The Stability of Non-Extremal Conifold Backgrounds with Sources

The Stability of Non-Extremal Conifold Backgroundswith Sources

Steve Young

The University of Texas at Austin

July 16, 2012

Steve Young (UT Austin) Non-Extremal Conifold Stability July 16, 2012 1 / 36

Outline

Outline

1 Intro: AdS/CFT and strongly coupled plasmas with fundamental flavor

2 U-duality as a solution generator: wrapped D5s → KS baryonic branch

3 Adding flavor: the CNP family

4 Non-extremal flavored backgroundsDeforming the flavored CNP familyRotating the solutions: non-extremal backgrounds with KS andFWRDC asymptoticsTemperature

5 Thermodynamics and stabilityThe ADM EnergyFirst LawSpecific Heat

6 Summary and Conclusions


Intro

Outline








Intro

Intro• In last decade AdS/CFT generalized:N = 4 SYM → less symmetric models.

• Real world application: Finite temperature QGP (RHIC/LHC)

• Simplest model: finite temperature N = 4 SYM

• T=0: QCD 6= N = 4 SYM

QCD N = 4 SYMconformal no yesconfining yes no

fundamentals yes noχral symmetry yes no

SUSY no yes

• T > 140 MeV: QCD ∼ N = 4 SYM

QCD N = 4 SYMconformal no T sets scaleconfining no no

fundamentals yes noχral symmetry no no

SUSY no no

• How can we do better?1 Finite temp deconfinement and χSB transitions

2 Add fundamentals


Intro

Intro: 1) Modeling deconfinement and χSB at finite temp

• Try to model finite temp deconfinement and χSB transitions

• Example: Duals to N = 1 theories (Klebanov-Strassler (KS)) atfinite temp (Buchel, Aharony, etc.)

• Caveat: black holes in KS may be unstable

• Other N = 1 duals (Maldacena-Nunez (MN)) known to be unstableat finite temp in deconfined, χ-symmetric phase (Gubser et al. ’01)→ dual to finite temp Little String Theory

• How can we get stable black hole backgrounds?


Intro

Intro: A stable non-extremal KS baryonic branch?

• Quick backtrack to zero temperature...

• MN background part of family of solutions describing D5 branesspanning R1,3 and wrapped on S2 of resolved conifold.

• (Maldacena et al. ’09):MN wrapped D5 family

U-duality⇐⇒ family dual to KS baryonic branch

• U-duality more general:• (Caceres et al. ’11) applied U-duality to non-extremal deformation of

MN family, produced non-extremal background with KS asymptotics

• Idea:

Non-extremal D5 familyU-duality⇐⇒

?stable non-extremal baryonic branch


Intro

Intro: 2) Modeling dynamical flavor

• QCD: Nc = Nf = 3. Nc >> 1 in AdS/CFT, could hope for Nf ∼ Nc

• Unquenched flavor (Nf ∼ Nc):• gluon plasma quark-gluon plasma

• (Field theory) Quarks can propagate in loops, affect β-fns, etc.

• (Gravity) Add flavor branes which backreact on geometry

• Veneziano scaling: Nc →∞, Nf →∞, Nf /Nc ∼ 1 fixed

• Seminal Nf ∼ Nc model: (Casero et al. ’06) (CNP)• Adds backreacted smeared flavor branes to MN background

• Dual in IR to N = 1 SQCD plus quartic operator (QQ)2

• (Gaillard et al. ’10):

(T = 0) CNP backgroundsU-duality

=⇒ flavored KS baryonic brancha.k.a. ‘flavored warped resolved deformed conifold’ (FWRDC).


Intro

Intro: Goal and Summary

• Goal: Combine flavored and non-extremal modifications, useU-duality to build dual to finite temp flavored KS baryonic branch→ model confinement, χSB, unquenched flavor at finite temperature

• To this end:• We construct new non-extremal flavored CNP backgrounds, use

U-duality to get non-extremal backgrounds with FWRDC asymptotics

• Also construct new unflavored non-extremal backgrounds, and relatednon-extremal deformations of the KS baryonic branch

• Results: Backgrounds are generically unstable→ U-duality procedure good for generating non-extremal decoupledSUGRA backgrounds, but doesn’t guarantee thermodynamic stability→ Reinforces doubt in stability of KS black holes


U-duality: Wrapped D5s→ KS baryonic branch

Outline









The Maldacena-Nunez and Klebanov-Strassler backgrounds

• MN wrapped D5 familyU-duality⇐⇒ family dual to KS baryonic branch

• Maldacena-Nunez (one point in wrapped D5 family):• Near-horizon geometry of D5 branes wrapped on S2 of resolved conifold

• In IR: N = 1 SYM coupled to KK tower of chiral and vector multiplets.

• KS baryonic branch:• Near-horizon geometry of N D3, M fractional D3 branes at tip of

deformed conifold, N = (kM | k ∈ Z+)

• Cascading N = 1 SU(N + M)× SU(N) quiver with twobifundamentals (two antibifund) and quartic superpotential.

• In IR, N = 1 SU(M) SYM with baryonic VEV from broken U(1)B



Two one-parameter families

• MN wrapped D5 familyU-duality⇐⇒ family dual to KS baryonic branch

• Baryonic VEV U parametrizes dual SUGRA family — KS is one point(Butti et al. ’04)• F3,H3,F5 fluxes and dilaton — dilaton constant in UV

• All backgrounds dual to field theories (decoupling limit)

• MN wrapped D5 family:• Only F3 flux and dilaton — dilaton constant in UV

• All backgrounds except MN are dual to field theories UV coupled togravity . MN’s UV completion is Little String Theory



U-duality procedure

• U-duality procedure: T-dualities, M-theory lift, boost by β,reduce to IIA, T-dualities, near brane limit

• Near brane limit: β →∞, rescale Minkowski coordinates

• Result: Wrapped D5 family (UV = gravity)→ KS baryonic branch (Field theory)

Final presentation Page 1

Figure: The U-duality procedure



U-duality procedure

• U-duality valid starting with any IIB background→ solution generating mechanism

• UV constant dilaton produces KS asymptotics after rotation

• Rotated background: dilaton, F5,F3,H3 in decoupling limit

Final presentation Page 1

Figure: The U-duality procedure


Adding flavor: the CNP family

Outline









An N = 1 SQCD dual

• (Casero et al. ’06) added Nf ∼ Nc backreacting flavor D5 branesources to MN solution → CNP backgrounds

• Space has topology R1,3 × Rρ × S2θ,ϕ × S3

θ,ϕ,ψ

• Flavor D5s span R1,3 and wrap two-cycle Σ2 along (ψ, ρ)

• Flavor branes in bulk realize global U(Nf ) in field theory

• Flavor D5s “smeared” across (θ, ϕ, θ, ϕ)→ Functions in ansatz only depend on ρ



Metric and F3 ansatze

• Metric and F3 have form (α′ = gs = 1,Nc absorbed into e2k , e2g , e2h):

ds210 = eφ(ρ)/2

[dx2

1,3 + e2k(ρ)dρ2 + e2h(ρ)

S2︷︸︸︷(dθ2 + sin2 θdϕ2) +

+e2g(ρ)

4

((ω1 + a(ρ)dθ)2 + (ω2 − a(ρ) sin θdϕ)2

)+

e2k(ρ)

4(ω3 + cos θdϕ)2

]

F3 =Nc

4

[− (ω1 + b(ρ)dθ) ∧ (ω2 − b(ρ) sin θdϕ) ∧ (ω3 + cos θdϕ)

+b′(ρ)dρ ∧ (−dθ ∧ ω1 + sin θdϕ ∧ ω2) + (1− b(ρ)2) sin θdθ ∧ dϕ ∧ ω3

]• ω1 ∧ ω2 ∧ ω3: volume form on S3

ω1 = cosψd θ + sinψ sin θdϕ, ω2 = − sinψd θ + cosψ sin θdϕ, ω3 = dψ + cos θdϕ



Action

• Action for background and smeared sources, S = SIIB + Ssources:

SIIB =1

2κ210

∫d10x

√|g10|

(R − 1

2(∂µφ)(∂µφ)− 1

12eφF 2

3

)

Ssources =T5Nf

(4π)2

(−∫

d10x sin θ sin θeφ/2√|g6|+

∫Vol(Y4) ∧ C6

)• Sources modify F3 Bianchi identity — defines smearing form Ξ4:

Ξ4 ≡ dF3 =Nf

4sin θ sin θdθ ∧ dϕ ∧ d θ ∧ dϕ︸︷︷︸

Vol(Y4)

• Solve by adding term to F3

F3 =Nc

4

[− (ω1 + b dθ) ∧ (ω2 − b sin θdϕ) ∧ (ω3 + cos θdϕ)

+ b′dρ ∧ (−dθ ∧ ω1 + sin θdϕ ∧ ω2) +

(1−b2− Nf

Nc

)sin θdθ ∧ dϕ ∧ ω3

]≡ Nc

4f3



Solutions: UV behavior

• EOMs: first order BPS equations — explicit solutions foundnumerically.

• Various classes of UV behavior. One has UV stabilized dilaton andreduces to unflavored wrapped D5 family as s ≡ Nf /Nc → 0:

e2k =2

3c+e

4ρ/3 + · · · e2h =1

4c+e

4ρ/3 + · · ·

e2g = c+e4ρ/3 + · · · e4φ = f10

(1− 3e−4ρ/3s

c++ · · ·

)a = 2e−2ρ + · · · b = e−2ρ(2 + 2Q0 − s − 2(−2 + s)ρ) + · · ·

• Integration constants Q0, c+, c−, f10

• c+ parametrizes SUGRA family. Related to baryonic VEV (c+ ∼ 1/U)s 6= 0→ flavored CNP family

• We will take a non-extremal deformation of these asymptotics


Non-extremal flavored backgrounds

Outline








Non-extremal flavored backgrounds

Non-extremal flavored backgrounds: Outline

1 Find horizon-containing solutions to non-extremal (finite temp)deformation of CNP family.• Deform ansatz and derive EOMs

• Determine UV behavior and solve EOMs numerically

2 Rotate solutions• Check consistency of rotation on non-extremal backgrounds

• Use rotation to construct new non-extremal backgrounds with FWRDCasymptotics

3 Study how temperature depends on UV parameters, amount of flavor


Non-extremal flavored backgrounds Deforming the flavored CNP family

Non-extremal Ansatz and Einstein equations

• Non-extremal metric deformation (Einstein frame):

ds210 = eφ(ρ)/2

[−e−8xdt2 + dxidx

i + ds26

]ds2

6 =[e8xe2kdρ2 +

e2k

4(ω3 + cos θdϕ)2 + e2h(dθ2 + sin2 θdϕ2)

+e2g

4

((ω1 + a dθ)2 + (ω2 − a sin θdϕ)2

)]• F3 = Nc

4 f3 unchanged

• Non-extremality breaks SUSY:BPS formalism Solve Einstein equations

• Get Einstein eqs via dimensional reduction method (Gubser et al. ’01)→ EOMs for φ, x , k , g , h, a, b as function of ρ only



Solving the Einstein equations: UV boundary conditions

• Solve EOMs numerically by shooting from UV→ need UV boundary conditions.

• Impose UV series expansion:

e2h =∞∑i=0

i∑j=0

hi,j ρj e4(1−i)ρ/3 e4φ =

∞∑i=1

i∑j=0

fi,j ρj e4(1−i)ρ/3

e2g =∞∑i=0

i∑j=0

gi,j ρj e4(1−i)ρ/3 e8x =

∞∑i=1

i∑j=0

xi,j ρj e2(1−i)ρ/3

e2k =∞∑i=0

i∑j=0

ki,j ρj e4(1−i)ρ/3 a =

∞∑i=1

i∑j=0

ai,j ρj e2(1−i)ρ/3

b =∞∑i=1

i∑j=0

bi,j ρj e2(1−i)ρ/3

• Demand coefficients satisfy EOMs→ 11 free coefficients.



Solving the Einstein equations: UV boundary conditions

• One free coefficient is x5,0: non-extremality parameterWe will instead call this C2

• Match other free coefficients to extremal CNP UV behavior as C2 → 0

• e8x asymptotics:

e8x = 1 + C2e−8ρ/3 − C2e−4ρs

2c++O(e−8ρ/3)

• Other function asymptotics modified by C2 at higher order

• Set c− = 0 for simplicity

• 3 remaining parameters: c+ (parameter on SUSY family)C2 (non-extremality)s (amount of flavor)



Solving the Einstein equations: Numerics

• Pick values of parameters c+,C2, s and shoot UV → IR

• Horizons only exist for sufficiently large C2

• e2k , e2g , e2h diverge near horizon (numerical error)→ Match to horizon series expansion to get good solutions

rh 4 5 6 7Ρ

200

400

600

Figure: Metric functions at s = 1,c+ = 50, C2 = 5000. e2k , e2g , e2h, e8x

rh 3.3 3.4 3.5 3.6Ρ

50

100

150

Figure: Metric functions at s = 1,c+ = 50, C2 = 5000. Near horizon region


Non-extremal flavored backgrounds Rotating the solutions

Rotating the solutions

• After IIA → M-theory uplift, interpretation of smeared sources unclear

• Rotation is well defined for smeared flavor D5s in N = 1 BPS case(Gaillard et al. ’10)→ rotation in space of Killing spinors, equivalent to U-duality

• Can’t use BPS formalism as our solutions are non-extremal

• Won’t address 11d interpretation; instead just verify non-extremalrotated backgrounds are solutions to EOMs of IIB plus sources



Rotating the solutions

• After rotation, backgrounds have the form

ds2IIB = Nc

[e−φ/2H−1/2(−e−8xdt2 + dxidx

i ) + e3φ/2H1/2ds26

],

F3 =Nc

4f3, H3 = − Nc

4e−4x e2φ ∗6 f3,

F5 = −N2c (1 + ∗10)

[Vol(4) ∧ d

(e−4x

H

)]with• H1/2 =

√e−2φ − e−8x , Nc ≡ Nc coshβ

• rescaled R1,3 coords: x1,3 →√

Nc coshβ x1,3



Rotated asymptotics

• ds2 = −g00dt2 +gxxdxidx

i +gρρdρ2 +gθθ(dθ2 + sin2 θdϕ2) +gθθ(ω2

1 + ω22) +gψψ ω

23

• Unflavored (Klebanov-Strassler asymptotics):

gtt , gxx ∼e4ρ/3

A(ρ)+ · · · gρρ ∼

2c+

3A(ρ) + · · ·

gθθ, gθθ ∼c+

4A(ρ) + · · · gψψ =

c+

6A(ρ) + · · ·

→ ds2 =u2

A(u)dx2

1,3 +3c+A(u)

2

(du2

u2+ ds2

T 1,1

)+O(u−2)

where u =3

2ln ρ, A(u) =

√9

2c2+

ln u + C2 −3

8c2+

• Flavored (KS asymptotics FWRDC asymptotics):

gtt , gxx ∼√

2c+

3se2ρ/3 + · · · gρρ ∼

√2c+s

3e2ρ/3 + · · ·

gθθ, gθθ ∼√

3c+s

32e2ρ/3 + · · · gψψ =

√c+s

24e2ρ/3 + · · ·


Non-extremal flavored backgrounds Temperature

Temperature

• Impose regularity of Euclidean metric at horizon. Temperature givenby horizon coefficients, unchanged by rotation

→ Tbef = Taft =1

4π

x1√k0

e−8x |ρ=ρh = x1(ρ− ρh) + · · ·

e2k |ρ=ρh = k0 + k1(ρ− ρh) + · · ·

• Temperature independent of flavor

(fix c+ = 50,C2 = 5000)

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

0 2 4 6 8s

0.002

0.004

0.006

0.008

T

• Temperature decreasing with

increasing C2 (fix c+ = 50, s = 1)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

10 000 20 000 30 000 40 000C2

0.003

0.004

0.005

0.006

0.007

0.008

T


Thermodynamics and stability

Outline








Thermodynamics and stability The ADM Energy

Thermodynamics and Stability

• Gravity backgrounds dual to finite temperature field theories need tobe thermodynamically stable:

Cv ≡ dE/dT > 0

• E given by conserved ADM energy:

E = − 1

8πG10

∫S∞t

√|g00|(8K −8 K0)dS∞t

• Evaluated on 9d constant time slice Σt of geometry

• 8K and 8K0: extrinsic curvatures of two (finite temp and referencebackground) 8d submanifolds St of Σt in ρ→∞ limit


Thermodynamics and stability The ADM Energy

Reference Backgrounds, Energy Density

• Reference background: one of the BPS flavored wrapped D5backgrounds with stabilized dilaton.• Has free parameters Qo , c+, c−, f1,0

• Also have freedom to set period νBPS of Euclidean time

• Adjust free parameters in BPS asymptotics to match finite temp andreference geometries in UV

• Field theory energy density e = E/V3: same before/after rotation,with/without flavor

ebef = eaft =5c2

+C2

96π4


Thermodynamics and stability First Law

First Law

• Check first law of thermodynamics:

• Entropy density s ≡ S/V3:

sbef =(Areahor )

4G10V3=

e2φe2h+2g+k

4π3

∣∣∣∣ρh

saft =e3φH1/2e2h+2g+k

4π3

∣∣∣∣ρh

= sbef

→ unchanged by rotation

• T also unchanged by rotation

• For de/ds = T to hold, we wouldnot expect e to change

• Thermodynamics invariant under

rotation

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææ

ææ

ææ

ææ

ææ

ææ

ææ

æ

æ

ææ

æ

æ

0 5.0 ´ 107 1.0 ´ 108 1.5 ´ 108 2.0 ´ 108s0.000

0.001

0.002

0.003

0.004

0.005

0.006

de�ds, T

Figure: de/ds and T vs. s(s = 1, c+ = 50)


Thermodynamics and stability First Law

First Law

• For large black holes, de/ds and T asymptote to a constant, so weexpect e = T s

• e/s and T are indeed converging for our (somewhat) large black holes

æææææ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

5.0 ´ 107 1.0 ´ 108 1.5 ´ 108 2.0 ´ 108sbef

100 000

200 000

300 000

400 000

ebef

Figure: e vs. s (s = 1, c+ = 50)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææ

ææ

ææ

ææ

ææ æ

ææ

ææ

æ

ææ

ææ

à

à

à

à

à

à

à

à

à

à

àà

àà

àà

àà à à à à

àà

àà

àà

àà

à

50 100 150 200 250 300 350Rh

0.002

0.004

0.006

0.008

0.010

ebef�sbef, T

Figure: e/s (blue) and T (red), vs. Rh

(s = 1, c+ = 50)


Thermodynamics and stability Specific Heat

Specific Heat

• Specific heat Cv = de/dT negative for our backgrounds(with/without flavor, before/after rotation)→ backgrounds generically unstable

ææ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0.003 0.004 0.005 0.006 0.007 0.008T

10 000

20 000

30 000

40 000

50 000

60 000

e

Figure: e vs. T . (s = 1, c+ = 50)

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012T

1000

2000

3000

4000

5000

e

Figure: e vs. T . (s = 1, c+ = 3)


Summary and Conclusions

Outline










• Trying to find finite temperature gravity dual to QCD-like theory withunquenched flavor — i.e. Nf ∼ Nc — studied by (Gaillard et al. ’10)

• Thermodynamics invariant under U-duality:unstable initial solutions → unstable rotated solutions

• What are finite temperature duals to theories of (Gaillard et al. ’10)?

• Even unflavored non-extremal baryonic branch backgrounds areunstable. Can we modify the non-extremal baryonic branch UVbehavior in some way to obtain stability (i.e. without using rotationprocedure)?

• Can we get flavor-dependent thermodynamics in non-extremalbackgrounds? Perhaps by starting from constructions of(Conde et al. ’11) that have KS asymptotics.


The Stability of Non-Extremal Conifold Backgrounds with Sources

Technology

nonextremal backgrounds

finite temperature n

finite temp deconfinement

ks baryonic branch uduality

symqcd n

unstableat finite temp

nf nc unquenched flavor

ks asymptotics idea