An overview of the ``fuzzball'' proposalpeople.roma2.infn.it/~stringhe/seminars/upload/file/giusto.pdf · An overview of the “fuzzball” proposal Stefano Giusto Università di

An overview of the “fuzzball” proposal

Stefano Giusto

Università di Padova

Tor Vergata, May 27, 2011

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Reviews:

S. Mathur: hep-th/0502050I. Bena, N. Warner: hep-th/0701216K. Skenderis, M. Taylor: arXiv:0804.0552V. Balasubramanian, J. de Boer, S. El-Showk, I. Messamah:arXiv:0811.0263B. Chowdhury, A. Virmani: arXiv:1001.1444

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Outline

1 The quest for a Black Hole Statistical Mechanics

2 Black Holes in String Theory

3 The fuzzball picture

4 Construction of microstate geometries2-charge3-charge

5 Outlook

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The quest for a Black Hole Statistical Mechanics

Thermodynamics

Classical General Relativity implies

first law: dM = k8πG dAHor + Ω dJ + φdQ

second law: ∆AH ≥ 0

Classical gravity coupled to a quantum field implies

Hawking radiation: TH = k2π

Hawking’s result together with first law imply

Black Hole entropy: SBH = AHor4G

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The quest for a Black Hole Statistical Mechanics

Statistical Mechanics?

Classically, there is a unique black hole for fixed macroscopicquantities (mass, charge, angular momentum)Where are the microstates responsible for black hole entropy?

SBH?= log(#states)

differences between the microstates could be quantum gravityeffects confined to r ∼ `Phowever, Hawking radiation is only sensitive to scalesr ∼ RHor `Phow can Hawking radiation carry information on the microstate?information paradox!

To address these questions one needs a quantum theory ofgravity⇒ String Theory

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Black Holes in String Theory

An example: D1-D5-P black hole in 5D

IIB string theory on R4,1 × T 4 × S1

T4

+ +S1

n1 npn5

D1 branes D5 branes momentum

gs<< 1

S1

n1 n5

effective string

at small gravitational coupling the bound state of D1 and D5branes is desribed by a CFT with target space (T 4)n1n5/Sn1n5

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Black Holes in String Theory

microstates of the effective CFT can be counted (Strominger,Vafa)

for BPS black holes, the counting reproduces exactly SBH

log(#microstates) = 2π√

n1n5nP = SBH

string theory captures all the degrees of freedom of black holes

What is the description of microstates for finite gravitational coupling?

the common lore is (Horowitz-Polchinski)

gs = 0 gs finite

D-brane microstate classical black hole7 / 30

The fuzzball picture

The fuzzball picture (Mathur et al.)

consider a coherent state of the CFTfollow its evolution as gs increasesits backreaction on the space-time is described by a classicalgeometry (microstate geometry)

gs = 0 gs finite

D-brane microstate microstate geometry

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Properties of microstate geometries

for r RHor they approach the black hole geometry:they carry the same charges and mass as the black holethey have no horizon, no singularity, no CTS’sthe region inside the horizon is replaced by a “smooth cap”which carries the information on the particular microstatefor generic microstates the size of the cap is ∼ RHor

stringy low mass degrees of freedom modify the classicalgeometry up to horizon scales !

rRHor → black hole

r∼RHor → cap

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The fuzzball program

Construct geometries for generic microstates:

completed for BPS 2-charge geometries (Lunin, Mathur;Kanitscheider, Skenderis, Taylor)many examples of BPS 3-charge geometries (S.G., Mathur,Saxena; Bena, Warner; Berglund, Gimon, Levi), but a systematicconstruction is still lackingvery few examples of non-BPS 3-charge geometries (Jejjala,Madden, Ross, Titchener)

Quantize the phase space of classical microstate geometries:

done for the BPS 2-charge case (Maoz, Richkov)done for a sub-family of BPS 3-charge geometries (deBoer,El-Showk, Messamah, Van den Bleeken)

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The fuzzball program (continued)

Define and verify the map between geometries and CFT states:

completely spelled out for BPS 2-charge geometries (Lunin,Mathur; Kanitscheider, Skenderis, Taylor)known only for a sub-set of BPS and non-BPS 3-chargegeometries, related by a symmetry (spectral flow) to 2-chargegeometries (S.G., Mathur, Saxena; Jejjala, Madden, Ross,Titchener)

How do classical black holes emerge from statistical average overmicrostate geometries?

CFT density matrix ?→ “averaged geometry” (Alday, de Boer,Messamah; Balasubramanian, de Boer, Jejjalla, Simon; Kraus,Shigemori)

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The fuzzball program (dynamics)

How do probes propagate in fuzzball (non-BPS) geometries?

E ∼ TH , t tcross ∼ RHor ⇒ Hawking radiation :

The radiation emitted by fuzzballs carries the information of themicrostate⇒ no information problem (Avery, Chowdhuri, Mathur)

E TH , t ∼ tcross ∼ RHor ⇒ Infall problem :

One expects the dynamics to be well-approximated by theclassical black hole (Mathur)

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Construction of microstate geometries

Construction of microstate geometries

In the following I will mainly describe one basic problem:the construction of (BPS) microstate geometries:

2-charge: the full (old) story;3-charge: attempts at a systematic construction.

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Construction of microstate geometries 2-charge

The 2-charge black hole (Lunin, Mathur)

P = 0 : Ramond ground states of the orbifold CFTthe 2-charge black hole is classically singular; in some cases (K3),the singularity is resolved by higher derivative correctionsthe 2-charge entropy is finite S = 2π

√2n1n5 (Sen)

use string dualities to simplify the problem:

D1− D5U−duality−→ F1− P

(n1,n5) −→ (np,nw )

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BPS states of the F1-P system

Quantum states:∏n

(αµ−n)mn |0 >,∑

n

nmn = npnw , µ = 1, . . . ,8

Coherent states:states with mn 1 are approximately eigenstates of the stringpositions xµ and are well described by giving the string profile

xµ = Fµ(t − y), y ∈ [0,2πnwR]

(we consider µ ∈ R4)

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The F1-P geometries

the classical geometry sourced by a fundamental string withprofile Fµ(t − y) is known (Dabholkar, Gauntlett, Harvey, Waldram;Callan, Maldacena, Peet)

for all non-trivial states the curve xµ = Fµ(t − y) extends in thetransverse directions

nw L

F(t-y)

size

geometries are not spherically symmetric (unlike the classical b.h.)geometries carry global F1 charge along S1 but also local F1charge along a transverse direction: dipole chargegeometries are singular on the curve xµ = Fµ(t − y)

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The D1-D5 geometries

the metric after U-duality

ds2 = H−1[−(dt − Aidx i)2 + (dy + Bidx i)2] + H dx idx i

H = 1 +QL

∫ L

0

dv|x − F (v)|2

, Ai = −QL

∫ L

0dv

Fi(v)

|x − F (v)|2, dB = ∗4dA

two length scales:√

Q and length of the curve F (v) ∼ a

Q

a

R4,1XS1

3AdS x S3

1/2

a→0→17 / 30


the metric after U-duality

ds2 = H−1[−(dt − Aidx i)2 + (dy + Bidx i)2] + H dx idx i

H = 1 +QL

∫ L

0

dv|x − F (v)|2

, Ai = −QL

∫ L

0dv

Fi(v)

|x − F (v)|2, dB = ∗4dA

H,Ai ,Bi are singular on the curve x i = F i(v)

the term dy + Bidx i describes the fiber of a KK-monopole: the S1

cycle vanishes smoothly

y

x=F(v)

|x|>>1

there is a local KK-monopole charge (dipole charge)18 / 30


General BPS D1-D5-P solution (Bena, Warner)

ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:

ds24 : 4D hyper-kahler euclidean space

a1,a2,a3 : 1-forms→ d1, d5, kkm dipole charges dai = ∗4dai

Z1,Z2,Z3 : 0-forms→ D1, D5, P global charges d ∗4 dZi = daj ∧dak

k : 1-form→ angular momentum dk + ∗4dk = Zidai

A linear system of equations !

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ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:






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ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:






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ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:






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ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:






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ds2 =1√

Z1Z2

[−(dt + k)2

Z3+ Z3

(dt + a3 −

dt + kZ3

+ dt)2]

+√

Z1Z2ds24

C(2) =(

a1 −dt + k

Z1

)∧ (dt + dy + a3) + γ2

Ingredients:






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D1-D5-P microstates?

What subset of the general class of solutions represent a bound stateof D1-D5-P?

we cannot use dualities to relate D1-D5-P to a simpler systemone can generate a class of 3-charge solutions by applyingsymmetries of the CFT (spectral flow) to 2-charge solutions (S.G.,Mathur, Saxena; Ford, S.G., Saxena)

it was found that the 4D metric ds24 is a non-flat, singular

hyper-kahler space with non constant signaturethe full 10D geometry is completely regular !in cases with U(1)× U(1) axially symmetry, ds2

4 is a 2-centerGibbons-Hawking space

in the presence of U(1)× U(1) axially symmetry, this class ofsolutions can be generalized: ds2

4 can be replaced with amulti-center Gibbons-Hawking space (Bena, Warner; Berglund,Gimon, Levi)

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Axially symmetric D1-D5-P microstates

the 4D metric (multi-center Gibbons-Hawking space):

ds24 = V−1(dψ + A)2 + V ds2

3 , ∗3dA = dV

V =∑

i

ni

|~x − ~xi |, ni ∈ Z ,

∑i

ni = 1

the signature of ds24 changes from (+ + ++) to (−−−−)

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Charges dissolved in fluxes

ds24 has topologically non-trivial 2-cycles

the “dipole 1-forms” ai have non-zero fluxes on the 2-cyclesdipole charges generate real charges via Chern-Simons couplings

L ∼ A(1)0 da2 ∧ da3

the geometry has global charges at infinity but no singular sourceselectric + magnetic charges generate angular momentumfluxes determine the positions of the GH centers(Denef; Bena, Warner)

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Generic D1-D5-P microstates?

the microstates known so far have axial symmetry and depend ona discrete number of parameters (the positions of the GH centersand the dipole fluxes)generic microstates are expected to have no symmetry anddepend on functions (generalization of the 2-charge profile F (v))axially symmetric microstates account for a small (measure zero)fraction of the 3-charge entropyhow to construct generic microstates?go back to the basics: perturbative description of D-branes asopen string boundary conditions

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Microstates from string amplitudes (S.G., Morales, Russo;

work in progress)

The amplitude

gµν

Bµν

C(2)µν

. . .

gives the correction around flat space (large distance behavior) of thegeometry sourced by the D-brane system associated with the givenboundary conditions on the disk.

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String amplitudes for the D1-D5-P system I

The amplitude for the D1-D5-P system has several contributions:1

D5 D1

terms of order 1/r2

contribute to Z1, Z2 ⇒ global D1, D5 charges

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String amplitudes for the D1-D5-P system II2

the symbol denotes insertion of the open vertex operatorVf = fi (v)∂X i + ∂v fi (v)ψvψi , with fi (v) the oscillation profile of theD1 or D5 brane

terms of order 1/r2 contribute to Z3 ⇒ global P charge

terms of oder 1/r3 contribute to a1, a2 ⇒ dipole d1, d5 charges

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String amplitudes for the D1-D5-P system III3

x

x

the symbol × denotes insertion of the twist verted operatorVµ = µA e−ϕ

2 SA ∆

the condensate µAµB = 13!cIJK (ΓIJK )AB is related to the moments of

the D1-D5 profile cvij ∼∫

dv Fi (v) ∂v Fj (v)

terms of order 1/r3 contribute to a3 ⇒ dipole kkm charge

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String amplitudes for the D1-D5-P system IV

4

x

x

this diagram involves all 3 charges

terms of order 1/r3 give corrections to ai and k

terms of order 1/r4 contribute to ds24 ⇒ non− flat 4D base !

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Outlook

Outlook I

What we know:

some microstates of D-brane bound states are described, in theregime of large gravitational coupling, by asymptotically flat,smooth, horinzonless geometriesthe geometries are dual to states of the CFTthese geometries look like the classical black hole geometry atlarge distances, but differ from it at scales of order of the horizonthe differences are non-perturbative (different topology); thegeometry at the horizon carries information on the microstate

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Outlook

Outlook II

Conjectures:

generic microstates are described by asymptotically flat, smooth,horinzonless geometriesthe number of states obtained by quantizing the moduli space ofclassical horinzoless geometries reproduces the full black holeentropydifferent point of view: smooth geometries (“hair”) give subleadingcorrections to black hole entropy (Sen)

the naive black hole geometry is a coarse-grained description ofan ensemble of microstatesit suggests a new surprising mechanism in quantum gravity:for bound states with a large number N of degrees of freedom,quantum gravity effects extend to macroscopic scales∼ Nα`p ∼ RHor

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Documents

An overview of the ``fuzzball'' proposalpeople.roma2.infn.it/~stringhe/seminars/upload/file/giusto.pdf · An overview of the “fuzzball” proposal Stefano Giusto Università di