Supersymmetric localization and black holes microstates Seyed - - PowerPoint PPT Presentation

Supersymmetric localization and black 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 interpretation

• f SBH for a class of asymptotically flat black holes.

[Strominger, Vafa’96]

Black holes → bound states of D-branes!

Seyed Morteza Hosseini (Kavli IPMU) 2 / 26

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.

Seyed Morteza Hosseini (Kavli IPMU) 3 / 26

Basics

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

◮ Case I has to rotate. ◮ Case II is topologically twisted and can be static.

◮ 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]

Seyed Morteza Hosseini (Kavli IPMU) 4 / 26

Counting microstates

BPS partition function Z(∆I, ωi) = TrQ=0 ei(∆IQI+ωiJi) =

• QI,Ji

dmicro(QI, Ji)ei(∆IQI+ωiJi) .

◮ It counts states w/ the same susy, charges, and angular momenta. ◮ 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) = log Z(∆I, ωi) − i(∆IQI + ωiJi) .

◮ ∂I(∆I, ωi)

∂∆I = ∂I(∆I, ωi) ∂ωi = 0 .

Seyed Morteza Hosseini (Kavli IPMU) 5 / 26

Counting microstates

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

Seyed Morteza Hosseini (Kavli IPMU) 6 / 26

Counting microstates

Problem AdS BHs preserve only two real supercharges while we have efficient tools for counting states preserving four.. Witten index (supersymmetric partition function) Zsusy

Md−1×S1(∆I, ωi) = TrHMd−1 (−1)F e−β{Q,Q†}ei(∆IQI+ωiJi) . ◮ Superconformal index for SCFTs on Sd−1 × S1

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

◮ 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 large cancellations between bosonic and fermionic ground states.

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

Seyed Morteza Hosseini (Kavli IPMU) 6 / 26

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]

◮ Preserve two real supercharges (1/16 BPS) ◮ Four electric and magnetic charges (pa, qa) under U(1)4 ⊂ SO(8),

• ne angular momentum J in AdS4.

◮ Only seven independent parameters:

twisting condition:

4

• a=1

pa = 2 − 2g . together with a charge constraint for having a regular horizon.

◮ SBH = O(N 3/2) . ◮ We focus on J = 0. ◮ Near horizon AdS2 × Σg .

Seyed Morteza Hosseini (Kavli IPMU) 7 / 26

Magnetic AdS black holes

Setting all qa = 0 SBH(p) = 2π 3 N 3/2

• F2 +

√ Θ , F2 ≡ 1 2

• a<b

papb − 1 4

4

• a=1

p2

a ,

Θ ≡ (F2)2 − 4p1p2p3p4 .

◮ Attractor mechanism:

SBH(pa, qa) = ipa ∂ W(∆a) ∂∆a − i∆aqa

• crit. .

◮ g-sugra prepotential:

W(∆a) = −2i√∆1∆2∆3∆4 .

◮

a ∆a = 2π: scalar fields at the horizon. [Ferrara, Kallosh, Strominger’ 06; Cacciatori, Klemm’08; Dall’Agata, Gnecchi’10]

Seyed Morteza Hosseini (Kavli IPMU) 8 / 26

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 .

◮ Magnetic background for global symmetries: Landau levels on S2. ◮ Twisting condition: 4 a=1 pa = 2 .

Dµǫ = ∂µǫ + 1 4ωab

µ γabǫ + i

Vµ

• i

4 ωab µ γab

ǫ = ∂µǫ ǫ = constant on S2.

Seyed Morteza Hosseini (Kavli IPMU) 9 / 26

Holographic microstates counting

Hp,σ QM A topologically twisted index ZS2×S1

β(va, pa) = TrHS2 (−1)F e−βHei 4 a=1 ∆aQa .

[Benini, Zaffaroni; 1504.03698]

◮ ∆a : chemical potentials for flavor symmetry charges Qa. ◮ σa : real masses. ◮ only states with 0 = H − σaJa contribute. ◮ electric charges qa can be introduced using ∆a. ◮ can be computed using supersymmetric localization.

The index is a holomorphic function of va with va = ∆a + iβσa. σa = 0 .

Seyed Morteza Hosseini (Kavli IPMU) 10 / 26

Supersymmetric localization

Consider a supersymmetric gauge theory on a compact manifold M. Partition function ZM ≡ Euclidean Feynman path integral =

• Dφ e−S[φ] .

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

Seyed Morteza Hosseini (Kavli IPMU) 11 / 26

Supersymmetric localization

Consider a supersymmetric gauge theory on a compact manifold M. Partition function ZM ≡ Euclidean Feynman path integral =

• Dφ e−S[φ] .

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

Localization argument

[Witten’88; Pestun’06]

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

ZM(t) =

• Dφ e−S[φ]−tδV ,

t ∈ R>0 .

Seyed Morteza Hosseini (Kavli IPMU) 11 / 26

Supersymmetric localization

Consider a supersymmetric gauge theory on a compact manifold M. Partition function ZM ≡ Euclidean Feynman path integral =

• Dφ e−S[φ] .

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

Localization argument

[Witten’88; Pestun’06]

◮ Let δ be a Grassmann-odd symmetry of our theories, i.e. δS = 0. ◮ 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

Supersymmetric localization

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

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

φ = φ0 + t−1/2 ˆ φ .

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

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

◮ Gaussian integration!

Localization formula ZM =

• (δV )|even=0

Dφ0 e−S[φ0]Z1-loop[φ0] .

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

Seyed Morteza Hosseini (Kavli IPMU) 12 / 26

A topologically twisted index

Localization formula

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

ZS2×S1(p, y) = 1 |W|

• m ∈ Γh
• C

Zint (m, x; p, y) ,

◮ x = eiu, ya = ei∆a. ◮ Classical piece:

Zcl = xkm .

◮ One-loop contributions:

Zχ

1-loop =

• ρ∈R

√xρya 1 − xρya ρ(m)−pa+1 , ZV

1-loop =

• α∈G

(1 − xα) . We are interested in the large N limit of the matrix integral.

Seyed Morteza Hosseini (Kavli IPMU) 13 / 26

TQFT and Bethe vacua

2D Reduction to two-dimensional theory w/ all KK modes on S1

[Witten’92; Nekrasov, Shatashvili’09]

◮ 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 written as a sum over Bethe vacua.

[Closset, Kim, Willett’17’18]

Seyed Morteza Hosseini (Kavli IPMU) 14 / 26

A topologically twisted index

Bethe sum formula: ZS2×S1(p, y) = (−1)rk(G) |W|

• x∗

Zint (m = 0, x∗; p, y)

• det

ij ∂i∂jW(x)

−1 .

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

Seyed Morteza Hosseini (Kavli IPMU) 15 / 26

A topologically twisted index

Bethe sum formula: ZS2×S1(p, y) = (−1)rk(G) |W|

• x∗

Zint (m = 0, x∗; p, y)

• det

ij ∂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

(˜ u2

i −u2 i )+ N

• i,j=1
• 4
• b=3

Li2

• ei(˜

uj−ui+∆b)

−

2

• a=1

Li2

• ei(˜

uj−ui−∆a)

.

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

ui = iN 1/2ti + vi , ˜ ui = iN 1/2ti + ˜ vi .

Seyed Morteza Hosseini (Kavli IPMU) 15 / 26

I-extremization principle

In the large N limit

[Benini, Hristov, Zaffaroni’15]

I(∆a, pa) ≡ log ZS2×S1(∆a, pa) − i

4

• a=1

∆aqa

• crit.

=

4

• a=1

ipa ∂ W(∆a) ∂∆a − i∆aqa

• crit.

.

◮ W(x∗) ≡

W(∆a) = 2i 3 N 3/2 2∆1∆2∆3∆4 .

◮ 4 a=1 ∆a = 2π with Re ∆a ∈ [0, 2π] .

Localization meets holography: W(x∗) ↔ prepotential of 4D N = 2 g-sugra . I-extremization ↔ attractor mechanism .

Seyed Morteza Hosseini (Kavli IPMU) 16 / 26

Generalizations

◮ 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: log ZS2×S1(∆a, pa) = −1 2

• a

pa ∂FS3(∆a) ∂∆a .

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

◮ 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]

◮ Localization in supergravity.

[Hristov, Lodato, Reys’17]

◮ 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]

◮ Black hole thermodynamics: log ZSCFT = Isugra

• n-shell.

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

Seyed Morteza Hosseini (Kavli IPMU) 17 / 26

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) .

◮ F(QI, Ji) = 0 ⇒ four independent conserved charges. ◮ They must rotate. ◮ 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(N 2) .

[Kim, Lee’06]

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

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

Seyed Morteza Hosseini (Kavli IPMU) 18 / 26

Entropy function for AdS5 black holes

BPS entropy function SBH(QI, Ji) = −πi(N 2 −1)∆1∆2∆3 ω1ω2 −2πi

• 3
• I=1

∆IQI −

2

• i=1

ωiJi

• crit.

.

◮ ∆1 + ∆2 + ∆3 − ω1 − ω2 = ±1 . ◮ Complex critical points but SBH(QI, Ji) is real at the extremum!

[SMH, Hristov, Zaffaroni’17]

Black hole thermodynamics:

◮ The critical points can be obtained by taking an appropriate zero

temperature limit of a family of supersymmetric Euclidean BHs.

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

−πi(N 2 − 1)∆1∆2∆3 ω1ω2 = Isugra

• n-shell .

Seyed Morteza Hosseini (Kavli IPMU) 19 / 26

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) .

◮ # of fugacities = # of conserved charges,

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

3

• I=1

yI = pq .

◮ For real fugacities log Z(∆I, ωi) = O(1).

[Kinney, Maldacena, Minwalla, Raju’05]

Localization formula

[e.g. Spiridonov, Vartanov’10]

Z(∆I, ωi) = A

• N−1
• i=1

dzi 2πizi

• 1≤j<j≤N

3

I=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−1

e

(yI; p, q) .

Seyed Morteza Hosseini (Kavli IPMU) 20 / 26

A puzzle!

Problem Large cancellations between bosonic and fermionic states.

◮ The critical points of the BPS entropy function are complex. ◮ Phases may obstruct the cancellations in the index. ◮ 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: log Z(∆I, ωi) ∼ −πiN 2 ∆1∆2∆3 ω1ω2 ,

2

• I=1

∆I −

2

• i=1

ωi = ±1 .

Seyed Morteza Hosseini (Kavli IPMU) 21 / 26

Generalizations

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

log Z ∼ 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 Gonz´ alez Lezcano, Pando Zayas; Lanir, Nedelin, Sela’19]

◮ BPS entropy functions for AdS7, AdS6, and AdS4 black holes.

[SMH, Hristov, Zaffaroni’18, Choi, Hwang, Kim, Nahmgoong’18; Cassani, Papini’19]

◮ Similar computations of the SCI in various dimensions.

[Choi, Kim, Kim, Nahmgoong’18; Choi, Kim’19; K´ antor, Papageorgakis, Richmond’19; Choi, Hwang, Kim’19]

◮ Near BPS entropy function.

[Larsen, Nian, Zeng’19]

Seyed Morteza Hosseini (Kavli IPMU) 22 / 26

What we have learned by now?

◮ A unique function, F(∆a), controls the entropy of both

KN-AdSd+1 and mAdSd+1 black holes/strings.

Seyed Morteza Hosseini (Kavli IPMU) 23 / 26

What we have learned by now?

◮ A unique function, F(∆a), controls the entropy of both

KN-AdSd+1 and mAdSd+1 black holes/strings. 4D N = 2 g-sugra F(∆a) ∝ FS3(∆a) , F ABJM

S3

(∆a) ∝

• ∆1∆2∆3∆4 .

◮ IeKN-AdS4(∆a, ω) ∝ F(∆a)

ω , w/

a ∆a − ω = 2. ◮ ImAdS4(∆a, pa) ∝

• a

pa ∂F(∆a) ∂∆a , w/

a ∆a = 2.

[See “Generalization” slides for references.]

Seyed Morteza Hosseini (Kavli IPMU) 23 / 26

What we have learned by now?

5D N = 2 g-sugra F(∆a) ∝ a4D(∆a) , aN =4

4D

(∆a) ∝ ∆1∆2∆3 .

◮ IKN-AdS5(∆a, ωi) ∝ F(∆a)

ω1ω2 , w/

a ∆a − ω1 − ω2 = 2. ◮ IAdS5 BS(∆a, pa) ∝

• a

pa ∂F(∆a) ∂∆a , w/

a ∆a = 2.

Seyed Morteza Hosseini (Kavli IPMU) 24 / 26

What we have learned by now?

5D N = 2 g-sugra F(∆a) ∝ a4D(∆a) , aN =4

4D

(∆a) ∝ ∆1∆2∆3 .

◮ IKN-AdS5(∆a, ωi) ∝ F(∆a)

ω1ω2 , w/

a ∆a − ω1 − ω2 = 2. ◮ IAdS5 BS(∆a, pa) ∝

• a

pa ∂F(∆a) ∂∆a , w/

a ∆a = 2.

F(4) g-sugra F(∆a) ∝ FS5(∆a) , F USp(2N)

S5

(∆a) ∝ (∆1∆2)3/2 .

◮ IKN-AdS6(∆a, ωi) ∝ F(∆a)

ω1ω2 , w/ ∆1 + ∆2 − ω1 − ω2 = 2.

◮ ImAdS6(∆a, pa) ∝ 2

• a,b=1

pasb ∂2F(∆a) ∂∆a∂∆b , w/ ∆1 + ∆2 = 2.

Seyed Morteza Hosseini (Kavli IPMU) 24 / 26

What we have learned by now?

7D N = 2 g-sugra F(∆a) ∝ a6D(∆a) , a(2,0)

6D (∆a) ∝ (∆1∆2)2 . ◮ IKN-AdS7(∆a, ωi) ∝ F(∆a)

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

◮ IAdS7 BS(∆a, pa) ∝ 2

• a,b=1

pasb ∂2F(∆a) ∂∆a∂∆b , w/ ∆1 + ∆2 = 2.

Seyed Morteza Hosseini (Kavli IPMU) 25 / 26

What we have learned by now?

7D N = 2 g-sugra F(∆a) ∝ a6D(∆a) , a(2,0)

6D (∆a) ∝ (∆1∆2)2 . ◮ IKN-AdS7(∆a, ωi) ∝ F(∆a)

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

◮ IAdS7 BS(∆a, pa) ∝ 2

• a,b=1

pasb ∂2F(∆a) ∂∆a∂∆b , w/ ∆1 + ∆2 = 2. Food for thought

◮ Attractor mechanism for black objects in various dimensions.

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

Seyed Morteza Hosseini (Kavli IPMU) 25 / 26

Outlook

◮ Other black holes in AdS5? ◮ Dyonic KN-AdS4 black holes.

[Hristov, Katmadas, Toldo’19]

◮ Black holes microstates in AdS4 × SE7. Problems w/ large N.. ◮ Rotating magnetic AdS4 black holes.

[Hristov, Katmadas, Toldo’18]

◮ Finite N corrections. ◮ . . .

Seyed Morteza Hosseini (Kavli IPMU) 26 / 26

Outlook

◮ Other black holes in AdS5? ◮ Dyonic KN-AdS4 black holes.

[Hristov, Katmadas, Toldo’19]

◮ Black holes microstates in AdS4 × SE7. Problems w/ large N.. ◮ Rotating magnetic AdS4 black holes.

[Hristov, Katmadas, Toldo’18]

◮ Finite N corrections. ◮ . . .

Thank you for your attention!

Seyed Morteza Hosseini (Kavli IPMU) 26 / 26