crystalline) “pairing”

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

• solution to the EOMs on D8-branes with

A ₁ (z = ±∞ ) = − x ₂ B:

B=1

Μ

ÑΠ⁰

-40 -20 0 20 40

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

zu_KK

gaugefields

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• Supercurrents in chirally broken phase (page 1/2)

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

→ anisotropic (or even crystalline) “pairing”

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

• solution to the EOMs on D8-branes with

A ₁ (z = ±∞ ) = − x ₂ B:

B=3

Μ

ÑΠ⁰

-40 -20 0 20 40

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

zu_KK

gaugefields

(20)

• Supercurrents in chirally broken phase (page 2/2)

• usual U (1) superfluid (e.g. ⁴ He): superflow via φ(x) = φ ₀ e ^iqz

superflow in z direction with velocity ∝ q (rotation in U(1) space)

• Sakai-Sugimoto: σ ∇ π ⁰ − π ⁰ ∇ σ ∝ µ (axial) supercurrent (rotation in chiral space)

• “chiral spiral” σ − iπ ⁰ ∝ e ^2iµz (rotation in chiral space)

G. Basar, G. V. Dunne, M. Thies, PRD 79, 105012 (2009)

• “quarkyonic chiral spiral” (rotation in spin space)

T. Kojo, Y. Hidaka, L. McLerran, R. D. Pisarski, arXiv:0912.3800 [hep-ph]

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• The Chiral Magnetic Effect (CME)

q

_L

q

_R

q

_R

momentum spin

B ⇒ electric current

parallel to B

J = e ² N _c

2π ² µ ₅ B

A.Y. Alekseev, V.V. Cheianov, J. Fr¨ohlich, PRL 81, 3503 (1998) K.Fukushima, D.E.Kharzeev, H.J.Warringa, PRD 78, 074033 (2008)

• analogously for axial current:

M.A. Metlitski, A.R. Zhitnitsky, PRD 72, 045011 (2005)

J ₅ = e ² N _c

2π ² µB

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• CME in heavy-ion collisions

+

−

B

• non-central collisions: eB ∼ 10 ¹⁷ G

V. Skokov et al., Int. J. Mod. Phys. A24, 5925 (2009)

• N ₅ from QCD axial anomaly dN ₅

dt = − g ² 16π ²

Z

d ³ x F _µν ^a F ˜ _a ^µν (large T : sphalerons)

• simplified description with µ ₅ : take “snapshot” with conserved N ₅

• “event-by-event” charge separation

B. I. Abelev et al. [STAR Collaboration], PRL 103, 251601 (2009)

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• CME in the Sakai-Sugimoto model

• maximal separation L = π/MKK

• one flavor N_f = 1

• holographic coordinate u → z ∈ [−∞,∞] (broken) z ∈ [0,∞] (symmetric)

• gauge choice A_z = 0

S = S _YM + S _CS

SYM = κM_KK² Z

d⁴x Z _∞

−∞

dz

k(z)F_zµF^zµ + h(z)

2M_KK² F_µνF^µν

SCS = N_c 24π²

Z

d⁴x Z _∞

−∞

dz A_µF_zνF_ρσǫ^µνρσ

k(z) ≡ 1 +z² κ ≡ λN_c

216π³

equations of motion δ ( L YM + L CS )

δA _µ = 0

boundary conditions

A ₀ (z = ±∞ ) = µ _L/R

A ₁ (x ₂ , z = ±∞ ) = − x ₂ B

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• Chiral currents

• YM and CS contributions to chiral currents

H. Hata, M. Murata, S. Yamato, PRD 78, 086006 (2008)

K. Hashimoto, T. Sakai, S. Sugimoto, Prog. Theor. Phys. 120, 1093 (2008)

S = S _YM + S _CS

⇒ J _L/R ^µ ≡ − δS

δA _µ (x, z = ±∞ ) = J _L/R,YM ^µ + J _L/R,CS ^µ

• sometimes CS contribution is ignored

H. U. Yee, JHEP 0911, 085 (2009)

D. T. Son, P. Surowka, PRL 103, 191601 (2009)

only YM part in asymptotics of A _µ :

A ^µ (x, z) = A ^µ (x, z = ±∞ ) ± J _L/R,YM ^µ 2κM _KK ²

1 z + O

1 z ²

.

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• Anomalies

• axial and vector anomalies [

^F^µν^L/R^(x) ≡ Fµν(x, z = ±∞)

]

∂ _µ J ₅ ^µ = N _c 24π ²

F _µν ^V F e _V ^µν + F _µν ^A F e _A ^µν

∂ _µ J ^µ = N _c

12π ² F _µν ^V F e _A ^µν

“consistent” anomaly

W.A. Bardeen, Phys. Rev. 184, 1848 (1969); C.T. Hill, PRD 73, 085001 (2006)

• need Bardeen’s counterterm

∆S = c Z

d ⁴ x(A ^L _µ A ^R _ν F _ρσ ^L + A ^L _µ A ^R _ν F _ρσ ^R )ǫ ^µνρσ (here interpreted as holographic renormalization)

→ determine c to get QED (“covariant”) anomaly

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• Correct anomaly with Bardeen’s counterterm

• Bardeen’s counterterm ∆S → new chiral currents J ¯ _L/R ^µ ≡ J _L/R,YM ^µ + J _L/R,CS ^µ + ∆ J _L/R ^µ

⇒

∂ _µ J ¯ ₅ ^µ = N _c

8π ² F _µν ^V F e _V ^µν + N _c

24π ² F _µν ^A F e _A ^µν

∂ _µ J ¯ ^µ = 0

“covariant” anomaly

J.S. Bell, R.Jackiw, Nuovo Cim. A60, 47 (1969); S.L. Adler, Phys. Rev. 177, 2426 (1969)

• conservation of vector current

& correct decay rate π ⁰ → 2γ

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• Use YM currents?

• notice: YM part alone gives

∂ _µ J _YM,5 ^µ = N _c 8π ²

F _µν ^V F e _V ^µν + F _µν ^A F e _A ^µν

∂ _µ J _YM ^µ = N _c

4π ² F _µν ^V F e _A ^µν

• seems OK for F _A = 0

• J _YM ^µ 6 = ¯ J ^µ even if F _A = 0

• J _YM ^µ not strictly conserved (need ∇ µ ₅ for charge separation

at RHIC!)

^µ⁵

µ₅= 0

= 0

B

−

+++

−

µ₅= 0

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• Results for currents

absence of CME

axial current

as expected

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• Possible problems in current approach – Need to consider conserved N ₅ ?

V. A. Rubakov, arXiv:1005.1888 [hep-ph]

A.Y. Alekseev, V.V. Cheianov, J. Fr¨ohlich, PRL 81, 3503 (1998)

– Careful distinction of chemical potential and source for currents

A. Gynther, K. Landsteiner, F. Pena-Benitez, A. Rebhan, arXiv:1005.2587 [hep-th]

– Need to work in canonical ensemble?

H.U. Yee, Talk at BNL Workshop, April 26 - 30, 2010 A. Rebhan, A. Schmitt, S.A. Stricker, work in progress

• Or take result seriously ...

– strong-coupling vector current differs from weak-coupling current (unlike axial current) – similar to NJL-like models

E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, PRC 80, 032801 (2009) K. Fukushima, M. Ruggieri, arXiv:1004.2769 [hep-ph]

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• Summary

• the Sakai-Sugimoto model provides a ”top-down” gauge/gravity duality for a non-supersymmetric, strongly-coupled SU (N _c ) gauge theory with confinement and chiral symmetry breaking ( → large-N _c QCD)

• for finite magnetic field and chemical potential, the model exhibits supercurrents (”chiral spirals”?) and finite baryon number in the chirally broken phase

• The Sakai-Sugimoto model

QCD at finite temperature and density from holography

• The Sakai-Sugimoto model

• T -µ phase diagram (and comparison to large-N c QCD)

• Effects of magnetic fields:

“chiral spirals”, “chiral magnetic effect”

• The gauge/gravity duality (page 1/2): basic idea

string theory

(in higher dimensions) ⇔ gauge theory

(on boundary)

original “AdS/CFT correspondence”:

string theory on AdS 5 × S 5 ⇔ N = 4 SU (N c ) SYM theory on R 3,1 R 4

ℓ 4 s = g YM 2 N c ≡ λ

ℓ s ≪ R

supergravity limit ( easy! )

⇔

λ ≫ 1

strong coupling limit

( difficult! )

• The gauge/gravity duality (page 2/2): overview

• QCD

– compare with N = 4 SYM

compute plasma properties ( → heavy-ion collisions)

∗

∗

∗

– find gravity dual of QCD

∗

∗

∗

• Other applications: cold atoms (unitary Fermi gas),

(high-T c ) superconductivity, etc ...

• The Sakai-Sugimoto model in two steps

1. Background geometry with D4-branes

2. Add flavor D8-branes

• The Sakai-Sugimoto model:

background geometry (page 1/3)

N c D4-branes compactified on circle x 4 ≡ x 4 + 2π/M KK

• 4-4 strings → adjoint scalars & fermions, gauge fields

• periodic x 4 → break SUSY by giving mass

∼ M KK to scalars & fermions

⇒ SU (N c ) gauge theory

λ = g 5 2 N c 2π/M KK

λ ≪ 1 λ ≫ 1 dual to large-N c QCD √

x (at energies ≪ M KK )

supergravity works x √

• Background geometry (page 2/3): two solutions Confined phase

Wick rotated regular geometry

Deconfined phase

Wick rotated black brane

• Background geometry (page 3/3):

deconfinement phase transition

T c = M KK

2π

• Add flavor (page 1/2)

• add N f D8- and D8-branes, separated in x 4

0 1 2 3 4 5 6 7 8 9

D4 x x x x x

D8/D8 x x x x x x x x x

• 4-8, 4-¯8 strings

→ fundamental, massless chiral fermions

under U (N f ) L × U (N f ) R

⇒ quarks & gluons

• Add flavor (page 2/2): Chiral symmetry breaking

• background geometry unchanged if N f ≪ N c (“probe branes”)

→ “quenched” approximation

• gauge symmetry on the branes → global symmetry at u = ∞

SU(N )

SU(N )

L x

D8

D8 L x

SU(N )

u

• chiral symmetry breaking

SU (N f ) L × SU (N f ) R → SU (N f ) L+R

• Conjectured QCD phase diagram in T -µ plane

“real” QCD

large-N c QCD

• More (conjectured) details

• T -µ phase diagram (and comparison to large-N _c QCD)

string theory on AdS ₅ × S ⁵ ⇔ N = 4 SU (N _c ) SYM theory on R ^3,1 R ⁴

ℓ ⁴ _s = g _YM ² N _c ≡ λ

ℓ _s ≪ R

(high-T _c ) superconductivity, etc ...

N _c D4-branes compactified on circle x ₄ ≡ x ₄ + 2π/M _KK

• periodic x ₄ → break SUSY by giving mass

∼ M _KK to scalars & fermions

⇒ SU (N _c ) gauge theory

λ = g ₅ ² N _c 2π/M _KK

λ ≪ 1 λ ≫ 1 dual to large-N _c QCD √

x (at energies ≪ M _KK )

T _c = M _KK

• add N _f D8- and D8-branes, separated in x ₄

under U (N _f ) _L × U (N _f ) _R

• background geometry unchanged if N _f ≪ N _c (“probe branes”)

SU (N _f ) _L × SU (N _f ) _R → SU (N _f ) _L+R

large-N _c QCD

• Try large-N _c (Sakai-Sugimoto) approach

• M _KK = 949 MeV (fit to ρ mass) ⇒ T _c ≃ 150 MeV

• deconfined, chirally broken phase for L < 0.3 π/M _KK

• “NJL limit” L ≪ π/M _KK

• homogeneous baryon number from 4-branes wrapped on S ⁴

• same diagram from large-N _c QCD, “quarkyonic matter”

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

A ₁ (z = ±∞ ) = − x ₂ B:

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

A ₁ (z = ±∞ ) = − x ₂ B:

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

A ₁ (z = ±∞ ) = − x ₂ B:

• “Stress” on h ψ ¯ _R ψ _L i through chemical potential µ

• Sakai-Sugimoto: magnetic field B needed to create σ ∇ π ⁰ − π ⁰ ∇ σ = µf (B ) ∝ µ (for large B )

A ₁ (z = ±∞ ) = − x ₂ B:

• usual U (1) superfluid (e.g. ⁴ He): superflow via φ(x) = φ ₀ e ^iqz

• Sakai-Sugimoto: σ ∇ π ⁰ − π ⁰ ∇ σ ∝ µ (axial) supercurrent (rotation in chiral space)

• “chiral spiral” σ − iπ ⁰ ∝ e ^2iµz (rotation in chiral space)

J = e ² N _c

2π ² µ ₅ B