#### Logarithmic corrections to black hole entropy from Kerr/CFT

Received: March
Logarithmic corrections to black hole entropy from Kerr/CFT
Oscar Varela 0 1 2 3 5
Open Access 0 1 3
c The Authors. 0 1 3
0 Am Muhlenberg 1 , D-14476 Potsdam , Germany
1 Santa Barbara , CA 93106 , U.S.A
2 Department of Physics, Utah State University
3 Cambridge , MA 02138 , U.S.A
4 Department of Physics , UCSB
5 Center for the Fundamental Laws of Nature, Harvard University
It has been shown by A. Sen that logarithmic corrections to the black hole area-entropy law are entirely determined macroscopically from the massless particle spectrum. They therefore serve as powerful consistency checks on any proposed enumeration of quantum black hole microstates. Sen's results include a macroscopic computation of the logarithmic corrections for a ve-dimensional near extremal Kerr-Newman black hole. Here we compute these corrections microscopically using a stringy embedding of the Kerr/CFT correspondence and nd perfect agreement.
cMax-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut)
1 Introduction 2 3 The
A Computation of kQ
B Change of ensemble
Introduction
ve-dimensional Kerr-Newman black hole
Near horizon, near extremal limit
Frolov-Thorne temperatures
Logarithmic correction to entropy
Microscopic computation
Match of the macroscopic and microscopic computations
The Bekenstein-Hawking area-entropy law universally applies to any self-consistent
quantum theory of gravity. E orts to understand how the former constrains the latter have led
to a wealth of insights.
Five years ago Sen et al. [1{6] pointed out that the leading corrections to this law,
which are of order log A (where A is the area) are also universal in that they depend
only on the massless spectrum of particles and are insensitive to the UV completion of
the theory. The basic reason for this is that the e ects of a particle of mass m can be
accounted for by integrating it out, which generates local higher derivative terms in the
e ective action. These lead to corrections to the entropy which are suppressed by inverse
powers of m2A and cannot give log A terms.
The macroscopically computed logarithms serve as a litmus test for any proposed
enumeration of quantum black hole microstates which is more re ned than the test provided
by the area law. Sen extensively analyzed a number of stringy examples all of which passed
the test with ying colors [7], and further noted that the macroscopic computation does not
match the loop gravity result. He also posed a matching of the logarithms as a challenge
for Kerr/CFT [8{11].
In this paper we show that the logarithms indeed match for one microscopic
realization [12, 13] of Kerr/CFT obtained by embedding a certain near-extremal ve-dimensional
Kerr-Newman black hole into a string compacti cation.1 The microscopic dual is a
twodimensional eld theory de ned as the IR
xed point of the worldvolume eld theory on
a certain brane con guration with scaled uxes. This IR limit is certainly nontrivial but
1The near-extremal regime sidesteps subtleties with logarithmic corrections at extremality.
is not a conventional 2D CFT. Its properties are incompletely understood and have been
studied in a variety of approaches: see e.g. [14{20].
The example of Kerr/CFT we chose is the simplest possible case. In the simplest
examples considered by Sen, the macro-micro match has a somewhat trivial avor: all
logs vanish in a certain thermodynamic ensemble, with logs generated on both sides of the
match by Legendre transforms to a di erent ensemble. However, later examples become
quite intricate and provide compelling tests of a variety of stringy constructions. In the
vedimensional Kerr/CFT match herein, all logs vanish in a certain thermodynamic ensemble,
and the character of the match is as in the simplest of Sen's examples. Non-trivial aspects
of our match reside in its reliance on the values of the Kerr/CFT central charges and
Kac-Moody levels which enter the microscopic computation. In particular, we
a necessary non-vanishing level of the current dual to the electric
eld is provided by a
Chern-Simons term which is crucially present in the e ective action obtained from string
While it is reassuring that the simplest case works, perhaps more challenging
matches such as the extremal 4D case may eventually provide more re ned and compelling
tests of Kerr/CFT.
This paper is organized as follows. In section 2 we describe the black hole solution,
take its near horizon limit, and determine the corresponding quantum mixed state in the
CFT. In section 3 we begin by stating the result of [6] for the logarithmic corrections
to the Bekenstein-Hawking entropy and then proceed to compute them microscopically in
the dual theory for two di erent enhancements of the global symmetries. We match the
Bekenstein-Hawking entropy of the near extremal black hole to rst order in the Hawking
temperature with the Cardy formula and, using the result of appendix A for the gauge
Kac-Moody level, we show that the logarithmic corrections also agree.
Previous work on logarithmic corrections to black hole entropy includes [21{31].
ve-dimensional Kerr-Newman black hole
We consider a charged and rotating black hole solution of ve-dimensional Einstein gravity
minimally coupled to a gauge eld. The dynamics of the latter is speci ed by the
YangMills-Chern-Simons Lagrangian, so that the complete action is,2
S5 =
Speci cally, we are interested in the following Kerr-Newman black hole solution to (2.1)
considered in [12],
(a2 + r^2)(a2 + r^2
ds52 =
A =
M0 sinh 2
(d ^2 + d ^2 + 2 cos d ^ d ^) +
2This coincides with the bosonic sector of minimal supergravity in ve dimensions.
where we have de ned the quantities
B = a2 + r^2
M0s4(2s2 + 3) ;
F = a(r^2 + a2)(c3 + s3)
= r^2 + a2 + M0s2 ;
f = r^2 + a2 ;
cosh . The geometry depends on three independent parameters
(a; M0; ) and the physical quantities of the black hole, i.e., its mass, angular momentum
and electric charge, are given in terms of those parameters by
M =
JL = aM0 (c3 + s3) ;
Q = M0sc :
In ve dimensions, it is possible to have a second angular momentum, JR, but we set
This black hole displays inner and outer horizons located at
r2 = 12 (M0
At the (outer) horizon, the angular velocities are
^ =
s3) + (c3 + s3)p1
4a2=M0
^ = 0 ;
SBH ext = 8 a3(c3
and the electric potential is
Finally, the Hawking temperature is given by
s2c + (c2s + s2c)p1
s3 + (c3 + s3)p1
4a2=M0
4a2=M0
TH =
4a2=M0
s3 + (c3 + s3)p1
4a2=M0
and the Bekenstein-Hawking entropy is
SBH =
2 M0 (c6 + s6)M0
The black hole approaches extremality in the limit M0 ! 4a2. In this limit, the
charges (2.5) become
Mext = 6a2 cosh 2 ;
JL ext = 4a3 (c3 + s3) ;
Qext = 4a2sc ;
and the angular velocity (2.7) and electric potential (2.8) become
2(c3 + s3)2a2 + (c4 + c2s2 + s4)pM0(M0
4a2) : (2.10)
L ext =
ext =
At extremality, the Bekenstein-Hawking entropy (2.10) reduces to
In this paper we are interested in the near-extreme case so we introduce a small parameter
this into (2.10) and keeping terms up to linear order in ^, the near extremal entropy is
SBH near ext = 8 a3(c3
s3) + 4 a3(c3 + s3) ^ + O(^2)
(6JL ext)
Near horizon, near extremal limit
Consider the coordinate transformation
t = 12
r =
= ^
= ^ :
Here, r+ is the location of the outer horizon given in (2.6) and
L ext is the extremal
angular velocity (2.12).
Making this coordinate transformation in the
one obtains the extremal near horizon geometry given in [12].
Here, we are interested in reaching the near horizon geometry of the black hole close,
but not exactly at, extremality. This is the analog of the so-called near-NHEK limit for
4D Kerr considered in [9]. In order to do this, we still make the coordinate
transformation (2.15), but now parametrize deviations from extremality with a parameter
de ned by
Then the metric (2.2) gives rise to
M0 = 4a2 + a2 2 2 :
ds52 =
r(r + 2 )
r(r + 2 )dt2 +
! 0 limit. Here, we have de ned
This notation will be clari ed in the next subsection. The location of the horizon in (2.17)
temperature by
When we identify
with the parameter ^ introduced in (2.14), the metric (2.17)
corresponds to the near horizon geometry of the black hole (2.2) close to extremality in
the following complementary sense as well. Making the coordinate transformation (2.15)
obtain (2.17) with
TR
The gauge eld corresponding to the near horizon, near extremal geometry is obtained
by accompanying the coordinate transformation (2.15) with the gauge transformation
Then the gauge eld (2.20) becomes
! 0 limit.
Frolov-Thorne temperatures
A =
ae tanh 2
+ cos d + e 2 (r + )dt
TR =
TL =
TQ =
We now move on to compute the Frolov-Thorne temperatures corresponding to the
nearextremal Kerr-Newman black hole, by adapting the strategy of [9] to our present context.
Consider a scalar eld
' = e i!t^+im ^ R^(r^) S( ) T ( ^)
on the the black hole background (2.2), with charge q under the gauge eld (2.3).
Zooming into the near horizon region requires performing the coordinate transformation (2.15)
combined with the gauge transformation (2.20). The charged scalar (2.22) thus becomes
' = eiq e inRt+inL R(r) S( ) T ( )
m = nL ;
! =
Now, the scalar eld is in a mixed quantum state whose density matrix has eigenvalues given
by the Boltzmann factor e T1H (! m L+q ), where TH is the Hawking temperature (2.9).
= e TL TR TQ
and using (2.24) we nd the following Frolov-Thorne temperatures:
TH =
TR =
TH =
TH =
s3 + (c3 + s3)p1
4a2=M0
4a2=M0
4a2=M0
Near extremality, M0 is given by (2.16) and (2.26){(2.28) become, in the
! 0 limit,
Recall that both TR and TL have already appeared in our discussion: the former as the
Hawking temperature (2.19) of the near-horizon, near-extremal metric (2.17) and the latter
as a parameter, (2.18), in that metric. The present analysis elucidates the names given
previously to those quantities.
TL =
TQ =
The logarithmic correction to the microcanonical entropy of a non-extremal, rotating
charged black hole in general spacetime dimension D has been computed by Sen in [6].
His result applies to the near extremal black hole considered in this paper, which has a
small but non zero Hawking temperature. Equation (1.1) in [6] for the correction to the
microcanonical entropy reads
Smc M; J~; Q~
= SBH
+ log a Clocal
where NC =
D 1 is the number of Cartan generators of the spatial rotation group and
nV is the number of vector elds in the theory. Clocal arises from one loop determinants of
massless elds
uctuating in the black hole background and vanishes in odd dimensions.
The remaining contribution in (3.1) comes from zero modes and Legendre transforms.
Smc = SBH
with, for the case at hand, SBH given by (2.10).
Microscopic computation
We now change gears and compute the logarithmic correction to the entropy of the
microscopic theory dual to the Kerr-Newman black hole. In [12] this solution was embedded
into string theory and the microscopic dual thereby shown to be the infrared
of a 1+1 eld theory living on the brane intersection. This xed point is a possibly
nonlocal deformation of an ordinary 1+1 conformal eld theory which preserves at least one
in nite-dimensional conformal symmetry. While the string theoretic construction implies
the existence of the xed point theory, it exhibits a new kind of 1+1 D critical behavior and
is only partially understood. The near horizon geometry (2.17) has a SL(2; R)R
isometry subgroup coming from the isometries of the AdS2 submanifold and the unbroken
SU(2)L rotation isometry respectively. Various in nite-dimensional enhancements
of this global isometry, involving di erent boundary conditions, have been extensively
considered in the literature, and may be relevant in di erent circumstances or for di erent
computations. See [20] for a recent discussion. We consider two of them which turn out to
both give the same log corrections.3
In this subsection we consider a CFT in which the global symmetries are enhanced as
SL(2; R)R
U(1)L ! V irR
where V irL and V irR are left and right moving Virasoro algebras with generators Ln and
Ln respectively. L0 generates
rotations and L0 generates AdS2 time translations.
3Had they been di erent, the matching of logarithmic corrections would have singled one out.
(EL; ER; p~) '
be evaluated by saddle point methods. The integrand reaches an extremum at
0 =
0 =
0i =
where the matrix kij is the inverse of kij and P
pipj kij . The leading contribution to
the entropy is obtained by evaluating (3.10) at the saddle (3.11). This gives
S = log 0 = 2
R). Standard modular invariance of this partition function
considering in this paper has an additional SU(2)
U(1) global symmetry, corresponding
to the SU(2) rotation isometry and the U(1) gauge symmetry. Turning on the associated
chemical potentials, the partition function becomes
and it obeys the modular transformation rule
Z( ; ; ~ ) = e
Here i are left chemical potentials associated with the left moving conserved charges P i
i j kij with kij the matrix of Kac-Moody levels of the left moving currents. In
our case i; j run from 1 to 2 but, for the sake of generality, we temporarily assume they
run from 1 to n. This partition function is related to the density of states, , at high
temperatures by
Z( ; ; ~ ) =
dEL dER dnp (EL; ER; p~) e2 i EL 2 i ER+2 i ipi ;
where EL; ER; pi are the eigenvalues of L0; L0; P i respectively. For small , (3.7)
imThen, inverting (3.8), we obtain the following expression for the density of states:
We put the CFT on a circle along
t and consider the ensemble
We assume that
Z( ; ) = Tr e2 i L0 2 i L0 :
we see that (3.14) matches the near-extremal Bekenstein-Hawking entropy (2.14) to linear
order in . This extends the match of [12] from the extremal to the near-extremal regime.
The logarithmic correction
S to the leading entropy (3.12) is generated by Gaussian
uctuations of the density of states (3.10) about the saddle (3.11):
S =
where A is the determinant of the matrix of second derivatives of the exponent in the
integrand of (3.10) with respect to , i
det A =
1 3
P2) n+23 ( ERv) 2 (4ER) 2 det kij :
U(1) current algebra corresponding to SU(2)
kQ is given in appendix A, the SU(2) level
ELv = ERv =
ER =
S =
rotations and the gauge eld. The SU(2)
/ Qext. The U(1) Kac-Moody level k22
following scalings,
Bringing (3.17) to (3.15), (3.16), we obtain
P2=4 ; ER
S =
In this subsection we consider a warped CFT, in which the global symmetries are
enhanSL(2; R)R
U(1)L ! U[(1)R
Here U[(1)R is a left moving Kac-Moody algebra whose zero mode R~0 generates the right
sector time translations in AdS2 and V irL is a left moving Virasoro algebra whose zero
mode L~0 generates the left sector U(1)L rotational isometry. The symmetry algebra of our
warped CFT is
hL~m; L~ni = (m
hR~m; R~ni =
n)L~m+n +
hL~m; R~ni =
where L~m and R~m are the Virasoro and Kac-Moody generators respectively. Putting the
theory on a circle along
, the partition function at inverse temperature
and angular
is given by Z( ; ) = Tr e
that by rede ning the charges as
R~0+i L~0 . On the other hand, in [18] it was shown
Ln = L~n
2 R~0R~n +
Rn =
2 R~0R~n
and putting the theory on the same circle but in the di erent ensemble4
the partition function obeys the usual CFT modular invariance:
Z( ; ) = Tr e2 i L0 2 i R0 ;
Z( ; ) = Z( 1= ; 1= ) :
i( L + R) we may then proceed as in the previous section replacing
L0 with R0 everywhere starting from equation (3.6) onwards. We thus arrive at the same
results for the leading entropy and its logarithmic correction.
It should be noted that the enhancement (3.19) is somewhat unusual in the context of
warped AdS3 [17]. A third more natural enhancement SL(2; R)R
in that context is also possible [20]. However, this case may not be treated as (3.19) above
because the arguments of [18] do not apply to the case when the identi cation in the
bulk (along
) is precisely anti-aligned with the action of L0 (along t). It is an
important outstanding problem in Kerr/CFT to generalize the arguments of [18] to accomodate
U(1)L ! V irR
Match of the macroscopic and microscopic computations
We have already exhibited the match, in the near-extremal regime, of the bulk and
microscopic results for the leading term of the entropy of the ve-dimensional Kerr-Newman
black hole under consideration: the Cardy formula (3.14) reproduces the near-extremal
Bekenstein-Hawking entropy (2.14).
We will now show that the logarithmic corrections also agree. In order to furnish a
sensible comparison, one must ensure that both results are given in the same ensemble.
This is not the case for the macroscopic, (3.2), and microscopic, (3.18), results given above.
The former assumes the entropy to be a function of the energy Q[@t^] conjugate to the
asymptotic time which features in the full black hole solution (2.2), while the latter is
instead a function of the energy Q[@t] conjugate to the near horizon time which appears
4A change of ensemble may result in di erent logarithmic corrections to the entropy.
explained in appendix B, the change of ensemble corresponding to the charge rede nitions (3.20) here does
not imply any change in the logarithmic correction to the entropy.
in (2.17). The transformation between the macroscopic and microscopic density of states
requires a Jacobian factor (appendix B),
Now, from the change of coordinates (2.15) and the expression for the extremal angular
velocity in (2.12), we see that this Jacobian scales like
bulk =
Sbulk =
S + log a ;
fQpm ; Qpn g =
im 24 2TL2ae tanh 2
which indeed is satis ed by (3.2) and (3.18).
Acknowledgments
We thank Monica Guica, Thomas Hartman, Maria J. Rodriguez and Ashoke Sen for useful
conversations. AP and AS are supported in part by DOE grant DE-FG02-91ER40654.
APP is supported by NSF grant PHY-1504541. OV is supported by the Marie Curie
fellowship PIOF-GA-2012-328798.
Computation of kQ
In this appendix we compute the level kQ of the U(1) Kac-Moody algebra associated with
the gauge eld A . We do not perform a full asymptotic symmetry group analysis here.
We expect that with appropriate boundary conditions on the gauge eld this Kac-Moody
is consistent with the rest of the asymptotic symmetries used in section 3. Here we are
particularly interested in deriving the scaling of the level kQ with a.
Thus we assume the U(1) current algebra is generated by
TL . In modes, the generators
satisfy the algebra
= (y~) ;
pn =
2 TL e iny~=(2 TL) ;
[pm; pn] = 0 :
Using the formulas in [33], one can compute the central extension in the corresponding
Dirac bracket algebra. We nd:
The central extension comes entirely from the Chern-Simons term in the action (2.1).
Passing to the commutators f ; g !
i[ ; ] we obtain the current algebra,
[Pm; Pn] =
kQ = 12 (2 TL)2 ae tanh 2 :
@(q1; q2; : : :)
@(q10; q20; : : :)
with level given by
Change of ensemble
appropriate Jacobian factor as
However, kR / c
change of ensemble.
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which scales like log a, often picks up contributions from the Jacobian factor above. We
have seen this explicitly in section 3.2 where the Jacobian (3.25) scales with a.
Another instance of a change of ensemble was mentioned in relation to the charge
rede nitions in (3.20). In this case the Jacobian is
@(L0; R0)
@(L~0; R~0)
= 2
R0 = 2
a3 [20] and R0
a3 so in this instance the Jacobian does not scale with
a and therefore the logarithmic correction to the entropy is left intact by this particular
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