Black hole microstate counting and their macroscopic counterpart - Ashoke Sen Warsaw, July 2013
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Black hole microstate counting and their
macroscopic counterpart
Ashoke Sen
Harish-Chandra Research Institute, Allahabad, India
Warsaw, July 2013Introduction
Bekenstein and Hawking gave a universal formula for
the black hole entropy:
A
SBH = (in ~ = 1, c = 1, kB = 1 unit)
4G
– valid in classical, two derivative theory of gravity.
This suggests that when the curvature at the horizon
is small compared to the Planck scale, the number
dmicro of microstates should be given by
dmicro ' exp[SBH ]
Can we verify this by direct counting?
2Counting black hole microstates directly is difficult.
Strategy in string theory: Work with supersymmetric
(BPS) black holes whose ‘degeneracy’ is independent
of the coupling constant.
Count microstates in the weak coupling limit when
gravity can be ignored, and compare the result with
exp[SBH ], computed when gravity is strong enough to
form a black hole.
3For a class of extremal supersymmetric black holes in
string theory this has been achieved. Strominger, Vafa; ...
SBH (q) ' ln dmicro (q)
SBH (q)= entropy of a supersymmetric black hole
carrying a given set of charges q ≡ {q1 , q2 , ...}.
dmicro (q): number of quantum states carrying the
same set of charges.
4The comparison between SBH (q) and ln dmicro (q) is
done in the limit of large charges q so that the
horizon has low curvature
⇒ higher derivative terms in the action and quantum
gravity corrections can be ignored.
One of our goals will be to relax this constraint on
charges.
5In actual practice, what one computes in the
microscopic theory is an index.
dmicro (q)= no of bosonic states of charge q
- number of fermionic states of charge q
It is the index dmicro , and not degeneracy, that is
independent of the coupling constant.
Formally
X
dmicro (q) = (−1)2J
states of charge q
J = angular momentum
6Questions
1. Can we justify comparing eSBH , which measures
degeneracy, with the index in the microscopic theory?
2. Can we find a prescription for computing the exact
black hole entropy taking into account higher
derivative and quantum corrections?
3. Can macroscopic analysis tell us about other
statistical properties of the microstates besides
degeneracy?
microscopic: gravity off, macroscopic: gravity on
7Note: Question 2 has been partially answerd by Wald
– a general formula for the black hole entropy taking
into account higher derivative corrections to the
classical effective action.
Swald = SBH for Einstein’s theory coupled to ordinary
matter fields.
Quantum corrections to the entropy are not
computed by Wald’s formula.
8Plan
1. Brief review of what is known on the microscopic
side.
2. General formalism for computing extremal black
hole entropy from macroscopic i.e. quantum gravity
analysis, generalizing Bekenstein-Hawking-Wald
formula.
3. Specific predictions of the macroscopic analysis
and their tests against microscopic results
– focus on the developments since my talk at MG12 in
2009.
9Microscopic results
10Exact microscopic results for the index are known for
a wide class of states in a wide class of theories with
32 or 16 unbroken supersymmetries:
1. Type II on T5 and T6 – 32 supersymmetries
2. Heterotic on T5 and T6 – 16 supersymmetries
3. Orbifolds of these theories (CHL models)
Chaudhury, Hockney, Lykken
– 16 supersymmetries
Dijkgraaf, Verlinde, Verlinde; Shih, Strominger, Yin; David, Jatkar, A.S.; Dabholkar, Gaiotto, Nampuri;
S. Banerjee, Srivastava, A.S.; Dabholkar, Gomes, Murthy; Govindarajan, Gopala Krishna; · · ·
11In each case the result is given in terms of Fourier
coefficients of some known functions with modular
properties.
(Siegel modular forms, Weak Jacobi forms etc.)
This allows us to compute the index dmicro as an exact
integer for a given set of charges carried by the black
hole.
12In each of these examples we can also systematically
calculate the behaviour of dmicro for large charges.
General strategy: Compute the Fourier integrals
using saddle point method.
The leading term of ln dmicro always agrees with the
Bekenstein-Hawking entropy of an extremal black
hole with the same charge
– will not be discussed in detail.
Our goal will be to understand to what extent
macroscopic analysis can explain the subleading
terms in ln dmicro .
13Example: type IIB string theory on T6
Result: For a class of supersymmetric black holes,
carrying electric charges {Qi } and magnetic charges
{Pi }, the index is
dmicro = (−1)Q·P+1 c(∆), ∆ ≡ Q2 P2 − (Q · P)2
Q2 , P2 , Q.P: certain bilinear combinations of charges
c(u) is defined through
X
−ϑ1 (z|τ )2 η(τ )−6 ≡ c(4k − l2 ) e2πi(kτ +lz)
k,l Shih, Strominger, Yin
ϑ1 : Jacobi theta function η: Dedekind eta function
14For large charges
√
log[dmicro ] = π ∆−2 ln ∆ + · · ·
Bekenstein-Hawking entropy SBH of a black hole
carrying the same charges is given by
√
π ∆
– agreement between ln dmicro and SBH at the leading
order.
Later we shall see the origin of the logarithmic term
from the macroscopic side.
15In many of these theories we can also define twisted
index.
Suppose the theory has a discrete ZN symmetry
generated by g which commutes with the unbroken
supersymmetries of the state.
Then X
(−1)2J g
states of charge q
is independent of the coupling constant.
These can also be computed in these theories and are
given by Fourier coefficients of Siegel modular forms.
16All these provide us with a wealth of ‘experimental
data’ against which we can test the predictions of
macroscopic analysis based on quantum gravity.
In the rest of the talk we shall describe to what extent
we have achieved this goal.
17Macroscopic analysis
18Supersymmetric black holes in string theory are
extremal.
In the extremal limit a black hole acquires an infinite
throat described by an AdS2 factor, separating the
horizon from the asymptotic space-time.
Full throat geometry:
AdS2 × K
K includes the angular coordinates of space, e.g.
polar and azimuthal angles in 3+1 dimensions, as well
as the 6 dimensional space on which we have
compactified string theory.
19The metric on the throat:
dr2
ds = a −r dt + 2 + ds2K
2 2 2 2
r
The original horizon is towards r→ 0.
Original asymptotic space-time is towards r→ ∞.
20Strategy: Analyze euclidean path integral on this
space-time.
Euclidean continuation of the throat metric: t → −iθ
dr2
ds = a r dθ + 2 + ds2K
2 2 2 2
r
Since we can change the period of θ by a rescaling
r → λr, θ → θ/λ, we choose
θ ≡ θ + 2π
The boundary of AdS2 is at r = ∞.
21Regularize the infinite volume of AdS2 by putting a
cut-off r ≤ r0 .
r = r0
This makes the AdS2 boundary have a finite length
L = 2π a r0
221. Define the partition function:
Z
Z = Dϕ exp[−Action − boundary terms]
ϕ: set of all fields in the theory
Boundary condition: For large r the field
configuration should approach the throat geometry of
the black hole.
r = r0
In the interior we allow all fluctuations including
fluctuations of the topology.
232. Z can be reinterpreted as
Z= Tr(e−LH ) → d0 e−L E0 as L → ∞
H: Hamiltonian, L: boundary length
(d0 , E0 ): (degeneracy, energy) of ground state.
We identify the quantum corrected entropy as
Smacro = ln d0
d
⇒ Smacro = limL→∞ 1 − L ln (Z)
dL
This gives the quantum generalization of the
Bekenstein - Hawking - Wald formula. A.S.
24Classical limit
The dominant saddle point that contributes to Z is the
euclidean black hole in AdS2 :
dr2
2 2 2 2
ds = a (r − 1)dθ + 2 + ds2K
r −1
= a2 (dη 2 + sinh2 η dθ2 ) + ds2K r = cosh η
r = r0
One can show that in the classical limit
Smacro ⇒ Swald
25Comparing macroscopic and microscopic
results
261. Degeneracy vs. index
AdS2 space has SL(2,R) isometry.
Supersymmetric black holes are also invariant under
4 supersymmetries.
Combined symmetry group: PSU(1,1|2)
– contains an SU(2) subgroup.
Thus supersymmetric black holes must be spherically
symmetric.
271. Spherical symmetry ⇒ vanishing average ~J.
2. Extremal black hole describes a microcanonical
ensemble of states with all states carrying same ~J
and other charges.
⇒ all states carry ~J = 0.
Thus A.S.; Dabholkar, Gomes, Murthy, A.S.
X X
Index = (−1)2J = (1) = degeneracy = exp[Smacro ]
states states
This justifies comparing black hole entropy to the log
of microscopic index.
28Index = degeneracy = exp[Smacro ] ⇒ Index > 0
In the microscopic counting, performed at weak
coupling, there may be both bosonic and fermionic
states.
However, in order to agree with macroscopic result,
the index should be positive, i.e. the number of
bosonic states must exceed the number of fermionic
states.
This has been tested in many supersymmetric string
theories where the microscopic index is known
exactly, even for finite charges. A.S.
29Some microscopic results for the index of black holes in
heterotic string theory on T6
(Q2 , P2 )\Q.P 2 3 4 5 6 7
(2,2) 648 0 0 0 0 0
(2,4) 50064 0 0 0 0 0
(2,6) 1127472 25353 0 0 0 0
(4,4) 3859456 561576 12800 0 0 0
(4,6) 110910300 18458000 1127472 0 0 0
(6,6) 4173501828 920577636 110910300 8533821 153900 0
(2,10) 185738352 16844421 16491600 0 0 0
No negative index
Blue entries represent states with Q2 P2 − (Q.P)2 < 0 for which
there are no classical black hole solutions.
30Positivity of the index is a non-trivial prediction for
the Fourier coefficients of modular forms which count
the black hole index.
This has now been proved for an infinite number of
cases. Bringmann, Murthy
Nevertheless this only covers a negligible fraction of
all the cases.
So far there are no counterexamples.
312. Logarithmic corrections to entropy
Typically the leading entropy SBH is a homogeneous
function of the various charges qi .
e.g. in D=4
SBH (Λq) =Λ2 SBH (q)
Logarithmic corrections: correction to the entropy
∝ ln Λ in the limit of large Λ.
These arise from one loop correction to the leading
saddle point result for Z from loops of massless
fields.
Banerjee, Gupta, A.S.; Banerjee, Gupta, Mandal, A.S.; A.S; Ferrara, Marrani; Bhattacharyya, Panda, A.S.
Mann, Solodukhin; Fursaev; ...
32Consider a spherically symmetric extremal black hole
in D=4 with horizon size a
dr2
2 2 2
ds = a 2 2 2
+ (r − 1)dθ + dψ + sin ψdφ + ds2compact
2
r2 − 1
When the charges scale uniformly by a large number
Λ then
a∼Λ
keeping other parameters, e.g. ds2compact , fixed.
33∆: kinetic operator of four dimensional massless
fields in the near horizon background.
Non-zero mode contribution to Z:
0 −1/2 1 0
(sdet ∆) = exp − ln sdet ∆
2
0
: remove zero mode contribution
We can use heat kernel expansion to determine the
terms in ln sdet0 ∆ proportional to ln a.
– arise from modes with eigenvaluesZero mode contribution
The zero modes are modes carrying zero eigenvalue
of ∆.
Integration over these modes gives the volume of the
space spanned by these modes rather than a
determinant.
By careful analysis we can determine the dependence
of this volume on the size ‘a’ of AdS2 and S2 .
The final result is obtained by combining the zero
mode and the non-zero mode contributions.
35Final results: S. Banerjee, Gupta, Mandal, A.S.; Ferrara, Marrani; A.S.
The theory logarithmic contribution microscopic
√
N = 4 in D=4 with nv matter 0
√
N = 8 in D=4 −8 ln Λ
N = 2 in D=4 with nV vector 1 (23 + nH − nV ) ln Λ ?
6
and nH hyper
N = 6 in D=4 −4 ln Λ ?
N = 5 in D=4 −2 ln Λ ?
N = 3 with nv matter in D=4 2 ln Λ ?
√
BMPV in type IIB on T5 /ZZN − 14 (nV − 3) ln Λ
or K3 × S1 /ZZN with nV vectors
√
BMPV in type IIB on T5 /ZZN − 14 (nV + 3) ln Λ
or K3 × S1 /ZZN with nV vectors
363. Twisted index A.S.
Suppose the theory has a discrete ZZN symmetry
generated by g.
What is the value of the g-weighted index
X
{(−1)2J g}
states
for a supersymmetric black hole?
By our earlier argument
X X
{(−1)2J g} = g
states states
37Macroscopic computation of the twisted index
⇒ evaluate the partition function Zg with a g-twisted
boundary condition along the boundary circle of
AdS2 .
X d
g = limL→∞ 1 − L ln Zg
states
dL
g
r = r0
The euclidean black hole solution is no longer
allowed saddle point since the boundary circle, along
which we have g-twist, is contractible. 38However there is a different saddle point, related to
the original saddle point by a ZZN orbifolding, which
contributes to the path integral.
dr2
ds = a (r2 − N−2 )dθ2 +
2 2
2 −2
+ ds2K
r −N
The corresponding action is 1/N times the action of
an Euclidean black hole.
This leads to the prediction:
X X
{(−1)2J g} = {g} ∼ exp[Swald /N]
states states
This has been verified in each of the cases where
microscopic result for the twisted index is known
⇒ a non-trivial test. 39Other applications
4. In CHL models in D=4, in the large charge limit
p !
p Q.P Q2 P2 − (Q.P)2
ln dmicro = π Q2 P2 − (Q.P)2 + f ,
P2 P2
+O(charge−2 )
f: a known function involving modular forms.
Cardoso, de Wit, Kappeli, Mohaupt; David, Jatkar, A.S.
p
π Q2 P2 − (Q.P)2 : Bekenstein-Hawking entropy.
√ 2 2
Q.P Q P −(Q.P)2
What about the first correction f P2 , P2
?
40On the macroscopic side, one loop correction to the
effective action produces a special class of terms of
the form
4 √
Z
1
Rµνρσ Rµνρσ − 4Rµν Rµν + R2 .
− 2
d x g f(a, S)
64π
(a,S): axion-dilaton fields
This precisely produces the required correction to the
entropy via Wald’s formula.
Caution: It has not been established conclusively that
to this order there is no other contribution to the
entropy.
415. Progress has been made towards evaluating Z
using localization techniques.
N. Banerjee, S. Banerjee, Gupta, Mandal, A.S.; Dabhokar, Gomes, Murthy
42Conclusion
Quantum gravity is capable of computing detailed
statistical properties of the microstates.
– a systematic procedure for computing corrections
to the leading Bekenstein-Hawking result
– sign of the index
– asymptotic growth of the twisted index
The macroscpic side of the analysis can be easily
generalized to non-supersymmetric black holes, but
supersymmetry seems necessary to keep the
microscopic analysis under control.
43Example: Entropy of extremal Kerr in pure gravity has
a correction
16
ln AH
45
Fursaev; Mann, Solodukhin; A.S.; Bhattacharyya, Panda, A.S.
The method can also be generalized to non-extremal
black holes, e.g. the Schwarzschild black hole
entropy has logarithmic correction
77
ln AH
90
The complexity of the coefficients ⇒ the microscopic
explanation of the entropy is likely to be more
complicated than those studied so far.
– challenge for any theory of quantum gravity.
44You can also read
