Large AdS black holes from QFT Joonho Kim (KIAS) Based on 1810.12067 - - PowerPoint PPT Presentation

Large AdS black holes from QFT

Joonho Kim (KIAS)

Based on 1810.12067 and 1811.08646 with 
 Sunjin Choi, Seok Kim, June Nahmgoong (SNU)

Black Holes & Holography @ TSIMF, 7 Jan 2019

Black Hole

• Classically, absolute zero temperature due to strong aSracTon force.
• Proper thermodynamic system with finite temperature and entropy:
• Thermodynamic entropy has staTsTcal, microscopic interpretaTon.
• Boltzmann interpretaTon of Bekenstein-Hawking entropy?
• Yes, for many black hole soluTons engineered in string theory.

[Strominger, Vafa][Breckenridge, Myers, Peet, Vafa][Dijkgraaf, Verlinde, Verlinde] …

• In this talk, I want to answer the quesTon for some AdS black holes.


[Benini, Hristov, Zaffaroni][Azzurli et al.][Hosseini, Hristov, Passias][Hosseini, Yaakov, Zaffaroni] …

T = κ 2π Hawking temperature SBH = A 4GN Bekenstein-Hawking entropy A 4GN = log Ωmicro

AdS-Schwarzschild Black Hole

• Example: AdS-Schwarzschild soluTon.

[Hawking, Page]

• Smooth and complete iff the t-coordinate has the periodicity:
• Two disTnct branches:
• Hawking-Page transiTon
• T < THP: thermal graviton in the global AdS.
• T > THP: large AdS-Schwarzschild black hole. F ~ 1/GN ~ N2.

ds2 = ✓r2 b2 + 1 − w2M r2 ◆ dt2 + dr2 r2

b2 + 1 − w2M r2

+ r2dΩ2

3

t ∼ t + β with β−1 = r+ πb2 + 1 2πr+ , r2

+ = −b2

2 + b 2 p b2 + 4w2M

T <E> Small BH: thermodynamically unfavored (positive free energy & negative specific heat) Large BH: dominates at high temperature

Deconfinement Phase TransiAon

• In the large N gauge theory, the Hawking-Page transiTon is realized

as the deconfinement phase transiTon.

[Witten ’98]

• Even in the weakly coupled regime of some large N gauge theory,

the phase diagram looks analogous to that of AdS black hole.

[Aharony, Marsano, Minwalla, Papadodimas, Van Raamsdonk]

• This talk: 1/16 BPS black hole in AdS5(7).

AdS5 [Gutowski, Reall] [Kunduri, Lucietti, Reall] [Chong, Cvetic, Lu, Pope] [Wu], AdS7 [Chong, Cvetic, Lu, Pope] [Chow]

• U(1)2 ⊂ SO(4) angular momenta, U(1)3 ⊂ SO(6) electric charges.
• Preserves 2 supercharges.
• Exhibits O(N2) entropy at O(N2) angular momenta and charges.

Q+++

−− , S−−− ++

− → E = J1 + J2 + Q1 + Q2 + Q3

S = 2π q Q1Q2 + Q2Q3 + Q3Q1 − 1

2N 2(J1 + J2) ∼ O(N 2)

Plan

• IntroducTon
• Large AdS5 black holes from N=4 SYM
• Deconfinement phase transiTon in N=4 SYM
• Conclusion

• Enumerate all BPS microstates preserving supercharges
• Index of N=4 Yang-Mills on S3 X R1 :

[Romelsberger] [Kinney, Maldacena, Minwalla, Raju]

• Chemical potenTals for SO(4) isometry and SO(6) R-symmetry.
• Independent of the regulator β, so formally take β → 0.
• SUSY protecTon. Can be evaluated from the free QFT calculus.
• Agrees with the BPS graviton index at low “temperature”.

Superconformal Index

P3

I=1 ∆I = P2 i=1 ωi + 2πi

• e−∆IQI−ωiJi, Q+++

−−

= 0

• e−∆IQI−ωiJi, S−−−

++

= 0

Z = I [dα] · exp 

N

X

a,b=1 ∞

X

n=1

1 n ✓ 1 + X

s1,s2,s3=±1

s1s2s3(−1)n−1e

nsI ∆I 2

2 sinh nω1

2

· 2 sinh nω2

2

◆ einαab

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Z = Tr h e−βE

3

Y

I=1

e−∆IQI

2

Y

i=1

e−ωiJii

E =

• Q+++

−− , S−−− ++

with

Q+++

−− , S−−− ++

• Does the index see the deconfinement at high “temperature”?
• Consider the asymptoTc free energy in the Cardy limit


where large angular momentum (“energy”) states dominate.

• Saddle point approximaTon:

is most important.

• Respects the 4πi periodicity of chemical potenTals.

AsymptoAc Free Energy

|ω1,2| ⌧ 1

α1 = · · · = αN

log Z = − N 2 ω1ω2 X

s1s2s3=+1

h Li3 ⇣ −e

sI ∆I 2

⌘ − Li3 ⇣ −e− sI ∆I

2

⌘i

log Z = N 2∆1∆2∆3 2ω1ω2 with P3

I=1 ∆I = P2 i=1 ωi + 2πi

Li3(−ex) − Li3(−e−x) = −x3 6 − π2x 6

Use tri-logarithm identity

−2π < Im(+∆1 + ∆2 + ∆3) < 2π, − 2π < Im(+∆1 − ∆2 − ∆3) < 2π −2π < Im(−∆1 + ∆2 − ∆3) < 2π, − 2π < Im(−∆1 − ∆2 + ∆3) < 2π

in the “canonical chamber”

AsymptoAc Free Energy

• AlternaTve derivaTon: the background field acTon in
• β corresponds to the circumference of the temporal circle.
• S3 effecTve acTon of background fields.

[Di Pietro, Komargodski]

• Most terms are suppressed. Only contribuTon comes from
• Gauge non-invariant CS terms: needed to match ’t Hoov anomaly
• Generalizable to non-Lagrangian QFTs, e.g., 6d (2,0) SCFT.

β/r ⌧ ω1,2 ⌧ 1

SCS = −iN 2 8π · β 2π Z

S3 CIJK

✓ AI

4AJ ∧ dAK + AI 4AJ 4 AK ∧ da + 1

3AI

4AJ 4 AK 4 a ∧ da

◆ = −N 2∆1∆2∆3 2ω1ω2

e−2Φ = 1 − r2 X

i

n2

i ω2 i

β2

dilaton

a = −i r2 P

i ωin2 i dφi

β(1 − P

i r2 n2

i ω2 i

β2 )

graviphoton

AI

4 = −i∆I

β

scalar

ds2

4 = r2

2 4dθ2 + X

i

n2

i dφ2 i +

r2(P

i ωin2 i dφi)2

β2(1 − r2 P

i n2

i ω2 i

β2 )

3 5 + e−2Φ (dτ + a)2

metric

AI = −AI

4a

gauge field

Large AdS Black Hole

• Having obtained the large N superconformal index, take an inverse

Laplace transformaTon to extract the degeneracy at given charges.

• AsymptoTc degeneracy at macroscopic charges can be evaluated by

the saddle point approximaTon, i.e., the Legendre transformaTon of the asymptoTc free energy in the Cardy limit.

[Hosseini, Hristov, Zaffaroni]

Ω(QI, Ji) = Z d∆I dωi Z(∆I, ωi) exp ⇣ P

I ∆IQI + P i ωiJi

⌘

S(∆I, ωi) = +N 2∆1∆2∆3 2ω1ω2 +

3

X

I=1

∆IQI +

2

X

i=1

ωiJi with P3

I=1 ∆I − P2 i=1 ωi = 2πi

S(∆I, ωi) = −N 3(∆1∆2)2 24ω1ω2ω3 +

2

X

I=1

∆IQI +

3

X

i=1

ωiJi with P2

I=1 ∆I − P3 i=1 ωi = 2πi

Large AdS Black Hole

• Further imposing the charge relaTon of known BPS black holes, 


it matches the known entropy formula of BPS black holes!

• AsymptoTc free energy = the free energy of the known BH soluTon.

AdS5 : AdS7 :

S = 2π q Q1Q2 + Q2Q3 + Q3Q1 − 1

2N 2(J1 + J2)

S = 2π s 3(Q2

1Q2 + Q1Q2 2) − N 3(J1J2 + J2J3 + J3J1)

3(Q1 + Q2) − N 3

Small BH: thermodynamically unfavored (positive free energy, negative specific heat) Large BH: dominates at high “temperature” Analogous to AdS-Schwarzschild solution.

Re(ωi)−1

free energy / N2 Hawking-Page transition point


• f the large black hole branch.

Plan

• IntroducTon
• Large AdS5 black holes from N=4 SYM
• Deconfinement phase transiTon in N=4 SYM
• Conclusion

Complex Fugacity

• Superconformal index clearly sees the deconfining phase at high

“temperature,” in contrast to what had been commonly believed: Massive boson/fermion cancela1on prevents the index 
 from seeing the O(N2) free energy of deconfining phase.

• B/F cancelaTon reduces the asymptoTc degeneracy at two levels.
• Case 1. the actual microstate degeneracy can be O(N0).
• Case 2. the degeneracy grows quickly but with alternaTng signs.



 
 
 Macroscopic charge approximaTon is insensiTve to the change by 1 or 2 charge units. Nearby terms are effecTvely smeared out. 
 The asymptoTc degeneracy becomes O(N0) aver “cancelaTon”.

1 + 3x2 − 2x3 + 9x4 − 6x5 + 11x6 − 6x7 + 9x8 + 14x9 − 21x10 + 36x11 − 17x12 − 18x13 + 114x14 − 194x15 + 258x16 − 168x17 − 112x18 + 630x19 − 1089x20 + 1130x21 − 273x22 − 1632x23 + 4104x24 − 5364x25 + 3426x26 + 3152x27 − 13233x28 + 21336x29 − 18319x30 − 2994x31 + 40752x32 − 76884x33 + 78012x34 − 11808x35 + · · ·

Complex Fugacity

• The macroscopic B/F cancelaTon can be obstructed by introducing

(relaTve) phase factors between nearby terms.

• Legendre transformaTon chooses an opTmal value to maximize the

asymptoTc degeneracy.

• Allowing the chemical potenTals to be complex-valued, I want to

trace how the deconfinement phase transiTon happens in the index.

[Aharony, Marsano, Minwalla, Papadodimas, Van Raamsdonk] [Kinney, Maldacena, Minwalla, Raju]

∆1 = ∆2 = ∆3 and ω1 = ω2 ⌧ 1

∆1 = ∆2 = ∆3 = 2

3πi

Confining Saddle Point

• Introducing the large N eigenvalue distribuTon, the index becomes
• Polyakov loop: an order parameter for (de)confinement transiTon.
• Confining saddle point:
• Most dominant saddle point at real-valued chemical potenTals.
• Gaussian integral = the supergraviton spectrum on AdS5 X S5.
• Becomes unstable if

with complex .

Z = Z Y

n6=0

dρn exp ✓ −

1

X

n=1

N 2 n Q

I(1 − en∆I)

Q

i(1 − enωi) ρnρn

◆ where ρn = 1 2πN

N

X

a=1

einαa

ρn = 0

Re Q

I(1 − e−∆I)

Q

i(1 − e−ωi)

• < 0

∆I, ωi

Confining Saddle Point

• At equal charges and angular momenta, 


the confining saddle point becomes unstable on the red line.

• Since true deconfinement should happen before/at the instability

point, this implies the existence of new black hole saddle point!

The earliest instability point: Earlier than the Hawking-Page transiTon point

• f known AdS5 black hole

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

exp(− 1

3Re(ωi))

− 1

3Im(ωi)

Re(ωi)−1 ∼ 0.663

Instability arises only at suitably chosen
 imaginary value of chemical potenTals. ∆1 = ∆2 = ∆3 and ω1 = ω2

Re(ωi)−1

Re(ωi)−1 ∼ 0.862

Conclusion

• Superconformal index sees the BPS black holes in AdS5/7.
• The microstate degeneracy of known BPS black hole soluTon was
• btained from the superconformal index at the Cardy limit.
• Imaginary chemical potenTals obstructs the massive B/F cancelaTon.
• Confining saddle point, dominant at low “temperature,” becomes

unstable at high “temperature” with imaginary chemical potenTals.

• The earliest deconfining saddle point is not of the known black hole,

thereby predicTng a new BPS black hole in asymptoTc AdS5.

• Many other saddle points showing O(N2) degeneracy was found.

Can we idenTfy them with some hairy BPS black hole soluTons?