supersymmetric localization and black holes …qft.web/2019/slides/hosseini.pdfintroduction no...

Supersymmetric localization andblack holes microstates

Seyed Morteza Hosseini

Kavli IPMU

YITP (Kyoto), August 19-23

Strings and Fields 2019

Seyed Morteza Hosseini (Kavli IPMU) 1 / 26

Introduction

Black holes have more lessons in store for us!

Bekenstein-Hawking entropy: SBH =Area

4GN.

The number of black hole microstates dmicro should then be given by

dmicro = eSBH .

But where are the microstates accounting for the black hole entropy?

String theory provides a precise statistical mechanical interpretationof SBH for a class of asymptotically flat black holes. [Strominger, Vafa’96]

Black holes → bound states of D-branes!


Introduction

No similar results for AdSd+1>4 black holes was known until recently![Benini, Hristov, Zaffaroni’15]

Holography + supersymmetric localization

Black hole entropy → counting states in the dual CFT

This talk

I will review recent progress for AdSd+1 BHs in diverse dimensions.


Basics

Stringy BPS black holes

I KN-AdS black holes ↔ SCFTd on Sd−1 × RtII magnetic AdS black holes ↔ SCFTd on Md−1 × Rt

I Case I has to rotate.

I Case II is topologically twisted and can be static.I Characterized by nonzero magnetic fluxes for the

graviphoton/R-symmetry:∫C⊂Md−1

F ∈ 2πZ .

Most manifest in AdS4 black holes w/ horizon AdS2 × S2. [Romans’92]


Counting microstates

BPS partition function

Z(∆I , ωi) = TrQ=0 ei(∆IQI+ωiJi) =

∑QI ,Ji

dmicro(QI , Ji)ei(∆IQI+ωiJi) .

I It counts states w/ the same susy, charges, and angular momenta.

I SBH(QI , Ji) = log dmicro(QI , Ji) ,

dmicro(QI , Ji) = eSBH(QI ,Ji) =

∫∆I , ωi

Z(∆I , ωi)e−i(∆IQI+ωiJi) .

Saddle point approximation (large charges)

SBH(QI , Ji) ≡ I(∆I , ωi) = logZ(∆I , ωi)− i(∆IQI + ωiJi) .

I∂I(∆I , ωi)

∂∆I=∂I(∆I , ωi)

∂ωi= 0 .


Counting microstates

Problem

AdS BHs preserve only two real supercharges while we have efficienttools for counting states preserving four..

Witten index (supersymmetric partition function)

ZsusyMd−1×S1(∆I , ωi) = TrHMd−1

(−1)F e−β{Q,Q†}ei(∆IQI+ωiJi) .

I Superconformal index for SCFTs on Sd−1 × S1

[Romelsberger’05; Kinney, Maldacena, Minwalla, Raju’05]

I Topologically twisted index for SCFTs on twisted Md−1 × S1

[Okuda, Yoshida’12; Nekrasov, Shatashvili’14; Gukov, Pei’15; Benini, Zaffaroni’15]

Lower bound on entropy. Index = entropy if there are no largecancellations between bosonic and fermionic ground states.

[Arguments for some asymptotically flat black holes by Sen’09]


Magnetic AdS black holes

Black holes in M-theory on AdS4 × S7:[Cacciatori, Klemm’08; Dall’Agata, Gnecchi’10; Hristov, Vandoren’10; Halmagyi14; Hristov, Katmadas, Toldo’18]

I Preserve two real supercharges (1/16 BPS)

I Four electric and magnetic charges (pa, qa) under U(1)4 ⊂ SO(8),one angular momentum J in AdS4.

I Only seven independent parameters:

twisting condition:

4∑a=1

pa = 2− 2g .

together with a charge constraint for having a regular horizon.

I SBH = O(N3/2) .

I We focus on J = 0.

I Near horizon AdS2 × Σg .


Magnetic AdS black holes

Setting all qa = 0

SBH(p) =2π

3N3/2

√F2 +

√Θ ,

F2 ≡1

2

∑a<b

papb −1

4

4∑a=1

p2a , Θ ≡ (F2)2 − 4p1p2p3p4 .

I Attractor mechanism:

SBH(pa, qa) = ipa∂W(∆a)

∂∆a− i∆aqa

∣∣∣crit.

.

I g-sugra prepotential: W(∆a) = −2i√

∆1∆2∆3∆4 .I∑

a ∆a = 2π: scalar fields at the horizon.

[Ferrara, Kallosh, Strominger’ 06; Cacciatori, Klemm’08; Dall’Agata, Gnecchi’10]


Holographic setup

ABJM on S2 × R w/ a twist on S2

N+k N−k

B2

A1

B1

A2 W = Tr(A1B1A2B2 −A1B2A2B1

),

∆1 + ∆2 + ∆3 + ∆4 = 2π ,

U(1)R × SU(2)1 × SU(2)2 ×U(1)top .

I Magnetic background for global symmetries: Landau levels on S2.

I Twisting condition:∑4a=1 p

a = 2 .

Dµε = ∂µε+1

4ωabµ γabε+ i Vµ︸︷︷︸

i4ω

abµ γab

ε = ∂µε

ε = constant on S2.


Holographic microstates counting

Hp,σ

QM

A topologically twisted index

ZS2×S1β(va, p

a) = TrHS2 (−1)F e−βHei∑4a=1 ∆aQa .

[Benini, Zaffaroni; 1504.03698]

I ∆a : chemical potentials for flavor symmetry charges Qa.

I σa : real masses.

I only states with 0 = H − σaJa contribute.

I electric charges qa can be introduced using ∆a.

I can be computed using supersymmetric localization.

The index is a holomorphic function of va with va = ∆a + iβσa.

σa = 0 .


Supersymmetric localization

Consider a supersymmetric gauge theory on a compact manifold M.

Partition function

ZM ≡ Euclidean Feynman path integral =

∫Dφ e−S[φ] .

I φ: the set of fields in the theory.I S[φ]: the action functional.

Localization argument [Witten’88; Pestun’06]

I Let δ be a Grassmann-odd symmetry of our theories, i.e. δS = 0.I Deform the theories by a δ-exact term.

ZM(t) =

∫Dφ e−S[φ]−tδV , t ∈ R>0 .

The partition function is independent of t!

∂ZM(t)

∂t= −

∫Dφ e−S[φ]−tδV δV = −

∫Dφ δ

(e−S[φ]−tδV V

)= 0 .

Hence we can evaluate ZM(t) as t→∞.



Consider a supersymmetric gauge theory on a compact manifold M.

Partition function

ZM ≡ Euclidean Feynman path integral =

∫Dφ e−S[φ] .

I φ: the set of fields in the theory.I S[φ]: the action functional.

Localization argument [Witten’88; Pestun’06]

I Let δ be a Grassmann-odd symmetry of our theories, i.e. δS = 0.I Deform the theories by a δ-exact term.

ZM(t) =

∫Dφ e−S[φ]−tδV , t ∈ R>0 .

The partition function is independent of t!

∂ZM(t)

∂t= −

∫Dφ e−S[φ]−tδV δV = −

∫Dφ δ

(e−S[φ]−tδV V

)= 0 .

Hence we can evaluate ZM(t) as t→∞.Seyed Morteza Hosseini (Kavli IPMU) 11 / 26


Localization locus

If (δV )|even ≥ 0 =⇒ the integral localizes to (δV )|even(φ0) = 0 .

I Let’s parameterize the fields around the localization locus by

φ = φ0 + t−1/2φ .

I For large t, we can Taylor expand the action around φ0:

S + δV = S[φ0] + (δV )(2)[φ] +O(t−1/2) .

I Gaussian integration!

Localization formula

ZM =

∫(δV )|even=0

Dφ0 e−S[φ0]Z1-loop[φ0] .

I Z1-loop[φ0]: the ratio of fermionic and bosonic determinants.



Localization formula [Benini, Zaffaroni’15; Closset, Kim, Willett’16]

ZS2×S1(p, y) =1

|W|∑

m∈Γh

∮CZint (m, x; p, y) ,

I x = eiu, ya = ei∆a .

I Classical piece:Zcl = xkm .

I One-loop contributions:

Zχ1-loop =∏ρ∈R

( √xρya

1− xρya

)ρ(m)−pa+1

, ZV1-loop =

∏α∈G

(1− xα) .

We are interested in the large N limit of the matrix integral.


TQFT and Bethe vacua

2D

Reduction to two-dimensionaltheory w/ all KK modes on S1

[Witten’92; Nekrasov, Shatashvili’09]

I Massive theory w/ a set of discrete vacua (Bethe vacua),

exp

(i∂W(x)

∂x

) ∣∣∣∣x=x∗

= 1 , W(x, ya) =∑ρ∈R

Li2(xρya) + . . . .

Many 3D and 4D supersymmetric partition functions can be writtenas a sum over Bethe vacua. [Closset, Kim, Willett’17’18]



Bethe sum formula:

ZS2×S1(p, y) =(−1)rk(G)

|W|∑x∗

Zint (m = 0, x∗; p, y)

(detij∂i∂jW(x)

)−1

.

[Okuda, Yoshida’12; Nekrasov, Shatashvili’14; Gukov, Pei’15; Benini, Zaffaroni’15; Closset, Kim, Willett’17]

For ABJM:

W =k

2

N∑i=1

(u2i−u2

i )+

N∑i,j=1

[ 4∑b=3

Li2(ei(uj−ui+∆b)

)−

2∑a=1

Li2(ei(uj−ui−∆a)

)].

I At large N one Bethe vacuum dominates the partition function.

ui = iN1/2ti + vi , ui = iN1/2ti + vi .


I-extremization principle

In the large N limit [Benini, Hristov, Zaffaroni’15]

I(∆a, pa) ≡ logZS2×S1(∆a, p

a)− i

4∑a=1

∆aqa

∣∣∣∣crit.

=

4∑a=1

ipa∂W(∆a)

∂∆a− i∆aqa

∣∣∣∣crit.

.

I W(x∗) ≡ W(∆a) =2i

3N3/2

√2∆1∆2∆3∆4 .

I∑4a=1 ∆a = 2π with Re ∆a ∈ [0, 2π] .

Localization meets holography:

W(x∗)↔ prepotential of 4D N = 2 g-sugra .

I-extremization ↔ attractor mechanism .


Generalizations

I Other AdS4 black holes in M-theory or massive type IIA.[SMH, Hristov, Passias’17; Benini, Khachatryan, Milan’17; Azzurli, Bobev, Crichigno, Min, Zaffaroni’17; Bobev, Min,

Pilch’18; Gauntlett, Martelli, Sparks’19; SMH, Zaffaroni’19]

An index theorem: logZS2×S1(∆a, pa) = −1

2

∑a

pa∂FS3(∆a)

∂∆a.

[SMH, Zaffaroni’16; SMH, Mekareeya’16]

I Subleading corrections in N .[Liu, Pando Zayas, Rathee, Zhao’17; Liu, Pando Zayas, Zhou’18; SMH’18; Gang, Kim, Pando Zayas’19; Bae, Gang,

Lee’19]

I Localization in supergravity. [Hristov, Lodato, Reys’17]

I Black holes and black strings in higher dimensions.[SMH, Nedelin, Zaffaroni’16; Hong, Liu’16; SMH, Yaakov, Zaffaroni’18; Crichigno, Jain, Willett’18; SMH, Hristov,

Passias, Zaffaroni’18; Suh’18; Fluder, SMH, Uhlemann’19; Bae, Gang, Lee’19]

I Black hole thermodynamics: logZSCFT = Isugra

∣∣∣on-shell

.

[Azzurli, Bobev, Crichigno, Min, Zaffaroni’17; Halmagyi, Lal’17; Cabo-Bizet, Kol, Pando Zayas, Papadimitriou, Rathee’17]


KN-AdS5 black holes

Solutions of 5D, N = 1 U(1)3 gauged supergravity

BPS black holes in AdS5 × S5 (w/ boundary S3 × Rt — no twist){Two angular momenta Ji in AdS5 U(1)2 ⊂ SO(4) ,Three electric charges QI in S5 U(1)3 ⊂ SO(6) .

I F (QI , Ji) = 0 ⇒ four independent conserved charges.

I They must rotate.

I Asymptotically global AdS5 → near horizon AdS2 ×w S3 .

[Gutowski, Reall’04; Chong, Cvetic, Lu, Pope’05; Kunduri, Lucietti, Reall’06]

SBH = 2π

√Q1Q2 +Q2Q3 +Q1Q3 −

π

4GN(J1 + J2) = O(N2) .

[Kim, Lee’06]

I dmicro = states of given Ji and QI in N = 4 super Yang-Mills.

[Hairy black hols by Markeviciute, Santos’16’18]


Entropy function for AdS5 black holes

BPS entropy function

SBH(QI , Ji) = −πi(N2−1)∆1∆2∆3

ω1ω2−2πi

( 3∑I=1

∆IQI−2∑i=1

ωiJi

)∣∣∣∣crit.

.

I ∆1 + ∆2 + ∆3 − ω1 − ω2 = ±1 .

I Complex critical points but SBH(QI , Ji) is real at the extremum!

[SMH, Hristov, Zaffaroni’17]

Black hole thermodynamics:

I The critical points can be obtained by taking an appropriate zerotemperature limit of a family of supersymmetric Euclidean BHs.

[Cabo-Bizet, Cassani, Martelli, Murthy’18]

−πi(N2 − 1)∆1∆2∆3

ω1ω2= Isugra

∣∣∣on-shell

.


A puzzle!

Superconformal index on S3 × S1[Romelsberger’05; Kinney, Maldacena, Minwalla, Raju’05]

Z(∆I , ωi) = TrHS3 (−1)F e−β{Q,Q†}e2πi(

∑I ∆IQI+

∑i ωiJi) .

I # of fugacities = # of conserved charges,

p = e2πiω1 , q = e2πiω2 , yI = e2πi∆I ,

3∏I=1

yI = pq .

I For real fugacities logZ(∆I , ωi) = O(1). [Kinney, Maldacena, Minwalla, Raju’05]

Localization formula [e.g. Spiridonov, Vartanov’10]

Z(∆I , ωi) = A∮ N−1∏

i=1

dzi2πizi

∏1≤j<j≤N

∏3I=1 Γe

(yI(zi/zj)

±1; p, q)

Γe((zi/zj)±1; p, q

) ,

A ≡((p; p)∞(q; q)∞

)N−1

N !

3∏I=1

ΓN−1e (yI ; p, q) .


A puzzle!

Problem

Large cancellations between bosonic and fermionic states.

I The critical points of the BPS entropy function are complex.

I Phases may obstruct the cancellations in the index.

I Stokes phenomena.

[Cardy limit by Choi, Kim, Kim, Nahmgoong’18]

[Modified index by Cabo-Bizet, Cassani, Martelli, Murthy’18]

[Large N using Bethe sum formula by Benini, Milan’18]

Final result:

logZ(∆I , ωi) ∼ −πiN2 ∆1∆2∆3

ω1ω2,

2∑I=1

∆I −2∑i=1

ωi = ±1 .


Generalizations

I 4D N = 1 gauge theories (equal charges)

logZ ∼ 2πi∆3

ω1ω2(3c− 2a) + 2πi

∆

ω1ω2(a− c) +O(1) ,

3∆− ω1 − ω2 = ±1 .

[Generalize Di Pietro, Komargodski’14][Kim, Kim, Song’19; Cabo-Bizet, Cassani, Martelli, Murthy’19; Amariti, Garozzo, Lo

Monaco’19][Large N by Gonzalez Lezcano, Pando Zayas; Lanir, Nedelin, Sela’19]

I BPS entropy functions for AdS7, AdS6, and AdS4 black holes.[SMH, Hristov, Zaffaroni’18, Choi, Hwang, Kim, Nahmgoong’18; Cassani, Papini’19]

I Similar computations of the SCI in various dimensions.[Choi, Kim, Kim, Nahmgoong’18; Choi, Kim’19; Kantor, Papageorgakis, Richmond’19; Choi, Hwang, Kim’19]

I Near BPS entropy function. [Larsen, Nian, Zeng’19]


What we have learned by now?

I A unique function, F(∆a), controls the entropy of bothKN-AdSd+1 and mAdSd+1 black holes/strings.

4D N = 2 g-sugra

F(∆a) ∝ FS3(∆a) , FABJMS3 (∆a) ∝

√∆1∆2∆3∆4 .

I IeKN-AdS4(∆a, ω) ∝ F(∆a)

ω, w/

∑a ∆a − ω = 2.

I ImAdS4(∆a, pa) ∝

∑a

pa∂F(∆a)

∂∆a, w/

∑a ∆a = 2.

[See “Generalization” slides for references.]



5D N = 2 g-sugra

F(∆a) ∝ a4D(∆a) , aN=44D (∆a) ∝ ∆1∆2∆3 .

I IKN-AdS5(∆a, ωi) ∝F(∆a)

ω1ω2, w/

∑a ∆a − ω1 − ω2 = 2.

I IAdS5 BS(∆a, pa) ∝

∑a

pa∂F(∆a)

∂∆a, w/

∑a ∆a = 2.

F (4) g-sugra

F(∆a) ∝ FS5(∆a) , FUSp(2N)S5 (∆a) ∝ (∆1∆2)3/2 .


ω1ω2, w/ ∆1 + ∆2 − ω1 − ω2 = 2.

I ImAdS6(∆a, pa) ∝

2∑a,b=1

pasb∂2F(∆a)

∂∆a∂∆b, w/ ∆1 + ∆2 = 2.



7D N = 2 g-sugra

F(∆a) ∝ a6D(∆a) , a(2,0)6D (∆a) ∝ (∆1∆2)2 .


ω1ω2ω3, w/ ∆1 + ∆2 − ω1 − ω2 − ω3 = 2.

I IAdS7 BS(∆a, pa) ∝

2∑a,b=1

pasb∂2F(∆a)

∂∆a∂∆b, w/ ∆1 + ∆2 = 2.

Food for thought

I Attractor mechanism for black objects in various dimensions.

[SMH, Hristov, Zaffaroni (work in progress)]


Outlook

I Other black holes in AdS5?

I Dyonic KN-AdS4 black holes. [Hristov, Katmadas, Toldo’19]

I Black holes microstates in AdS4 × SE7. Problems w/ large N ..

I Rotating magnetic AdS4 black holes. [Hristov, Katmadas, Toldo’18]

I Finite N corrections.

I . . .

Thank you for your attention!


supersymmetric localization and black holes …qft.web/2019/slides/hosseini.pdfintroduction no...

Documents