Subleading Correction to Statistical Entropy for BMPV Black Hole

Abstract

We study higher derivative corrections to the statistical entropy function and the statistical entropy for five dimensional BMPV black holes by doing the asymptotic expansion of the partition function. This enables us to evaluate entropy for a large range of charges, even out of Cardy (Farey tail) limit. Comment: 0+11 pages, V2, one section added, typos corrected

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arXiv:0807.1314v2 [hep-th] 8 Sep 2008
Preprint typeset in JHEP style - HYPER VERSION
Subleading Correction to Statistical Entropy
for BMPV Black Hole∗
Nabamita Banerjee
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019,
INDIA
Email: nabamita@mri.ernet.in
Abstract: We study higher derivative corrections to the statistical entropy function
and the statistical entropy for five dimensional BMPV black holes by doing the
asymptotic expansion of the partition function. This enables us to evaluate entropy
for a large range of charges, even out of Cardy (Farey tail) limit.
Keywords: Higher derivative, Degeneracy, Statistical Entropy.
∗Based on talks given at LPTHE, Jussieu, AEI, Potsdam, ETH, Zurich, INFN Rome.

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Contents
1. Introduction1
2. Degeneracy Function For 5-dimensional BMPV Black Holes
2.1Degeneracy Function of 4-dimensional Dyonic Black holes
2.2 Degeneracy for BMPV Black Holes
2
3
5
3. Correction to The Statistical Entropy Function5
4. Correction to Statistical Entropy8
5. Degeneracy for More General 5D Black Holes9
6. Discussion9
1. Introduction
Counting of 1/4 BPS dyonic states in four dimensional N = 4 supersymmetric string
theories has been studied in great detail in last few years[1],[2],[3],[4],[5],[6],[7],[8]. We
now have a good understanding of the degeneracy formula, its moduli dependence and
the wall crossing formulae. Large charge asymptotic expansion of these degeneracy
formulae exactly capture the dyonic black hole entropy including certain subleading
corrections due to higher derivative corrections to the supergravity.
Five dimensional spinning (BMPV) black holes[9] is a close cousin of the four
dimensional dyonic black hole. These black holes were first constructed in [9], as
a spinning generalization of [10]. These are charged, spinning, 5-dimensional black
holes with constant dilation and constant moduli in type IIB theory on K3 × S1.
The microscopic configurations of these black holes can be described as p-solitonic
states (Dp- branes) in type IIB theory on K3 × S1, for p = 1,3,5. The states
also carry certain momenta along the S1circle and angular momenta along the non
compact directions. This microscopic description is much similar to that of the four
dimensional dyonic black holes. In fact,when described in terms of D-branes, the
BMPV black hole consists of D1-D5-p system, whereas the four dimensional dyonic
black hole in addition has a KK monopole background. It is therefore natural to
study BMPV black hole entropy in terms of the four dimensional dyon degeneracy
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formula without KK monopole contribution. A general degeneracy formula for D1-
D5-p system is then easy to write down. However, we are interested in finding out
subleading correction to the BMPV black hole entropy due to higher derivative terms
in the effective action. Higher derivative correction to five dimensional black holes has
been computed in [11] and their entropy has been computed [12]. In this paper we will
take a different approach to this problem. Determination of subleading correction
is done in a most effective fashion using the statistical entropy function and the
effective action formalism. Using the statistical entropy function one can write down
a one dimensional effective action, and using the Feynman diagram technique one can
obtain systematic large charge asymptotic expansion of the statistical entropy. This
method correctly reproduces subleading correction to the entropy of four dimensional
1/4 BPS dyonic black holes.
This particular feature of the statistical entropy function gives the motivation
to compute similar subleading correction to the five dimensional BMPV black hole.
The exactness of the statistical entropy (or the statistical entropy function) suggests
that we can evaluate the entropy (or the entropy function) to any order.
The rests of the paper is divided into three sections. In the first section, we
present a different form of the degeneracy function of the five dimensional BMPV
black holes, based on the degeneracy of the four dimensional dyonic black holes. In
the next section we discuss the first subleading (O(Q0)) correction to the statistical
entropy function and statistical entropy of these black holes. In the last section, we
have some discussions on our results.
As this paper was being written a paper [13] appeared in the arXiv that discusses
similar issues.
2. Degeneracy Function For 5-dimensional BMPV Black Holes
In this section, we rewrite the degeneracy function of the BMPV black holes in a
different form. The microscopic description for this black hole is a particular D-
brane configuration in type IIB theory compactified on K3 × S1. This contains Q1
number of D1 branes, Q5 number of D5 branes, −n units of momenta along S1
circle and angular momenta J1 and J2 along the non-compact spatial directions.
This configuration, however does not contain any D3 branes. For extremal black
holes, the corresponding microscopic configuration requires the modulus of the two
angular momenta to be same |J1| = |J2| = J. The microscopic computation for the
leading entropy was first done in [9]. We will write α′(inverse string tension) exact
degeneracy function for this configuration, from the knowledge of the degeneracy
function of 4-dimensional dyonic black holes in N = 4 supersymmetric string theory.
Here we will sketch in brief how the degeneracy function was obtained for these
4-dimensional black holes [14],[15].
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2.1 Degeneracy Function of 4-dimensional Dyonic Black holes
Let us consider type IIB theory compactified on K3 × S1טS1. Following a chain
of duality transformations, one can look at the same theory as a heterotic string
theory compactified on T6. These theories have dyonic black-hole solutions. Let us
consider a specific configuration in this compactified type IIB theory : Q1number
of D1-branes wrapped along S1, Q5number of D5-brane wrapped along K3 × S1,
a single Kaluza-Klein monopole associated with˜S1circle,−n units of momentum
along S1direction and J units of angular momentum along˜S1direction. In the dual
Heterotic picture, this represents dyonic black hole solutions. If we stay in a region of
the moduli space where the type IIB theory is weakly coupled, the partition function
of the entire system can be obtained by considering three weakly interacting sources:
1. the relative motion of the D1-brane in the plane of D5-brane, carrying certain
momenta −L along S1and J′ ˜S1directions,
2. the center of mass motion of D1-D5 system in the KK-monopole background
carrying momenta −l0along S1and j0along˜S1directions,
3. excitations of the KK-monopole carrying −l′
0momentum along S1,
with n = L + l0+ l′
˜S1directions respectively. Hence, in the weak coupling limit, the partition function
f(˜ ρ, ˜ σ, ˜ v) of the configuration can be expressed as,
0and J = J′+ j0 being the sum of momenta along S1and
f(˜ ρ, ˜ σ, ˜ v) = −1
64
?
?
Q1,L,J′
(−1)J′dD1(Q1,L,J′)e2πi(˜ σQ1/N+˜ ρL+˜ vJ′)
?
??
l0,j0
(−1)j0dCM(l0,j0)e2πil0˜ ρ+2πij0˜ v
?
?
l′
0
dKK(l′
0)e2πil′
0˜ ρ

, (2.1)
where dD1(Q1,L,J′) is the degeneracy of source (1), dCM(l0,j0) is the degeneracy
associated with source (2) and dKK(l′
(3). The factor of 1/64 accounts for the fact that a single quarter BPS supermultiplet
has 64 states. Evaluating these three pieces separately, the full partition function of
the system looks like,
0) denotes the degeneracy associated with source
f(˜ ρ, ˜ σ, ˜ v) = e−2πi(˜ ρ+˜ v)
?
k′∈z z+r,l∈z z,j∈2z z
k′,l≥0,j<0 for k′=l=0
?
1 − e2πi(˜ σk′+˜ ρl+˜ vj)?−c(4lk′−j2)
. (2.2)
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Then we define the degeneracy function˜Φ(˜ ρ, ˜ σ, ˜ v) and degeneracy of states
d(?Q,?P) as,
f(˜ ρ, ˜ σ, ˜ v) =
e2πi˜ σ
˜Φ(˜ ρ, ˜ σ, ˜ v),
d(?Q,?P) = (−1)Q·P+1h
?1
2Q2,1
2P2,Q · P
?
, (2.3)
where (?Q,?P) are the charge vectors carried by the black holes:
Q =





0
−n
0
−1




,P =





Q5(Q1− Q5)
−J
Q5
0




. (2.4)
and h(m,n,p) are the coefficients of Fourier expansion of the function 1/˜Φ(˜ ρ, ˜ σ, ˜ v):
1
˜Φ(˜ ρ, ˜ σ, ˜ v)=
?
m,n,p
g(m,n,p)e2πi(m ˜ ρ+n ˜ σ+p ˜ v).(2.5)
At this point it is worth noting from equations (2.2) and (2.5) that the power
series gets a contribution e−2πi˜ v(1 − e−2πi˜ v)−1from k′= l = 0 term and one can
expand the series either in e−2πi˜ vor in e2πi˜ v. There is an ambiguity in the expansion,
we will come back to this point in the last section .
To evaluate the degeneracy of a state associated with charges (?Q,?P), we need
to invert equation (2.5) as,
d(?Q,?P) = (−1)Q·P+1
?
C
d˜ ρd˜ σd˜ ve−πi(˜ ρQ2+˜ σP2+2˜ vQ·P)
1
˜Φ(˜ ρ, ˜ σ, ˜ v), (2.6)
where C is a three real dimensional subspace of the three complex dimensional space
labeled by (˜ ρ, ˜ σ, ˜ v), given by
˜ ρ2= M1,
0 ≤ ˜ ρ1≤ 1,
˜ σ2= M2,
0 ≤ ˜ σ1≤ 1,
˜ v2= −M3,
0 ≤ ˜ v1≤ 1.(2.7)
M1, M2 and M3 are large but fixed positive numbers with M3 << M1,M2. The
choice of the Mi’s is determined from the requirement that the Fourier expansion is
convergent in the region of integration.
The N = 4 supersymmetric string theories discussed above are invariant under
O(6,22,Z) T-duality and SL(2,Z) S-duality symmetry. The T-duality invariants are
given as,
Q2= 2n,P2= 2Q5(Q1− Q5),
The function˜Φ actually behaves as a modular form of weight k = 10 under the
S-duality group SL(2,Z).
Q · P = J.(2.8)
– 4 –