The Stability of Non-Extremal Conifold Backgrounds with Sources

The Stability of Non-Extremal Conifold Backgrounds
with 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 backgrounds
Deforming the flavored CNP family
Rotating the solutions: non-extremal backgrounds with KS and
FWRDC asymptotics
Temperature
5 Thermodynamics and stability
The ADM Energy
First Law
Specific Heat
6 Summary and Conclusions


Intro

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat


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 = N = 4 SYM • T > 140 MeV: QCD ∼ N = 4 SYM
QCD N = 4 SYM QCD N = 4 SYM
conformal no yes conformal no T sets scale
confining yes no confining no no
fundamentals yes no fundamentals yes no
χral symmetry yes no χral symmetry no no
SUSY no yes 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)) at
finite temp (Buchel, Aharony, etc.)

• Caveat: black holes in KS may be unstable

• Other N = 1 duals (Maldacena-N`nez (MN)) known to be unstable
u˜
at 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 branes
spanning R1,3 and wrapped on S 2 of resolved conifold.

• (Maldacena et al. ’09):
U-duality
MN wrapped D5 family ⇐⇒ family dual to KS baryonic branch
• U-duality more general:
• (C`ceres et al. ’11) applied U-duality to non-extremal deformation of
a
MN family, produced non-extremal background with KS asymptotics

• Idea:
U-duality ?
Non-extremal D5 family ⇐⇒ 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):
U-duality
(T = 0) CNP backgrounds =⇒ flavored KS baryonic branch
a.k.a. ‘flavored warped resolved deformed conifold’ (FWRDC).

Intro

Intro: Goal and Summary

• Goal: Combine flavored and non-extremal modifications, use
U-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 related
non-extremal deformations of the KS baryonic branch

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


U-duality: Wrapped D5s → KS baryonic branch

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat



The Maldacena-N´nez and Klebanov-Strassler backgrounds
u˜

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

• Maldacena-N´nez (one point in wrapped D5 family):
u˜
• Near-horizon geometry of D5 branes wrapped on S 2 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 two
bifundamentals (two antibifund) and quartic superpotential.

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



Two one-parameter families

U-duality
• MN wrapped D5 family ⇐⇒ 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 to
gravity . 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 )



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


Adding ﬂavor: the CNP family

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat



An N = 1 SQCD dual

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

• Space has topology R1,3 × Rρ × S 2 × S 3
θ,ϕ ˜˜
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 ans¨tz only depend on ρ
a



Metric and F3 ans¨tze
a

• Metric and F3 have form (α = gs = 1, Nc absorbed into e 2k , e 2g , e 2h ):
S2

ds10 = e φ(ρ)/2 dx1,3 + e 2k(ρ) dρ2 + e 2h(ρ) (dθ2 + sin2 θdϕ2 ) +
2 2

e 2g (ρ) e 2k(ρ)
+ (ω1 + a(ρ)dθ)2 + (ω2 − a(ρ) sin θdϕ)2 + (ω3 + cos θdϕ)2
4 4
Nc
F3 = − (ω1 + b(ρ)dθ) ∧ (ω2 − b(ρ) sin θdϕ) ∧ (ω3 + cos θdϕ)
4
+b (ρ)dρ ∧ (−dθ ∧ ω1 + sin θdϕ ∧ ω2 ) + (1 − b(ρ)2 ) sin θdθ ∧ dϕ ∧ ω3

• ω1 ∧ ω2 ∧ ω3 : volume form on S 3
˜ ˜ ˜
ω1 = cos ψd θ + sin ψ sin θd ϕ, ˜ ˜ ˜
ω2 = − sin ψd θ + cos ψ sin θd ϕ, ˜ ˜
ω3 = dψ + cos θd ϕ



Action
• Action for background and smeared sources, S = SIIB + Ssources :
1 1 1
SIIB = d 10 x |g10 | R − (∂µ φ)(∂ µ φ) − e φ F3
2
2κ2
10 2 12
T5 Nf ˜
Ssources = − d 10 x sin θ sin θe φ/2 |g6 | + Vol(Y4 ) ∧ C6
(4π)2
• Sources modify F3 Bianchi identity — deﬁnes smearing form Ξ4 :
Nf ˜ ˜
Ξ4 ≡ dF3 = sin θ sin θdθ ∧ dϕ ∧ d θ ∧ d ϕ
˜
4
Vol(Y4 )

• Solve by adding term to F3
Nc
F3 = − (ω1 + b dθ) ∧ (ω2 − b sin θdϕ) ∧ (ω3 + cos θdϕ)
4
Nf Nc
+ b dρ ∧ (−dθ ∧ ω1 + sin θdϕ ∧ ω2 ) + 1−b 2 − sin θdθ ∧ dϕ ∧ ω3 ≡ f3
Nc 4



Solutions: UV behavior

• EOMs: first order BPS equations — explicit solutions found
numerically.
• Various classes of UV behavior. One has UV stabilized dilaton and
reduces to unflavored wrapped D5 family as s ≡ Nf /Nc → 0:
2 1
e 2k = c+ e 4ρ/3 + · · · e 2h = c+ e 4ρ/3 + · · ·
3 4
3e −4ρ/3 s
e 2g = c+ e 4ρ/3 + · · · e 4φ = f10 1 − + ···
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 = 0 → flavored CNP family

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

Non-extremal ﬂavored backgrounds

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat


Non-extremal flavored backgrounds

Non-extremal flavored backgrounds: Outline

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

• 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 FWRDC
asymptotics

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


Non-extremal ﬂavored backgrounds Deforming the ﬂavored CNP family

Non-extremal Ans¨tz and Einstein equations
a

• Non-extremal metric deformation (Einstein frame):
2
ds10 = e φ(ρ)/2 −e −8x dt 2 + dxi dx i + ds6
2

2 e 2k
ds6 = e 8x e 2k dρ2 + (ω3 + cos θdϕ)2 + e 2h (dθ2 + sin2 θdϕ2 )
4
e 2g
+ (ω1 + a dθ)2 + (ω2 − a sin θdϕ)2
4

• F3 = Nc f3 unchanged
4

• 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:
∞ i ∞ i
e 2h = hi,j ρj e 4(1−i)ρ/3 e 4φ = fi,j ρj e 4(1−i)ρ/3
i=0 j=0 i=1 j=0
∞ i ∞ i
e 2g = gi,j ρj e 4(1−i)ρ/3 e 8x = xi,j ρj e 2(1−i)ρ/3
i=0 j=0 i=1 j=0
∞ i ∞ i
e 2k = ki,j ρj e 4(1−i)ρ/3 a= ai,j ρj e 2(1−i)ρ/3
i=0 j=0 i=1 j=0
∞ i
b= bi,j ρj e 2(1−i)ρ/3
i=1 j=0

• Demand coeﬃcients satisfy EOMs
→ 11 free coeﬃcients.


Solving the Einstein equations: UV boundary conditions

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

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

• e 8x asymptotics:

C2 e −4ρ s
e 8x = 1 + C2 e −8ρ/3 − + O(e −8ρ/3 )
2c+
• 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
• e 2k , e 2g , e 2h diverge near horizon (numerical error)
→ Match to horizon series expansion to get good solutions
150

600

100

400

50
200

Ρ Ρ
rh 4 5 6 7 rh 3.3 3.4 3.5 3.6

Figure: Metric functions at s = 1, Figure: Metric functions at s = 1,
c+ = 50, C2 = 5000. e 2k , e 2g , e 2h , e 8x 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-extremal
rotated backgrounds are solutions to EOMs of IIB plus sources



Rotating the solutions

• After rotation, backgrounds have the form
2
dsIIB = Nc e −φ/2 H−1/2 (−e −8x dt 2 + dxi dx i ) + e 3φ/2 H1/2 ds6 ,
2

Nc Nc −4x 2φ
F3 = f3 , H3 = − e e ∗6 f 3 ,
4 4
2 e −4x
F5 = −Nc (1 + ∗10 ) Vol(4) ∧ d
H

with
√
• H1/2 = e −2φ − e −8x , Nc ≡ Nc cosh β

• rescaled R1,3 coords: x1,3 → Nc cosh β x1,3



Rotated asymptotics
• ds 2 = −g00 dt 2 + gxx dxi dx i + gρρ dρ2 + gθθ (dθ2 + sin2 θdϕ2 ) + gθθ (˜ 1 + ω2 ) + gψψ ω3
˜˜ ω
2
˜2 ˜2

• Unﬂavored (Klebanov-Strassler asymptotics):
e 4ρ/3 2c+
gtt , gxx ∼ + ··· gρρ ∼ A(ρ) + · · ·
A(ρ) 3
c+ c+
gθθ , gθθ
˜˜ ∼ A(ρ) + · · · gψψ = A(ρ) + · · ·
4 6
u2 3c+ A(u) du 2
→ ds 2 = 2
dx1,3 + 2
+ dsT 1,1 + O(u −2 )
A(u) 2 u2
3 9 3
where u = ln ρ, A(u) = 2
ln u + C2 − 2
2 2c+ 8c+

• Flavored (KS asymptotics FWRDC asymptotics):
2c+ 2ρ/3 2c+ s 2ρ/3
gtt , gxx ∼ e + ··· gρρ ∼ e + ···
3s 3
3c+ s 2ρ/3 c+ s 2ρ/3
gθθ , gθθ ∼
˜˜ e + ··· gψψ = e + ···
32 24

Non-extremal flavored backgrounds Temperature

Temperature
• Impose regularity of Euclidean metric at horizon. Temperature given
by horizon coefficients, unchanged by rotation
1 x1 e −8x |ρ=ρh = x1 (ρ − ρh ) + · · ·
→ Tbef = Taft = √
4π k0
e 2k |ρ=ρh = k0 + k1 (ρ − ρh ) + · · ·

• Temperature independent of flavor • Temperature decreasing with
(fix c+ = 50, C2 = 5000) increasing C2 (fix c+ = 50, s = 1)
T
T

0.008

0.008

0.006 0.007

0.006
0.004

0.005

0.002 0.004

0.003
s C
0 2 4 6 8 10 000 20 000 30 000 40 000 2


Thermodynamics and stability

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat


Thermodynamics and stability The ADM Energy

Thermodynamics and Stability

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

Cv ≡ dE /dT > 0

• E given by conserved ADM energy:

1
E =− |g00 |(8 K −8 K0 )dSt∞
8πG10 St∞

• Evaluated on 9d constant time slice Σt of geometry

8
• K and 8 K0 : extrinsic curvatures of two (finite temp and reference
background) 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 D5
backgrounds 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 and
reference geometries in UV

• Field theory energy density e = E /V3 : same before/after rotation,
with/without flavor
2
5c+ C2
ebef = eaft =
96π 4


Thermodynamics and stability First Law

First Law
• Check ﬁrst law of thermodynamics:
• Entropy density s ≡ S/V3 :

(Areahor ) e 2φ e 2h+2g +k
sbef = = de ds, T

4G10 V3 4π 3 ρh 0.006

e 3φ H1/2 e 2h+2g +k 0.005
saft = = sbef
4π 3 ρh 0.004

0.003
→ unchanged by rotation
0.002

• T also unchanged by rotation 0.001

• For de/ds = T to hold, we would 0.000
0 5.0 107 1.0 108 1.5 108 2.0 108
s

not expect e to change
Figure: de/ds and T vs. s
(s = 1, c+ = 50)
• Thermodynamics invariant under
rotation

Thermodynamics and stability First Law

First Law
• For large black holes, de/ds and T asymptote to a constant, so we
expect e = T s
• e/s and T are indeed converging for our (somewhat) large black holes
ebef sbef , T
ebef

0.010
400 000

0.008

300 000

0.006

200 000
0.004

100 000 0.002

Rh
sbef 50 100 150 200 250 300 350
5.0 107 1.0 108 1.5 108 2.0 108

Figure: e/s (blue) and T (red), vs. Rh
Figure: e vs. s (s = 1, c+ = 50)
(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

e e

60 000
5000
50 000
4000
40 000
3000
30 000

2000
20 000

10 000 1000

T T
0.003 0.004 0.005 0.006 0.007 0.008 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012

Figure: e vs. T . (s = 1, c+ = 50) Figure: e vs. T . (s = 1, c+ = 3)


Summary and Conclusions

Outline
FWRDC asymptotics
Temperature
The ADM Energy
First Law
Speciﬁc Heat




• Trying to find finite temperature gravity dual to QCD-like theory with
unquenched 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 are
unstable. Can we modify the non-extremal baryonic branch UV
behavior in some way to obtain stability (i.e. without using rotation
procedure)?

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


The Stability of Non-Extremal Conifold Backgrounds with Sources

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