text:
\section{Introduction}

Recent studies  has led to a good understanding of
the spectrum of $1/4$ BPS dyonic
states in a class of ${\cal N}=4$ supersymmetric string
theories \cite{Dijkgraaf:1996it,LopesCardoso:2004xf,Shih:2005uc,Gaiotto:2005hc,
Shih:2005he,Jatkar:2005bh,David:2006ji,Dabholkar:2006xa,
David:2006yn,David:2006ru,David:2006ud,Dabholkar:2006bj,Banerjee:2007ub}.
These theories are a class of generic ${\cal N}=4$ supersymmetric
${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_N$ orbifold of type IIA string theory on $K3\times T^2$ or
$T^6$. In each example studied so far, the statistical entropy computed
by taking the logarithm of the degeneracy of states agrees with the
entropy of the corresponding black hole for large charges.
This is not only in the leading order, but also in the first
sub-leading order \cite{LopesCardoso:2004xf,Jatkar:2005bh,
David:2006yn,David:2006ru,David:2006ud}.
On the black hole side this requires taking into account the effect of the
Gauss-Bonnet term in the low energy effective action of the theory,
and the use of the Wald's generalized formula for the black hole entropy in the
presence of higher derivative corrections
\cite{Wald:1993nt,Iyer:1994ys,Sen:2005wa}.
These $1/4$ BPS dyons are known to have regions of marginal stability,
the degeneracies of the dyons jump across these regions of marginal stability.
These changes are precisely captured by
the same dyon partition function but with different
choices of the integration contour, the moduli dependence of the
contour of integration is known precisely
\cite{Sen:2007vb,Dabholkar:2007vk,Sen:2007pg,Cheng:2007ch,
Mukherjee:2007nc,Sen:2007nz,Mukherjee:2007af}.
For a recent review on these topics see \cite{Sen:2007qy}.

So far similar studies in ${\cal N}=2$ theories has been lacking.
Because of reduced  supersymmetry the Gauss-Bonnet term
in the low energy effective action is not just a term dependent on the
axion and dilaton, but also on the other moduli of the vector multiplets.
The precise agreement
of the asymptotic degeneracy of the dyons in ${\cal N}=4$
theory to the first sub-leading order depended crucially on just
the axion-dilaton dependence of the Gauss-Bonnet term.
Any similar proposal for dyons in ${\cal N}=2$  theories should
address the question the moduli dependence of the other vector multiplets.

The study of 1/2 BPS dyons in a
generic ${\cal N}=2$ theory would be a hard task. In this paper we focus
on two ${\cal N}=2$ theories which are closely related to
${\cal N}=4$ theories.
The first  theory is
the STU model obtained by a  freely acting orbifold action of
type IIA on $T^4\times T^2$. There are three vector multiplets in this
theory, the $S$, $T$ and the $U$.
This model was constructed and studied in \cite{Sen:1995ff,Gregori:1999ns},
the coefficient of the Gauss-Bonnet term in this model is known exactly
and it is a sum of contributions from the S, T and the U-moduli.
In this case the exact moduli space of the vector multiplets is
given by
\be{vecstum}
{\cal M}_V = \left. \frac{SU(1,1)}{U(1)}\right|_S \times
\left. \frac{SU(1,1)}{U(1)}\right|_T \times
\left. \frac{SU(1,1)}{U(1)}\right|_U.
\end{equation}
where each $SU(1,1)/U(1)$ factor parameterizes the respective moduli.
We  study  the entropy function for this model and evaluate the expectation
values of the moduli at the attractor point.
This enables us to extract the S, T, U duality invariants constructed
out of bilinears of the electric and magnetic charges
which characterize the entropy at the next leading order.
Using these invariants we  propose a partition function for dyons
in this model. This partition function involves the product of three
Siegel modular forms of weight zero, it has all the duality symmetries of the
STU model. The statistical entropy obtained from the partition function
reproduces correctly the entropy of a dyonic black hole to the
first sub-leading order for large values of charges \footnote{
For the  reader who is interested in the formula for the partition function,
please  see \eq{stupart}.}.

Another well known example of a ${\cal N}=2$ model closely related
to a parent ${\cal N}=4$ theory is that of the self mirror
Calabi-Yau constructed in \cite{Ferrara:1995yx}.
On the type IIA side it is constructed by a freely acting orbifold of
type IIA on $K3\times T^2$, the resulting Calabi-Yau
is  a self-mirror manifold. Therefore the moduli space of this theory is
known exactly. The vector multiplet moduli space is given by
\be{vecmoduli}
{\cal M}_V = \frac{SU(1,1)}{U(1)}  \times \frac{SO(2,10)}{S0(2) \times SO(10)}.
\end{equation}
One of the important property of this moduli space is that the dilaton-axion
moduli factorizes from the rest of the vector multiplet moduli.
The dilaton-axion parameterizes the coset $SU(1,1)/U(1)$ while the
rest of the moduli(the T-moduli)
 parameterize the coset $SO(2,10)/SO(2)\times SO(10)$. Because of this
factorization, the coefficient of the Gauss-Bonnet term can be computed
exactly \cite{Harvey:1996ts}. From the analysis of the entropy function for
the FHSV model and the attractor values of the T-moduli
we  show that this factorization allows one to
parametrically separate the subleading
contribution to the entropy of dyons in this model into two parts.
The contribution from the axion-dilaton dependence of the Gauss-Bonnet
term is dominant when the electric charges are much larger than the
magnetic charges. We then write down a Siegel modular form of weight
4 which captures the dependence of the entropy function on the axion-dilaton
moduli.

The organization of the paper is as follows: In section 2. we review the
construction of the STU model
as well as the FHSV model and recall the
the coefficient of the Gauss-Bonnet term in these models.
Section 3. contains the entropy function
analysis of these models. We explictly solve for the attractor moduli
in both these models. This enables us to evaluate the
sub-leading contribution
to the black hole entropy of dyons from the coefficient of the Gauss-Bonnet
term in these models. It also helps us to determine the charge bilinears
characterizing the entropy at the subleading order.
In section 4. we propose a partition function for dyons in the STU model
and show that it has the  required duality invariance and reproduces
correctly the entropy of a dyonic black hole to the first sub-leading
order for large values of the charges.
In section 5. we write down an approximate partition function for
dyon in the FHSV model which captures the entropy of the corresponding
black hole for large values of the charges but with electric charges
much larger than the magnetic charges.
The appendices contain
conventions regarding the 't Hooft symbols for SO(2,2) and
the properties of the Siegel modular forms used in the paper.
Appendix B shows the systematic method by which
the attractor equations for the FHSV model is solved. This procedure
can be used for any model which has the following vector multiplet
moduli space
\be{genvecmodi}
{\cal M}_V = \frac{SU(1,1)}{U(1)} \times \frac{SO(2,n)}{SO(2) \times SO(n)}.
\end{equation}

\section{Two ${\cal N}=2$ theories}

In this section we review the construction of the two ${\cal N}=2$ models
that
we will be studying.  The dyons which will be the focus of our interest
preserve $1/2$  of the $8$ supersymmetries of these theories.

\subsection{The STU model}

This model is best described in terms of a freely acting ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2\times
{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$  orbifold of
type IIB \cite{Sen:1995ff,Gregori:1999ns} (example C in \cite{Sen:1995ff}).
Consider Type IIB compactified on $T^4\times S^1 \times \tilde S^1$,
the first ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$, $g_1:$ acts
by $(-1)^{F_L}$ together with a half shift on $S^1$,
the second ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$, $g_2$ acts
as an inversion of the coordinates on $T^4$ together with a half shift
on $\tilde S^1$. The theory with only with the
$g_1$ action is the same as the
one considered in \cite{David:2006ru}, this is a ${\cal N}=4$ theory.
The second ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$ action $g_2$ further breaks the supersymmetry down
to ${\cal N}=2$.
This theory has $3$ vector multiplets and $4$ hyper multiplets.
The $T$-duality symmetry of this theory is
$SO(2,2; {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$. In the type IIA description of this orbifold the
dilaton belongs to the
hypermultiplet, while in the type IIB description \footnote{
Here the duality symmetry
which relates the type IIA and the type IIB description of the theory
is not the conventional T-duality but the one
which is part of the U-duality group and
is a strong-weak duality symmetry \cite{Sen:1995ff}}
it belongs
to the vector multiplet \cite{Sen:1995ff,Gregori:1999ns}.
Therefore the vector multiplet moduli space is not corrected
by quantum corrections and it is given by
\be{vecmstu1}
{\cal M}_V = \left. \frac{SU(1,1)}{U(1)} \right|_{S} \times
\left. \frac{SU(1,1)}{U(1)} \right|_{T}\times \left. \frac{SU(1,1)}{U(1)}
\right|_{U},
\end{equation}
where $S$ refers to the axion dilaton moduli and $T$ and $U$ refer to the
K\"{a}hler and complex structure of the torus $S\times \tilde S$. This theory
is invariant under the symmetry $\Gamma_S(2)\times \Gamma_T(2)\times
\Gamma_U(2)$ where $\Gamma(2)$ is defined as
\be{defgamm2}
\left(\begin{array}{cc}
a & b \\
c & d
\end{array} \right)
\quad ad - bc =1, \quad b, c \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}},\; a, d \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1.
\end{equation}
It also has the triality invariance
$S\leftrightarrow T\leftrightarrow -1/U$.

The dyons we consider are the twisted sector
dyons of the parent ${\cal N}=4$ theory which
are preserved by the second orbifold projection $g_2$. Thus these dyons have
electric, and magnetic charges  only along $S^1$ and $\tilde S^1$ directions.
Since these dyons are $1/4$ BPS states in the parent  theory they
will preserve $1/2$ of the supersymmetries of the daughter theory.
For the purposes of obtaining the subleading corrections to the entropy of the
dyons we will need the
coefficient of the Gauss-Bonnet term. The Gauss-Bonnet term is given
 by following combinations of
4 derivative terms made of the curvature tensor
\be{defgb}
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} - 4 R_{\mu\nu} R^{\mu\nu} + R^2.
\end{equation}
The coefficient of this term for the STU model  was evaluated in
\cite{Gregori:1999ns},
this is given by
\bea{cofstugb}
 & & \frac{1}{128\pi^2} \left(  4\ln\left( \frac{M_p^2}{p^2} \right) -
4\log \left[\vartheta_2 (\tau)^2\vartheta_2(-\bar\tau)^2 (\tau_2) \right]
\right. \\ \nonumber
& & \left. - 4\log \left[\vartheta_2(y^+)^2  \vartheta_2(-\bar y^+)^2
    (y^+_2)
 \right]
-4\log \left[\vartheta_4(y^-)^2\vartheta_4(-\bar y^-)^2 (y^-_2) \right]
\right),
\end{eqnarray}
where $\tau$ refers to the complex combination of the  axion
and dilaton moduli given by
\be{defs}
\tau = -a + i S,
\end{equation}
where $a$ is the axion and $S= \exp(-2\phi)$ is the dilaton.
 $y^+$ and
$y^-$ refer to the K\"{a}hler and complex structure of the
$S\times\tilde S$ torus, which we have denoted as the $T$ and $U$
moduli
\footnote{Throughout the paper, the subscripts $1$ and $2$ on
complex moduli refer to its real and imaginary parts respectively.}.
The normalization of the  Gauss-Bonnet term is determined as follows
\cite{Sen:2005pu}: Let
the axion-dilaton dependence of the Gauss-Bonnet term be given by
\be{normgb}
-\frac{{\cal K}}{128\pi^2}
\ln\left[ \tau_2 f(\tau) f(-\bar\tau) \right].
\end{equation}
The functional dependence of the term in the square  bracket is invariant
under the S-duality symmetry  $\Gamma(2)_S$.
Then the coefficient
${\cal K}$ is given by the number of harmonic $p$ forms of $T^4$ left
invariant under the action of $g_1$ and $g_2$
weighted by $(-1)^p$. $g_2$ projects retains
only the even forms which results in $8$ forms. Since $g_1$ reduces the
supersymmetry by projecting out the left moving fermions, it project out
the $3$ self-dual 2-forms and a combination of the zero form and $4$ form.
Thus
out of the $8$ forms $4$ are left invariant. We therefore
conclude  ${\cal K}=4$. Once
this normalization is determined the dependence of the axion-dilaton
moduli and the other moduli is obtained from the calculation of
\cite{Gregori:1999ns}, equation (5.3). The dependence of the
coefficient of the Gauss-Bonnet term on $\ln(M_p^2/p^2)$
where $M_p$ is the Planck's constant and $p^2$ is the
graviton momentum is due to the
trace anomaly of the theory.
Though the coefficient of the trace anomaly does
not play a role in the
evaluation of the sub-leading contribution to the entropy, its
contribution to the coefficient of the Gauss-Bonnet term
for a general ${\cal N}=2$ theory is given by
\be{coeftracean}
-\frac{1}{128\pi^2} \left( \frac{23 + n_h - n_v}{ 6}\right) \ln
\left( \frac{M_p^2}{\Lambda^2}\right),
\end{equation}
where $n_h$ is the number of hypermultiplets and $n_v$ is the number
of vector multiplets, for the STU model $n_h-n_v=1$.
Note that the coefficient of the Gauss-Bonnet term \eq{cofstugb} is
invariant under the triality symmetry
$\tau \leftrightarrow y^+\leftrightarrow -1/y^-$ as well as
$\Gamma_S(2) \times \Gamma_T(2) \times \Gamma_U(2)$, which is
the symmetry of the model.
This completes the explantation of all the terms in \eq{cofstugb}.

\subsection{The FHSV model}

For our purposes it is easiest to describe the FHSV model as a freely
acting orbifold of the heterotic $E_8\times E_8$ theory.
Consider the heterotic string on the following
even, self-dual Lorentzian lattice $\Gamma^{(22,6)}$ of the form
\be{hetlat}
\Gamma^{(9,1)} \oplus \Gamma^{(9,1)}\oplus\Gamma^{(1,1)}\oplus
\Gamma^{(1,1)}\oplus\Gamma^{(2,2)}.
\end{equation}
Here the two $\Gamma^{(9,1)}$ factors are isomorphic.
We now orbifold by a ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$ action which exchanges the first
two factors and act as $-1$ on the third $\Gamma^{(1,1)}$ and
the last $\Gamma^{(2,2)}$ together with the half shift in
the fourth $\Gamma^{(1,1)}$ factor.
The T-duality group of this heterotic string is $SO(2,10 ;{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$,
the moduli space of this theory is known exactly \cite{Ferrara:1995yx}.
For purposes of this paper it is sufficient to focus on the vector multiplet
moduli space. There are $11$ vector multiplets in this
theory and it moduli space  is given by
\be{vmodfshv}
{\cal M}_V = \frac{SU(1,1)}{U(1)} \times \frac{SO(2, 10)}{SO(2)\times SO(10)}.
\end{equation}
Note that the coset $SU(1,1)/U(1)$ which is parameterized by the
axion-dilaton moduli
factorizes from the rest of the vector multiplet
moduli, we call these moduli the T-moduli. The $S$ duality of this
theory is $\Gamma_S(2)$.

We focus on a class of
dyon configurations which preserves $1/2$ of the remaining symmetry
in the FHSV model. These class of dyons are
 obtained by embedding the $1/4$ BPS dyon of
the parent ${\cal N }=4$ theory  such that the dyon configuration
is preserved by the
FHSV orbifold action.
We now describe the charge configuration of such a dyon:
Let us call the unprojected combination of the two $\Gamma^{(9,1)}$ lattices as
$\Gamma^{(9,1)}_I$, we denote the fourth
$\Gamma^{(1,1)}$ as $\Gamma^{(1,1)}_S$.
We focus on dyons which have electric and magnetic charges values on the
lattices $\Gamma^{(9,1)}_I\oplus \Gamma^{(1,1)}_S$ which have twisted sector
charges along the $\Gamma^{(1,1)}_S$ lattice.
It is clear that such charge configurations are preserved by the
FHSV orbifold action,
since these configurations do not have charges in the
lattices $\Gamma^{(1,1)}\oplus \Gamma^{(2,2)}$ on which the
FHSV orbifold acts as an inversion.

We will  study the macroscopic entropy of these dyons using
the entropy function formalism to the first subleading term.
For this purpose we will need the coefficient of the Gauss-Bonnet term
in this model. The dilaton-axion and the T-moduli
dependence of the Gauss-Bonnet coefficient of this
model was evaluated exactly in  \cite{Harvey:1996ts}.
We now review their result:
The coefficient of this term for ${\cal N}=2$ compactifications is given
by
\be{cofgb}
\frac{1}{128\pi^2} \left( 4   \ln(\frac {M_p^2}{p^2} )
 +  {\cal F}_1 (\tau,  \bar\tau, y, \bar y )\right).
\end{equation}
In \eq{cofgb}
${\cal F}_1(\tau , \bar\tau)$ is the modular function determined by the
threshold calculation and $\tau$ is the complex combination of the
axion and the dilaton given in \eq{defs}.
For the FHSV model, the $\tau$  and the T-moduli dependence of the
threshold corrections is given by \cite{Harvey:1996ts}
\bea{threshfs}
{\cal F}_1(\tau, \bar \tau, y, \bar y) &=&
-12\log \left[ \eta(2\tau)) (\eta(-2\bar\tau)  (\tau_2) \right]\cr
& &  - \log \left[ \Phi_{{\rm BE}} (y)
\Phi_{{\rm BE}} (-\bar y)  ( 4 y^{+}_2 y^{-}_2 - \vec y_2^2)^4 \right],
\end{eqnarray}
where $\Phi_{{\rm BE}}(y)$ is the Borcherds-Enriques
modular form of weight four on the
discrete group $E_8\times \Gamma^{1,1}(-2)\times \Gamma^{1,1}(-2)$.
$y$ refer to the $10$ complex T-moduli,
$y=\{ y^+, y^-, y^i\}, i = 1, \cdots 8$.
The subscript $2$ on the various moduli in
\eq{threshfs} stands for the imaginary values
the  moduli. For the purposes of this paper will not need the details of the
Borcherds-Enriques form, please see \cite{Harvey:1996ts} for the
details.
Just as in the STU model we fix the normalization of the
coefficient of the Gauss-Bonnet term by examining the
axion-dilaton dependence. This is given by
\eq{normgb}, here ${\cal K}$ is the number of p-forms invariant on
the dual description of the the FHSV orbifold in type IIA theory on
$K3$ weighted by $(-1)^p$. This turns out to be $12$, the FHSV orbifold
action projects out $12$ of the $24$ forms of $K3$.
The dependence of the axion
dilaton and the remaining T-moduli dependence is then read out from
\cite{Harvey:1996ts}. The trace anomaly dependence is given
by \eq{coeftracean} with $n_v =11$ and $n_h=12$ for the FHSV model.

\section{Entropy function for dyons}

In this section we will evaluate the first sub-leading contribution to the
Hawking-Bekenstein entropy due to Gauss-Bonnet term in the
effective action for dyons in both the STU model and the FHSV model
 discussed in the previous
section. We will follow the entropy function approach developed in
\cite{Sen:2005iz}. In this approach
to find the first sub-leading contribution to the entropy we will need both the
coefficient of the Gauss-Bonnet term as well as the attractor values of
vector multiplet moduli at the two derivative level.
We first evaluate the attractor values of all the vector multiplet moduli
to both these models and then substitute these values in the
coefficient of the respective Gauss-Bonnet term in these models to
evaluate the sub-leading contribution to the Hawking-Bekenstein entropy.

To make the discussion self contained we briefly review the
entropy function formalism.
Consider an extremal black hole solution in any of the ${\cal N}=2$
theories of interest in this paper. The near horizon geometry of this
black holes is given by
\bea{defsoln}
ds^2 = v_1 \left( -r^2 dt^2 + \frac{dr^2}{r^2} \right) + v_2 (d\theta^2 +
\sin^2 \theta d\phi^2 ), \cr
S =u_S, \qquad a=u_a, \qquad \tilde M_{ij} = u_{M ij}, \cr
F_{rt}^{(i)} = e_i,  \quad
F_{\theta\phi}^{(i)} = \frac{p_i}{4\pi}.
\end{eqnarray}
Here $i =1, \cdots n+2 $ and the near horizon vector multiplet moduli,
$n=10$ for the FHSV model and  $n=2$ for the STU model.
$u_{M {ij}}$ satisfies
\be{modcond}
u_M^T L u_M = L, \quad u_M^T=u_M,
\end{equation}
where $L$ is the Lorentzian metric with $n$ negative signatures and
$2$ positive signatures; $L = {\rm Dia} ( -1, -1, \cdots -1, +1, +1 )$.
Substituting the solution \eq{defsoln} in the supergravity action we obtain
\cite{Sen:2005iz}
\bea{nhsugra}
& &f(u_s, u_a, u_M, \vec v, \vec e, \vec p) \equiv
\int d\theta d\phi \sqrt{-{\rm det} G } {\cal L} \\ \nonumber
&=& \frac{1}{8} v_1v_2 u_S \left[
-\frac{2}{v_1} + \frac{2}{v_2} + \frac{2}{v_1^2} e_i ( L u_M L)_{ij} e_j
- \frac{1}{8\pi^2 v_2^2 } p_i (Lu_M L)_{ij} p_j +
\frac{u_a}{\pi u_S v_1 v_2} e_i L_{ij} p_j
\right].
\end{eqnarray}
We then obtain the electric charges $q_i$ given by
\be{elchrg}
q_i \equiv \frac{\partial f}{\partial e_i}
= \frac{v_2 u_S}{2 v_1} ( L u_M L)_{ij} e_j + \frac{u_a}{8\pi} L_{ij} p_j.
\end{equation}
Evaluating the Legendre transform with respect to the variables $e_i$ we
obtain the entropy function $F$
\bea{entfn}
F(u_S, u_a, u_M, \vec v, \vec q, \vec p)
&\equiv& 2\pi ( e_i p_i - f(u_S, u_a, u_M, \vec v, \vec e, \vec p) ) \cr
&=& 2\pi \left[ \frac{u_S}{4} ( v_2 -v_1) + \frac{v_1}{v_2 u_S}
q^T u_M q + \frac{v_1}{64\pi^2 v_2 u_s} ( u_S^2 + u_a^2) L u_M L p \right. \cr
&\;& \left. - \frac{ v_1}{4\pi v_2 u_S} u_a q^T u_M L p \right].
\end{eqnarray}
We define the charge vectors
\be{defnechr}
Q_i = 2 q_i, \qquad P_i = \frac{1}{4\pi} L_{ij} p_j,
\end{equation}
so that $P_i$ and $Q_i$ are integers. Substituting these definitions of
charges in the entropy function we obtain
\be{neentfn}
F = \frac{\pi}{2} \left[ u_S( v_2 -v_1) + \frac{ v_1}{v_2 u_S}
\left( Q^T u_M Q + ( u_S^2 + u_a^2) P^T u_M P - 2 u_a Q^T u_M P \right)
\right].
\end{equation}
Now to evaluate the Hawking-Bekenstein entropy we
 need to find the the extremum of the above function with respect to
the moduli $u_S, u_a, u_{M ij}, v_1, v_2$.
For this we can first minimize the entropy function in \eq{neentfn} with
respect to $v_1$ and $v_2$. It is easy to see from the dependence of the
entropy function on $v_1$ and $v_2$, the minimum occurs at $v_1 = v_2$.
We then organize the axion and the dilaton moduli in terms of the complex
scalar $\tau$ defined as
\be{deftau}
\tau = -u_a +  i u_S.
\end{equation}
Using $\tau$ and the parameterization and $v_1=v_2$, the entropy function
\eq{neentfn} reduces to
\be{entfns}
F = \frac{\pi}{2\tau_2} \left[ ( Q + \tau P)^{T} u_M (Q +\tau P) \right].
\end{equation}
In the next two subsections we use the above entropy function and
to further minimize with respect to the T-moduli parameterized by
$u_M$ and the axion dilaton moduli $\tau$ for the STU model and the
FHSV model. In both the cases we obtain the attractor values of these
moduli in terms of the charges explictly. We then use the attractor
values of the moduli to evaluate the subleading correction due to the
Gauss-Bonnet term.

\subsection{The two derivative entropy function: STU model}

For the case of the STU model the moduli matrix $u_M$ is a $4\times 4$ matrix
which parameterizes the coset $SO(2,2)/(SO(2)\times SO(2))$. It satisfies
the conditions
\be{condstu}
u_M^T = u_M, \qquad u_M^TL u_M = u_M,
\end{equation}
where $L$ is the diagonal metric $L = {\rm Dia} ( 1, 1, -1, -1)$.
From the conditions in \eq{condstu}
it can be easily seen that there are $4$ independent variables
which parameterize the matrix $u_M$. A convenient parameterization is
as follows:
We first introduce 4 complex numbers satisfying
\be{consteq1}
w_1^2 + w_2^2 - w_3^2 - w_4^2  = 0,
\end{equation}
together with the identification $w_I \sim c w_I$ where $c$ is a complex
number. Thus $w_I$'s parameterize the coset $SO(2,2)/(SO(2) \times SO(2))$.
Using the scaling symmetry we can solve the constraint \eq{consteq1} by
introducing complex numbers $y^+$ and $y^-$ and
writing $w_I$'s as
\bea{relwystu}
& w_1 = -1 + y^+ y^-, \qquad w_2 = y^+ + y^-, \\ \nonumber
& w_3 = y^+ -y^-, \qquad w_{4} =  1+  y^+ y^-.
\end{eqnarray}
Note that we have use the scaling symmetry of $w_I$'s to set
$w_4 - w_1 =2$. Using the above solution of the constraint \eq{consteq1}
it can be seen
\be{modconst}
|w_1|^2 + |w_2|^2 - |w_3|^2 - |w_4|^2 = 2Y,
\end{equation}
where $Y = 4y_2^+y_2^-$
is related to the K\"{a}hler potential on the moduli space by
\be{realkpot1}
K = -\log Y.
\end{equation}
Now we can parameterize the moduli matrix $u_M$ by
\be{defmodmat1}
u_M = L \tilde U L -L,
\end{equation}
where $\tilde U$ is given by
\be{partu1}
\tilde U = \frac{w_I\bar w_J + \bar w_I w_J}{Y}.
\end{equation}
It can be seen that using \eq{relwystu}, \eq{modconst} and the
definitions \eq{defmodmat1} and \eq{partu1} the conditions on
$u_M$ given in \eq{condstu} are satisfied. Note that now
we have parameterized the matrix $u_M$ in terms of $y^+$ and $y^-$.
Substituting this parameterization in the entropy function
\eq{entfns} we obtain
\be{entfnstu}
F = \frac{\pi}{2} \left[ \frac{|(Q+ \tau P) \cdot w|^2}{\tau_2 Y}
+  \frac{|(Q+ \bar\tau P) \cdot w|^2}{\tau_2 Y}
- \frac{( Q+ \tau P ) \cdot (Q + \bar\tau P)}{ \tau_2} \right],
\end{equation}
where dot product $\cdot$ is with respect to the metric $L$.
To determine the values of the T and U-moduli
($y^+, y^-$ respectively ) at the attractor point it is
sufficient to focus terms on the first two terms in \eq{entfnstu}.
The first two terms are identical except for the exchange of
$\tau\leftrightarrow \bar\tau$ in the numerator.
Our strategy for minimizing with respect to $y^+$ and $y^-$ is as follows:
We will first just focus on the first term
\be{tenfstu1}
F_T = \frac{\pi}{2} \left[ \frac{|(Q+ \tau P) \cdot w|^2}{\tau_2 Y} \right],
\end{equation}
and minimize this term with respect to $y^+$ and $y^-$. We will see that
the attractor values of the moduli $y^+$ and $y^-$ are independent of
the axion-dilaton moduli $\tau$. Therefore these values of the
attractor moduli minimize the second term in \eq{entfnstu} simultaneously,
since the second term is same as the first term with $\tau\rightarrow
\bar\tau$. Thus to minimize with respect to the $y^+$ and $y^-$ moduli
it is sufficient to focus on \eq{tenfstu1}.

To write out \eq{tenfstu1} in terms of the moduli $y^+$  and $y^-$
it is convenient to define the following variables
\bea{nwinvar1}
Q^1 + Q^4 = N_1, \quad Q^1-Q^4 = W_1, \quad Q^2 + Q^3 = N_2, \qquad Q^2 - Q^3
= W_2, \cr
P^1 + P^4 = \tilde N_1,
\quad P^1-P^4 = \tilde W_1, \quad P^2 + P^3 = \tilde N_2, \qquad P^2 - P^3
= \tilde W_2.
\end{eqnarray}
We also define the various components of the complex combination $Q+ \tau P$
as follows
\bea{nwvar2}
Q^1 + Q^4 + \tau( P^1 + P^4) =  N_1 + \tau \tilde N_1 = n_1, \quad
Q^1 - Q^4 + \tau( P^1 -P^4) = W_1 + \tau \tilde W_1 = w_1', \cr \nonumber
Q^2 + Q^3 + \tau( P_2 + P_3) =  N_2 + \tau \tilde N_2 = n_2, \quad
Q^2 - Q^3 + \tau( P_2 -P_3) = W_2 + \tau \tilde W_2 = w_2'.
\\
\end{eqnarray}
Using the variables the T-moduli dependent part of the entropy function
in \eq{entfnstu} is given by
\be{entfnstut}
F_{T} = \frac{\pi}{8\tau_2 y^+_2 y^-_2}
\left| -n_1 + y^+ y^-  w_1' + y^+ w_2'
+ y^- n_2 \right|^2,
\end{equation}
where the subscript $T, U$ indicates the term dependent on the
$T, U$ moduli of the entropy function in \eq{entfnstu}.
Minimizing with respect to $y^+$ and $y^-$ we obtain the following equations
respectively
\bea{mineqyy}
 y^- \bar y^+ w_1' +  w_2' \bar y^+  +
   y^- n_2 - n_1 = 0, \\ \nonumber
 y^+ \bar y^- w_1' +  n_2 \bar y^-
+  y^+ w_2'  - n_1. =0
\end{eqnarray}
Eliminating $y^-$ from the above equations we obtain the following
quadratic equation for $y^+$
\be{eqy+}
( \tilde W_2 W_1 - W_2 \tilde W_1) (y^+ )^2
+ [ ( \tilde W_2 N_2 - W_2 \tilde N_2) - ( \tilde N_1 W_1 - N_1 \tilde W_1) ]
y^+ - ( \tilde N_1 N_2 - N_1 \tilde N_2 ) = 0.
\end{equation}
Note that in this quadratic equation which determines the moduli $y^+$, the
axion dilaton dependence completely drops out.
To obtain the solution for  $y^+$ we can simplify the
discriminant
\bea{discrim}
D &=& \left[ ( \tilde W_2 N_2 - W_2 \tilde N_2) - ( \tilde N_1 W_1 - N_1 \tilde
  W_1)\right]^2
+ 4 (\tilde W_2 W_1 - W_2 \tilde W_1 ) ( \tilde N_1 N_2 - N_1 \tilde N_2)
\nonumber,\\
 &=&  4 ( ( Q\cdot P)^2 - Q^2 P^2).
\end{eqnarray}
Here we have re-written the charges in terms of $Q$'s and $P$'s using the
relations in \eq{nwinvar1}. Since we are looking at supersymmetric black holes
we have $Q^2 P^2 - (Q\cdot P)^2>0,   Q^2>0, P^2>0$,
this implies the discriminant $D$ is
always negative. Thus the solution for $y^+$ is always complex, the
real and imaginary parts of $y^+$ are given by
\bea{soly+}
 y^+_1 & =& - \frac{  ( \tilde W_2 N_2 - W_2 \tilde N_2)
    -( \tilde N_1 W_1 - N_1 \tilde W_1) }{
2 ( \tilde W_2 W_1 - W_2 \tilde W_1 ) }, \\ \nonumber
 y^+_2 &=&
\frac{\sqrt{ Q^2 P^2 - ( Q\cdot P)^2}}{  ( \tilde W_2 W_1 - W_2 \tilde W_1 )}.
\end{eqnarray}
Note that finally we  have to choose the  solution with $y_2^+>0$, we choose
$\tilde W_2W_1 - W_2\tilde W_1>0$.
For later purpose it is convenient to write the above solution more
suggestively as follows.
The electric and magnetic charges are 4-vectors in $SO(2,2; {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$.
Let us consider the  antisymmetric combination of these vectors
\be{anstdual}
T^{ij} = Q^i P^j - Q^j P^i,
\end{equation}
where $i, j =1, \cdots 4$,  these 6 components are S-duality invariants.
We can project them into components which transform as a ${\bf{3}}$ of
the left $SO(2,1;{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ of $SO(2,2;{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ and
${\bf{3}}$ of the right $SO(2,1;{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ using the self dual and anti-self dual
't Hooft symbols of $SO(2,2)$ respectively.
From the definition of the 't Hooft symbols in \eq{defseld}
and \eq{adefseld} we see that the solution
\eq{soly+} can be written as
\be{susoly+}
y_1^+ = -\frac{\eta_{1;ij} T^{ij} }{ \eta_{+i;j} T^{ij} }, \quad
y_2^+ = \frac{ \sqrt{ \eta_{a;ij} \eta^a_{\,kl} T^{ij}T^{kl} }}{
\eta_{+;ij} T^{ij} }.
\end{equation}
where $a=1,2,3$. Note that we have used the identity
\eq{a6} to rewrite the invariant $Q^2P^2 - (Q\cdot P)^2$  in the
above equation.
A similar analysis for the $y^-$ yields the quadratic equation
\bea{eqy-}
( \tilde N_2 W_1 - N_2 \tilde W_1) (y^-)^2
+ \left[ (\tilde N_2 W_2 - N_2 \tilde W_2) + ( \tilde W_1N_1 - \tilde N_1 W_1)
 \right]y^-   + ( \tilde W_2 N_1 - W_2\tilde N_1) =0, \nonumber \\
\end{eqnarray}
here again the axion dilaton dependence drops out.
The discriminant of this quadratic equation is given by
\bea{discrim1}
D&=& \left[ ( \tilde N_2W_2 - N_2 \tilde W_2) + ( \tilde W_1 N_1 - \tilde N_1
  W_1) \right]^2 - 4 ( \tilde N_2 W_1 - N_2 \tilde W_1) ( \tilde W_2 N_1 - W_2
\tilde N_1), \nonumber \\
&=& 4 (Q\cdot P)^2 - Q^2 P^2.
\end{eqnarray}
Thus the real and the imaginary parts of $y^-$ are given by
\bea{soly-}
 y_1^- &=& - \frac{ (\tilde N_2W_2 - N_2 \tilde W_2) + (
  \tilde W_1 N_1 - \tilde N_1 W_1) }{2 ( \tilde N_2 W_1 -N_2 \tilde W_1)},
\\ \nonumber
 y_2^- &=&
\frac{\sqrt{ Q^2 P^2 - (Q\cdot P)^2}}{ \tilde N_2 W_1 - N_2 \tilde W_1}.
\end{eqnarray}
Again we have to choose the solution with $y_2^- >0$ for that we
have $\tilde N_2 W_1 - N_2 \tilde W_1>0$. Rewriting the solution
\eq{soly-} in terms of the 't Hooft symbols we obtain
\be{susoly-}
y_1^- = -\frac{\tilde\eta_{1;ij} T^{ij} }{ \tilde\eta_{-;ij} T^{ij} },
\qquad
y_2^- = \frac{ \sqrt{ \tilde\eta_{a;ij} \tilde\eta^a_{\,kl} T^{ij}T^{kl} }}{
\tilde\eta_{-;ij} T^{ij} }.
\end{equation}
For later convenience we also write down the real and imaginary
parts of $-1/y^-$
\bea{susoly-1}
-\left(\frac{1}{y^-} \right)_1 &=& \frac{ (\tilde N_2W_2 - N_2 \tilde W_2) + (
  \tilde W_1 N_1 - \tilde N_1 W_1) }{2 ( \tilde W_2 N_1 -W_2 \tilde N_1)}
=
\frac{\tilde\eta_{1;ij} T^{ij} }{ \tilde\eta_{+;ij} T^{ij} },
\\ \nonumber
-\left(\frac{1}{y^-} \right)_2& =&
\frac{\sqrt{ Q^2 P^2 - (Q\cdot P)^2}}{ \tilde W_2 N_1 - W_2 \tilde N_1} =
\frac{ \sqrt{ \tilde\eta_{a;ij} \tilde\eta^a_{\,kl} T^{ij}T^{kl} }}{
\tilde\eta_{+;ij} T^{ij} }.
\end{eqnarray}
The solutions given in \eq{soly+} and \eq{soly-} are independent of the
axion-dilaton moduli and thus minimize
both the terms which depend on $y^+$ and $y^-$ in \eq{entfnstu} and
therefore are the  attractor values for the entropy function.
The solution for the attractor values of the moduli has been derived earlier
for special charge configurations \cite{Sen:2005iz,Sahoo:2006rp}.
The analysis given above is
the complete solution for arbitrary charge configurations, it reduces to the
solutions found in \cite{Sen:2005iz,Sahoo:2006rp}
for the respective charge configurations.
The general analysis  enabled us
to write down the attractor values of the moduli in a  manifestly symmetric
form given in \eq{susoly+}, \eq{susoly-}.

Substituting the attractor
values of the $T$ and $U$ moduli given in \eq{susoly+} and
\eq{susoly-} in the entropy function
\eq{entfnstu} we obtain
\be{finalent}
F|_{T,U, {\rm{min}}}
= \frac{\pi}{2\tau_2} \left[ (Q + \tau P)\cdot (Q + \bar \tau P) \right].
\end{equation}
We now can further minimize with respect to the axion dilaton moduli
to obtain
\be{soltau}
\tau_1 = -\frac{Q\cdot P}{P^2}, \qquad
\tau_2 = \frac{\sqrt{Q^2 P^2 - (Q\cdot P)^2}}{ P^2}.
\end{equation}
Finally substituting the values of all the vector multiplet moduli
in the entropy function we get the usual Hawking-Bekenstein entropy
\be{hbent}
F|_{{\rm min}}= \pi \sqrt{Q^2P^2 -(Q\cdot P)^2}.
\end{equation}
Note that all the vector multiplets $y^+, y^-, \tau$ are fixed at the
attractor values, therefore this attractor point has no flat directions.

\subsection{The two derivative entropy function: FHSV model}

Let us call all the vector multiplet moduli other than the
axion-dilaton moduli as T-moduli. The T-moduli in the FHSV model
parameterizes the coset
\be{vecmod}
{\cal M}_T = \frac{SO(10, 2)}{ SO(10)\times SO(2) },
\end{equation}
To explictly find the values of the T-moduli at the attractor point
we first need to parameterize the $12\times 12$ moduli matrix $u_M$ which
satisfies the condition
\be{condfshv}
u_M^T = u_M, \qquad u_M^T L u_M = L.
\end{equation}
where $L = {\rm{Dia}}( -1, \cdots, -1, 1, 1)$ is the Lorentzian  metric
with 10 negative 1's and 2 positive 1's  .
From the conditions in \eq{condfshv}
the number of independent variables required to parameterize $u_M$ is
$20$.
Just as in the STU model we first  introduce $10+2$ complex numbers satisfying
\bea{consteq}
-\sum_{I=1}^{10} w_I^2 +w_{11}^2 + w_{12}^2 = 0,
\end{eqnarray}
together
with the identification $w_I \sim c w_I$, where $c$  is a complex number.
Note that the  constraint in \eq{consteq}
and the identification of $w$'s upto complex scalings
reduce the number of independent parameters to $20$ which is the required
number of variables to parameterize the moduli space in \eq{vecmod}.
Using the scaling degree of freedom
 the constraints in \eq{consteq} can be solved by
introducing the $10$ complex numbers $(y^+, y^-, \vec y)$ where
$\vec y$ is a $8$ dimensional vector.
These variables are related to the
$12$ $w_I$'s by
\bea{relwy}
w_I = y_I, \;\;I =1 \cdots 8, &\quad& w_{9} = \frac{1}{\sqrt{2}} ( y^+ -  y^-),
\cr
w_{10} = 1+ \frac{ y^2}{4}, &\quad&
w_{11} = \frac{1}{\sqrt{2}} ( y^+ +  y^-), \cr
w_{12} = -1 + \frac{y^2}{4},
&\quad& y^2 = 2 y^+ y^- + \vec y^2.
\end{eqnarray}
On substituting these values of $w_I$ in \eq{consteq} it is easy to see that
the constraint is satisfied. The above parameterization amounts to
scaling $w_{10}- w_{12} $ such that its value is constant given by
$ w_{10}- w_{12}=2$. Using the above solution of  the constraint
\eq{consteq} it can be seen that
\bea{mconstfs}
-\sum_{I=1}^{10} |w_I|^2 + |w_{11}|^2 + |w_{12}|^2 = 2Y, \\ \nonumber
\hbox{where}\; Y = ({\rm Im} y)^2 = 2 y_2^+ y_2^- - \vec y_2^2.
\end{eqnarray}
 $Y$ is
related to the K\"{a}hler potential on the moduli space
by
\be{relkpot}
K = -\log Y.
\end{equation}
The variables $y$ to parameterize the vector multiplet moduli was introduced in
\cite{Harvey:1996ts,Klemm:2005pd}.
We now parameterize the vector multiplet moduli matrix as
\be{defmodmat}
u_M = L \tilde U L - L.
\end{equation}
The conditions on the moduli matrix given in \eq{modcond} result in
the following conditions on $\tilde U$
\be{contu}
\tilde U^T = \tilde U, \qquad
\tilde U L \tilde U - 2 \tilde U =0.
\end{equation}
We can now further parameterize  $\tilde U$ as
\be{partu}
\tilde U = \frac{ w_I \bar w_J + \bar w_I w_J}{ Y},
\end{equation}
here $I =1 \cdots 12$
It is easy to see that given the constraints \eq{consteq} on
the $w$'s we see that the conditions on $\tilde U$ in \eq{contu} are
satisfied.

Using $\tau$ to parameterize
the axion dilaton moduli as in
\eq{deftau} and  parameterizing  the moduli matrix $u_M$ in
terms of $w$ from \eq{defmodmat},
\eq{partu} in the entropy function
\eq{entfns} we can write the entropy function as
\bea{compent}
F &=& \frac{\pi}{2\tau_2 } \left[ ( Q+ \tau P)u_M ( Q+ \bar\tau P) \right], \\
\nonumber
& =&  \frac{\pi}{2} \left[  \frac{ | ( Q + \tau P) \cdot w|^2}{\tau_2 Y}  +
\frac{ | ( Q + \bar\tau P) \cdot w|^2}{\tau_2 Y}
- \frac{( Q+ \tau P)\cdot (Q+ \bar \tau P) }{\tau_2} \right].
\end{eqnarray}
Note that in the above the dot product is taken with the metric  $L$,
From here it is easy to see that $F$ is both $SL(2,R)$ as well as $SO(10,2;R)$
invariant.
We first  minimize the entropy function
with respect to the T-moduli $(y^+, y^-, \vec y)$ for right moving charges.
Right moving charges are defined by the condition $Q\cdot Q>0, P\cdot P>0$.
Generic supersymmetric dyons are right moving.
Details of the minimization procedure are provided in the appendix.
To write down the solution we define the following combination of
charges
\bea{combchg}
Q^{12} + Q^{10} = N_1, \quad Q^{12} -Q^{10} = W_1, \quad Q^{11} + Q^{9} =
N_2, \quad Q^{11} - Q^{9} = W_2, \\ \nonumber
P^{12} + P^{10} = \tilde N_1,
\quad P^{12} -P^{10} = \tilde W_1, \quad P^{11} + P^{9} =
\tilde N_2, \quad P^{11} - P^{9} = \tilde W_2.
\end{eqnarray}
We now write down the solution of all the T-moduli: $y^i$,
$i=1, \cdots 8$  and $y^-$ are determined in
terms of $y^+$ from the equation
\bea{solyiy-}
y^i& =& \sqrt{2} y^+
\frac{Q^i \tilde W_1 - P^i W_1}{ N_2 \tilde W_1 - \tilde N_2 W_1}
+ 2 \frac{\tilde N_1 Q^i - N_2P^i}{ N_2 \tilde W_1 - \tilde N_2 W_1},
\\ \nonumber
y^- &=& y^+ \frac{W_2\tilde W_1 - \tilde W_2  W_1}
{N_2 \tilde W_1 - \tilde N_2 W_1} + \sqrt{2}
\frac{\tilde N_2 W_2 - N_2 \tilde W_2}{N_2 \tilde W_1 - \tilde N_2 W_1},
\end{eqnarray}
$y^+$ is determined from the solution of the following quadratic equation
\bea{solyfy+}
& &A\left( \frac{y^+}{\sqrt{2}}\right)^2 + B \frac{y^+}{\sqrt{2}}
+ C = 0, \\ \nonumber
& &A = \tilde W_2 W_1 - W_2\tilde W_1 + \sum_{i=1}^8
\frac{(Q^i \tilde W_1 - P^i W_1)^2 }{ N_2 \tilde W_1 - \tilde N_2 W_1}, \\
\nonumber
& & B = (\tilde W_2 N_2 - W_2 \tilde N_2) -( \tilde N_1 W_1 - N_1 \tilde W_1)
+ \sum_{i=1}^8
\frac{( \tilde N_2 Q^i -P^i N_2)(\tilde W_1 Q^i - P^i W_1) }{
 N_2 \tilde W_1 - \tilde N_2 W_1},
\\ \nonumber
& &C= N_1 \tilde N_2  - \tilde N_1 N_2.
\end{eqnarray}
Note that the coefficients which determine the attractor values  of the moduli
$y^+$ and $y^i$ in \eq{solyiy-} and \eq{solyfy+}
are all functions of the S-duality invariants $Q^I P^J - P^IQ^J$, with
$I, J =1, \cdots 12$. Therefore the attractor values of the T-moduli
do not transform under the S-duality symmetry of the FHSV model.
As a small check on the above solutions for the T-moduli, note that
on setting  the charges $Q^i=0, P=0$ the equations \eq{solyfy+} reduce
to \eq{eqy+} with the $y^+ \rightarrow \sqrt{y^+}$.
Furthermore from the solutions it is easy to see that under the
scaling $Q^I \rightarrow \lambda Q^I,\; P^I \rightarrow \tilde\lambda P^I$
the attractor values of the T-moduli remain invariant. Thus for independent
scalings of the electric and the magnetic charges the attractor values of the
T-moduli are invariant. From \eq{solyfy+}, it is see that
the attractor values of $y^+$ are functions of ratios of
$B/A, C/A$, thus they are of ${\cal O}(Q^0, P^0 )$ even if the charges
electric and magnetic charges are scaled with
$\lambda >>1, \tilde \lambda>>1$.
We now substitute the values of the T-moduli into the entropy function
given in \eq{compent}, this gives
\be{tminentfn}
F = \frac{\pi}{2\tau_2} \left[ (Q + \tau P)\cdot (Q + \bar\tau P)\right].
\end{equation}
Now we proceed with minimizing the above function with respect to the
axion-dilaton moduli, this results in the following attractor values
\be{soltauf}
\tau_1 = -\frac{Q\cdot P}{P^2}, \qquad
\tau_2 = \frac{\sqrt{Q^2P^2 - (Q\cdot P)^2}}{P^2}.
\end{equation}
Finally substituting all the attractor values of the
vector multiplet moduli into the
entropy function we obtain the following Hawking-Bekenstein entropy
\be{finentfs}
F= \pi\sqrt{Q^2P^2 -(Q\cdot P)^2}.
\end{equation}

\subsection{Including the Gauss-Bonnet correction}

In this subsection we include the contribution of the Gauss-Bonnet
term to the entropy function and evaluate its  contribution to the black hole
entropy. We will retain terms to ${\cal{ O}}(Q^0, P^0)$ terms in charges and
neglect contributions at ${\cal{O}}(1/Q^2, 1/P^2)$.
The coefficient of the
 Gauss-Bonnet in both the STU model \eq{cofstugb} and the FHSV model
\eq{cofgb} contains a term proportional to the trace anomaly which
depends on graviton momentum $p^2$. Since this term does not
depend on the moduli of the theory the contribution to the
Gauss-Bonnet term  can be neglected.
Retaining this term in the entropy function formalism just shifts the
Entropy by a charge independent constant which we do not keep track of.
Therefore we can restrict our attention to the moduli dependent
terms in the coefficient of the Gauss-Bonnet term in \eq{cofstugb}
and \eq{cofgb}.
Consider first a coefficient of the Gauss-Bonnet term to be
\be{excofgb}
\frac{1}{128\pi^2} {\cal F}(\mu, \bar\mu),
\end{equation}
where $\mu$ refers to the axion-dilaton moduli together with the  rest of
the vector multiplet moduli.
The change in the entropy function due to the above coefficient is
\eq{excofgb} is
\be{chentfn}
\Delta F = \frac{1}{2} {\cal F} (\mu, \bar \mu).
\end{equation}
The total entropy function is now given by
\be{totentfnex}
F = \lambda^2 F^{(2)}( \mu, \bar\mu ) + \frac{1}{2}{\cal  F}
(\mu, \bar\mu),
\end{equation}
where $F^{(2)}$ refers to the two derivative entropy function
discussed in the previous section. The two derivative term is
proportional to ${\cal O}(Q^2, P^2)$, this is indicated by the
coefficient $\lambda^2$ in \eq{totentfnex}.
We can now minimize with respect to the moduli $\mu$ and solve
for the moduli in powers of $\frac{1}{\lambda^2}$, we denote this
correction as
\be{coremu}
\mu = \mu^{*} + \frac{1}{\lambda^2} \delta\mu + {\cal O}(1/\lambda^4),
\end{equation}
where $\mu^*$ solves the attractor equations of the
two derivative entropy function $F^{(2)}$ given by
\be{2derateq}
\left.
\frac{\partial F^{(2)}( \mu, \bar\mu) }{\partial \mu}
\right|_{\mu =\mu^*} = 0, \qquad
\left.
 \frac{\partial F^{(2)}( \mu, \bar\mu) }{\partial \bar\mu}
\right|_{\mu = \mu^*} = 0.
\end{equation}
Since the
attractor point has no flat directions one can obtain a unique solution
for $\delta \mu$. Substituting the solution back into $F$ given by
\eq{totentfnex} and using \eq{2derateq}  it is easy to see that to
${\cal{O}}(Q^0, P^0)$ in charges  the entropy is given by
\be{valentmin}
F = \lambda^2 F^{(2)} (\mu^*, \bar\mu^*) +
\frac{1}{2} {\cal  F}(\mu^*, \bar\mu^*) + {\cal O}(1/\lambda^2).
\end{equation}
Thus to evaluate the correction to entropy due to the Gauss-Bonnet term
to ${\cal O}(Q^0, P^0)$ in charges  it is sufficient to
evaluate its coefficient at the attractor values of the moduli.

\vspace{.5cm}
\noindent
{\emph{STU model}}
\vspace{.5cm}

To evaluate the correction to the entropy including the Gauss-Bonnet
term to ${\cal O}(Q^0, P^0)$ in charges we substitute the values of the
vector multiplet at the attractor point given in
\eq{susoly+}, \eq{susoly-} and  \eq{soltau} into the
coefficient of the Gauss-Bonnet term and using \eq{valentmin}. We obtain
\bea{stusolentfn}
 F &=& \pi\sqrt{Q^2P^2 - (Q\cdot P)^2}  \\ \nonumber
& &-\ln\left[\vartheta_2 (\tau)^4 \vartheta_2 (-\bar\tau)^4(\tau_2)^2
\right]|_*
-\ln\left[\vartheta_2 (y^+)^4 \vartheta_2 (-\bar y^+)^4(\tau_2)^2
\right]|_* \\ \nonumber
& &-\ln\left[\vartheta_2(-1/y^-)^4 \vartheta_2 (1/\bar y^-)^4((-1/y^-)_2)^2
\right]|_* + {\cal{O}}(1/Q^0, 1/P^0).
\end{eqnarray}
Here the subscript $*$ refers to the fact that we have substitute the
attractor
values for the moduli given in \eq{susoly+}, \eq{susoly-} and  \eq{soltau}.
Note that using a modular transformation we have written the
$\vartheta_4(y^-)$ in terms of $\vartheta_2(-1/y^-)$ in the
coefficient of the Gauss-Bonnet term.

From the expressions for the moduli in \eq{susoly+}, \eq{susoly-} and
\eq{soltau} and the fact that the leading term in the entropy is
invariant under triality,  it is easy to see that
the corrected entropy given in \eq{stusolentfn} is invariant under the
following triality symmetries.
\be{entrisym1}
{\cal T}_1: W_1\leftrightarrow \tilde N_2,    \qquad ;
W_2\leftrightarrow -\tilde N_1.
\end{equation}
Under this exchange of charges the attractor
value of the moduli transform as
\be{atexsy1}
\tau_*\leftrightarrow y^+_*,
\quad  y^-_*\; \hbox{invariant}.
\end{equation}
Similarly  under the exchange of the charges
\be{entrisym2}
{\cal T}_2: W_1 \leftrightarrow N_1, \qquad ;\tilde
 W_1 \leftrightarrow \tilde N_1,
\end{equation}
the attractor values of the  moduli transform as
\be{atexsy2}
y^+\leftrightarrow_* -\frac{1}{y^-_*},
\qquad   \tau\; \hbox{invariant}.
\end{equation}
Finally under the exchange of charges
\be{entrisym3}
{\cal T}_3: \tilde N_2 \leftrightarrow N_1\qquad ;
\tilde W_1\rightarrow -W_2,
\end{equation}
the attractor values of the moduli transform as
\be{atexsy3}
\tau_*\rightarrow -\frac{1}{y^-_*},  \qquad
y^+\; \hbox{invariant}.
\end{equation}

In the next section we write down a partition function which for dyons
in the STU model which captures the subleading coefficient obtained by
considering the Gauss-Bonnet term. The partition function manifestly has
all the symmetries of the model.

\vspace{.5cm}
\noindent
{\emph{The FHSV model}}
\vspace{.5cm}

To obtain the entropy of the dyonic black hole in the FHSV model with the
 Gauss-Bonnet  term to ${\cal O}(Q^0,P^0)$ in charges we substitute the
 attractor values of the the moduli given in \eq{solyiy-}, \eq{solyfy+} in
 \eq{valentmin}. The coefficient of the Gauss-Bonnet term is obtained from
 \eq{threshfs}. Performing this we obtain
\bea{fshvf1}
F &= &\pi \sqrt{Q^2P^2 -(Q\cdot P)^2} -
\ln\left( \eta(2\tau)^{12} \eta( -2\bar \tau)^{12} \tau_2^{6} \right)|_* \\ \nonumber
& & -\frac{1}{2}\ln\left( \Phi_{\rm{BE}}(y) \Phi_{\rm{BE}}
(-\bar y) ( 2y_2^+ y_2^-  -\vec y^2)^4 \right)|_* + {\cal{O}}( 1/Q^2, 1/P^2).
\end{eqnarray}
In this model there is a scaling limit of the charges in which the contribution of
entropy from the
the T-moduli becomes negligible compared that of the axion-dilaton moduli.
Let us consider the following scaling of the charges
\bea{scalimit}
& &Q^I\rightarrow \lambda Q^I,\qquad P^I\rightarrow \lambda'P^I  \quad {\hbox{with}}\;
\lambda>>1, \; \lambda'>>1 \\ \nonumber
& & {\hbox{ and}} \;  \lambda>>\lambda'.
\end{eqnarray}
From the solution of the T-moduli given in \eq{solyiy-}, \eq{solyfy+}, it is
easy to see that in this scaling limit the T-moduli are of order 1,
${\cal O}(\lambda^0, \lambda^{'0})$. The axion dilaton moduli on the other
hand scales as
\be{scaleaxdi}
\tau_1\rightarrow \frac{\lambda}{\lambda'} \tau_1, \qquad
\tau_2\rightarrow \frac{\lambda}{\lambda'} \tau_2.
\end{equation}
Thus in \eq{fshvf1} we can neglect the contribution of the T-moduli and retain
the leading contribution from the axion-dilaton moduli. Therefore  \eq{fshvf1}
reduces to
\bea{scalentfn}
F &=& \pi \sqrt{Q^2P^2 -(Q\cdot P)^2} + 2\pi \tau_2|_* - 6\ln \tau_2|_ *+
{\cal{ O}}( Q^{0}, P^0)  \\ \nonumber
& =& \pi \sqrt{Q^2P^2 -(Q\cdot P)^2} + 2 \pi \frac{ \sqrt{Q^2P^2 -(Q\cdot P)^2} }{P^2}
-6 \ln \left( \frac{ \sqrt{Q^2P^2 -(Q\cdot P)^2} }{P^2}\right) \\ \nonumber
& & + {\cal{ O}}( Q^{0}, P^0).
\end{eqnarray}
In section 4. we will write down an approximate dyon  partition function involving a
Siegel modular form of weight 4 which captures the degeneracy of the dyons in the
FHSV model in the limit when the electric charges are much larger than the
magnetic charges as in \eq{scalimit}. Note that in this limit though we loose most
of the details of the correction to the entropy from the Gauss-Bonnet term,
the information of the weight of the $SL(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ modular form which captures
the axion-dilaton dependence is present in the coefficient of the logarithm
in \eq{scalentfn}.

\section{Partition function for the STU model}

In this section we propose a partition function for the dyons
of the  STU model. The partition function is given in terms
of product of three Siegel modular forms of weight zero.
For the purposes of extracting the degeneracy from the
partition function we need to define the charge bilinears
which occur in the Fourier expansion of the partition function.
From the consideration of the entropy function in the pervious
section we have seen that that the following charge bilinears
characterize the black hole entropy of the dyons to the next leading
order. There are three sets of charge bilinears, each set is invariant
with respect to any  two of the three $\Gamma(2)$ symmetries of the STU
model. The T-duality and U-duality invariants are given by
\be{tdualinv}
M_{\sigma^1} = \frac{1}{3} Q\cdot Q ,
\qquad M_{\rho^1}  =  \frac{1}{3} P\cdot P, \qquad
M_{v^2} = \frac{1}{3}  Q\cdot P. \\ \nonumber
\end{equation}
The S-duality invariants are given by following anti-symmetric second
rank tensor $T^{ij}$ of $SO(2,2)$
\be{sdualinv}
T^{ij} = Q^iP^j - Q^j P^i
\end{equation}
where $i, j = 1, \cdots 4$. From the above 6 components of the
S-duality invariant
the  charge bilinears  we can project into the self-dual or the
anti-self dual combinations which are further invariant under
either T-duality or the U-duality symmetries. This projection
is done by the 't Hooft symbols of $SO(2,2)$.   They are given by
\bea{sdualcomb}
M_{\sigma^2} &=&\frac{1}{3} \eta_{-;ij}  T^{ij}, \qquad
M_{\rho^2} = \frac{1}{3}  \eta_{+;ij }  T^{ij}, \qquad
M_{v^2} = \frac{1}{3}\eta_{1;ij} T^{ij}, \\ \nonumber
M_{\sigma^3} &=& \frac{1}{3} \tilde \eta_{-;ij} T^{ij}, \qquad
M_{\rho^3} =\frac{1}{3}  \tilde \eta_{+;ij}  T^{ij}, \qquad
M_{v^3} = -\frac{1}{3} \tilde \eta_{1;ij} T^{ij}.
\end{eqnarray}
Here $\eta_{a, ij}$ and $\tilde \eta_{a, ij}$ are the 't Hooft symbols which
decomposes the second rank anti-symmetric tensor $T^{ij}$ of $SO(2,2)$ and
as $({\bf{3}},0)$ and $(0,{\bf{3}})$ of $SU(1,1)\times SU(1,1)$. $a, =1, \cdots 3$.
These 't Hooft symbols are defined in  appendix A.

We now propose the partition function for this model:
The partition function involves the product of inverses of three $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$
modular forms of weight zero and is given by
\bea{stupart}
d(M_{\rho}, M_{\sigma}, M_v)
&=&\prod_{\alpha =1}^3  I_{\alpha} (M_{\rho^\alpha}, M_{\sigma^\alpha}, M_{v^\alpha}),
\\ \nonumber
 I_{\alpha} (M_{\rho^\alpha}, M_{\sigma^\alpha}, M_{v^\alpha}) &=&
\frac{K}{4}\exp{(i\pi  M_v)}\int _{{\cal{C}^\alpha}}
d\tilde\rho^\alpha d\tilde \sigma^\alpha d\tilde v^\alpha \frac{1}{\widetilde{\Phi}_0(\tilde\rho^\alpha,\tilde\sigma^\alpha,\tilde v^\alpha)} \\ \nonumber
& \times& \exp\left(-2\pi i \left[ \frac{M_{\rho^\alpha}}{2} \tilde\rho^\alpha
+\frac{M_{\sigma^\alpha} }{2}\tilde\sigma^\alpha  + M_{v^\alpha} \tilde v
\right] \right).
\end{eqnarray}
The $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ modular form $\widetilde{\Phi}_0(\tilde\rho, \tilde\sigma, \tilde v)$
which occurs in the partition function is given
by
\be{defphi0}
\widetilde{\Phi}_0 = \widetilde{\Phi}_2 \sqrt{ \frac{\widetilde{\Phi}_2'}{\widetilde{\Phi}_6} }.
\end{equation}
Here $\widetilde{\Phi}_6$ is the Siegel modular form of weight 6 under the subgroup
$\tilde G$ of $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ which captures the degeneracy of dyons in
the ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$ CHL model \cite{Jatkar:2005bh,David:2006ji}
 $\widetilde{\Phi}_2$ is the Siegel modular form
of weight $2$ under the same subgroup $\tilde G$ of $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ which
captures the degeneracy of dyons in the ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$ orbifold of type IIB theory
\cite{David:2006ru}.
Finally $\widetilde{\Phi}_2'$ is  the modular form of weight $2$ related to $\widetilde{\Phi}$ by
\be{relphi2s}
\widetilde{\Phi}_2'(\tilde\rho,\tilde,\sigma, \tilde v) =
\widetilde{\Phi}_2(\frac{\tilde\sigma}{2}, 2\tilde\rho, \tilde v).
\end{equation}
From property 1 of $\widetilde{\Phi}_2'$ listed in the appendix C, we see that
$\widetilde{\Phi}_2'$ is also a Siegel modular form of weight 2 under the subgroup
$\tilde G$ of $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$.
It is clear from the definition of $\widetilde{\Phi}_0$ that it is a modular form
of weight 0, under the same subgroup $\tilde G$ of $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$.
All the modular forms given in \eq{defphi0} are explicitly constructed in the
appendix C in their product representation from appropriate
threshold integrals.
In \eq{stupart} the three dimensional contour ${\cal{C}}^\alpha$ is given by
\bea{phi0cont}
{\rm Im}\, \tilde\rho^\alpha = M_1^\alpha, \;
{\rm Im}\, \tilde\sigma^\alpha = M_2^\alpha, \;
{\rm Im}\, \tilde v^\alpha = M_3^\alpha,\\ \nonumber
0\leq {\rm Re} \,\tilde\rho^\alpha \leq 2, \;
0\leq {\rm Re}\, \tilde\sigma^\alpha \leq 2, \;
0\leq {\rm Re} \,\tilde v^\alpha \leq 1.
\end{eqnarray}
The constant $K = -2^{-10}$.
From the Fourier expansion of the
partition function it can be shown that
the  degeneracy formula in \eq{stupart} is
valid with
\be{validdeg}
M_{\rho^\alpha} \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, \quad
M_{\sigma^\alpha} \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, \quad
M_{v^\alpha} \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}.
\end{equation}
We now list the properties of $\widetilde{\Phi}_0$ which are proved in the
appendix C.

\vspace{.5cm}
\noindent
{\emph{Properties of $\widetilde{\Phi}_0$}}
\vspace{.5cm}
\begin{enumerate}
\item
$\widetilde{\Phi}_0$ is a modular form of weight 0 under the subgroup
$\tilde G$ os $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$.
\item
$\widetilde{\Phi}_0(\tilde\rho,\tilde\sigma, v)$ is an analytic function with second order zeros at
\bea{zphi0}
& &n_2(\tilde\sigma,\tilde\rho -\tilde v^2) + b \tilde v + n_1\tilde\sigma -\tilde\rho m_1 + m_2 =0, \\ \nonumber
& & {\hbox{for}}\; m_1n_1+ m_2 n_2 + \frac{b^2}{4} = \frac{1}{4},\; m_1 \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, n_1\in{\hskip-3pt\hbox{ Z\kern-1.6mm Z}},
b\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1 , m_2, n_2 \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}.
\end{eqnarray}
It has second order poles at
\bea{pphi0}
& &
n_2(\tilde\sigma,\tilde\rho -\tilde v^2) + b \tilde v + n_1\tilde\sigma -\tilde\rho m_1 + m_2 =0, \\ \nonumber
& & {\hbox{for}}\; m_1n_1+ m_2 n_2 + \frac{b^2}{4} = \frac{1}{4},\; m_1 \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1, n_1\in{\hskip-3pt\hbox{ Z\kern-1.6mm Z}},
b\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1 , m_2, n_2 \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}.
\end{eqnarray}
and first order poles at
\bea{p1phi0}
& & n_2(\tilde\sigma,\tilde\rho -\tilde v^2) + b \tilde v + n_1\tilde\sigma -\tilde\rho m_1 + m_2 =0, \\ \nonumber
& & {\hbox{for}}\; m_1n_1+ m_2 n_2 + \frac{b^2}{4} = \frac{1}{4},\; m_1 \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, n_1\in{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}
+\frac{1}{2},
b\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1 , m_2, n_2 \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}.
\end{eqnarray}
\item
$\widetilde{\Phi}_0$ is invariant under the following $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformations
\bea{sp2zt}
\Phi_0(\tilde\rho', \tilde\sigma', \tilde v') =
\Phi_0(\tilde\rho,\tilde\sigma,\tilde v),
\\
\label{tt1}
\hbox{for}\; \tilde\rho'= \tilde\rho + 4\tilde\sigma + 4 \tilde v -2, \quad
\tilde\sigma' = \tilde\sigma, \quad \tilde v' = 2\tilde\sigma + \tilde v,
\\
\label{tt2}
\hbox{and} \; \tilde\rho'= \tilde\rho, \quad
\tilde\sigma' = 4\tilde\rho +\tilde\sigma+ 4\tilde v -2 ,
\quad \tilde v' = 2\tilde\sigma + \tilde v.
\end{eqnarray}
This property is due to the fact that $\widetilde{\Phi}_0$ is invariant under the
subgroup of $\tilde H$ defined in \eq{subgroup}. The transformation
\eq{tt1} corresponds to the choice $a= -1, b=2, c=0, d=-1$ in \eq{sghtrans}
and  \eq{tt2} corresponds to the choice $a =1, b =0, c=-2, d=1$ in
\eq{sghtrans}. The above $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformations are
essentially the generators of $\Gamma(2)$.
Note that  $\widetilde{\Phi}_0$ is invariant under the
subgroup \eq{subgroup} which is $\Gamma_0(2)$ which contains the
group $\Gamma(2)$.
\item
 The expansion of $1/\widetilde{\Phi}_0$ in terms of Fourier coefficients  given by
\be{fexp}
\frac{K}{\widetilde{\Phi}_0(\tilde\rho,\tilde\sigma, \tilde v)} =
\sum_{\stackrel{m, n, p}{m\geq -1/2, n \geq 1/2}}
e^{2\pi i( m \tilde\rho + n\tilde\sigma + p\tilde v)}
g(m , n, p),
\end{equation}
with $g(m, n, p)$ being integers,  $p\in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}$ while $m, n\in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}/2$
\item
For small $\tilde\rho\tilde\sigma - \tilde v^2 + \tilde v$  $\widetilde{\Phi}_0$ factorizes as
\be{facphi0}
\widetilde{\Phi}_0(\tilde\rho,\tilde\sigma, \tilde v) =
4\pi^2 ( 2v - \rho - \sigma)^2 v^2 f^{(0)}(\rho) f^{(0)}(\sigma)
+ {\cal O} (v^2),
\end{equation}
where
\be{deff2}
f^{(0)} (\rho) = \vartheta_2(\rho)^4.
\end{equation}
The variables $(\rho, \sigma, v)$ and
$(\tilde\rho, \tilde\sigma, \tilde v)$ are related by the
$Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformation
\be{chgvar}
\rho= \frac{\tilde\rho\tilde\sigma  - \tilde v^2}{\tilde \sigma}, \qquad
\sigma = \frac{\tilde\rho\tilde\sigma - (\tilde v -1)^2}{\tilde \sigma},
\qquad
v = \frac{\tilde\rho\tilde\sigma - \tilde v^2 + \tilde v}{\tilde \sigma},
\end{equation}
or the inverse relations
\be{chgvar2}
\tilde\rho = \frac{v^2 - \rho\sigma}{2 v - \rho -\sigma}, \qquad
\tilde\sigma = \frac{1}{2 v -\rho - \sigma},
\qquad
\tilde v = \frac{v -\rho}{2 v - \rho -\sigma}.
\end{equation}
\end{enumerate}

We now perform several check on the proposal \eq{stupart}, we show
that it has all the duality symmetries of the STU model and then
show that for large charges the statistical entropy obtained from
the partition function reproduces the degeneracy of dyonic
black holes in the STU model to the first subleading order in
charges.

\subsection{Consistency checks}

In this section we will subject our proposal given in \eq{stupart} to various
consistency checks. We first verify that the integrand in \eq{stupart}  has
STU triality symmetry and then show that it has
$\Gamma(2)_S\times \Gamma(2)_T\times \Gamma(2)_U$
symmetry.

\vspace{.5cm}
\noindent
{\emph{STU triality symmetry}}
\vspace{.5cm}

From the definitions of the charge bilinears it is easy to show the following
transformation properties:
Under
\bea{dtrail1}
{\cal T}_1: \;W_1\leftrightarrow \tilde N_2,  \qquad
W_2\leftrightarrow -\tilde N_1,
\end{eqnarray}
the following charge bilinears transform as
\bea{trail1}
M_{\rho^1} \leftrightarrow M_{\rho^2}, \quad
\quad M_{\sigma^1}\leftrightarrow M_{\sigma^1},
\quad M_{v^1} \leftrightarrow M_{v^2},
\end{eqnarray}
while the charge bilinears $M_{\rho^3}, M_{\sigma^3}, M_{v^3}$
remains invariant. It is now easy to see that
under this  transformation ${\cal T}_1$the integrand in the partition
function \eq{stupart} is manifestly invariant.
Similarly under
\bea{dtrail2}
{\cal T}_2: \;W_1\leftrightarrow  N_1, \qquad
\tilde W_1\leftrightarrow \tilde N_1,
\end{eqnarray}
that is the exchange of momentum and winding on  the first circle,
the following charge bilinears transform as
\bea{trail2}
M_{\rho^2} \leftrightarrow M_{\rho^3}, \quad
 M_{\sigma^2}\leftrightarrow M_{\sigma^3}, \quad
 M_{v^2} \rightarrow M_{v^3},
\end{eqnarray}
while the charge bilinears $M_{\rho^1}, M_{\sigma^1}, M_{v^1}$
remains invariant. Thus it is easy to see that
the integrand in the partition
function \eq{stupart} is manifestly
invariant under the transformation ${\cal T}_2$.
Finally the transformation
\bea{dtrail3}
{\cal T}_3: \;\tilde N_2\leftrightarrow  N_1, \qquad
 \tilde W_1\leftrightarrow -W_2,
\end{eqnarray}
the following charge bilinears transform as
\bea{trail3}
M_{\rho^1} \leftrightarrow M_{\rho^3}, \quad
 M_{\sigma^1}\leftrightarrow M_{\sigma^3},
\quad M_{v^1} \rightarrow M_{v^3},
\end{eqnarray}
while the charge bilinears $M_{\rho^2}, M_{\sigma^2}, M_{v^2}$
remains invariant. Thus it is easy to see that
the integrand in the partition
function \eq{stupart} is manifestly
invariant under the transformation ${\cal T}_3$.
Therefore ignoring considerations of the contour ${\cal{C}^\alpha}$ the
degeneracy is invariant under the
triality symmetries ${\cal T}_1, {\cal T}_2, {\cal T}_3$.

\vspace{.5cm}
\noindent
{\emph{$\Gamma(2)_S\times \Gamma(2)_T\times \Gamma(2)_U$ symmetry}}
\vspace{.5cm}

We first look at $\Gamma(2)_S$ action, which acts on the
electric and magnetic charges as
\be{sdualtra}
Q\rightarrow Q' = a Q+ b P, \quad
P \rightarrow P' = cQ + d P , \quad
\left(
\begin{array}{cc}
a & b \\
c & d
\end{array}
\right) \in \Gamma(2).
\end{equation}
The generators of $\Gamma(2)$ are given by the following matrices
\be{gengamma2}
M_{\infty} = \left(
\begin{array}{cc}
-1 & 2 \\
0 & -1
\end{array} \right), \qquad
M_{1} = \left(
\begin{array}{cc}
1 & 0 \\
-2 & 1
\end{array}
\right).
\end{equation}
Using \eq{sdualtra} with the generator $M_\infty$ we find that the
T-duality invariants $M_{\rho^1}, M_{\sigma^1}, M_{v^1}$ transform as
\be{tstrans}
\left(
\begin{array}{c}
M_{\sigma^{1}}'\\
M_{\rho^{1}}'\\
M_{v^{1}}'
\end{array}
\right)
=
\left(\begin{array}{ccc}
1 & 4 & -4 \\
0 & 1 & 0 \\
0 & -2 & 1
\end{array}\right)
\left(
\begin{array}{c}
M_{\sigma^1}\\
M_{\rho^1}\\
M_{v^1}
\end{array}
\right).
\end{equation}
Now following \cite{Jatkar:2005bh} let us define
\be{defgamm2om}
\tilde\Omega^{1' }\equiv
\left(\begin{array}{cc}
\tilde\rho^{1'} &\tilde v^{1'}  \\
\tilde v^{1'}  & \tilde\sigma^{1\prime}
\end{array}
\right)
= (\tilde A\tilde\Omega^1 + \tilde B) ( \tilde C\tilde\Omega^1  + \tilde D)^{-1}, \quad
\left(
\begin{array}{cc}
\tilde A & \tilde B \\
\tilde C & \tilde D
\end{array}
\right)
= \left(
\begin{array}{cccc}
a & -b & b &0 \\
-c & d & 0 & c \\
0 & 0& d & c \\
0 & 0 & b & a
\end{array}
\right).
\end{equation}
For the matrix $M_\infty$ we obtain
\bea{minftytr}
\tilde\rho^{1' }&=& \tilde\rho^1 + 4\tilde\sigma^1 + 4 \tilde v^1 -2, \cr
\tilde\sigma^{1'} &=& \tilde\sigma^1, \cr
\tilde v^{1'} &=& 2\tilde\sigma^1 + \tilde v^1.
\end{eqnarray}
Using the transformations \eq{tstrans} and \eq{minftytr} we
obtain the following transformations
\bea{mestrans}
& & \exp\left[2\pi i \left( \frac{M_{\rho^1}'}{2} \rho^{1'} +
 \frac{M_{\sigma^1}'}{2} \sigma^{1'} + M_{v^1}' v^{1'} \right)\right] \\ \nonumber
 & & \;\;\;\;\;\;\;\;\; \rightarrow
 \exp\left[2\pi i \left( \frac{M_{\rho^1}}{2} \rho^{1} +
 \frac{M_{\sigma^1}}{2} \sigma^{1} + M_{v^1}v^{1} \right)\right]
 \exp{(2\pi i M_{\rho^1})},
 \\ \nonumber
& &  \exp{(i\pi M_{v^1}')}
 \rightarrow \exp{(-2\pi  i M_{\rho^1})} \exp{(i\pi M_{v^1})}.
 \end{eqnarray}
 Furthermore from \eq{minftytr} and  \eq{sp2zt} we see that
\be{invarphi}
\widetilde{\Phi}_0(\tilde\rho^{1'}, \tilde\sigma^{1'}, \tilde v^{1'} ) = \widetilde{\Phi}_0
(\tilde\rho^1, \tilde\sigma^1,
\tilde v^1).
\end{equation}
 Finally one can show that
\be{mesutrans}
d\tilde\rho^{1'}d \tilde\sigma^{1'}d \tilde v^{1'} =d\tilde\rho^1d\tilde\sigma^1d
\tilde v^1.
\end{equation}
Combining \eq{mestrans}, \eq{mesutrans} and \eq{invarphi} we see that the
the integrand in $I_1$ is invariant under $M_\infty$. Since the transformations in
\eq{tstrans} are S-duality transformations they leave the
leave the the remaining  charge bilinears in \eq{sdualcomb} invariant.
Therefore  the integrand in the partition function \eq{stupart} remains
invariant under the S-duality action of $M_\infty$.
Using the same argument one can show that the integrand in
\eq{stupart} is invariant under the S-duality action of $M_1$ under which
the charge bilinears transform as
\be{tstrans2}
\left(
\begin{array}{c}
M_{\sigma^{1}}'\\
M_{\rho^{1}}'\\
M_{v^{1}}'
\end{array}
\right)
=
\left(\begin{array}{ccc}
1 & 0 & 0 \\
4 & 1 & -4 \\
-2 & 0 & 1
\end{array}\right)
\left(
\begin{array}{c}
M_{\sigma^{1}}\\
M_{\rho^{1}}\\
M_{v^{1}}
\end{array}
\right),
\end{equation}
and the $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ variables transform as the equations in the third line
of \eq{sp2zt}.
We can thus conclude that the integrand is invariant under $\Gamma(2)_S$.
In fact since the form $\widetilde{\Phi}_0$ is invariant under
$\Gamma_0(2)$ (see property 1 of $\widetilde{\Phi}_0$ in appendix C.),
the partition function is invariant under this larger symmetry.

The action of the two generators $M_\infty$ and
$M_1$ of $\Gamma(2)_T$ on the charge bilinears $M_{\rho^2}, M_{\sigma^2},
M_{v^2}$ is given by the the same equations \eq{tstrans} and \eq{tstrans2}
with the $1$ replaced by $2$.
To show that the integrand is
invariant under $\Gamma(2)_T$ is is sufficient to use
the fact that the integrand is invariant under the
triality symmetry ${\cal{T}}_1$.
From the action of ${\cal T}_1$ on the charge bilinears
given in \eq{dtrail1}, \eq{trail1} we have the following
relation between the action of $\Gamma(2)_T$ and $\Gamma(2)_S$
\be{relts}
\Gamma(2)_T = {\cal T}_1 \Gamma(2)_S {\cal T}_1.
\end{equation}
Since the integrand is invariant under $\Gamma(2)_S$ and the triality symmetry
${\cal T}_1$ we see that the $\Gamma(2)_S$ is a symmetry of the integrand.
Finally the action of the two generators $M_\infty$ and
$M_1$ of $\Gamma(2)_U$ on the charge bilinears $M_{\rho^2}, M_{\sigma^2},
M_{v^2}$ is given by the the same equations \eq{tstrans} and \eq{tstrans2}
with the $1$ replaced by $3$. Using \eq{dtrail3} and \eq{trail3} we have
\be{relts1}
\Gamma(2)_U = {\cal T}_3  \Gamma(2)_S {\cal T}_3.
\end{equation}
Since the integrand is invariant under $\Gamma(2)_S$ and the triality symmetry
${\cal T}_3$ we see that the $\Gamma(2)_S$ is a symmetry of the integrand.
Thus we conclude that ignoring considerations of the contour
the degeneracy given by  \eq{stupart} is invariant under
$\Gamma(2)_S\times \Gamma(2)_T\times\Gamma(2)_U$.

\vspace{.5cm}
\noindent
{\emph{Integrality of the $d(M_\rho, M_{\sigma}, M_v)$}}
\vspace{.5cm}

From property \eq{fexp} of $\widetilde{\Phi}_0$ that Fourier coefficients $d(M_\rho,
M_\sigma, M_v)$ are all integers. From our definition of the charge bilinears
in \eq{sdualcomb}  and from the moding of the Fourier expansions in
\eq{fexp} we see that
\be{chgquant}
M_{\rho^\alpha} = {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, \qquad
M_{\sigma^\alpha} = {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, \qquad
M_{v^\alpha} = {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}.
\end{equation}
Thus the class of dyons the partition function given in \eq{stupart} is
applicable has the above quantization conditions.

\subsection{Statistical entropy of the STU model}

In this section we obtain the asymptotic degeneracies for dyons
for large charges $M_{\rho^\alpha}, M_{\sigma^\alpha}, M_{v^\alpha}>>0$
with $Q^2P^2-(Q\cdot P)^2 >>0$.
We begin with the formula \eq{stupart} for the degeneracy of dyons in the
STU model.  The degeneracy is obtained as a product of three integrals
given by
\bea{statint}
 I_{\alpha} (M_{\rho^\alpha}, M_{\sigma^\alpha}, M_{v^\alpha}) &=&
\frac{K}{4}\exp{(i\pi  M_{v^\alpha})}\int _{{\cal{C}^\alpha}}
d\tilde\rho^\alpha d\tilde \sigma^\alpha d\tilde v^\alpha \frac{1}
{\widetilde{\Phi}_0(\tilde\rho^\alpha,\tilde\sigma^\alpha,\tilde v^\alpha)} \\ \nonumber
& \times& \exp\left[-2\pi i \left( \frac{M_{\rho^\alpha}}{2} \tilde\rho^\alpha
+\frac{M_{\sigma^\alpha} }{2}\tilde\sigma^\alpha  + M_{v^\alpha} \tilde v
\right) \right].
\end{eqnarray}
This formula is identical in form to equation (3.29) of \cite{David:2006yn}
with the substitution
$Q_e^2 \rightarrow M_{\sigma^\alpha},\; Q_m^2 \rightarrow
M_{\rho^\alpha}, \;Q_e\cdot Q_m \rightarrow M_{v^\alpha}$.
Following \cite{David:2006yn} one can show that the dominant contribution
to this integral comes form the residue at the pole at
\be{dompole}
\tilde\rho^{\alpha} \tilde\sigma^\alpha - (\tilde v^{\alpha})^2 +
\tilde v^\alpha = 0.
\end{equation}
The behavior of $\widetilde{\Phi}_0$ near this zero is given by
\eq{facphi0}, is identical to the corresponding relation (4.17) in
\cite{David:2006yn} with $k\rightarrow 0$ and
$f^{(k)}(\rho)  \rightarrow f^{(0)}(\rho)$
given in \eq{deff2}. Thus following an analysis identical
to that in \cite{David:2006yn} we can conclude that for large charges the
contribution to the statistical entropy form the integral $I_{\alpha}$
defined as the log of the contribution of the degeneracy
 $I_{\alpha} (M_{\rho^\alpha}, M_{\sigma^\alpha}, M_{v^\alpha})$ is obtained
by minimizing the statistical entropy function
\bea{entstu}
-\tilde\Gamma_B^\alpha (\vec\gamma^\alpha) &=&
\frac{\pi}{2\gamma_2^\alpha}
\left( M_{\sigma^\alpha} + 2\gamma_1^\alpha M_{v} + \tau^\gamma\bar\tau^\gamma
 M_{\rho^\alpha} \right)\\ \nonumber
& & -\ln\left[ \vartheta_2(\gamma^\alpha)^4 \vartheta_2(-\bar\gamma^\alpha)^4
(2\gamma_2^\alpha)^2 \right] + \hbox{constant} + {\cal O}(1/Q^2),
\end{eqnarray}
with respect to the real and imaginary parts of $\gamma^\alpha$.
To order ${\cal O}(1/Q^2)$ it is sufficient to obtain the value
of $\tau^\alpha $ at the minimum by minimizing the ${\cal O}(Q^2)$ term in
\eq{entstu}. This is given by
\bea{mintau}
\gamma_1^\alpha|_*
 &=& -\frac{M_{v^{\alpha}}}{M_{\rho^\alpha}}, \\ \nonumber
\gamma_2^\alpha|_{*}  &=& \frac{
\sqrt{ M_{\rho^\alpha} M_{\sigma^\alpha} - M_{v^\alpha}^2 }}
{M_{\rho^\alpha}}, \\ \nonumber
&=& \frac{ \sqrt{Q^2P^2 - (Q\cdot P)^2} }{3 M_{\rho^\alpha}}.
\end{eqnarray}
We can
substituting these values in the statistical entropy function \eq{entstu}
to obtain the value of the statistical entropy from
$I_\alpha$ to ${\cal O}(1/Q^2)$, this is given by
\bea{valminsent}
-\tilde\Gamma_B(\vec\gamma^\alpha)|_*
&=& \pi
\sqrt{  M_{\rho^\alpha} M_{\sigma^\alpha} - M_{v^\alpha}^2 } \\ \nonumber
 & &
-\left. \ln\left[ \vartheta_2(\gamma^\alpha)^4
\vartheta_2(-\bar\gamma^\alpha )^4
(2\gamma_2^\alpha)^2 \right]\right|_* + \hbox{constant} + {\cal O}(1/Q^2),
\\ \nonumber
&=& \frac{\pi}{3}
\sqrt{Q^2P^2 - (Q\cdot P)^2 } \\ \nonumber
 & &
-\left. \ln\left[ \vartheta_2(\gamma^\alpha)^4
\vartheta_2(-\bar\gamma^\alpha )^4
(2\gamma_2^\alpha)^2 \right]\right|_* + \hbox{constant} + {\cal O}(1/Q^2).
\end{eqnarray}
The total statistical entropy is then given by
\be{stattot}
-\tilde\Gamma_B(\vec\gamma^1, \vec\gamma^2, \vec\gamma^3)
= -\sum_{\alpha =1}^3 \tilde\Gamma_B(\vec\gamma^\alpha).
\end{equation}
From comparing \eq{soltau}, \eq{soly+}, \eq{susoly-1}
to the minimum values in \eq{mintau}  and
using the definition of the charge bilinears in \eq{tdualinv} and
\eq{sdualcomb} it is seen that we obtain the following equations
\bea{mincorresp}
\gamma^1|_{*} = \tau|_{*}, \qquad
\gamma^2|_{*} = y^+|_{*}, \qquad
 \gamma^3|_{*} = -\frac{1}{y^-}|_{*}.
\end{eqnarray}
From \eq{stattot} and \eq{valminsent} we see that to ${\cal O}(1/Q^2)$,
the total statistical entropy is given by
\bea{totmin}
-\tilde\Gamma_B(\vec\gamma^1, \vec\gamma^2, \vec\gamma^3)|_*
&=&
\pi \sqrt{Q^2P^2 - (Q\cdot P)^2 } \\ \nonumber
 & &
-\sum_{\alpha =1}^3\left. \ln\left[ \vartheta_2(\gamma^\alpha)^4
\vartheta_2(-\bar\gamma^\alpha )^4
(2\gamma_2^\alpha)^2 \right]\right|_* + \hbox{constant} + {\cal O}(1/Q^2).
\end{eqnarray}
Using \eq{mincorresp} and \eq{totmin} we see that the statistical
entropy coincides with the entropy of the black hole to the next leading
order.

\section{Approximate partition function for the FHSV model}

The subleading corrections to the entropy from the the coefficient of the
Gauss-Bonnet in the FHSV model depends on the attractor values of the
T-moduli through the Bocherds-Enriques form \eq{fshvf1}.
The complete dyon partition function should capture this dependence on
the T-moduli. Here we will focus on the scaling limit \eq{scalimit}
in which the electric charges are much larger than the magnetic charges
and write down an approximate dyon partition function which captures
the degeneracy in this limit.
\be{prtfn1}
d(Q, P) = \frac{K'}{2} \exp{i\pi(Q\cdot P)}\int_C
d\tilde\rho d\tilde\sigma d\tilde v
\frac{1}{\widetilde{\Phi}_4(\tilde\rho, \tilde\sigma, \tilde v)} \exp\left[
-i\pi \left( \tilde\sigma Q^2 + \tilde\rho P^2 + 2 \tilde v Q\cdot P\right)
\right],
\end{equation}
where $Q^2\equiv Q\cdot Q$, $P^2 \equiv P\cdot P$,
$\widetilde{\Phi}_4$ is a function to be
specified, and $C$ is a three real dimensional subspace of the three complex
dimensional space labeled by $(\tilde\rho, \tilde\sigma, \tilde v)$ given by
\bea{domains}
{\rm Im}\, \tilde \rho = M_1 \quad  {\rm Im}\,  \tilde\sigma = M_2,\quad
\quad {\rm Im}\, \tilde v = M_3,
\cr
0 \leq {\rm Re}\, \tilde\rho \leq 1,
\quad  0 \leq{\rm Re}\, \tilde\sigma \leq 2, \quad
0\leq{\rm Re}\, \tilde v \leq 1.
\end{eqnarray}
$M_1, M_2, M_3$ being fixed large positive numbers. The normalization
constant in \eq{prtfn1} is given by $K' = 2^{-6}$.
The $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ modular form  $\Phi_4$ which occurs
in \eq{prtfn1} is  by
\be{reltphi4}
\widetilde{\Phi}_4 (\tilde\rho, \tilde\sigma, \tilde v) = \left(
\widetilde{\Phi}_6(\tilde\rho, \tilde\sigma, \tilde v)
\widetilde{\Phi}_2(\tilde\rho, \tilde\sigma, \tilde v) \right)^{\frac{1}{2}},
\end{equation}
where $\widetilde{\Phi}_6$ is the modular form of
weight $6$ which captures the degeneracy
of dyons for the $N=2$ CHL orbifold
discussed in \cite{Jatkar:2005bh,David:2006ji}
Similarly  $\widetilde{\Phi}_2$ is the modular form
of $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ of weight $2$ which capture degeneracy of
dyons for the ${\hskip-3pt\hbox{ Z\kern-1.6mm Z}}_2$ orbifold of type II theory
constructed in \cite{David:2006ru}.
From the definition of $\widetilde{\Phi}_4$ in \eq{reltphi4} it is easily seen that
it is a modular form of weight $4$ under the subgroup
$\tilde G$ of $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ defined in \cite{Jatkar:2005bh}.
Thus we have
\be{modtranp}
\widetilde{\Phi}_4\left( A\Omega + B)(C\Omega + D)^{-1} \right) = \hbox{det}\,
(C\Omega +D)^4 \widetilde{\Phi}_4(\Omega), \qquad
\left(
\begin{array}{cc}
A & B \\
C & D
\end{array}
\right) \in \tilde G.
\end{equation}
We now list some properties of $\widetilde{\Phi}_4$ which are discussed in the
the appendix in detail.

\vspace{.5cm}
\noindent
{\emph{Properties of ${\widetilde{\Phi}}_4$}}
\vspace{.5cm}

\begin{enumerate}
\item
$\widetilde{\Phi}_4(\tilde\rho, \tilde\sigma, \tilde v)$ is an analytic function
in $\tilde\rho, \tilde\sigma, \tilde v$ with second order zeros at
\bea{loczero}
\left( n_2(\tilde \sigma\tilde\rho -\tilde v^2) +
 b\tilde v + n_1 \tilde\sigma -
\tilde\rho m_1 + m_2\right) =0, \\ \nonumber
{\rm for}\; m_1\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, m_2, n_2\in{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, b\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1,
m_1n_1 +m_2n_2 + \frac{b^2}{4} = \frac{1}{4}.
\end{eqnarray}
It has simple poles at
\bea{locpol}
& &\left( n_2(\tilde \sigma\tilde\rho -\tilde v^2) +
 b\tilde v + n_1 \tilde\sigma -
\tilde\rho m_1 + m_2\right) =0, \\ \nonumber
& &{\rm for}\, m_1 \in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1, m_2, n_2 \in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, n_1\in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}, b\in 2{\hskip-3pt\hbox{ Z\kern-1.6mm Z}}+1,
m_1n_1 +m_2n_2 + \frac{b^2}{4} = \frac{1}{4}.
\end{eqnarray}
\item
$\widetilde{\Phi}_4$ is invariant under the following $Sp(2,{\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformations
\bea{sp2ztf}
\widetilde{\Phi}_4(\tilde\rho', \tilde\sigma', \tilde v') =
\widetilde{\Phi}_4(\tilde\rho,\tilde\sigma,\tilde v),
\\ \label{ttt1}
\hbox{for}\; \tilde\rho'= \tilde\rho + 4\tilde\sigma + 4 \tilde v -2, \quad
\tilde\sigma' = \tilde\sigma, \quad \tilde v' = 2\tilde\sigma + \tilde v, \\
 \label{ttt2}
\hbox{and} \; \tilde\rho'= \tilde\rho, \quad
\tilde\sigma' = 4\tilde\rho +\tilde\sigma+ 4\tilde v -2 , \quad
\tilde v' = 2\tilde\sigma + \tilde v.
\end{eqnarray}
This property is due to the fact that $\widetilde{\Phi}_4$ is invariant under the
subgroup of $\tilde H$ defined in \eq{subgroup}. The transformation
\eq{ttt1} corresponds to the choice $a= -1, b=2, c=0, d=-1$ in \eq{sghtrans}
and  \eq{ttt2} corresponds to the choice $a =1, b =0, c=-2, d=1$ in
\eq{sghtrans}.  The above $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformations essentially
correspond to generators of $\Gamma(2)$, but $\widetilde{\Phi}_4$ is invariant
under the subgroup \eq{subgroup}
which is $\Gamma_0(2)$, this contains the group
$\Gamma(2)$.
\item
 The expansion of $1/\widetilde{\Phi}_4$ in terms of Fourier coefficients  given by
\be{fexpf}
\frac{K'}{\widetilde{\Phi}_4(\tilde\rho,\tilde\sigma, \tilde v)} =
\sum_{\stackrel{m, n, p}{m\geq -1, n \geq -1/2}}
e^{2\pi i( m \tilde\rho + n\tilde\sigma + p\tilde v)}
g(m , n, p),
\end{equation}
with $g(m, n, p)$ being integers.
$m , p\in {\hskip-3pt\hbox{ Z\kern-1.6mm Z}}$ while $n$ runs over integer multiples
of $1/2$
\item
To determine the asymptotic properties of the partition
function given in \eq{prtfn1} for large
charges we need the behavior of $\widetilde{\Phi}_4$ near
$\tilde\sigma\tilde\rho -\tilde v^2 + \tilde v =0$.
Near this surface $\widetilde{\Phi}_4$ factorizes as
\be{facprop3}
\widetilde{\Phi}_4(\tilde\rho, \tilde\sigma, \tilde v)
= 4\pi^2 ( 2 v-\rho - \sigma)^2 v^2 f(\rho) f(\sigma) + {\cal O}(v^4),
\end{equation}
where
\be{deffrho}
f^{(4)} (\rho) = \eta(2\rho)^{12},
\end{equation}
and the relationship between the variables $\rho, \sigma, v$
and $\tilde\rho, \tilde\sigma, \tilde v $ is given by
the $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformation \eq{chgvar} and \eq{chgvar2}.
\end{enumerate}

Just as in the case of $\widetilde{\Phi}_0$ in the previous section, following the
same logic and using the fact that $\widetilde{\Phi}_4$ is also invariant under
the $Sp(2, {\hskip-3pt\hbox{ Z\kern-1.6mm Z}})$ transformations given in \eq{ttt1}and \eq{ttt2}, we see that
the integrand in \eq{prtfn1} is invariant under $\Gamma(2)_S$ symmetry.
This is what is expected of a partition function which aims to capture the
dependence of the axion-dilaton moduli dependence of the subleading terms
in the entropy function. In fact from property since $\widetilde{\Phi}_4$ is invariant
under the subgroup \eq{subgroup}, the partition function \eq{prtfn1} is
invariant under the larger group $\Gamma_0(2)$.

We shall now compute the statistical entropy from the
approximate partition function for the FHSV model given in \eq{prtfn1}.
We will see that the statistical entropy from this partition function
captures the axion-dilaton moduli dependence of the entropy
\eq{fshvf1}. To obtain the statistical entropy we will follow the
method of \cite{David:2006yn}.
of the dyons in the FHSV model. The value of this function at its
extremum gives the statistical entropy, the logarithm of the degeneracy
of states corresponding to the given set of charges.
We see that the black hole entropy function evaluated in section 4
agrees precisely with the statistical entropy.
Following \cite{David:2006yn,Dijkgraaf:1996it,LopesCardoso:2006bg}
one can show that the dominant contribution to the above integral comes
form the pole at
\be{poleq}
\tilde\sigma\tilde\rho - \tilde v^2 + \tilde v =0.
\end{equation}
The behavior of $\widetilde{\Phi}_4$ near this zero is given by \eq{facprop3}, which
is identical to the corresponding relation $(4.17)$ with $k\rightarrow 4$ and
$f^{(k)}(\rho) \rightarrow f(\rho)$.
Thus following an analysis identical
to that in \cite{David:2006yn} we can conclude that for large
charges the statistical entropy
$S_{\rm{stat}} (Q, P)$ defined as the logarithm of the degeneracy $d(Q, P)$
is obtained by minimizing the statistical entropy function
\bea{statentrop}
-\tilde\Gamma_B(\vec\tau) &=& \frac{\pi}{2\tau_2}| Q + \tau P|^2
-\ln (f^{(4)} (\tau)) - \ln( f^{(4)}(-\bar\tau)) - 6 \ln (2\tau_2) +
{\hbox{constant}} \nonumber \\
& &  +
{\cal O}(1/Q^2, 1/P^2),
\end{eqnarray}
where
\be{deffnct}
f^{(4)}(\tau) = \eta(2\tau)^{12}.
\end{equation}
We can then obtain the statistical entropy by first minimizing the the function
$-\tilde\Gamma_B$ with respect to the real and
imaginary parts of $\tau$ and then
evaluating the value of the statistical entropy function at this
critical point.
This gives
\be{fshventm}
-\tilde\Gamma_B(\vec\tau)|_* = \pi \sqrt{Q^2P^2 - (Q\cdot P)^2}
- \ln\left[ f^{(4)}(\tau)f^{(4)}(-\bar\tau) (2\tau_2)^6 \right]|_*,
\end{equation}
where
\be{crtvtau}
\tau_1|_* = -\frac{Q\cdot P}{P^2}, \qquad
\tau_2|_* = \frac{\sqrt{Q^2P^2 - (Q\cdot P)^2}}{P^2}.
\end{equation}
Comparing \eq{fshvf1} and \eq{fshventm} using \eq{soltauf} and
\eq{crtvtau} we see that the approximate partition function given in
\eq{prtfn1} captures the contribution of the axion-dilaton moduli in
the subleading terms of the black hole entropy obtained from the
Gauss-Bonnet term. Therefore in the scaling limit given in \eq{scalimit}
where the T-moduli contribution to the entropy is subleading, the approximate
partition function given in \eq{prtfn1} agrees with
the black hole entropy \eq{scalentfn} ignoring ${\cal O}(Q^0, P^0)$
terms.

We note that the vector multiplet moduli space of the FHSV model
factorizes into the axion-dilaton dependence and the T-moduli dependence.
We also have seen that  the partition function in \eq{prtfn1} captures the
subleading contribution to the dyon entropy form the axion-dilaton dependence
of the Gauss-Bonnet term.
Thus we can conclude that a product of $\widetilde{\Phi}_4$ with a suitable
function should capture the degeneracy of dyons in the FHSV model.
It will be interesting to determine this function using clues from the
corrections to the entropy function given in \eq{fshvf1} and the
attractor values of the T-moduli given in \eq{solyiy-} and \eq{solyfy+}.

\acknowledgments

The author wishes to thank N. Banerjee, A. Dabholkar, R. Gopakumar,
 B. Pioline,  B. Sahoo for useful comments and
D. Jatkar,   A. Sen for extensive discussions and for a careful reading
of the manuscript.
He thanks A. Dabholkar and LPTHE,
Jussieu  for hospitality  during an extended visit  where part of this
work was done. Support from the XIth  plan of the DAE
(Project No: 11-R\&D-HRI-5.02-0304) is acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Archives for A Place at the Table
Talk of the Towns 6/14/13
Program Topic: A Place at the Table—Responding to Hunger in our Midst
a) You have chosen to screen a recent documentary, A Place at the Table, to highlight the issue of hunger and inequality… how and why did that come about?
b) What do we know about hunger and food insecurity in this part of Maine?
c) What factors contribute to persistent hunger and food insecurity?
d) What programs are in place to address these issues, and what are the gaps? (SNAP, WIC, School Lunch program, Commodity foods)
e) What are individual communities and community organizations (yours and others) doing to address these issues and the gaps?
f) Tell some stories about who is affected, their circumstances, their strengths…
g) What do you hope the screening of A Place at the Table will accomplish to raise awareness and stimulate individual and community response?
h) What do you find particularly compelling about the documentary? What would you tell a friend or neighbor who you wanted to attend?
i) Against this backdrop, Congress is debating the reauthorization of the Farm Bill, which includes funding for food assistance… what is being proposed? Is there anything happening with the state budget that addresses or makes worse food insecurity?
j) Attending to food insecurity in the short term is important… what are ways forward to address the broader inequality in our country, where some have food and some do not?
k) Details about tonight's screening and other contact info…
A) Adam Lacher, United Way of Eastern Maine
B) Sister Lucille MacDonald, Emmaus Center, Ellsworth
C) Lisa Weisker
D) Rick Traub, Tree of Life Food Pantry, Blue Hill
E) Zoe Scott, Executive Director, The Grand, Ellsworth

redpajama_set_name: RedPajamaCommonCrawl


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text:
Thursday Movie Picks: Just One Day
You know how it is. Some nights, you go somewhere, the night doesn't go quite as planned, so you try to leave. Here's the thing: leaving this place and getting home is a whole lot tougher than you thought it would be. You quite literally have to fight your way out. Thursdays make me feel that way. They almost never go the way I want them to and I wind up fighting just to make it to Friday. So yeah, for this week's Thursday Movie Picks hosted by Wanderer at Wandering Through the Shelves, I'm picking movies that echo this experience. The official topic is movies in which all the action takes place on just one day. Hopefully, your day goes a lot smoother than it does for these guys.
Cyrus is the leader of the biggest gang in New York. Instead of just flexing his muscle and overrunning the competition, he wants to unite all the gangs. He invites nine delegates from every gang in the city to call for a truce. Of course, he gets shot dead. And everyone thinks the Warriors did it. Our boys in the brown leather vests have to bop their way from the Bronx all the way home to Coney Island. It's one of my favorite movies of all-time, and I'm not even joking. I've literally watched it at least once almost every year since it's release and often more than once.
The Raid: Redemption
Rama is a rookie cop with a baby on the way. He's also part of a unit of 20 officers going into an apartment building to drag out a crime lord. There's just one little thing. The building is loaded with bad guys and every one of them is armed to the teeth...or can fight his ass off...or both. what ensues is one of the best action movies to ever make it to the screen. And possibly one of the bloodiest. And I love every second of it. (My full review)
Let's take the plot from my previous pick, except we're going to trim the number of officers going into this ever-so-dangerous apartment building from 20 down to 2. Next, let's make it a post-apocalyptic setting in which people are herded into these rather large dwellings in the few inhabitable places on Earth and make the cops legally empowered to serve as judge, jury, and executioner. What we get is another rousing action flick that was overwhelmingly ignored despite its dual status as a superhero flick. Of course, it was an R-rated one before Deadpool and, unfairly in my opinion, had to deal with the stigma of the dreddfull 1995 movie about the same character starring Sylvester Stallone. (My full review)
Labels: Action, Dredd, Just One Day, The Raid, The Raid Redemption, The Warriors, Thursday Movie Picks
Brittani Burnham March 8, 2018 at 8:52 AM
Nice theme! I liked Dredd, that was way better than I expected it to be. My husband loves Warriors, but I always found it annoying. I never saw either Raid movie.
Dell March 12, 2018 at 12:22 AM
"My husband loves Warriors, but I always found it annoying."
The first half of this sentence fills me with joy. The second half turns it into sadness. I'll let you make it up to me by watching The Raid movies. I think that's a win-win.
joel65913 March 8, 2018 at 9:40 AM
Love that you made it a sort of get to morning alive theme within the theme. :-)
WARRIORS!!!! I haven't thought of that movie in years but it was SO popular when cable was just starting. It was everywhere for a while but now is hardly run at all. I don't have the slavish devotion to it that you do but I enjoyed it.
I haven't seen your other two. I avoided Dredd specifically because I slogged my way though the first Judge Dredd which despite the presence of my beloved Diane Lane was a piece of garbage.
I didn't go as dark with my three.
High Noon (1952)-Just married to his Quaker bride Amy (Grace Kelly) that morning Sherriff Will Kane (Gary Cooper who won the Oscar for this performance) has quit his post and plans a quiet farm life away from the town. Word comes that a killer he had imprisoned 5 years ago has been paroled and is arriving with his gang on the noon train intent on a showdown. Ignoring advice from all to flee, including successful businesswoman Helen Ramirez (an outstanding Katy Jurado) his former mistress, Will tries to gather help to stand up to the outlaws but finds all the townsfolk turning their backs on him and he must face them alone at High Noon. A thinly veiled allegory for the blacklist which affected many involved with the film.
Cleo from 5 to 7 (1962)-A young singer, Cleo (Corinne Marchand) must wait two hours for the results of a biopsy and drifts around Paris seeing friends and considering the exigencies of life. Part of the Nouvelle Vague movement this isn't for everyone but it has its charms.
Ferris Bueller's Day Off (1986)-"Life moves pretty fast. If you don't stop and look around once in a while, you could miss it." That's the credo of uber popular high school student Ferris Bueller (Matthew Broderick) who decides that today is too perfect to be wasted in school. After convincing his parents he's sick he gathers up his best friend Cameron (Alan Ruck) and girlfriend Sloane (Mia Sara) for a day of adventure in the city while his suspicious principal Mr. Rooney (Jeffrey Jones) tries to catch him red-handed. Broderick is madly charismatic in the lead and surrounded by a fantastic cast including Edie McClurg who is hysterical as Rooney's secretary.
All any of us can do is try to get to morning alive, Joel.
Dredd won't change your life or anything like that, but it is a very solid action flick that doesn't really feel like a superhero movie.
High Noon and Ferris Bueller's Day Off are both fantastic. I still need to see Cleo from 5 to 7.
SJHoneywell March 8, 2018 at 11:48 AM
I have no complaints about any of your three choices--love them all. I'll also second two of Joel's picks. High Noon is a classic, and Cleo from 5 to 7 is a wonderful film written and directed by Agnes Varda, the grandmother of French Nouvelle Vague (and nominee at the last Oscars for Faces Places). For what it's worth, I use Cyrus's speech from The Warriors to discuss elements of nonverbal communication in my comms classes. Pitch, cadence, pacing, gesture, tone, etc. It's a great and powerful example.
I've never been a huge Ferris Bueller fan.
As it happens, the review I posted yesterday is a film that takes place in about 12 hours. Also, if memory serves, Rebel Without a Cause takes place within 24 hours as well, as does Training Day. In fact, if you replace The Warriors with Training Day, you'd have a cop trio.
Cleo has been on my watchlist for a while. I need to get it crossed off one of these days.
That is so cool about Cyrus's speech. Props to you for using it.
I'll be watching Rebel later this year as part of the Blind Spot Challenge.
I thought about picking Training Day, but wasn't thinking along the lines of a cop trio, but I guess I should've.
Birgit March 8, 2018 at 12:21 PM
I haven't seen any of these and only heard of Dredd but, you are right, I thought it would be the same as the awful one made with Stallone
Thankfully, it's not.
Sonia Cerca March 8, 2018 at 4:26 PM
I haven't seen any of these but I like the theme!
thevoid99 March 8, 2018 at 4:49 PM
Dredd is the one film in that list that I haven't seen but it's been in my watchlist for years as I'm waiting for it to be on TV or available on DVD in my local library. The other 2 picks are perfect, especially The Raid: Redemption which is a top-tier action film.
I think you'll enjoy Dredd. Hope you get to watch it soon.
s. March 8, 2018 at 5:35 PM
Dredd is amazing. There is actually this news from this week that Alex Garland was actually the one who directed it.
Dell March 12, 2018 at 1:01 AM
Daniel March 9, 2018 at 9:28 AM
I've heard such good things about The Warriors; I really need to see it one of these days. I've heard good things about The Raid, too, but very mixed things about Dredd.
Yes, yes, and makes sense.
Steph March 9, 2018 at 7:38 PM
Great theme within a theme! But damn, I haven't seen any of these movies.
Sounds like you've got some movies to watch.
Wanderer March 15, 2018 at 1:47 PM
I like Dredd too. It was surprisingly a good action movie.
Dell March 19, 2018 at 7:21 PM
Yes, it was.
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Even though CHA is still a few days away, I'm already compiling my "loves" list!
Here is one of my favorite new tag sets--can't wait to play!
And these wonderful loopy thickers!
I'm also loving the cute apple tags from Bits from the Daydream collection.
And this has been the talk of the town, but I adore the Amy Tangerine/American Crafts line! The motifs are so fun and whimsical!
And I can't wait to try out this cool ampersand mask!
Well, there yo have it--some highlights and things I'm coveting. Think my pocketbook is about to get a bit lighter!
What are YOUR favorite items from what you've seen so far?
Oooh, lots of eye candy. Those buttons are a blast!
I want those AC tags too!
love your highlights. I see a lot of things I need to have too. i don't know how I missed those huey masks. very cool.
I started a pin board and all these are on it- the colors this season are so GREAT!! Super cheerful and lots of cool texture too.
The American Crafts has caught my eye too! Can't wait to get my hands on it!!!
Based on this post I think we could be best friends! You covered all the bases for me.
I agree with all of yours! Also looking forward to Jillibean Soup, October Afternoon and Echo Park. I can't wait for it all to hit the stores!
Love the new JBS, Studio Calico,Crate and American Crafts. Going to have to check all of Elle's new releases out too. Have fun at CHA!
Fab stuff! Thanks for sharing:) My list is growing more and more with each passing hour:) Have a great weekend!
Girl, you and me both on that AC Garden Cafe!