Holography, Localization, Black Holes

Holography, localization, black holes

Alberto Zaﬀaroni

Milano-Bicocca

Paris, June 2016 2nd Workshop on String Theory and Gender

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 1 / 39 Introduction Introduction

Lot of recent activity in the study of supersymmetric and superconformal theories in various dimensions in closely related contexts

A deeply interconnected web of supersymmetric theories arising from branes, • most often strongly coupled, related by various types of dualities.

Progresses in the evaluation of exact quantum observables. Related to • localization and the study of supersymmetry in curved space.

Many results on indices and counting problems that can be also related to • BH physics.

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 2 / 39 Introduction Introduction A large superconformal zoo

(2,0) M5

Global symmetry enhancement 5d SCFT

AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories Several non conformal defs

Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT

2d SCFT

1d SCFT Black Hole entropy

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 3 / 39 Introduction Introduction A large superconformal zoo with holographic duals

(2,0) M5 AdS7

Global symmetry enhancement 5d SCFT AdS6

AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories AdS5 Several non conformal defs

Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4

2d SCFT AdS3

1d SCFT Black Hole entropy AdS2

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 4 / 39 Introduction Introduction A large superconformal zoo with holographic duals

(2,0) M5 AdS7

Global symmetry enhancement 5d SCFT AdS6

AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories AdS5 Several non conformal defs

Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4

2d SCFT AdS3

1d SCFT Black Hole entropy AdS2

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 5 / 39 Introduction Introduction

2 Supersymmetric black holes have horizon AdS2 S ×

CFT1 living at the horizon of black holes •

density of states = BH entropy •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 6 / 39 Introduction Introduction

Conformal algebra in 1d involves SL(2, R)

H , D, K

with various superconformal extensions.

AdS2/CFT1 still poorly understood. •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 7 / 39 Introduction Introduction Supersymmetric AdS black holes are also related by holography to states of CFT3 and CFT4

(2,0) M5

Global symmetry enhancement 5d SCFT

AGT correspondence N=4 SYM 4d SCFT AdSN=2/N=15 rotating non Lagrangian BH theories Several non conformal defs

Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4 static BH

2d SCFT

1d SCFT Black Hole entropy

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 8 / 39 Introduction Introduction

Entropy of AdS BH should be related to the counting of supersymmetric states in CFT:

We still don’t know where to look for AdS5: people tried with • superconformal index, 1/4-BPS partition functions, ...

We have now an answer for AdS4: the index of the topologically twisted • theory

Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 9 / 39 In between a pletora of possibilities. TopologicalSCFT on Spheres Field Theories[Pestun 2007][Witten, 1988] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] n n JustN = use2 theories conformal in 4d map with fromSU(2)R RtoR-symmetryS can be consistently defined • on any M4 euclidean manifoldδψ (x) = + 0 µ ∇µ · · · ≡ The resulting theory is topological: independent of metric. • Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. T = Q, µν { ···} [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...]

Euclidean SCFT Euclidean SCFT in ﬁnite volume

To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold

Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 In between a pletora of possibilities. SCFT on Spheres [Pestun 2007] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] Just use conformal map from Rn to S n • δψ (x) = + 0 µ ∇µ · · · ≡ Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry.

[klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...]

Euclidean SCFT Euclidean SCFT in ﬁnite volume

To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold

Topological Field Theories [Witten, 1988]

N = 2 theories in 4d with SU(2)R R-symmetry can be consistently deﬁned • on any M4 euclidean manifold The resulting theory is topological: independent of metric. • T = Q, µν { ···}

Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 In between a pletora of possibilities. Topological Field Theories [Witten, 1988] A general method is to couple to supergravity and find solution of • [Festuccia-Seiberg] N = 2 theories in 4d with SU(2)R R-symmetry can be consistently defined • on any M4 euclidean manifoldδψ (x) = + 0 µ ∇µ · · · ≡ The resulting theory is topological: independent of metric. • Backgrounds with twisted (conformal) Killing spinors have been classified in various dimensions and with various amount of supersymmetry. T = Q, µν { ···} [klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...]

Euclidean SCFT Euclidean SCFT in ﬁnite volume

To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold

SCFT on Spheres [Pestun 2007]

Just use conformal map from Rn to S n •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 10 / 39 TopologicalSCFT on Spheres Field Theories[Pestun 2007][Witten, 1988]

n n JustN = use2 theories conformal in 4d map with fromSU(2)R RtoR-symmetryS can be consistently deﬁned • on any M4 euclidean manifold The resulting theory is topological: independent of metric. • T = Q, µν { ···}

Euclidean SCFT Euclidean SCFT in ﬁnite volume

To regularize IR physics we can put the theory on a Euclidean compact manifold. Supersymmetry restricts the manifold

In between a pletora of possibilities. A general method is to couple to supergravity and ﬁnd solution of • [Festuccia-Seiberg]

δψ (x) = + 0 µ ∇µ · · · ≡ Backgrounds with twisted (conformal) Killing spinors have been classiﬁed in various dimensions and with various amount of supersymmetry.

[klare,tomasiello,A.Z.;dumitruescu,festuccia,seiberg;Closser,Dumitrescu,Festuccia,Komargodski; ...]

Alberto Zaffaroni (Milano-Bicocca) CFT june 2016 10 / 39 R for theories with more supercharges and in different dimensions, Aa is • promoted to a non-abelian gauge field, there can be other backgrounds tensor fields and extra conditions.

Euclidean SCFT Superconformal theories on curved spaces

For SCFT, the variation of the gravitino can be written as a (twisted) conformal Killing equation

R A 1 A ( a iA )+ + γa− = 0 = + = γa + ∇ − a ⇒ ∇a d ∇

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 11 / 39 Euclidean SCFT Superconformal theories on curved spaces

For SCFT, the variation of the gravitino can be written as a (twisted) conformal Killing equation

R A 1 A ( a iA )+ + γa− = 0 = + = γa + ∇ − a ⇒ ∇a d ∇

R for theories with more supercharges and in different dimensions, Aa is • promoted to a non-abelian gauge field, there can be other backgrounds tensor fields and extra conditions.

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 11 / 39 Euclidean SCFT Superconformal theories on curved spaces

A possible solution is to have covariantly constant spinors

A 1 A A + = γa + solved by + = 0 ∇a d ∇ ∇a

Topological Field Theories can be realized this way by canceling the spin connection with a background R-symmetry

A 1 αβ αβ R + = ∂a+ + ω Γ + + A + = ∂a+ ∇a 4 a a

which can be solved by + = constant on any manifold

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 12 / 39 Euclidean SCFT Superconformal theories on curved spaces

The well know examples of Topological Field Theories back in the eighties

A 1 αβ αβ R + = ∂a+ + ω Γ + + A + = ∂a+ ∇a 4 a a

N = 2 theories on any M4: SU(2)R background gauge ﬁeld • + transform in the (2, 0) representation of the local Euclidean group SO(4) = SU(2) × SU(2)

A twist in 2d on any Riemann surface Σg : U(1)R background gauge ﬁeld •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 13 / 39 Euclidean SCFT Superconformal theories on curved spaces

A possible solution is to have Killing spinors

A 1 A + = γa + solved by a+ = γa+ ∇a d ∇ ∇

Realized for Theories on Spheres, with zero background ﬁeld AR = 0.

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 14 / 39 The path integral on S d S 1 can be written as a trace over a Hilbert space × P F −β{Q,S}+ ∆aJa Z d 1 = Tr( 1) e S ×S − This is the superconformal index: it counts BPS states graded by dimension and charges

[Romelsberger; Kinney,Maldacena,Minwalla,Rahu]

computed for large class of 4d SCFT [Gadde,Rastelli,Razamat,Yan]

Euclidean SCFT Superconformal theories on curved spaces

Or combinations: Killing spinors on S d and R- and ﬂavor holonomies on S 1

A 1 A + = γa + solved by a+ = γa+ , At = i ∇a d ∇ ∇

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 15 / 39 Euclidean SCFT Superconformal theories on curved spaces

Or combinations: Killing spinors on S d and R- and ﬂavor holonomies on S 1

A 1 A + = γa + solved by a+ = γa+ , At = i ∇a d ∇ ∇

The path integral on S d S 1 can be written as a trace over a Hilbert space × P F −β{Q,S}+ ∆aJa Z d 1 = Tr( 1) e S ×S − This is the superconformal index: it counts BPS states graded by dimension and charges

[Romelsberger; Kinney,Maldacena,Minwalla,Rahu]

computed for large class of 4d SCFT [Gadde,Rastelli,Razamat,Yan]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 15 / 39 1 The path integral on Σg S can be written as a Witten index (elliptic genus) × F F iJF A −βH Z 1 = TrH ( 1) e e Σg ×S −

2 F Q = H − σ JF holomorphic in AF + iσF

of the dimensional reduced theory on Σg and counts ground states graded by charges. This is the topologically twisted index.

[Benini,AZ; Closset-Kim]

Euclidean SCFT Superconformal theories on curved spaces

1 2 Or combinations: a topological twist on Σg and ﬂavor holonomies on S (or T )

A 1 A A R + = γa + solved by + = 0 , A = ωa ∇a d ∇ ∇a a

The integral R F R gives a magnetic charge for R-symmetry Σg

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 16 / 39 Euclidean SCFT Superconformal theories on curved spaces

1 2 Or combinations: a topological twist on Σg and ﬂavor holonomies on S (or T )

A 1 A A R + = γa + solved by + = 0 , A = ωa ∇a d ∇ ∇a a

The integral R F R gives a magnetic charge for R-symmetry Σg

1 The path integral on Σg S can be written as a Witten index (elliptic genus) × F F iJF A −βH Z 1 = TrH ( 1) e e Σg ×S −

2 F Q = H − σ JF holomorphic in AF + iσF

of the dimensional reduced theory on Σg and counts ground states graded by charges. This is the topologically twisted index.

[Benini,AZ; Closset-Kim]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 16 / 39 Euclidean SCFT Superconformal theories on curved spaces

The two indices are quite diﬀerent

the superconformal index counts the BPS • states on S 2/operators

the topologically twisted index counts the • ground states/Landau levels of the theory on S 2 with magnetic charges for R and ﬂavor symmetries

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 17 / 39 Euclidean SCFT Localization

Exact quantities in supersymmetric theories with a charge Q2 = 0 can be obtained by a saddle point approximation

Z Z ¯ detfermions Z = e−S = e−S+t{Q,V } = e−S|class t1 × detbosons

Z −S+t{Q,V } ∂t Z = {Q, V }e = 0

Very old idea that has become very concrete recently, with the computation of partition functions on spheres and other manifolds supporting supersymmetry.

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 18 / 39 N1 N2 Q 2 ui −uj 2 vi −vj Z sinh sinh 2 2 Y Y i

ABJM, 3d Chern-Simon theories, [Kapustin,Willet,Yakoov;Drukker,Marino,Putrov]

Euclidean SCFT Localization

Localization ideas apply to path integral of Euclidean supersymmetric theories

Compact space provides IR cut-off, making path integral well defined • Localization reduces it to a finite dimensional integral, a matrix model •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 19 / 39 Euclidean SCFT Localization

Localization ideas apply to path integral of Euclidean supersymmetric theories

Compact space provides IR cut-off, making path integral well defined • Localization reduces it to a finite dimensional integral, a matrix model •

N1 N2 Q 2 ui −uj 2 vi −vj Z sinh sinh 2 2 Y Y i

ABJM, 3d Chern-Simon theories, [Kapustin,Willet,Yakoov;Drukker,Marino,Putrov]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 19 / 39 Euclidean SCFT Localization

Carried out recently in many cases

many papers on topological theories • S 2, T 2 • 3 3 2 1 S , S /Zk , S S , Seifert manifolds • × 4 4 3 1 S , S /Zk , S S , ellipsoids • × S 5, S 4 S 1, Sasaki-Einstein manifolds • × with addition of boundaries, codimension-2 operators, ...

Pestun 07; Kapustin,Willet,Yakoov; Kim; Jaﬀeris; Hama,Hosomichi,Lee, too many to count them all ···

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 20 / 39 Euclidean SCFT Localization

In all cases, it reduces to a ﬁnite-dimensional matrix model on gauge variables, possibly summed over diﬀerent topological sectors

X Z ZM (y) = dx Zint(x, y; m) m C

with diﬀerent integrands and integration contours.

When backgrounds for ﬂavor symmetries are introduced, ZM (y) becomes an interesting and complicated function of y which can be used to test dualities

Sphere partition function, Kapustin-Willet-Yakoov; ··· • Superconformal index, Spironov-Vartanov; Gadde,Rastelli,Razamat,Yan; ··· • Topologically twisted index, Benini,AZ; Closset-KIm; ··· •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 21 / 39 Euclidean SCFT Free energy on S d Successfully testing holographic prediction at large N: large N matrix model techniques

(2,0) M5

Global symmetry enhancement 5d SCFT

AGT correspondence N=4 SYM 4d SCFT N=2/N=1 non Lagrangian theories Several non conformal defs

Non trivial dynamics ABJM N=4/N=2 mirror symmetry 3d SCFT

2d SCFT

1d SCFT Black Hole entropy

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 22 / 39 Euclidean SCFT Free energy on S d

The free energy on S 3 ﬁlled a gap

(2,0) M5 (2,0) M5

Global symmetry enhancement 5d SCFT Global symmetry enhancement 5d SCFT

AGT correspondence N=4 SYM 4d SCFT AdSN=2/N=15 rotating AGT correspondencenon Lagrangian BH theories N=4 SYM 4d SCFT a N=2/N=1maximization non LagrangianSeveral theories non conformal defs Several non conformal defs Non trivial dynamics Non trivial dynamics ABJM N=4/N=2 mirror symmetry ABJM N=4/N=2 mirror symmetry 3d SCFT AdS4 static BH 3d SCFT F 3 extremization S 2d2d SCFT SCFT c maximization Black Hole entropy 1d1d SCFT SCFT Black Hole entropy?

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 23 / 39 Euclidean SCFT Free energy on S d

3 The free energy on S ﬁlled a gap [Jaﬀeris,Klebanov,Pufu,Safdi;Casini,Huerta]

(2,0) M5 (2,0) M5

Global symmetry enhancement 5d SCFT Global symmetry enhancement 5d SCFT

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 24 / 39 Euclidean SCFT Free energy on S d

The free energy on S 3 ﬁlled a gap

(2,0) M5 (2,0) M5

Global symmetry enhancement 5d SCFT Global symmetry enhancement 5d SCFT

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 25 / 39 AdS4 black holes

AdS4 black holes

A nice arena where many of the previous ingredients meet is the counting of microstates of asymptotically AdS4 BPS black holes [Benini,Hristov, AZ]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 26 / 39 AdS4 black holes

AdS4 black holes

I’m talking about BPS asymptotically AdS4 static black holes

c 2 e−K(X )dr 2 s2 = eK(X ) gr + t2 e−K(X )r 2 s2 d d 2 d Σg 2gr − c − gr + 2gr

vacua of N = 2 gauged supergravities arising from M theory truncations • 1 R supported by magnetic charges on Σg : n = 2 Σ2 F • π g preserving supersymmetry via an R-symmetry twist • ( iA ) = ∂ = = cost ∇µ − µ µ ⇒

[Cacciatori,Klemm; Gnecchi,Dall’agata; Hristov,Vandoren;Halmagyi;Katmadas]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 27 / 39 AdS4 black holes Holographic Perspective

General vacua of a bulk eﬀective action

1 µν = + FµνF + V ... L −2R with a metric

2 2 dr 2 2 ds = + (r ds + O(r)) A = AM + O(1/r) d+1 r 2 Md d

and a gauge fields profile, correspond to CFTs on a d-manifold Md and a non trivial background field for the R- or global symmetry

µ LCFT + J Aµ

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 28 / 39 This is the Witten index of the QM obtained by reducing Σ2 S 1 S 1. The BH entropy is related to a Legendre Transform of the indexg × [Benini-Hristov-AZ]→ magnetic charges n are not vanishing at the boundary andd appear in the • HamiltonianSBH (q, n) Re (∆) = Re(log Z(n, ∆) i∆q) , I = 0 ≡ I − d∆ electric charges can be introduced using chemical[similar potentials to Sen’s formalism, ∆ OSV, etc] •

AdS4 black holes

The boundary theory is topologically twisted, with a magnetic charge for the R-symmetry and for the global symmetries of the theory

twisted F iJ∆ −βHn ZΣ2 ×S1 (n, ∆) = TrH ( 1) e e g −

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 29 / 39 The BH entropy is related to a Legendre Transform of the index [Benini-Hristov-AZ] d SBH (q, n) Re (∆) = Re(log Z(n, ∆) i∆q) , I = 0 ≡ I − d∆ [similar to Sen’s formalism, OSV, etc]

AdS4 black holes

The boundary theory is topologically twisted, with a magnetic charge for the R-symmetry and for the global symmetries of the theory

twisted F iJ∆ −βHn ZΣ2 ×S1 (n, ∆) = TrH ( 1) e e g −

This is the Witten index of the QM obtained by reducing Σ2 S 1 S 1. g × → magnetic charges n are not vanishing at the boundary and appear in the • Hamiltonian electric charges can be introduced using chemical potentials ∆ •

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 29 / 39 This is the Witten index of the QM obtained by reducing Σ2 S 1 S 1. g × → magnetic charges n are not vanishing at the boundary and appear in the • Hamiltonian electric charges can be introduced using chemical potentials ∆ •

AdS4 black holes

The boundary theory is topologically twisted, with a magnetic charge for the R-symmetry and for the global symmetries of the theory

twisted F iJ∆ −βHn ZΣ2 ×S1 (n, ∆) = TrH ( 1) e e g −

The BH entropy is related to a Legendre Transform of the index [Benini-Hristov-AZ] d SBH (q, n) Re (∆) = Re(log Z(n, ∆) i∆q) , I = 0 ≡ I − d∆ [similar to Sen’s formalism, OSV, etc]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 29 / 39 AdS4 black holes 7 Black holes in AdS4 S × The ABJM twisted index is

Z N N 1 X Y dxi dx˜i Y xi x˜i Z = x kmi x˜−kme i 1 1 (N!)2 2 ix 2 ix˜ i i x x˜ N π i π i × − j − j × m,me∈Z i=1 i6=j

q xi q xi N y1 mi −me j −n1+1 y2 mi −me j −n2+1 Y x˜j x˜j xi xi × 1 y1 1 y2 i,j=1 − x˜j − x˜j q q x˜j m −m −n +1 x˜j m −m −n +1 y3 e j i 3 y4 e j i 4 xi xi x˜j x˜j 1 y3 1 y4 − xi − xi Q P i yi = 1 , ni = 2(1 − g)

i∆i where m, me are the gauge magnetic ﬂuxes, yi = e are fugacities and ni the magnetic ﬂuxes for the three independent U(1) global symmetries

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 30 / 39 AdS4 black holes 7 Black holes in AdS4 S Strategy: ×

Re-sum geometric series in m, m. • e Z dxi dx˜i f (xi , x˜i ) Z = N N 2πixi 2πix˜i Q (eiBi 1) Q (eiBej 1) j=1 − j=1 −

˜ Step 1: ﬁnd the zeros of denominator eiBi = eiBj = 1 at large N • Step 2: evaluate the residues at large N • (0) (0) X f (x , x˜ ) Z i i ∼ det I B

[Benini-Hristov-AZ]

[extended to other model Hosseini-AZ; Hosseini-Mekareeya]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 31 / 39 AdS4 black holes 7 Black holes in AdS4 S × Step 1: solve the large N Limit of algebraic equations giving the positions of poles

x˜j x˜j x˜j x˜j N 1 − y3 1 − y4 N 1 − y3 1 − y4 k Y xi xi k Y xi xi 1 = xi =x ˜j −1 x˜j −1 x˜j −1 x˜j −1 x˜j j=1 1 − y 1 − y i=1 1 − y 1 − y 1 xi 2 xi 1 xi 2 xi

Bethe Ansatz Equations - derived by a potential VBA(xi , x˜i ) with an ansatz

log xi = i√Nti + vi , logx ˜i = i√Nti +v ˜i

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 32 / 39 AdS4 black holes 7 Black holes in AdS4 S ×

The index is obtained from BA √∆1∆2∆3∆4: V ∼ 1 n 3/2 X p i i∆i (∆) = N 2k∆1∆2∆3∆4 i∆i qi yi = e I 3 i − ∆i −

This function can be extremized with respect to the ∆i and

Re crit = BH Entropy(ni , qi ) I|

i ∆i crit X (rh) | ∼ [Benini-Hristov-AZ]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 33 / 39 AdS4 black holes A. Attractor mechanism

The BPS equations at the horizon imply that the gauge supergravity quantity

Λ Λ ∂ = FΛn X qΛ , FΛ = F R − ∂X Λ

with (q, n) electric and magnetic charges, is extremized with respect to the scalar ﬁelds at the horizon and its critical value gives the entropy Under X Λ ∆Λ → 0 1 2 3 = 2i√X X X X BA(∆) F ≡ V

X nΛ 0 1 2 3 Λ i = √X X X X iX qΛ (∆) R −X Λ − ≡ I

[Benini-Hristov-AZ; Hosseini-AZ]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 34 / 39 AdS4 black holes B. R-symmetry extremization

Recall the cartoon

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 35 / 39 AdS4 black holes B. R-symmetry extremization

The extremization reﬂects exactly what’s going on in the bulk. The graviphoton ﬁeld strength depends on r

K/2 Λ Tµν = e X FΛ,µν

suggesting that the R-symmetry is diﬀerent in the IR and indeed

i ∆i crit X (rh) | ∼

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 36 / 39 AdS4 black holes B. R-symmetry extremization

Some QFT extremization is at work? symmetry enhancement at the horizon AdS2

QM CFT1 1 →

The twisted index depends on ∆i because we are computing the trace

F i∆i Ji R TrH( 1) e TrH( 1) − ≡ −

where R = F + ∆i Ji is a possible R-symmetry of the system.

R is the exact R-symmetry at the superconformal point • Natural thing to extremize: in even dimensions central charges are • extremized, in odd partition functions...

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 37 / 39 AdS4 black holes C. One dimension more

In AdS5 there are two interesting objects boundary bulk

AdS5 rotating black hole; where • the entropy comes from?

AdS5 black string; horizon • AdS3 Σg . 2d central charge of the CFT× matched with gravity. c- extremization for R-symmetry [Benini-Bobev]

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 38 / 39 Thank you for the attention !

Conclusions Conclusions

The world of SCFT in 1 d 6 is a lot of fun. ≤ ≤ Many unexpected progresses recently, many expected in the future.

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 39 / 39 Conclusions Conclusions

The world of SCFT in 1 d 6 is a lot of fun. ≤ ≤ Many unexpected progresses recently, many expected in the future.

Thank you for the attention !

Alberto Zaﬀaroni (Milano-Bicocca) CFT june 2016 39 / 39