Supersymmetric localization and black holes …qft.web/2019/slides/hosseini.pdfIntroduction No...
Transcript of 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!
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 × 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]
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) .
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 .
Seyed Morteza Hosseini (Kavli IPMU) 5 / 26
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]
Seyed Morteza Hosseini (Kavli IPMU) 6 / 26
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]
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]
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 .
Seyed Morteza Hosseini (Kavli IPMU) 7 / 26
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]
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 .
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.
Seyed Morteza Hosseini (Kavli IPMU) 9 / 26
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 .
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] .
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
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→∞.
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] .
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
Supersymmetric localization
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.
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
∮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.
Seyed Morteza Hosseini (Kavli IPMU) 13 / 26
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]
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)
(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 .
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)
(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 .
Seyed Morteza Hosseini (Kavli IPMU) 15 / 26
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 .
Seyed Morteza Hosseini (Kavli IPMU) 16 / 26
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]
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) .
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]
Seyed Morteza Hosseini (Kavli IPMU) 18 / 26
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
.
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) .
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) .
Seyed Morteza Hosseini (Kavli IPMU) 20 / 26
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 .
Seyed Morteza Hosseini (Kavli IPMU) 21 / 26
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]
Seyed Morteza Hosseini (Kavli IPMU) 22 / 26
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.]
Seyed Morteza Hosseini (Kavli IPMU) 23 / 26
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.]
Seyed Morteza Hosseini (Kavli IPMU) 23 / 26
What we have learned by now?
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 .
I IKN-AdS6(∆a, ωi) ∝F(∆a)
ω1ω2, w/ ∆1 + ∆2 − ω1 − ω2 = 2.
I 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?
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 .
I IKN-AdS6(∆a, ωi) ∝F(∆a)
ω1ω2, w/ ∆1 + ∆2 − ω1 − ω2 = 2.
I 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 .
I IKN-AdS7(∆a, ωi) ∝F(∆a)
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)]
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 .
I IKN-AdS7(∆a, ωi) ∝F(∆a)
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)]
Seyed Morteza Hosseini (Kavli IPMU) 25 / 26
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!
Seyed Morteza Hosseini (Kavli IPMU) 26 / 26
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!
Seyed Morteza Hosseini (Kavli IPMU) 26 / 26