Black hole microstate counting&Picard-Lefschetz theory

Masazumi Honda(本多正純)

Ref: MH, work in progress
Related works:
Choi-Kim-Kim-Nahmgoong, arXiv:1810.12067; MH, arXiv:1901.08091; Ardehali, arXiv:1902.06619
Benini-Milan, arXiv:1812.09613; Cabo-Bizet-Murthy, arXiv:1909.09597;
Copetti-Grassi-Komargodski-Tizzano, arXiv:2008.04950; Ardehali-Hong-Liu, arXiv:1912.04169, etc…
25th, Sep., 2020

Potential Toolkit to Attack Nonperturbative Aspects of QFT- Resurgence and related topics - @YITP, Kyoto

[Bekenstein ’72, Hawking ’75]

Entropy:=Black hole

Thermodynamic?

Area
𝑆BH =4𝐺[Event Horizon Telescope
https://www.eso.org/public/images/eso1907 ]Test of Quantum Gravity
Bekenstein-Hawking entropy:

Area
𝑆BH =4𝐺[Bekenstein ’72, Hawking ’75]

macroscopic

What is a microscopic origin of the entropy?
Candidates of quantum gravity should answer∋ String Theory!

Answer from string theory for a specific black hole
[Strominger-Vafa ’96 ]For an asymptotically flat supersymmetric (SUSY) charged BH,

𝑆𝐵𝐻 = log

# of states in string theory
w/ the same quantum numbers

statistical explanation of BH entropy!

Another challenge for asymptotically AdS BH[Maldacena ’97 ]AdS/CFT correspondence:dual

d-dim. CFTGravity in AdSd+1

Does the CFT side give a microscopic explanationof BH entropy on the gravity side?

Another challenge for asymptotically AdS BH[Maldacena ’97 ]AdS/CFT correspondence:dual

d-dim. CFTGravity in AdSd+1

Does the CFT side give a microscopic explanationof BH entropy on the gravity side?
A natural expectation:

𝑆𝐵𝐻 = log

# of states in CFT
w/ the same quantum numbers??

This talkSet up:4d 𝑆𝑈 𝑁 𝒩 = 4
Super Yang-Mills theory

IIB string on 𝐴𝑑𝑆5 × 𝑆 5

This talkSet up:4d 𝑆𝑈 𝑁 𝒩 = 4
Super Yang-Mills theory

IIB string on 𝐴𝑑𝑆5 × 𝑆 5∃black holes on the string side (rotating charged):[Gutowski-Reall ’04 etc.]

This talkSet up:4d 𝑆𝑈 𝑁 𝒩 = 4
Super Yang-Mills theory

IIB string on 𝐴𝑑𝑆5 × 𝑆 5∃black holes on the string side (rotating charged):

Recent progress: (explained later)[Gutowski-Reall ’04 etc.][Choi-Kim-Kim-Nahmgoong, Benini-Milan, MH,
Ardehali, Cabo-Bizet-Murthy, etc.
]

The BH entropy is captured by superconformal index
but ∃subtle points w/ smells of Stokes phenomena
Resurgence can help?

This talkSet up:4d 𝑆𝑈 𝑁 𝒩 = 4
Super Yang-Mills theory

IIB string on 𝐴𝑑𝑆5 × 𝑆 5∃black holes on the string side (rotating charged):

Recent progress: (explained later)[Gutowski-Reall ’04 etc.][Choi-Kim-Kim-Nahmgoong, Benini-Milan, MH,
Ardehali, Cabo-Bizet-Murthy, etc.
]

The BH entropy is captured by superconformal index
but ∃subtle points w/ smells of Stokes phenomena
Resurgence can help?
Related aspects:

・(de)confinement transition [Copetti-Grassi-Komargodski-Tizzano] (⇔ Hawking-Page transition)
・partial deconfinement [Ardehali-Hong-Liu]

Contents
1. Introduction
2. Recent progress & Subtle points

3. Resurgence to the rescue (?)
4. Summary

Black holes on the gravity side
4d 𝑆𝑈(𝑁) 𝒩 = 4 SYM

IIB string on 𝐴𝑑𝑆5 × 𝑆 5
・2 angular momenta: ( 𝐽1 , 𝐽2 )∃Black holes on the string side w/・3 electric charges: (𝑄1 , 𝑄2 , 𝑄3 )・2 supercharges (1/16 BPS)
・mass: 𝑔( 𝐽1 + 𝐽2 + 𝑄1 + 𝑄2 + |𝑄3 |)[Gutowski-Reall ’04 etc.]

Black holes on the gravity side
4d 𝑆𝑈(𝑁) 𝒩 = 4 SYM

IIB string on 𝐴𝑑𝑆5 × 𝑆 5
・2 angular momenta: ( 𝐽1 , 𝐽2 )∃Black holes on the string side w/

・3 electric charges: (𝑄1 , 𝑄2 , 𝑄3 )・2 supercharges (1/16 BPS)
・mass: 𝑔( 𝐽1 + 𝐽2 + 𝑄1 + 𝑄2 + |𝑄3 |)[Gutowski-Reall ’04 etc.]

Bekenstein-Hawking entropy:withIs the entropy realized by counting CFT states?

Grand canonical partition function & Index
Q. Black hole entropy from counting CFT states?

Grand canonical partition function:

AdS/CFT→
difficult to computeGrand canonical partition function & Index
Q. Black hole entropy from counting CFT states?

Grand canonical partition function:

AdS/CFT→
difficult to compute

Superconfomal index:[Kinney-Maldacena-Minwalla-Raju’06 ]under control

Grand canonical partition function & Index
Q. Black hole entropy from counting CFT states?

Grand canonical partition function:

AdS/CFT→
difficult to compute

Superconfomal index:[Kinney-Maldacena-Minwalla-Raju’06 ]under control

Superconformal index in SU(N) N=4 SYM
In 4d 𝒩 = 1 language,

4d 𝒩 = 4 SYM = 1 vector + 3 adjoint chirals Φ1,2,3Partition function on

Mathematical description: finite dim. integral
It is known that superconformal index is 1-loop exact:1

𝐼=𝑝;𝑝 𝑁 𝑞;𝑞 𝑁 2 𝑁−1𝑥 ς𝑖≠𝑗 Γ𝑒 (𝑥𝑖𝑗 + 𝜎 + 𝜏; 𝜎, 𝜏)‫ ׬‬1𝑑𝑁!
−2× Γ𝑒 𝑥𝑖𝑗 + 𝑚2 +131

ς𝑖,𝑗 Γ𝑒 𝑥𝑖𝑗 + 𝑚1 + 𝜎 + 𝜏 ; 𝜎, 𝜏3𝜎 + 𝜏 ; 𝜎, 𝜏 Γ𝑒 𝑥𝑖𝑗 − 𝑚1 − 𝑚2 +ⅇ 2𝜋𝑖𝑥𝑗 : holonomy along 𝑆1 , 𝑥𝑖𝑗 = 𝑥𝑖 − 𝑥𝑗
𝑚1 , 𝑚2 : chemical potential for 𝑈(1) flavor symmetries
𝑝 = ⅇ 2𝜋𝑖𝜎 , q= ⅇ 2𝜋𝑖𝜏 ,𝑘𝑎; 𝑞 = ς∞
𝑘=0(1 − 𝑎𝑞 ),1 − ⅇ −2𝜋𝑖𝑥 𝑝 𝑗+1 𝑞 𝑘+1
Γ𝑒 𝑥; 𝜎, 𝜏 = ෑ
1 − ⅇ 2𝜋𝑖𝑥 𝑝 𝑗 𝑞 𝑘𝑗,𝑘≥0

1
3𝜎 + 𝜏 ; 𝜎, 𝜏“Lore” on superconformal index until 2018
[ Kinney-Maldacena-Minwalla-Raju’06, etc. ]1

𝐼=𝑝;𝑝 𝑁 𝑞;𝑞 𝑁 2‫ ׬‬1 𝑑 𝑁−1 𝑥 ς𝑖≠𝑗 Γ𝑒 (𝑥𝑖𝑗 + 𝜎 + 𝜏; 𝜎, 𝜏)𝑁!
−2× Γ𝑒 𝑥𝑖𝑗 + 𝑚2 +1
3

1

ς𝑖,𝑗 Γ𝑒 𝑥𝑖𝑗 + 𝑚1 + 𝜎 + 𝜏 ; 𝜎, 𝜏
3𝜎 + 𝜏 ; 𝜎, 𝜏 Γ𝑒 𝑥𝑖𝑗 − 𝑚1 − 𝑚2 +1
3Let us consider
𝑁 → ∞, 𝑚1 = 𝑚2 = 0, 0 < 𝑝, 𝑞 < 1𝜎 + 𝜏 ; 𝜎, 𝜏(𝑝 = ⅇ 2𝜋𝑖𝜎 , q= ⅇ 2𝜋𝑖𝜏 )“Lore” on superconformal index until 2018
[ Kinney-Maldacena-Minwalla-Raju’06, etc. ]1

𝐼=𝑝;𝑝 𝑁 𝑞;𝑞 𝑁 2‫ ׬‬1 𝑑 𝑁−1 𝑥 ς𝑖≠𝑗 Γ𝑒 (𝑥𝑖𝑗 + 𝜎 + 𝜏; 𝜎, 𝜏)𝑁!
−2× Γ𝑒 𝑥𝑖𝑗 + 𝑚2 +1
3

1

ς𝑖,𝑗 Γ𝑒 𝑥𝑖𝑗 + 𝑚1 + 𝜎 + 𝜏 ; 𝜎, 𝜏
3𝜎 + 𝜏 ; 𝜎, 𝜏 Γ𝑒 𝑥𝑖𝑗 − 𝑚1 − 𝑚2 +1
3Let us consider

𝜎 + 𝜏 ; 𝜎, 𝜏(𝑝 = ⅇ 2𝜋𝑖𝜎 , q= ⅇ 2𝜋𝑖𝜏 )𝑁 → ∞, 𝑚1 = 𝑚2 = 0, 0 < 𝑝, 𝑞 < 1
Then, the famous result:

𝐼 = ⅇ 𝒪(1)independent of N!?

“Lore” on superconformal index until 2018
[ Kinney-Maldacena-Minwalla-Raju’06, etc. ]1

𝐼=𝑝;𝑝 𝑁 𝑞;𝑞 𝑁 2‫ ׬‬1 𝑑 𝑁−1 𝑥 ς𝑖≠𝑗 Γ𝑒 (𝑥𝑖𝑗 + 𝜎 + 𝜏; 𝜎, 𝜏)𝑁!
−2× Γ𝑒 𝑥𝑖𝑗 + 𝑚2 +1
3

1ς𝑖,𝑗 Γ𝑒 𝑥𝑖𝑗 + 𝑚1 + 𝜎 + 𝜏 ; 𝜎, 𝜏
3

𝜎 + 𝜏 ; 𝜎, 𝜏 Γ𝑒 𝑥𝑖𝑗 − 𝑚1 − 𝑚2 +13

Let us consider

𝜎 + 𝜏 ; 𝜎, 𝜏(𝑝 = ⅇ 2𝜋𝑖𝜎 , q= ⅇ 2𝜋𝑖𝜏 )𝑁 → ∞, 𝑚1 = 𝑚2 = 0, 0 < 𝑝, 𝑞 < 1
Then, the famous result:independent of N!?

𝐼 = ⅇ 𝒪(1)Obviously, this does not capture the BH entropy:

Standard interpretation was∃many cancelations due to−1 𝐹

Loophole of the “lore”[Choi-Kim-Kim-Nahmgoong, Cabo-Bizet-Cassani-Martelli-Murthy, etc. ]To pick up # of states, we have to extract coefficients:
𝑑𝑝

𝑑𝑞

𝑑𝑣𝑑𝑣

(# of states) = ‫𝑃𝑁𝑝 ׯ‬+1 𝑁𝑞+1 𝑁𝑣11+1 𝑁𝑣22+1 𝐼(𝑝, 𝑞, 𝑣1 , 𝑣2 )𝑞

𝑣1𝑣2

Loophole of the “lore”[Choi-Kim-Kim-Nahmgoong, Cabo-Bizet-Cassani-Martelli-Murthy, etc. ]To pick up # of states, we have to extract coefficients:
𝑑𝑝

𝑑𝑞

𝑑𝑣𝑑𝑣

(# of states) = ‫𝑃𝑁𝑝 ׯ‬+1 𝑁𝑞+1 𝑁𝑣11+1 𝑁𝑣22+1 𝐼(𝑝, 𝑞, 𝑣1 , 𝑣2 )𝑞

𝑣1𝑣2𝑝, 𝑞, 𝑣1 , 𝑣2

x 𝐼 = ⅇ 𝒪(1)

Loophole of the “lore”[Choi-Kim-Kim-Nahmgoong, Cabo-Bizet-Cassani-Martelli-Murthy, etc. ]To pick up # of states, we have to extract coefficients:
𝑑𝑝𝑑𝑞𝑑𝑣𝑑𝑣(# of states) = ‫𝑃𝑁𝑝 ׯ‬+1 𝑁𝑞+1 𝑁𝑣11+1 𝑁𝑣22+1 𝐼(𝑝, 𝑞, 𝑣1 , 𝑣2 )
𝑞

𝑣1

𝑣2

𝑝, 𝑞, 𝑣1 , 𝑣2

x 𝐼 = ⅇ 𝒪(1)Need information for complex fugacities!

Recent updates(details explained in next pages)[Choi-Kim-Kim-Nahmgoong, Benini-Milan, MH,Ardehali, Cabo-Bizet-Murthy, etc.
]

Loophole: complex fugacities may give different results
In the shrinking limit of 𝑆1 (“Cardy limit”),

𝐼=ⅇ

𝒪(𝑁2 )(for complex fugacities)

Recent updates(details explained in next pages)[Choi-Kim-Kim-Nahmgoong, Benini-Milan, MH,
Ardehali, Cabo-Bizet-Murthy, etc.
]

Loophole: complex fugacities may give different results
In the shrinking limit of 𝑆1 (“Cardy limit”),

𝐼=ⅇ

𝒪(𝑁2 )(for complex fugacities)

Picking up (# of states) from this formula w/ 𝑁 → ∞ gives

SUSY Cardy formula for N=4 SYM[Di Pietro-Komargodski,
Ardehali ’15,Di Pietro-MH ’16]

SUSY Cardy formula for N=4 SYM[Di Pietro-Komargodski,
Ardehali ’15,Di Pietro-MH ’16]

Cardy limit 𝛽 → 0 𝜏, 𝜎 → 0 :1/2

𝐼𝑆1 ×𝑆3 → න−1/2

𝑑𝑁−1𝑖𝜋𝑖𝜋 𝜏+𝜎𝑉2 𝑎 +𝑉1 (𝑎)2𝜏𝜎𝑎 ⅇ 6𝜏𝜎Effective potential analysis
1/2

𝐼𝑆 1 ×𝑆 3 → න𝑖𝜋𝑖𝜋 𝜏+𝜎𝑑 𝑁−1 𝑎 ⅇ 6𝜏𝜎 𝑉2 𝑎 + 2𝜏𝜎 𝑉1 (𝑎)−1/2

Let’s focus on the regime→ minimize[MH ’19]Effective potential analysis
1/2

𝐼𝑆 1 ×𝑆 3 → න𝑖𝜋𝑖𝜋 𝜏+𝜎𝑑 𝑁−1 𝑎 ⅇ 6𝜏𝜎 𝑉2 𝑎 + 2𝜏𝜎 𝑉1 (𝑎)−1/2Let’s focus on the regime→ minimize[MH ’19]Effective potential analysis
1/2

𝐼𝑆 1 ×𝑆 3 → න𝑖𝜋𝑖𝜋 𝜏+𝜎𝑑 𝑁−1 𝑎 ⅇ 6𝜏𝜎 𝑉2 𝑎 + 2𝜏𝜎 𝑉1 (𝑎)−1/2

Let’s focus on the regime

𝑓 𝑥 has the minimum at the origin:Thus,→ minimize[MH ’19]

Entropy from index

Step 1: Redefine chemical potentials:[cf. Cabo-Bizet-CassaniMartelli-Murthy ’18]

Entropy from index

Step 1: Redefine chemical potentials:[cf. Cabo-Bizet-CassaniMartelli-Murthy ’18]

Step 2: Pick up “degeneracy” via Legendre transformation

Entropy from index (cont’d)Pick up “degeneracy”:

Entropy from index (cont’d)Pick up “degeneracy”:

Entropy w/ large quantum numbers:

Entropy from index (cont’d)Pick up “degeneracy”:

Entropy w/ large quantum numbers:

BH entropy:

Agrees in the large-N limit!!

Quantum black hole entropy?
𝑁2 −1
In terms of the central charge 𝑐 = 4 ,(True also for other gauge group and orbifold N=4 SYM)

Quantum black hole entropy?
𝑁2 −1
In terms of the central charge 𝑐 = 4 ,(True also for other gauge group and orbifold N=4 SYM)

Somehow this is exactly the same as the BH formula
if we slightly modify the AdS/CFT dictionary:Non-renormalization against quantum corrections?

Other regime
1/2

𝐼𝑆 1 ×𝑆 3 → න𝑖𝜋𝑑 𝑁−1 𝑎 ⅇ 6𝜏𝜎 𝑉2 𝑎 +𝑖𝜋 𝜏+𝜎𝑉1 (𝑎)
2𝜏𝜎

−1/2

So far, Re𝑖<0.
𝜏𝜎What if Re𝑖𝜏𝜎>0 ? → maximize 𝑉2 (𝑎)Other regime
1/2

𝐼𝑆 1 ×𝑆 3 → න𝑖𝜋𝑑 𝑁−1 𝑎 ⅇ 6𝜏𝜎 𝑉2 𝑎 +𝑖𝜋 𝜏+𝜎𝑉1 (𝑎)
2𝜏𝜎

−1/2

So far, Re𝑖<0.
𝜏𝜎What if Re𝑖𝜏𝜎>0 ? → maximize 𝑉2 (𝑎)minima are not isolated& ∃flat directions
Stokes phenomena?

Subtleties
・In general, ∃complex saddle points
so we need to do proper saddle point analysis

・But “action” in the strict Cardy limit isnon-differentiable(∵ “action” is described by 𝜅 𝑥 = {𝑥}(1 − {𝑥})(1 − 2{𝑥}))・It seems ∃Stokes phenomena
as changing phases of chemical potentials
・Other works reported (de)confinement transition& partial deconfinement[Copetti-Grassi-Komargodski-Tizzano][Ardehali-Hong-Liu]

Oops!
Next slides include preliminary results!