An overview of the ``fuzzball''...
Transcript of An overview of the ``fuzzball''...
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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The fuzzball picture
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 picture
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 picture
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 picture
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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Construction of microstate geometries 2-charge
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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Construction of microstate geometries 2-charge
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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Construction of microstate geometries 2-charge
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
Construction of microstate geometries 2-charge
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
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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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Construction of microstate geometries 3-charge
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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