\vspace*{-4ex} D1-D5-J system, small black ring in five dimensions,\\ and the 4D-5D connection
hep-th0506215.tex
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%\title{\vspace*{-4ex} D1-D5-J system, small black ring in five dimensions,\\ and the 4D-5D connection}
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 \vspace*{-2cm} 
 hep-th/0506215\\
 TIFR-TH-05-26\\
 CALT-68-2565
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\begin{center}
\vspace*{2cm} 
{\bf
{\LARGE  A Note on D1-D5-J System \\[1ex]
 and 5D Small Black Ring}\\% and the 4D-5D connection}\\
\vspace*{1.3cm}
Norihiro Iizuka$^1$ and Masaki Shigemori$^2$\\
}
\vspace*{1.0cm} 
{\it
$^1$ Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India\\[1ex]
$^2$ California Institute of Technology 452-48, Pasadena, CA 91125, USA\\
}
\vspace*{0.8cm}
{\tt iizuka@theory.tifr.res.in},
{\tt {sh{}}i{}ge{}@t{}h{{}}e{ory}.c{}a{}l{{}}te{{}}ch.e{{}}d{}u} % avoid spam!
\end{center}
\vspace*{1.5cm}

%%
%% abstract
%%

\baselineskip=18pt % a la harvmac


\noindent
%
The ``small'' black ring in 5D obtained by giving angular momentum to
the D1-D5 system compactified on $S^1\times {\rm K3}\,$ is a very
interesting object in the sense that it does not have an event horizon
in the supergravity limit whereas it microscopically has a finite
entropy.
%
The microscopic origin of this small black ring can be analyzed in
detail since it is constructed by adding angular momentum to the
well-studied D1-D5 system.  On the other hand, its macroscopic,
geometrical picture is difficult to study directly.
%
%
In this note, by duality transformations and the 4D-5D connection, we
relate this 5D small black ring to a 4D small non-rotating black hole,
where the latter is known to develop a non-vanishing horizon due to
stringy $R^2$ corrections to the supergravity action.  
%
%
%
%%% the following is version 1 -- M.S.
%This gives an indirect evidence that a non-vanishing horizon is formed
%for the 5D small black ring.  We also show that the entropy of the 4D
%small black hole agrees with the microscopic entropy of the 5D small
%black ring, which supports that the 4D-5D connection is indeed valid
%even for small black objects.
%
%%% the following is version 2 -- M.S.
The entropy of the 4D small black hole agrees with the microscopic
entropy of the 5D small black ring, which supports the validity of the
4D-5D connection even for small black objects.  These results give an
indirect evidence that a non-vanishing horizon is formed for the 5D
small black ring.


%
%This horizon of the 5D small black ring can be interpreted as the
%``stretched horizon'' by Lunin and Mathur in hep-th/0202072.

%
%In this note, by using a duality chain and the 4D-5D connection, we
%point out that one can relate this 5D small black ring to a 4D small
%black hole.  The latter is known to develop a finite horizon because of
%stringy $R^2$ corrections to the supergravity action, and its entropy is
%well-understood both macroscopically and microscopically.
%%
%Using this relation, we reproduce the microscopic entropy of the 5D
%small black ring from the entropy of the 4D small black hole.
%%
%These analyses give indirect evidences of a horizon showing up in the
%5D small black ring.
%%
%This black ring horizon can be interpreted as ``stretched horizon'' by
%Lunin and Mathur in hep-th/0202072.
%

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%\tableofcontents


%%
%% main text
%%


\section{Introduction}

The D1-D5-P system, made of $N_1$ D1-branes, $N_5$ D5-branes and $N_P$
units of momentum, has been an ideal arena for studying microscopic
physics of black holes in string theory \cite{Strominger:1996sh}.
%
The $N_P=0$ case (the D1-D5 system) is not classically a black hole
because its horizon vanishes at the supergravity level, whereas its
microscopic entropy computed from the dual CFT is finite:
$S=2\sqrt{2}\pi\sqrt{N}$ ($T^4$ compactification) and
$S=4\pi\sqrt{N}$ (K3 compactification), where $N\equiv N_1 N_5$.
%
In a recent beautiful paper by Dabholkar \cite{Dabholkar:2004yr}, it was
shown that heterotic 4D black hole with classically vanishing horizon,
which is dual to the above D1-D5 system compactified on $T^2\times {\rm
K3}$, becomes a black hole with string-size event horizon once stringy
$R^2$ corrections to the supergravity action are taken into account, and
that the $R^2$-corrected macroscopic entropy\footnote{The
$R^2$-corrected macroscopic entropy was derived in a sequence of papers
\cite{deWitetal}.}  of such ``small'' black hole agrees with the
microscopic entropy, as predicted by Sen \cite{Sen:1995in}.  Further
references for these developments include
\cite{Dabholkar:2004dq}\cite{afterDabholkar}.



If we consider the D1-D5 system compactified instead on $S^1\times{\rm
K3}$ and add angular momentum $J=\CO(N)$ to it, the brane worldvolume starts to
look like a ring \cite{Lunin:2001fv}\cite{Lunin:2002qf} from the 5D
viewpoint, rather than a pointlike object.
Therefore, this D1-D5-J system is expected to be described by a
``small'' version of the black ring in 5D supergravity
\cite{Elvang:2004rt}, once we consider stringy corrections to the
supergravity action.  However, a systematic framework for studying
stringy corrections to the 5D supergravity, like the one for 4D
\cite{deWitetal}, has not been available so far,
%\cite{Guica:2005ig}\cite{Bena:2005ae}
so we cannot directly study the
stringy corrections to this system.
%       (cf.\ in
%       \cite{Lunin:2002qf} it was shown that ${A_{\rm
%       stretched}/4G}\sim\sqrt{N_1N_5-|J|}$ as expected from microscopic
%       state counting, where $A_{\rm stretched}$ is the area of the
%       stretched horizon, defined as the surface on which individual
%       microstate geometries start to differ.)



Recently, a new connection between 4D and 5D black objects was proposed
\cite{Gaiotto:2005gf}\cite{Elvang:2005sa}\cite{Gaiotto:2005xt}\cite{Bena:2005ni},
which relates the partition function of a 5D black hole/ring with that
of a 4D black hole.
%
Although this ``4D-5D connection'' is based on supergravity analysis, it
is expected to hold even if we take into account stringy corrections,
since it is natural to expect that a BPS solution interpolating 4D and
5D objects exists even if stringy corrections are included, and the
continuous-moduli-independence of entropy is the property of the
microscopic theory, not of supergravity.
%
%a 4D black hole is subject to higher curvature corrections
%implies that a 5D black hole/ring is also subject to corresponding
%higher curvature corrections \cite{Guica:2005ig}\cite{Bena:2005ae}.
Indeed, the microscopic entropy of 5D black ring
\cite{Bena:2004tk}\cite{Cyrier:2004hj} agrees with that of 4D black hole
\cite{Maldacena:1997de} including quantum corrections
\cite{Guica:2005ig}\cite{Bena:2005ae}.  This suggests that 5D black
holes/rings which classically have no horizon will generally develop
finite small horizon.
%This enables one to study 5D black objects using the
%techniques developed for 4D.


Therefore, one naturally wants to apply the 4D-5D connection to the
D1-D5-J ring system on $S^1\times{\rm K3}$ to study its properties using
4D techniques.  However, the 4D-5D connection is not directly
applicable, since one cannot get to the duality frame in which one can
make use of the 4D-5D connection via simple $S$-duality and $T$-duality
along $S^1$ directions in type II\@.
%
In this note, we map by $U$-duality the D1-D5-J system into a duality
frame in which the 4D-5D connection is applicable, and relate it with a
small non-rotating black hole in 4D\@.  
%
The microscopic entropy counting is well-understood for both these 4D
and 5D configurations.  On the macroscopic side, the horizon geometry of
the 4D small non-rotating black hole is well-understood whereas the
geometry of the original D1-D5-J system, which is expected to be a 5D
small black ring, is not well-understood.  By connecting this D1-D5-J
system with the geometry of the 4D small black hole that has a
small horizon, we give an indirect evidence of event horizon
showing up in the 5D small black ring by stringy corrections to the
supergravity action.



%% version 1
%The organization of this paper is as follows. In section 2, we study the
%microscopic entropy of the D1-D5-J system, and clarify the limit in
%which it can be regarded as a ring with a well-defined profile.  In
%section 3, we present the duality chain which connects the D1-D5-J
%system and a 4D non-rotating black hole, where the latter has a
%non-vanishing horizon.  This suggests that event horizon appears
%for the small black ring by stringy corrections to the supergravity
%action.

%% version 2
The organization of this paper is as follows. In section
\ref{sec:D1-D5-J}, we study the microscopic entropy of the D1-D5-J
system, and clarify the limit in which it can be regarded as a ring with
a well-defined profile.  In section \ref{sec:4D-5D_conn}, we present a
duality chain which relates the D1-D5-J system and a 4D non-rotating
small black hole via the 4D-5D connection. 
%This 4D small black hole has non-vanishing horizon by stringy corrections 
%to the supergravity action. 
%
The entropy of the 4D small black hole agrees with the microscopic
entropy of the D1-D5-J system, which justifies applying the 4D-5D
connection for the small black objects.
%
Also, this 4D small black hole has non-vanishing horizon because of
stringy corrections to the supergravity action.
%
These results suggest that event horizon
appears for the 5D small black ring by stringy corrections to the
supergravity action.
%
In section \ref{sec:concl}, We conclude with some comments on future
directions.

\medskip
The D1-D5-J small black ring system and its entropy was also studied
recently by Kraus and Larsen \cite{Kraus:2005vz} from a different point
of view.


%\section{``Large'' black ring}
%
%{\bf (This section isn't perhaps necessary)}
%
%The five-dimensional supersymmetric black ring was constructed in
%\cite{Elvang:2004rt}\cite{Bena:2005ni}, and its microscopic counting of
%entropy was done in \cite{Bena:2004tk} from the D1-D5 CFT viewpoint and
%in \cite{Cyrier:2004hj} from the M-theory viewpoint.  Later, 4D-5D
%connection
%\cite{Gaiotto:2005gf}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}\cite{Elvang:2005sa}
%gave another perspective to entropy counting.  
%
%Let us review the construction of the supersymmetric black ring.  One
%considers M-theory compactified on $T^6$.  The remaining five dimensions
%are $\bbR_t\times \bbR^{4}$.  Take an $S^1$ in the $\mathbb
%R^{4}$, and call the $S^1$ direction $\psi$.  Now wrap M5-branes on the
%$\psi$ direction and three $T^4$'s in $T^6$.  Turn on worldvolume fields
%on M5, which is equivalent to having M2-branes wrapping $T^2$'s in
%$T^6$.  Furthermore, give the system angular momentum $J$ in the $\psi$
%direction, which keeps the ring from collapsing.  Explicitly, the
%configuration of the branes is as follows:
%\begin{align}
% \begin{array}{cc||c|cccccc}
%  && \psi& 5 & 6&7&8&9&10\\
%  \cline{2-9}
%q_1&   M2 & \sim     & \bigcirc & \sim     & \sim     & \sim     & \sim     &\bigcirc\\
%q_2&   M2 & \sim     & \sim    & \bigcirc     & \bigcirc     & \sim     & \sim     &\sim\\
%q_3&   M2 & \sim     & \sim & \sim     & \sim     & \bigcirc     & \bigcirc  &\sim\\
%p^1&  m5 & \bigcirc & \sim & \bigcirc & \bigcirc & \bigcirc & \bigcirc &\sim\\
%p^2&  m5 & \bigcirc & \bigcirc & \sim & \sim & \bigcirc & \bigcirc &\bigcirc\\
%p^3&  m5 & \bigcirc & \bigcirc & \bigcirc & \bigcirc & \sim & \sim &\bigcirc\\
%J_\psi&  p & \bigcirc & \sim & \sim &\sim &\sim &\sim &\sim\\
% \end{array}
%\end{align}
%This configuration leaves 4 supersymmetries out of the 32
%supersymmetries.  From the 5D point of view, this is a black ring, with
%$\psi$ being the ring direction.  The macroscopic, Bekenstein--Hawking entropy
%$S_{macro}$ of the black ring can be computed from the area of the horizon
%\cite{Elvang:2004rt}\cite{Bena:2004de}:
%\begin{align}
% S_{macro}&=2\pi\sqrt{{D^2\over 4}-D J_\psi-{1\over 4}\left[(p^1q_1)^2+(p^2q_2)^2+(p^3q_3)^2\right]
% +{D\over 2}\left({q_1q_2\over p^3}+{q_2q_3\over p^1}+{q_3q_1\over p^2}\right)},
% \label{Smacro_BR}
%\end{align}
%where $q_A$ and $p^A$ are the number of M2-branes and M5-branes, and
%$D=p^1p^2p^3$.
%
%Microscopic account for the entropy was done in \cite{Bena:2004tk} from
%the D1-D5 CFT viewpoint and in \cite{Cyrier:2004hj} from the M-theory
%viewpoint.  Later, using the 4D-5D connection
%\cite{Gaiotto:2005gf}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}\cite{Elvang:2005sa},
%it was realized that the microscopic entropy counting can be reduced
%exactly to the entropy counting of 4D black holes by Maldacena,
%Strominger, and Witten \cite{Maldacena:1997de}.
%
%
%The 4D-5D connection says, by embedding the black ring in the Taub-NUT
%geometry, that the contractible $S^1$, i.e.\ the $\psi$ direction along
%which the black ring is wound, can be exchanged with a non-contractible
%$S^1$, which can be taken to be the M-circle.  After this switching, the
%system becomes a special case of the system considered in
%\cite{Maldacena:1997de}, with $J$ now regarded as the momentum along the
%M-circle, i.e., the D0 number.  In \cite{Maldacena:1997de}, they
%considered M-theory compactified on $CY_3\times S^1$, with M5-branes
%wrapping 4-cycles in $CY_3$ and $S^1$, and M2-branes wrapping 2-cycles
%in $CY_3$.  Their result for microscopic entropy was
%\begin{align}
% S_{micro}&=2\pi\sqrt{{1\over 6}|\qh_0|C_{ABC}p^A p^Bp^C}.
% \label{MSW_cl}
%\end{align}
%Here $C_{ABC}$ is the triple intersection number of $CY_3$. In the
%present case where $CY_3$ is $T^6$, $C_{ABC}=|\epsilon_{ABC}|$.  This
%microscopic entropy \eqref{MSW_cl} agrees with the macroscopic entropy
%\eqref{Smacro_BR} \cite{Cyrier:2004hj}.


%\section{``Small'' black ring}

%Can we use the above black ring to see whether even in 5D, a supergravity
%solution that has no classical horizon can develop a finite
%horizon by stringy effects?  The MSW entropy formula that includes
%quantum correction is
%\begin{align}
% S_{micro}&=2\pi\sqrt{{1\over 6}|\qh_0|(C_{ABC}p^A p^Bp^C+c_{2A}p^A)}.
% \label{MSW}
%\end{align}
%The first term in the square root corresponds to the classical entropy
%\eqref{MSW_cl} while the second term is the quantum correction.
%In general, the second term is nonzero.  
%%
%However, in the black ring we considered above, the $CY_3$ is $T^6$ and
%hence the second Chern class vanishes, $c_{2A}\equiv 0$.  Therefore, if
%the classical entropy is zero, the whole entropy \eqref{MSW} vanishes,
%and we cannot hope to be able to see that a finite horizon is formed.
%
%So we need to come up with a more nontrivial situation.  Of course, the
%hint is that Dabholkar considered heterotic string on $T^4$ which is
%dual to type IIA on K3 to successfully reproduce the ``small'' black
%hole entropy.


\section{D1-D5-J system}
\label{sec:D1-D5-J}

IN order to understand the charges and dipole charges that appear in the
D1-D5-J system, it is easiest to start from heterotic string
compactified in $\bbR^{1,4}\times S^1\times T^4$.  Let us take
$\bbR^{1,4}$, $S^1$, and $T^4$ directions to be $01234$, $5$, and $6789$
directions, respectively.  We will write the $S^1$ as $S^1_5$
henceforth. We wrap $N_F$ F1's along $S^1_5$ and put on it $N_P$ units
of linear momentum along $S^1_5$.  From Virasoro constraint and BPS
condition, we should impose $N_L=1-N_F N_P$, where $N_L$ is the
left-moving oscillation number.  Since $N_L>0$, we should choose $N_F
N_P<0$.  Furthermore, we give $J=n_p$ units of angular momentum to the
system in the $\psi$ direction, where $\psi$ is the angular direction on
the 1-2 plane.  Now the F1 worldvolume is a helix or a coil wound on a
cylinder spanned by $\psi$ and $x^5$, moving upwards along the $x^5$
axis.  From the 5D (012345) point of view, the system looks like a ring
in the $\psi$ direction. The configuration is as follows:
\begin{align}
 \label{cfg_BR_het}
 \begin{array}{c@{~~~~}c|cccc}
  && \psi& S^1_5 & T^4\\
  \cline{2-5}
  N_P & {P } & \sim & \bigcirc& \sim    \\
  N_F & {F1} & \sim & \bigcirc& \sim    \\
  n_p & {p } & \bigcirc& \sim & \sim    \\
  n_f & {f1} & \bigcirc& \sim  &\sim    
 \end{array}
\end{align}
%\begin{align}
% \label{cfg_BR_het}
% \begin{array}{cc||c|c|cccc}
%  && \psi& 5 & 6&7&8&9\\
%  \cline{2-8}
%  N_P & {P } & \sim & \bigcirc& \sim    & \sim    & \sim    & \sim    \\
%  N_F & {F1} & \sim & \bigcirc& \sim    & \sim    & \sim    & \sim    \\
%  n_p & {p } & \bigcirc& \sim & \sim    & \sim    & \sim    & \sim    \\
%  n_f & {f1} & \bigcirc& \sim  &\sim    & \sim    & \sim    & \sim    
% \end{array}
%\end{align}
Here, ``$\bigcirc$'' denotes wrapped directions, while ``$\sim$''
denotes smeared directions.  $P,p$ denote momenta and $N_P,n_p$ are the
corresponding momentum numbers.  $F1,f1$ denote fundamental string and
$N_F,n_f$ are the corresponding winding numbers.  The lowercase letters
mean that they are along the contractible direction $\psi$.  The
momentum number $n_p$ along $\psi$ is nothing but angular momentum
number $J$.  We will refer to this system \eqref{cfg_BR_het} as the FP
system.

The maximum angular momentum we can have is $J_{\rm max}=N=|N_F
N_P|=-N_F N_P$, which is the case when the F1 worldvolume is a perfect
helix of radius $\sim\sqrt{J_{\rm max}}$.  In the corresponding
classical solution \cite{Lunin:2001fv}, the F1 winds once around the
$\psi$ circle while it winds $N_F$ times the $x^5$ direction $S^1_5$.
Therefore, $n_f=1$ if $J=J_{\rm max}$.  For $J<J_{\rm max}$, the F1
worldvolume starts to fluctuate around the perfect helix, and it is not
obvious that the winding number $n_f$ is well-defined.  We will discuss
later in what limit $n_f$ is well-defined.


The system \eqref{cfg_BR_het} is dual to the well-studied D1-D5 system
in type IIB compactified on $S^1\times{\rm K3}$, by the following chain
of dualities: heterotic/IIA duality, $T(5)\,$\footnote{$T$-duality along
the axis of such helical objects was discussed in \cite{Mateos:2002yf}.}, and
then $S$.  The resulting configuration is:
\begin{align}
 \begin{array}{c@{\,}c@{\,}c@{~~~~}c|cccc}
  &&&& \psi& S^1_5 & {\rm K3}\\
  \cline{4-7}
  N_1&=& N_P  & D1 & \sim & \bigcirc& \sim    & \\
  N_5&=&-N_F  & D5 & \sim & \bigcirc& \bigcirc& \\
  n_p&=&n_p   & p  & \bigcirc& \sim & \sim    & \\
  n_{kk}&=&-n_f  & kk & \bigcirc& S^1  & \bigcirc& 
 \end{array}\label{D1D5smallring}
\end{align}
%\begin{align}
% \begin{array}{c||c|c|cccc}
%  & \psi& 5 & 6&7&8&9\\
%  \hline
%  D1 & \sim & \bigcirc& \sim    & \sim    & \sim    & \sim    \\
%  D5 & \sim & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \bigcirc\\
%  p  & \bigcirc& \sim & \sim    & \sim    & \sim    & \sim    \\
%  kk & \bigcirc& S^1  & \bigcirc& \bigcirc& \bigcirc& \bigcirc
% \end{array}\label{D1D5smallring}
%\end{align}
Now the 6789 direction is K3, and ``$S^1$'' means the special circle of
the KK monopole.  The numbers of D1- and D5-branes are $N_1=N_P$ and
$N_5=-N_F$. Note that the relation between the F1 number in heterotic
string and the NS5 number in type IIA involves a minus sign
\cite{Witten:1995ex}\cite{Kiritsis:1998hy}.  It is consistent with the
fact that we need $N_1 N_5>0$ for susy on the type II side.  This system
is dual to a 1+1 dimensional $\CN=(4,4)$ CFT, which is sigma model with
target space ${\rm K3}^N/S_N$, $N=N_1 N_5$ \cite{Vafa:1995zh}.

Now, let us count the entropy of the D1-D5-J system
\eqref{D1D5smallring}, or equivalently, the FP system
\eqref{cfg_BR_het}.  Let us here take the FP description, which is a
special case of the computation done in \cite{Russo:1994ev}.  We need to
count the number of states generated by 24 left-moving transverse bosons
$\alpha_{n}^i$, $i=1,...,24$, at level $N=|N_F N_P|=-N_F N_P$ that have
angular momentum $J=n_p$ in the 1-2 plane.

If we write
\begin{align}
 \alpha^{i=1}_n&=\sqrt{n/ 2}(a_n^+ + a_n^-),\qquad
 \alpha^{i=2}_n=i\sqrt{n/ 2}(a_n^+ - a_n^-),\qquad n=1,2,\dots,
\end{align}
then the level $N$ and angular momentum $J$ are
\begin{align}
 N
 &=\sum_{n=1}^\infty 
 \sum_{i=1}^{24}\alpha_{-n}^i\alpha_{n}^i
 =\sum_{n=1}^\infty 
 \left[ n(a_{n}^+{}^\dagger a_{n}^+ + a_{n}^-{}^\dagger a_{n}^-)
 + \sum_{i=3}^{24}\alpha_{-n}^i\alpha_{n}^i\right],\label{levef_def}\\
 J
 &=-i\sum_{n=1}^\infty {1\over n}(\alpha_{-n}^1\alpha_n^2-\alpha_{-n}^2\alpha_n^1)
 =\sum_{n=1}^\infty \left[ a_{n}^+{}^\dagger a_{n}^+ - a_{n}^-{}^\dagger a_{n}^-\right].
\end{align}
We can compute the entropy $S(N,J)$ by studying the partition function
\begin{align}
 Z={\rm Tr}[e^{-\beta(N+\lambda J)}]
 =\sum_{N,J}d_{N,J}q^N z^J.
\end{align}
Here $\lambda$ is the chemical potential conjugate to $J$, and
$q=e^{-\beta},z=e^{\beta\lambda}$.  The evaluation of $d_{N,J}$ was done
in \cite{Russo:1994ev}, and the leading term of the entropy for $N\gg
1$, $J=\CO(N)$ is
% \footnote{ More geneally, if there are $N_B$ bosons, two
% of which have angular momentum $J=\pm 1$, and $N_F$ fermions, which 
% may or may not have angular momentum, the entropy is
% \begin{align*}
%  S&=
%  {a(2N-|J|)\over \sqrt{N-|J|}}-2\log\cosh{aJ\over 2\sqrt{N-|J|}}
%  -{N_B+3\over 4}\log(N-|J|)\\
%  &\sim2a\sqrt{N-|J|}-{N_B+3\over 4}\log(N-|J|),
% \end{align*}
% where $a=\pi\sqrt{c/6}$, $c=N_B+N_F/2$.  The result does not depend on
% whether fermions carry spin or not.}
\begin{align}
 S&=\log d_{N,J}
 = 4\pi\sqrt{N-|J|} 
% -{27\over 4}\log(N-|J|) +{\rm const.}
% +\CO(e^{-{2\pi|J|\over \sqrt{N-|J|}}})
 .
 \label{S(N,J)}
\end{align}
%In appendix \ref{entropy_w_fermions}, we generalize this result to the
%case with fermions, which is relevant for the D1-D5-J system on $T^4$.

%Note that this result does not depend at all on $n_f$, the number of
%fermions that carry spin.  There are overwhelmingly larger number of
%states in which $J$ is carried by bosons than the number of states in
%which $J$ is carried by fermions.


One sees from \eqref{S(N,J)} that the only effect of $J\neq0$ is to
replace $N$ in the entropy formula with $N-J$. Here we assumed $J>0$,
$J=\CO(N)$. This means that, Bose--Einstein condensation of $J$
$a_{n=1}^+$ particles occurs and the whole angular momentum $J$ is
carried by the condensate. The remaining particles have level $N-J$ but
no net angular momentum.  In other words, in the ensemble with level $N$
and angular momentum $J>0$, $J=\CO(N)$, the states that contribute to entropy are of
the form
\begin{align}
 \underbrace{(a_{n=1}^+{}^\dagger)^J
 \raisebox{-3.25ex}{\rule{0pt}{1ex}}
 }_
 {\begin{minipage}{11ex}\scriptsize
   Bose--Einstein con\-den\-sate
  \end{minipage}}
 \times\,\,
 \underbrace{\prod_{n=1}^\infty \left[\prod_{i=\pm,3...24}(\alpha_{-n}^{i})^{N_{ni}}\right]\ket{0}}_
 {\begin{minipage}{24ex}\scriptsize 
   states that are responsible for entropy of the ensemble
  with level $N-J$ and no angular  momentum 
  \end{minipage}}
 .\label{hlro26May05}
\end{align}
The state $(a_{n=1}^+{}^\dagger)^J\ket{0}$, $J>0$ represents
(classically) a fundamental string that goes once around the circle in
the 1-2 plane of radius $\sim \sqrt{J}$, which is large if $J\gg 1$
\cite{Lunin:2001fv}. Clearly, this state has $n_f=1$.  Acting on this
state by the operator $\prod_{n=1}^\infty \left[\prod_{i=\pm,3...24}
(\alpha_{-n}^{i})^{N_{ni}}\right]$ in \eqref{hlro26May05} makes the
fundamental string fluctuate around this circle.  If this fluctuation is
much smaller than the radius of the circle $\sim\sqrt{J}$, the winding
number $n_f$ is well-defined and $n_f=1$.  Because
$\sum_{ni}N_{ni}=N-J$, statistical mechanics tells us that $N_{ni}\sim
\sqrt{N-J}$ for $n=\CO(1)$.  This means that the size of the fluctuation
is of order $(N-J)^{1/4}$, which is much smaller than the radius of the
circle $\sqrt{J}$ if
\begin{align}
 N,J\gg 1&, \qquad J=\CO(N). \label{cond_N,J}
\end{align}
In this limit, the system is expected to become a small black
ring,\footnote{More detailed analysis shows that, if $J=\CO(N^\gamma),$
$1/2<\gamma\le 1$, Bose--Einstein condensation occurs and the system is
expected to become a small black ring in 5D\@.  On the other hand, if
$\gamma\le 1/2$, Bose--Einstein does not occur and the system is
expected to become a small rotating black hole in 5D
\cite{Horowitz:1995tm}. } and talking about winding number along $\psi$
makes sense, so we can say $n_f=1$.  This relation between the black
hole/ring transition and Bose--Einstein condensation is very reminiscent
of the microscopic description of the large black ring in
\cite{Bena:2004tk}.


If $J<0$, then the above argument all goes through if we replace
$(a_{n=1}^+{}^\dagger)^J$ with $(a_{n=1}^-{}^\dagger)^{|J|}$.  In this case,
$n_f=-1$.  So, the $|J|$ in \eqref{S(N,J)} can be replaced with $
|n_p|=n_f n_p$.

Therefore, the entropy of the system \eqref{cfg_BR_het} or
\eqref{D1D5smallring} can be written as
\begin{align}
 S_{\rm micro}&=
 4\pi\sqrt{-N_F N_P-n_f n_p}
 =4\pi\sqrt{N_1 N_5+n_p n_{kk}},
 \label{S(N,J)-2}
\end{align}
where in the last equality we used the relation between charges listed
in \eqref{D1D5smallring}.

%{\bf (Need to insert comments on the case with $|n_f|>1$?)}

The entropy counting can be done also in the D1-D5 frame
\eqref{D1D5smallring} using the dual CFT\@.  If we replace the
oscillators $\alpha_{-n}^i$ with the chiral primaries of the $\CN=(4,4)$
CFT \cite{Lunin:2001pw}, the counting can be done in a completely
identical way.

\section{4D-5D connection}
\label{sec:4D-5D_conn}

Now, we would like use the 4D-5D connection
\cite{Gaiotto:2005gf}\cite{Elvang:2005sa}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}
in order to deform the 5D small black ring \eqref{cfg_BR_het} (or
equivalently \eqref{D1D5smallring}) into a 4D small black hole.


The obstacle to doing that in a straightforward manner is that
$T$-duality along the $S^1_5$ direction and $S$-duality in type II
string will not take the system \eqref{D1D5smallring} to a duality frame
in which the 4D-5D connection as derived in
\cite{Gaiotto:2005gf}\cite{Elvang:2005sa}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}
is applicable; one cannot avoid having unwanted nonzero charges.
%The 4D-5D connection is based on $\CN=2$ supergravity in
%four and five dimensions, which is obtained by compactifying type IIA or
%M-theory on a generic $CY_3$.  The 4D/5D gauge fields come from wrapping
%D-branes or M-branes on the even-dimensional cycles in the $CY_3$.
%However, in the present case, the internal manifold involves K3 and
%$S^1$, and there can be 4D/5D gauge fields coming from objects wrapping
%odd-dimensional cycles, such as KK momentum along $S^1$ or NS5 wrapping
%$S^1\times {\rm K3}$.  These cannot directly be incorporated in the
%framework of
%\cite{Gaiotto:2005gf}\cite{Elvang:2005sa}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}.
%Indeed, even if we take $S$-dual or $T$-dual along the $S^1_5$ direction
%of the D1-D5-J system \eqref{D1D5smallring}, we cannot avoid having such
%unwanted charges, so cannot use the 4D-5D connection directly.
Fortunately, type IIA string on $S^1\times {\rm K3}$ has $O(5,21;\bbZ)$
symmetry as a part of the $U$-duality group
\cite{Hull:1994ys}\cite{Witten:1995ex}\footnote{The full U-duality group
is $O(5,21;\bbZ) \times \bbZ_2$
\cite{Hull:1994ys}\cite{Witten:1995ex}.}, which interchanges charges so
that we can use the 4D-5D connection. Or equivalently, we start from
heterotic configuration \eqref{cfg_BR_het}, where we have $O(5,21;\bbZ)$
symmetry as $T$-duality.  After $T$-dualizing the charges appropriately
in heterotic string, we can then go to type II by heterotic/IIA duality.

Either way, after such a chain of dualities, we end up with the
following configuration in type IIA:
\begin{align}
 \begin{array}{r@{~~~~}c|ccc}
  && \psi& S^1_5 & {\rm K3}\\
  \cline{2-5}
  N_P  &{D2} & \sim & \sim& \alpha_2    \\
  -N_F  &{D2} & \sim & \sim& \alpha_3    \\
  n_p  &{p } & \bigcirc& \sim & \sim    \\
  -n_f &{ns5}& \bigcirc& \sim  &\bigcirc
 \end{array}
 \label{cfg_D2D2}
\end{align}
Here $\alpha_a$ are 2-cycles in K3, i.e., $\alpha_{a}\in H_2({\rm K3})$,
$a=2,3,\dots, 23$ (the reason for reserving the index 1 will become
clear shortly).  The $F,P$ in \eqref{cfg_BR_het} have been mapped into
D2-branes wrapping particular two 2-cycles, $\alpha_2$ and $\alpha_3$.
%Note that the relation between the F1 charge $N_F$ in heterotic string
%and the D2 charge in IIA does not involve a minus sign (cf.\
%\eqref{D1D5smallring}), because the F1 on $S_1^5$ was $T$-dualized into
%some other charge in heterotic string, which in turn was mapped to D2
%\cite{Witten:1995ex}\cite{Kiritsis:1998hy}.  
%

Which particular two 2-cycles $\alpha_2$, $\alpha_3$ should the
D2-branes wrap?  This question can be answered as follows.
%
First, let us recall how the $O(5,21;\bbZ)$ $T$-duality group arises in
heterotic string.  In heterotic string theory on $S^1\times T^4$, there
are 21 left- and 5 right-moving momenta in the internal directions.
These momenta are quantized electric charges from the 5D point of view.
They form the 26-dimensional Narain lattice with signature (5,21), and
the $O(5,21;\bbZ)$ $T$-duality rotates the charges in this lattice.
$N_P$ and $N_F$ in \eqref{cfg_BR_het} are two such electric charges of
heterotic string, and in the 2-dimensional sublattice of the Narain
lattice in which these charges live, the metric is proportional to
$({0~1\atop 1~0})$ (recall that the left- and right-moving momenta in
the $x^5$ direction are $p_{L,R}^5={N_P/ R}\pm{N_F R/ \alpha'}$, and the
invariant form contains $(p_L^5)^2-(p_R^5)^2={4N_P N_F/\alpha'}$
\cite{Polchinski:1998rq}).
%
Correspondingly, there is $O(5,21;\bbZ)$ $U$-duality group on the IIA
side.  This $O(5,21;\bbZ)$ group contains $O(3,19;\bbZ)$ subgroup that
interchanges the D2-branes wrapping 2-cycles in K3.  Because there are
22 2-cycles $\alpha_a,a=2,\dots,23$ in K3, the D2-brane charges live in
22-dimensional lattice $H_2({\rm K3},\bbZ)$.  The metric for this charge
lattice is the intersection number $C_{ab}$ of 2-cycles in K3, which is
known to have signature $(3,19)$ \cite{Aspinwall:1994rg}.
%
%The corresponding structure on the IIA side is the lattice $H_2({\rm
%K3},\bbZ)$ spanned by $\alpha_a$, $a=2,\dots, 23$ (the reason for
%reserving the index 1 will become clear shortly), which is called the
%Picard lattice.  This lattice is 22-dimensional, therefore it
%corresponds to a sublattice of the 26-dimensional Narain lattice on the
%heterotic side.  The natural metric on the Picard lattice is the
%intersection number $C_{ab}$, which is known to have signature $(19,3)$
%(see e.g.\ \cite{Aspinwall:1996mn}).
%
So, the answer to the question above is: we should choose the 2-cycles
$\alpha_2,\alpha_3$ so that, in the 2-dimensional sublattice of the
lattice $H_2({\rm K3},\bbZ)$ in which they live, the metric should be
$({0~1\atop 1~0})$.  In other words, the intersection numbers $C_{ab}$
should satisfy
\begin{align}
 C_{22}=C_{33}=0,\qquad C_{23}=1.\label{intersect_num}
\end{align}


Now we are ready to use the 4D-5D connection. Uplifting \eqref{cfg_D2D2}
to M-theory,
\begin{align}
 \begin{array}{c@{\,}c@{\,}c@{~~~~}c|cccc}
  &&&& \psi& S^1_5 & {\rm K3} & S^1_{10}\\
  \cline{4-8}
  q_2 &=&N_P &{M2} & \sim & \sim& \alpha_2 & \sim   \\
  q_3 &=&-N_F &{M2} & \sim & \sim& \alpha_3 & \sim   \\
  2J_L^3&=&n_p &{p } & \bigcirc& \sim & \sim     & \sim  \\
  p^1 &=&-n_f &{m5}& \bigcirc& \sim  &\bigcirc  & \sim
 \end{array}
 \label{5DBH_before_4D5D}
\end{align}
where $J_L^3$ is the $SU(2)_L\subset SO(4)$ charge.  This can be thought
of as a 5D small black ring (internal directions are $S^1_5,{\rm
K3},S^1_{10}$), and is simply a special case of the configurations
considered in
\cite{Gaiotto:2005gf}\cite{Elvang:2005sa}\cite{Gaiotto:2005xt}\cite{Bena:2005ni}.
So, by the (string corrected) 4D-5D connection, there is a way to
continuously deform this 5D ring, by way of a Taub-NUT geometry, into a
5D black string which from the 4D point of view is a black hole.  The
entropy of the 5D ring and that of the 4D hole are identical since
entropy cannot change in such an adiabatic process.  In this process,
the role of the Taub-NUT is just a ``catalyst'' to wind the ring around
the M-circle and it can be removed when we have reached the 4D
configuration, since the entropy is independent of the D6 charge.
Practically, one can say that we can reinterpret the contractible circle
$\psi$ as a non-contractible M-circle, if the relation between 4d and 5d
charges (Eq.\ \eqref{hdyq6Aug05} below) is correctly taken into account.
Compactifying now on this non-contractible M-circle, we obtain a 4D
small non-rotating black hole:
\begin{align} 
 \begin{array}{c@{\,}c@{\,}c@{~~~~}c|cccc}
  &&&& S^1_5 & {\rm K3} & S^1_{10}\\
  \cline{4-7}
  q_2&=&N_P &{D2} & \sim& \alpha_2 & \sim   \\
  q_3&=&-N_F &{D2} & \sim& \alpha_3 & \sim   \\
  q_0&=&n_p &{d0} & \sim & \sim     & \sim  \\
  p^1&=&-n_f &{d4}& \sim  &\bigcirc  & \sim
 \end{array}
 \label{4DBH}
\end{align}
where we used the relation between 5D and 4D charges
\cite{Bena:2004tk}\cite{Bena:2005ni}\cite{Bena:2005ae}\cite{Gaiotto:2005gf},
\begin{align}
 p^A_{4D}=p^A_{5D},\qquad
 q_A^{4D}=q_A^{5D}-3D_{ABC}p^Bp^C=q_A^{5D},\qquad
 q_0^{4D}=2J_L^3.\label{hdyq6Aug05}
\end{align}
The reserved index 1 is now for the 2-cycle $T^2=S^1_5\times S^1_{10}$.

The object with charges \eqref{4DBH} does not have a classical horizon,
but develops a finite horizon when $R^2$ corrections are taken into
account \cite{Dabholkar:2004yr}\cite{Dabholkar:2004dq}.  The
macroscopic (Bekenstein-Hawking-Wald) entropy is
\begin{align}
 S_{\rm macro}=2\pi \sqrt{\qh_0c_{2A}p^A\over 6},\qquad
 \qh_0=q_0+{1\over 12}\hat D^{AB}q_A q_B,\qquad
 D_{AB}=D_{ABC}p^C.\label{S_macro}
\end{align}
Here $c_{2A}$ are the coefficients of the second Chern class, and
$D_{ABC}={1\over 6}C_{ABC}$, where $C_{ABC}$ is the
triple intersection number. The matrix $\hat D^{AB}$ is the inverse of
$D_{AB}$ in the subspace orthogonal to its kernel.

In the present case, $c_{2,A=1}=24$ and $C_{1ab}=C_{ab}$, $a,b=1,\dots,23$
with the only relevant values given in \eqref{intersect_num}.  Therefore,
the macroscopic entropy \eqref{S_macro} is computed as
\begin{align}
 S=4\pi\sqrt{q_0p^1+q_2q_3}=4\pi\sqrt{-N_P N_F - n_p n_f}
 \label{S_macro_SBH}
\end{align}
Thus we reproduced the microscopic entropy (Eq.\ \eqref{S(N,J)-2}) of
the D1-D5-J system from the macroscopic entropy of the 4D small black
hole.  This supports the validity of the 4D-5D connection that we used
to arrive at the 4D configuration \eqref{4DBH}, even when stringy
corrections are crucial.



%\begin{align}
% \begin{array}{c||c|c|cccc}
%  & \psi& 5 & 6&7&8&9\\
%  \hline
%  D1 & \sim & \bigcirc& \sim    & \sim    & \sim    & \sim    \\
%  D5 & \sim & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \bigcirc\\
%  p  & \bigcirc& \sim & \sim    & \sim    & \sim    & \sim    \\
%  kk & \bigcirc& S^1  & \bigcirc& \bigcirc& \bigcirc& \bigcirc
% \end{array}\label{D1D5smallring}
%\end{align}
%Now $x^{6,7,8,9}$ is K3.  ``$S^1$'' means the KK-circle.  The numbers of
%D1- and D5-branes are $N_1=N_P$ and $N_5=N_F$.  This system is dual to the
%D1-D5 CFT, which is sigma model with target space ${\rm K3}^N/S_N$,
%$N=N_1 N_5$.  Entropy counting can be done in this CFT, too.
%
%
%Now let's T-dualize \eqref{D1D5smallring} along $x^5$ and lift to
%M-theory, to get
%\begin{align}
% \begin{array}{c||c|c|cccc|c}
%  & \psi& 5 & 6&7&8&9&10\\
%  \hline
%  P  & \sim & \sim    & \sim    & \sim    & \sim    & \sim    & \bigcirc\\
%  M5 & \sim & \sim    & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \bigcirc\\
%  p  & \bigcirc& \sim & \sim    & \sim    & \sim    & \sim    & \sim    \\
%  m5 & \bigcirc& \sim & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \sim    
% \end{array}
%\end{align}
%By using the 4D-5D connection, we can switch the contractible circle
%$\psi$ and the non-contractible M-circle ({\bf not very
%sure}\footnote{I'm not very sure if Strominger's argument applies here.
%Namely, is the unbroken supersymmetry of this small black ring solutino
%aligned with that of Taub-NUT?  Also, I don't know if the can we really
%move the D1-D5 infinitely away from the center of the Taub-NUT and
%ignore the NUT.  The NUT has magnetic charge, and the magnetic field may
%still interact with the D1-D5.}).  If we then descend to IIA by
%compactifying on $\psi$,
%\begin{align}
% \begin{array}{c||c|cccc|c}
%  & 5 & 6&7&8&9&10\\
%  \hline
%  P  &  \sim    & \sim    & \sim    & \sim    & \sim    & \bigcirc\\
%  NS5&  \sim    & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \bigcirc\\
%  d0 &  \sim & \sim    & \sim    & \sim    & \sim    & \sim    \\
%  d4 & \sim & \bigcirc& \bigcirc& \bigcirc& \bigcirc& \sim    
% \end{array}
%\end{align}
%Now $d0,d4$ are not wrapping a contractible circle, but we are still
%writing them in lowercase.
%
%This can now be thought of as 4D black hole.  We would like to dualize
%this system into a duality frame in which the formalism of de Wit et
%al.\ is applicable, and we would like to see that a finite horizon is
%formed.  That would mean that the original ring \eqref{D1D5smallring}
%also has a finite horizon, namely it's a small black ring.
%
%After $T(5),S,T(5)$, and heterotic/IIA duality,
%\begin{align}
% \begin{array}{c||c|cccc|c}
%  & 5  &6&7&8&9& 10\\
%  \hline
%  P & \sim     & \sim    & \sim    & \sim    & \sim   & \bigcirc \\
%  F1& \sim     & \sim    & \sim    & \sim    & \sim   & \bigcirc \\
%  p & \bigcirc & \sim    & \sim    & \sim    & \sim   & \sim     \\
%  f1& \bigcirc & \sim    & \sim    & \sim    & \sim   & \sim
% \end{array}
%\end{align}
%Now $x^{6,7,8,9}$ is $T^4$ again.  This is simply a two-direction
%version of the system Dabholkar studied.
%
%One can study the entropy of this system macroscopically using de Wit et
%al.'s formalism.  Because of stringy corrections, a ``small'',
%$\alpha'$-sized horizon will be formed.  From the 5D point of view in
%which $x^{0,1,2,3,4,10}$ are spacetime directions, this is a 5D
%``small'' black string wound around the $x^{10}$ circle. This 5D
%``small'' black string has a finite, $\alpha'$-sized horizon in the 5D
%sense too.  Because this $x^{10}$ direction is related the $\psi$
%direction of the ring of \eqref{cfg_BR_het} by the 5D/4D connection,
%this means that the original ring \eqref{cfg_BR_het} also had a
%$\alpha'$-sized horizon.  Namely, it is a ``small'' black ring.
%
%
%According to \cite{Sen:2004dp}, the macroscopic entropy of the system
%Dabholkar studied is, when we include nonholomorphic correction to the
%generalize prepotential,
%\begin{align}
% S_{BH}&=\sqrt{|nw|}-6\ln|nw|+\CO(1).
%\end{align}
%(see Eq.\ (3.82) of \cite{Sen:2004dp}).  Since entropy should be
%T-duality ($O(22,6,\bbZ)$) invariant, the entropy of the
%two-direction version of this system should have $nw$ should be replaced
%by an invariant quantity, which is $n_{10}w_{10}+n_5w_5=N_F N_P+n_f n_p$. Therefore
%\begin{align}
% S_{BH}&=\sqrt{|N_F N_P+n_f n_p|}-6\ln|N_F N_P+n_f n_p|+\CO(1).
%\end{align}
%{\bf (The leading term agree with \eqref{S(N,J)-2}), but the coeffcient
%of the log term does not agree.  In \cite{Sen:2004dp}, Sen had to
%consider an ensemble where a chemical potential was introduced for $nw$,
%not just for $J=n_p=n_5$.  However, it's not clear what it means to
%introduce a chemical potential for $n_5w_5+n_{10}w_{10}=N_F N_P+n_f n_p$
%in our original small black ring system.  We perhaps have to content
%ourselves with agreement of the leading terms.)}
%
%\newpage
%\section{Memo: 4D/5D connection}
%
%A 4D BH with (D6, D4, D2, D0) charges $(1,0,q_A,q_0)$ can be lifted
%to 5D BH with M2 charge $q_A$ and $SU(2)_L$ charge $J_L^3$, embedded in
%the Taub-NUT geometry in M-theory.  The relation is
%\begin{align}
% q_0&={2J_L}.
%\end{align}
%\begin{itemize}
% \item 5D limit: spinning black hole with $(q_A,J_L^3)$, sitting in
%       center of Taub-NUT
% \item 4D limit : black hole with $(1,0,q_A,q_0)$
%\end{itemize}
%How can we include D4 charge? If we have $p^0=1$, turn on D6 worldvolume
%field strength $F^{(2)}_{world}=p^A\alpha_A$, where $\alpha_A\in
%H^2(X)$.  This induces (D6, D4, D2, D0) charges $(1,~p^A,~3p^B p^C
%D_{ABC},~-D)$.  If we lift this to M-theory, we have Taub-NUT with
%4-form flux $F^{(4)}=\omega_{NUT}\wedge (p^A \alpha_A)$.  This induces
%M2 charge $q_A=3p^B p^C D_{ABC}$ and angular momentum $q_0=-D$.  
%
%The above configuration does not have entropy.  Now, we consider
%Taub-NUT with flux $F^{(4)}=\omega_{NUT}\wedge p^A\alpha_A$, and place a
%spinning 5D BH with $J_L^3,q_A^{5D}$.  Then this descends to 4D BH with
%$(1,p^A, q_A,q_0)$ with
%$$q_A=3p^B p^C D_{ABC}+q_A^{5D},\qquad q_0=-D-p^A q_A^{5D}+2J_L^3)$$
%\begin{itemize}
% \item 5D limit: spinning black hole with $(q_A,J_L^3)$, sitting in
%       center of Taub-NUT with  $F^{(4)}$
% \item 4D limit : black hole with $(1,p^A,q_A,q_0)$
%\end{itemize}
%

\section{Conclusion and Discussion}
\label{sec:concl}

The D1-D5-J system is expected to develop a finite horizon once stringy
$R^2$ corrections to the supergravity action are taken into account, and
become a 5D small black ring.  We clarified the limit, $|J|\gg
\sqrt{N}$, in which the D1-D5-J system can be regarded as a ring with a
well-defined profile, and showed that we can relate this system by
duality transformations and the 4D-5D connection to a 4D small
non-rotating black hole, whose horizon structure is well-understood.
This gives an indirect evidence that event horizon appears in the 5D
small black ring by stringy corrections.  We also checked that the
entropy of the 4D small black hole agrees with the microscopic entropy
of the D1-D5-J system, which can be regarded as a consistency check.



A novel idea for understanding black hole physics was put forward by
Mathur and collaborators (for a review see \cite{Mathur:2005zp}), who
conjectured that a black hole is not a singularity surrounded by empty
space and horizon, but an ensemble of smooth but complicated (possibly
quantum) geometries inside the stretched horizon.  This bold conjecture
has been established for the BPS D1-D5 system, for which microstate
geometries have been explicitly written down at the supergravity level
\cite{Lunin:2001jy}\cite{Lunin:2002iz}.\footnote{The supergravity
microstate solutions obtained in \cite{Lunin:2001jy}\cite{Lunin:2002iz}
include fluctuations in the noncompact $\bbR^4$ directions, but
not in the internal $T^4$ or K3 directions.}  
%
%The small black hole of \cite{Dabholkar:2004yr} is believed to be
%interpretable as a ``coarse-grained'' geometry that effectively
%describes the ensemble of $\sim e^S$ smooth microstate geometries,
%although what ``coarse-graining'' means is to be better understood.
%
In this Mathur picture, the entropy of a black hole can be estimated
from the area of the ``stretched horizon,'' defined as the surface on
which the microstate geometries start to differ from each other.
%
In \cite{Lunin:2002qf}, an ensemble of D1-D5 microstate geometries with
angular momentum $J$ were studied, and the entropy was correctly
estimated to be $\sim\sqrt{N_1N_5-|J|}\,$ from this ``stretched horizon''
area.  
%In this Mathur picture of black holes, the 5D small black ring we
%studied in this note should also be interpretable as an effective
%geometry describing an ensemble of D1-D5 microstate geometries with
%angular momentum $J$.  Indeed, in \cite{Lunin:2002qf}, these microstate
%geometries were studied and the entropy was correctly estimated to be
%$\sim\sqrt{N_1N_5-|J|}$ from the ``stretched horizon'' area.  
It is interesting to study the relation between their ``stretched
horizon'' and the non-vanishing horizon that we argued to appear in the
5D small black ring by stringy corrections.

The above definition of the stretched horizon by Mathur is a qualitative
one, and in order to determine the entropy of a black hole from an area
law including the numerical factor, it is desirable to define the
stretched horizon in a more precise way.  It is also interesting to
study the relation between this definition of the stretched horizon and
other definitions, for example, as the distance at which a probe D-brane
starts to become indistinguishable from the D-branes that compose the
background black hole, due to thermal effect \cite{Iizuka:2001cw}.

%as the radius where
%stringy degrees of freedom start to become thermally excited {\bf (more
%explanation?)}.


Another aspect of the D1-D5-J system is that, being dual to a
configuration of fundamental heterotic string, it is an example of
systems for which the microscopic entropy can be computed very precisely
\cite{Dabholkar:2004yr}\cite{Dabholkar:2005by}.  It is interesting to
analyze this system further to study the recently proposed relation
between the black hole partition function and the topological string
partition function
\cite{Ooguri:2004zv}\cite{Verlinde:2004ck}\cite{Dabholkar:2005by}.


\section*{Acknowledgments}

It is pleasure to thank Atish Dabholkar and Ashoke Sen for helpful
conversations.  We also would like to thank Per Kraus for useful
comments on the early draft of this note.
%
N.I. would like to thank friends in Harvard and Caltech string theory
group for their very nice hospitality where this work was done.
%
The work of M.S. was supported in part by Department of Energy grant
DE-FG03-92ER40701 and a Sherman Fairchild Foundation postdoctoral
fellowship.
%
Finally we are happy to thank the people of India for their generosity.

%\appendix
%
%\section{Entropy}
%\label{entropy_w_fermions}
%
%Here we generalize the result \eqref{S(N,J)} to the case with fermions,
%which is relevant for the D1-D5-J system on $T^4$.
%
%Let us suppose that we have $N_B$ bosons and $N_F$ fermions.  Assume
%that $n_B$ bosons and $n_F$ fermions have $J=1$, and another $n_B$
%bosons and $n_F$ fermions have $J=-1$.  The remaining $N_B-2n_B$ bosons
%and $N_F-2n_F$ fermions carry no spin.  For the D1-D5 system on $T^4$,
%$N_B=N_F=8,n_B=n_F=1$, while for the D1-D5 system on $K3$,
%$N_B=24,n_B=1,N_F=n_F=0$.  The partition function is
%\begin{align}
% Z(\beta,\mu)
% &=\sum_{N,J}d_{N,J}q^N z^J
% =\prod_{n=1}^\infty 
% {(1+zq^n)^{n_F}(1+z^{-1}q^n)^{n_F}(1+q^n)^{N_F-2n_F}
% \over
% (1-zq^n)^{n_B}(1-z^{-1}q^n)^{n_B}(1-q^n)^{N_B-2n_B} },\label{gajx9Jun05}
%\end{align}
%where $q=e^{-\beta},z=e^{\beta\mu}$.  This can be written in terms of
%theta functions. For large $N$ or equivalently for small $\beta$, using
%modular properties of theta functions, one readily shows
%\begin{align}
% Z(\beta,\mu)\sim \beta^{N_B\over 2}\left({\mu\over\sin\pi\mu}\right)^{n_B}e^{\pi^2c/6\beta},
% \qquad c=N_B+{N_F\over 2}.
%\end{align}
%
%For $n_B=1$, by inverting the relation \eqref{gajx9Jun05} using saddle
%point approximation \cite{Russo:1994ev},
%%\footnote{Note that Eq.\ (2.16)
%%of \cite{Russo:1994ev} is incorrect; the power of the first factor
%%should be $-(D+1)/4$, not $-(D+3)/4$.}
%\begin{align*}
% S&=
% {a(2N-|J|)\over \sqrt{N-|J|}}-2\log\cosh{aJ\over 2\sqrt{N-|J|}}
% -{N_B+3\over 4}\log(N-|J|)\\
% &\approx 2a\sqrt{N-|J|},
%\end{align*}
%where $a=\pi\sqrt{c/6}$.  Note that this result does not depend on
%whether fermions carry spin or not; almost always the spin is carried
%only by bosons, and the typical state is still of the form of
%\eqref{hlro26May05}.

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\end{document}

hep-th0506215.extracted_bib
\begin{thebibliography}{99}
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\end{thebibliography}