Attractors with Large Complex Structure
for One-Parameter Families of
Calabi-Yau Manifolds
Philip Candelas1, Pyry Kuusela2
and
Joseph McGovern3
00footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Mathematical
Institute
University of Oxford
Andrew Wiles Building
Radcliffe Observatory Quarter
Oxford, OX2 6GG, UK
Abstract

The attractor equations for an arbitrary one-parameter family of Calabi-Yau
manifolds are studied in the large complex structure region. These equations
are solved iteratively, generating what we term an $N$-expansion, which is a
power series in the Gromov-Witten invariants of the manifold. The coefficients
of this series are associated with integer partitions. In important cases we
are able to find closed-form expressions for the general term of this
expansion. To our knowledge, these are the first generic solutions to
attractor equations that incorporate instanton contributions. In particular,
we find a simple closed-form formula for the entropy associated to rank two
attractor points, including those recently discovered. The applications of our
solutions are briefly discussed. Most importantly, we are able to give an
expression for the Wald entropy of black holes that includes all genus 0
instanton corrections.

###### Contents

  1. 1 Introduction
    1. 1.1 Preamble
    2. 1.2 Outline of the paper
    3. 1.3 Notation
  2. 2 Solving the Attractor Equations: a General Overview
    1. 2.1 Particular brane systems
    2. 2.2 General comments concerning the alignment equations
    3. 2.3 Setting up the iteration
  3. 3 Partitions
    1. 3.1 Basic definitions and notation
    2. 3.2 Operations on partitions
    3. 3.3 Series expansions
  4. 4 The D4-D2-D0 System
    1. 4.1 The orthogonality equation
    2. 4.2 The alignment equations
  5. 5 The D6-D2-D0 System
    1. 5.1 The orthogonality equation
    2. 5.2 The alignment equations
  6. 6 Solutions
    1. 6.1 The D6-D2-D0 alignment equations
    2. 6.2 The D4-D2-D0 orthogonality equation
    3. 6.3 Entropy at rank two attractor points
    4. 6.4 An example: a rank two attractor point on AESZ34
  7. 7 Applications
    1. 7.1 Classification of attractor points
    2. 7.2 Comparison to microscopic entropy
  8. Acknowledgements
  9. A Symplectic Transformations
  10. B Monodromy
    1. B.1 D4-D2-D0 system
    2. B.2 D6-D2-D0 system
  11. C Microscopic Charge Vector
  12. D The $\sigma$-Expansion for the D6 Orthogonality Equation
  13. E The $\sigma$-Expansion for the D6 Alignment Equations
  14. F Convergence of Solutions
    1. F.1 Bound for $a_{\mathfrak{p}}$

##  1 Introduction

###  1.1 Preamble

Ever since the discovery of the attractor mechanism [1], the study of
supersymmetric black holes in $\mathcal{N}=2$ theories in four dimensions has
been intimately linked to the study of attractor points on Calabi-Yau
manifolds. This has led to many interesting conjectures and results, such as
the intricate connections to number theory, discussed by Moore in [2, 3] and
the BPS state existence conjecture [4], which links the existence of certain
type of attractor points, in the $d=4,\,\mathcal{N}=2$ supergravity theories,
to the existence of BPS states in the quantum theory of gravity.

While the attractor mechanism applies to any four-dimensional $\mathcal{N}=2$
supergravity theory coupled to vector multiplets, an especially interesting
class of examples is provided by compactifying 10-dimensional type IIA and IIB
supergravities on Calabi-Yau threefolds (see [5, 6] and references cited
therein). In this case, special geometry is used to describe the complex
structure or complexified Kähler moduli space. The scalars in $\mathcal{N}=2$
vector multiplets can be interpreted as coordinates on the moduli space, and
the attractor mechanism can be viewed as specifying, on the black hole
horizon, a particular Calabi-Yau manifold in the moduli space as a function of
the charges.

In this paper, we study attractor equations in $d=4,\,\mathcal{N}=2$
supergravity theories that have been obtained by compactifying a type IIA or
IIB supergravity on a one-parameter family of Calabi-Yau manifolds. We
concentrate on the region of the moduli space that is near the large complex
structure point. In this region, it is possible to solve the instanton-
corrected attractor equations using perturbative methods.

What is even more surprising than the existence of this kind of solution is
the fact that for some important special cases, we have been able to find a
closed-form for the instanton corrections. With the help of these solutions,
we are able to give interesting formulae for locations of attractor points,
among other physical quantities of interest. For example, we are able to give
a formula for the Wald entropy of a black hole that, in the large charge
limit, is fully quantum- and instanton-corrected.

Our aim in this paper is to develop what we term $N$-expansions, by which we
mean expansions in terms of the Gromov-Witten invariants $N^{\mbox{\tiny
GW}}_{k}$. These include all perturbative and non-perturbative terms. The
prototypical quantity that exhibits such an $N$-expansion is the prepotential,
which is a key ingredient in formulating the attractor equations. Recall that
the prepotential is introduced by first choosing a homology basis
$A^{a},B_{b}\in H_{3}(X,\mathbbl{Z})$ and a dual cohomology basis
$\alpha_{a},\beta^{b}\in H^{3}(X,\mathbbl{Z})$, for a Calabi-Yau manifold $X$.
The holomorphic three-form $\Omega$ can be expressed in terms of the
cohomology basis

$\displaystyle\Omega\;=\;z^{a}\alpha_{a}-\mathcal{F}_{b}\beta^{b}~{},\qquad
a,b\;=\;0,\dots,h^{2,1}(X)~{},$ (1.1)

where [7] the $\mathcal{F}_{b}$ are the derivatives
$\frac{\partial\mathcal{F}}{\partial z^{b}}$ of a prepotential $\mathcal{F}$.
In the body of this paper, we will concentrate, for simplicity, on Calabi-Yau
manifolds $X$ that lie in one-parameter families.

It is believed that near the large complex structure point there is a choice
of a symplectic basis such that, in the one-parameter case, the prepotential
$\mathcal{F}(z^{0},z^{1})$ takes the form

$\displaystyle\begin{split}\mathcal{F}\,=-\frac{1}{3!}Y_{abc}\frac{z^{a}z^{b}z^{c}}{z^{0}}-(z^{0})^{2}\mathcal{I}\left(\frac{z^{1}}{z^{0}}\right)\,=-(z^{0})^{2}\left(\frac{1}{6}Y_{111}t^{3}+\frac{1}{2}Y_{110}t^{2}+\frac{1}{2}Y_{100}t+\frac{1}{6}Y_{000}+\mathcal{I}\left(t\right)\\!\right)\\!,\end{split}$
(1.2)

where in the second relation $t=x+\text{i}y=z^{1}/z^{0}$.

The terms proportional to $Y_{100},\,Y_{110},$ and $Y_{000}$ come from
perturbative worldsheet corrections in $\alpha^{\prime}$ to the string sigma
model. The term $\mathcal{I}$ incorporates the nonperturbative genus-0
instanton contributions. Apart from the imaginary part of $Y_{000}$, all of
these perturbative corrections can be added or removed by performing a Peccei-
Quinn transformation and in this manner their contribution to attractor point
locations was analysed in [8]. The $Y_{100}$ terms, in particular, were
included in [9].

The Yukawa coupling $Y_{111}$ is related to the topological quantity

$Y_{111}\;=\;\int_{\widetilde{X}}e_{1}\wedge e_{1}\wedge e_{1}~{},$ (1.3)

where $e_{1}$ is a basis vector of the one-dimensional space
$H^{2}(\widetilde{X},\mathbbl{Z})$, and $\widetilde{X}$ is the mirror manifold
of $X$. It is believed that a basis $(\alpha_{a},\beta^{b})$ in (1.1) can be
chosen such that [7, 10, 11, 12]

$\displaystyle\begin{split}Y_{110}\;\in\;\left\\{0,\frac{1}{2}\right\\}~{},\ \
\ \ Y_{100}\;=\;-\frac{1}{12}\int_{\widetilde{X}}c_{2}\wedge e_{1}~{},\ \ \ \
Y_{000}\;=\;-3\frac{\zeta(3)}{(2\pi i)^{3}}\chi(\widetilde{X})~{}.\end{split}$

In the one-parameter case the component $Y_{110}$ can be taken to be zero if
the integer $Y_{111}$ is even, and $1/2$ if $Y_{111}$ is odd. The quantity
$\mathcal{I}(t)$ corresponds to the exponentially small terms, and has the
form

$\displaystyle\mathcal{I}(t)\;=\;\frac{1}{(2\pi\text{i})^{3}}\sum_{k=1}^{\infty}n_{k}\text{Li}_{3}(\text{e}^{2\pi\text{i}kt})\;=\;\frac{1}{(2\pi\text{i})^{3}}\sum_{k=1}^{\infty}\frac{N_{k}}{k^{3}}\text{e}^{2\pi\text{i}kt}~{},$
(1.4)

where the integral coefficients $n_{k}$ are the instanton numbers or
Gopakumar-Vafa invariants [13, 14] and in the second expression we have
written the instanton sum in terms of what we shall call the _scaled Gromov-
Witten invariants_ $N_{k}$, which are given by

$\displaystyle N_{k}\;=\;\sum_{d|k}d^{3}n_{d}~{},\qquad
N_{k}\;=\;k^{3}N_{k}^{GW}~{},$ (1.5)

with $N_{k}^{GW}$ the usual Gromov-Witten invariants. The form of the
prepotential has long been known. However, we have recalled this to emphasise
that $\mathcal{F}$ has a _universal_ form. Given the topological quantities
$Y_{ijk}$ and the instanton numbers $n_{k}$ corresponding to a given manifold
$X$, one can write down the prepotential and this $N$-expansion converges in a
neighbourhood of the large complex structure point. What we seek here are
similar universal expansions for quantities of interest such as the coordinate
$t$ and the central charge corresponding to an attractor point. In the general
case, the expressions for these quantities are somewhat detailed. However, in
simple cases, they can be concisely stated. For example, for the case of a
D6-D0 brane system with attractor point on the imaginary axis and nonvanishing
central charge, which we specify completely in §6, we have

$\displaystyle\begin{split}t=\text{i}y,\qquad\text{ where
}\quad\frac{y}{y_{0}}=1-\sum_{m=1}^{\infty}\;\sum_{\mathfrak{p}\in\text{pt}(m)}a_{\mathfrak{p}}N_{\mathfrak{p}}\left(\frac{m}{2\pi
y_{0}Y_{111}}\right)^{\bm{\ell}(\mathfrak{p})}\bm{k}_{\bm{\ell}(\mathfrak{p})-1}(2\pi
my_{0}).\end{split}$ (1.6)

In this expression $y_{0}$ is the perturbative solution for $y$, obtained by
solving the perturbative form attractor equations, where the non-perturbative
terms $\mathcal{I}$ are set to zero. The quantity $m$ is an integer while
$\mathfrak{p}$ runs over the partitions of $m$. If
$\mathfrak{p}=\\{j^{\mu_{j}}\\}$, with $\sum_{j}\mu_{j}j=m$, is such a
partition, then $\bm{\ell}(\mathfrak{p})=\sum_{j}\mu_{j}$ is the length of the
partition. Surprisingly, the quantity $\bm{k}_{n}$ that arises here is the
modified spherical Bessel function
$\bm{k}_{n}(z)=\smash{\sqrt{\frac{\scriptstyle 2}{\scriptstyle\pi
z}}K_{n+\frac{1}{2}}(z)}$. The coefficient $a_{\mathfrak{p}}$ is a
combinatorial factor related to the partition, and the quantity
$N_{\mathfrak{p}}$ is a combination of scaled Gromov-Witten invariants

$\displaystyle
a_{\mathfrak{p}}\;=\;\prod_{j}\frac{1}{\mu_{j}!\,j^{2\mu_{j}}}~{},\qquad
N_{\mathfrak{p}}\;=\;\prod_{j}N_{j}^{\mu_{j}}~{}.$ (1.7)

Our results apply, in particular, to one of the rank two attractor points of
[15]. In that case, $y$ is known explicitly to high precision, and this
formula has been checked for integers $m$ up to 75 and so for over 80 million
partitions, giving an accuracy of over 50 decimal places. The sum also
responds well to the techniques of accelerated convergence and using an
iterated Shanks transformation of order 11, the agreement can be extended to
80 decimal places. The difficulty in checking this to a higher degree of
accuracy lies in the need to store large numbers of partitions.

A similarly concise expression can be given for $|\mathcal{Z}|^{2}$, where
$\mathcal{Z}$ is the central charge associated to a certain class of rank two
attractor points,

$\displaystyle|\mathcal{Z}(Q_{1})|^{2}\;=\;\frac{\kappa^{2}y}{2}|\Upsilon{-}\Lambda|~{},$
(1.8)

where $y$ is as in the expression given above and the rank two attractor point
corresponds to the two charge vectors

$\displaystyle Q_{1}\;=\;\kappa\begin{pmatrix}0\\\ \Upsilon\\\ 1\\\
0\end{pmatrix},\qquad Q_{2}\;=\;\lambda\begin{pmatrix}\Lambda\\\ 0\\\ 0\\\
1\end{pmatrix}.$

In (1.8), we have given an expression for $|\mathcal{Z}(Q_{1})|^{2}$. The
corresponding expression for $|\mathcal{Z}(Q_{2})|^{2}$ is

$\displaystyle|\mathcal{Z}(Q_{2})|^{2}\;=\;\frac{\lambda^{2}}{2y}|\Upsilon{-}\Lambda|~{}.$

Though the expression (1.8) appears simpler, it is clear that the quantity
$1/y$ can be re-expressed as an $N$-expansion.

These $N$-expansions have a compelling interpretation in terms of microstate
counting, and this is particularly clear in virtue of the appearance of the
instanton numbers. The $\mathcal{N}{\;=\;}4$ case was famously studied by
Strominger and Vafa [16]. Important early papers on microstate counting of
black holes in $\mathcal{N}{\;=\;}2$ theories are Maldacena, Strominger and
Witten [17], the papers [18, 19, 20, 21, 6, 22, 23, 24] and the extensive
research of Sen (see for example [25] and references therein). These
calculations take into account the leading term with the coefficient $Y_{111}$
of the prepotential and also, in the case of [17], the term with coefficient
$Y_{100}$, which is important owing to its relation to the genus one
correction to the prepotential. What we wish to do here is to take into
account also the remaining terms, namely the perturbative $Y_{000}$ term, and
the instanton corrections, whose importance for the application to counting is
manifest.

The utility of the prepotential for us is that, as in (1.1), we can write the
holomorphic 3-form in terms of its derivatives. This immediately implies that
the period vector $\Pi$ can also be written in this form.

$\displaystyle\Pi(t)=\begin{pmatrix}\frac{\partial}{\partial
z^{0}}\mathcal{F}\\\\[3.0pt] \frac{\partial}{\partial
z^{1}}\mathcal{F}\\\\[3.0pt] z^{0}\\\
z^{1}\end{pmatrix}=z^{0}\begin{pmatrix}\frac{1}{6}Yt^{3}-\frac{1}{2}Y_{100}t-\frac{1}{3}\text{i}Y\sigma-2\mathcal{I}(t)+t\mathcal{I}^{\prime}(t)\\\\[3.0pt]
-\frac{1}{2}Yt^{2}-Y_{110}t-\frac{1}{2}Y_{100}-\mathcal{I}^{\prime}(t)\\\\[3.0pt]
1\\\\[3.0pt] t\end{pmatrix}.$ (1.9)

Here we have used the fact that the period vector is only defined up to a
scale, and is a homogeneous of degree one as a function of $z_{0},z_{1}$. For
brevity, we shall write in the following

$\displaystyle Y_{111}\;=\;Y\qquad\text{and}\qquad
Y_{000}\;=\;\text{i}Y\sigma~{}.$

Black holes in type II supergravity theories compactified on Calabi-Yau
manifolds can be viewed as consisting of D0-, D2-, D4-, and D6-branes wrapped
on 0-, 2-, 4-, and 6-cycles in type IIA theory, or alternatively as D3-branes
wrapping 3-cycles in type IIB theory. This implies that the charge vector of
the black hole can be viewed as an element of
$H^{0}(X,\mathbbl{Z}){\oplus}H^{2}(X,\mathbbl{Z}){\oplus}H^{4}(X,\mathbbl{Z}){\oplus}H^{6}(X,\mathbbl{Z})$
or $H^{3}(X,\mathbbl{Z})$. We will concentrate on one-parameter families of
Calabi-Yau manifolds and usually choose a basis for these vector spaces so
that we can think of the charge vector $Q$ as an element of $\mathbbl{Z}^{4}$
and the period vector $\Pi$ as an element of $\mathbbl{C}^{4}$.

A generic element $\Gamma\in
H^{0}(X,\mathbbl{Z}){\oplus}H^{2}(X,\mathbbl{Z}){\oplus}H^{4}(X,\mathbbl{Z}){\oplus}H^{6}(X,\mathbbl{Z})$
can be written as

$\displaystyle\Gamma\;=\;p^{0}+p\,e_{1}+q\,\frac{e_{1}^{2}}{Y}+q_{0}\,\frac{e_{1}^{3}}{Y}~{}.$

The corresponding charge vector can be written as

$Q\;=\;\left(\begin{matrix}{}_{\phantom{0}}q_{0}\\\
{}_{\phantom{0}}q_{\phantom{0}}\\\ {}_{\phantom{0}}p^{0}\\\
{}_{\phantom{0}}p^{\phantom{0}}\end{matrix}\right)\;=\;\left(\begin{matrix}q_{\text{\tiny
D0}}\\\ q_{\text{\tiny D2}}\\\ q_{\text{\tiny D6}}\\\ q_{\text{\tiny
D4}}\end{matrix}\right),$ (1.10)

where the last equality reminds us of the relation of the components to the
D-brane charges111In these conventions the D-brane charges do not include the
charges induced on various branes by the coupling to the world-volume
curvature. Often it is useful to switch to conventions where these charges are
included. For the relation between the two conventions, see appendix C.. This
choice of basis coincides with the choice that resulted in the expression
(1.9) for the period vector, so that we are allowed to think of the charge and
period vectors both as elements of same vector space isomorphic to
$\mathbbl{C}^{4}$, although due to charge quantization the elements of $Q$ are
integral.

Mirror symmetry implies that the IIB theory compactified on a Calabi-Yau
manifold $X$ is equivalent to IIA theory compactified on $\widetilde{X}$, the
mirror of $X$. The mirror symmetry exchanges the complex structure moduli
space $\mathcal{M}_{\mathbbl{C}}$ with coordinate $\varphi$ and complexified
Kähler moduli space $\mathcal{M}_{\mathbbl{C}K}$ with coordinate $t$ of the
two manifolds with each other. At the level of supergravity, the equivalence
is that the vector multiplet scalars in IIB theories take values in
$\mathcal{M}_{\mathbbl{C}}$, while the hypermultiplet scalars take values in
$\mathcal{M}_{\mathbbl{C}K}$, with the opposite correspondence in IIA
theories. In the following, we will focus mostly on the IIA perspective.

#### The large complex structure region

We use the term large complex structure region to refer to a region in the
Calabi-Yau moduli space where our solutions, such as (1.6), converge. We will
later in appendix F argue that such a region does exist and contains an open
neighbourhood of the point of maximal unipotent monodromy which we will refer
to simply as the large complex structure point.

There are also at least two other closely related, but ultimately different,
regions in the moduli space that could reasonably be called large complex
structure regions. One is the region where the instanton sum $\mathcal{I}(t)$
converges. This is determined by the asymptotic growth of the Gromov-Witten
invariants (see for example [10, 26, 27, 28])

$\displaystyle\log N^{\text{\tiny{GW}}}_{k}\;\sim\;2\pi y_{\text{sing}}k~{},$

where $y_{\text{sing}}$ is the coordinate of the singularity nearest to the
large complex structure point. Thus the instanton sum $\mathcal{I}(t)$
converges if and only if $\text{Im}\,t>y_{\text{sing}}$.

It is also possible to study this region using the mirror symmetry relation to
the complex structure moduli space on the mirror manifold. On the complex
structure moduli space, the coordinates can be chosen so that the point of
maximal unipotent monodromy is located at $\varphi=0$. The period vector is
most conveniently expressed in terms of a basis of solutions
$\widehat{\varpi}_{i}$ to a Picard-Fuchs equation around $\varphi=0$, where
close to the large complex structure point the vector with components these
solutions, $\widehat{\varpi}$, has asymptotics

$\displaystyle\widehat{\varpi}\;\sim\;\left(\begin{matrix}1\\\ t\\\ t^{2}\\\
t^{3}\end{matrix}\right)~{}.$

After identifying
$(z_{0},z_{1}){\;=\;}(\widehat{\varpi}_{0},\widehat{\varpi}_{1})$, one
expresses the complexified Kähler moduli space coordinate as follows:

$\displaystyle
t\;=\;\frac{\widehat{\varpi}_{1}(\varphi)}{\widehat{\varpi}_{0}(\varphi)}\;=\;\frac{1}{2\pi\text{i}}\log\varphi+\mathcal{O}(\varphi)~{},$
(1.11)

with the last equality following from the specifics of the Picard-Fuchs
equation [29, 15]. One should note the slightly different notation of each
reference, and that we use the same notation as [15]. The change of basis from
this vector to the period vector $\Pi$ that we work with is given by

$\displaystyle\Pi\;=\;-\left(\begin{matrix}\frac{1}{3}Y_{000}&\frac{1}{2}Y_{100}&0&-\frac{1}{6}Y_{111}\\\\[2.0pt]
\frac{1}{2}Y_{100}&Y_{110}&\frac{1}{2}Y_{111}&0\\\ -1\phantom{-}&0&0&0\\\
0&-1\phantom{-}&0&0\end{matrix}\right)\widehat{\varpi}~{}.$

The solutions $\widehat{\varpi}_{i}$ have a radius of convergence determined
by the singularities of the Picard-Fuchs operator. This determines yet another
region that could be reasonably called the large complex structure region.
Applying the mirror map (1.11), this region gets mapped to a neighbourhood of
$t\to\infty$, bounded by a circle around the nearest singularity, unless the
Picard-Fuchs operator has apparent singularities. However, this image does not
in general coincide with the $\text{Im}\,t>y_{\text{sing}}$ region, as the
correction terms of order $\mathcal{O}(\varphi)$ affect the relation. In
Figure 1, we have sketched the region around $\varphi=0$, its image under the
mirror map, and image of the resulting region under the map $t\to 1/t$.

i ii iii $\boxed{\varphi}$ $\boxed{t}$ $\boxed{\frac{1}{t}}$ Figure 1: Sketch
of three different views on the large complex structure region. Figure i
displays the large complex structure region on the mirror manifold. The region
is centred around the large complex structure singularity at $\varphi=0$ and
bounded by a circle through the singularity nearest to the $\varphi=0$
singularity. Under the mirror map, this gets mapped to the subset of the $t$
upper half plane displayed in Figure ii, bounded from below and extending to
imaginary infinity. This figure contains multiple copies of the fundamental
domain, corresponding to the multiples branches of the mirror map. This can be
more conveniently displayed by making a transformation $t\to 1/t$, resulting
in the region displayed in Figure iii. Note that the exact shapes of these
regions depend on the manifold in question, but the qualitative features of
these regions are similar for any one-parameter family of Calabi-Yau
manifolds.

#### The attractor mechanism

In [30, 1], the authors studied a four-dimensional $\mathcal{N}{\;=\;}2$
supergravity coupled to $\mathcal{N}{\;=\;}2$ vector multiplets, which can be
conveniently formulated in terms of special geometry [31]. For special
geometry itself, see [32, 7], and for a short summary in the style of the
present paper, see [15]. For a pedagogical introduction to the attractor
mechanism, see for example [33, 34, 35]. It was noticed that, for the
supersymmetric black hole solutions in these theories, the special geometry
moduli have to satisfy a pair of relatively simple differential equations.
These equations can be interpreted as requiring that the central charge of the
supergravity theory has a critical point at the horizon of the black hole. In
this section, we will look at things from the IIB point-of-view.

If the black hole solution in question is spherically symmetric and static, we
can write the line element

$\text{d}s^{2}\;=\;e^{2U(r)}\text{d}t^{2}-e^{-2U(r)}(\text{d}r^{2}+r^{2}\text{d}\Omega^{2}),$

where the radial coordinate $r$ is zero on the horizon. A key quantity is the
central charge $\mathcal{Z}$ of the superymmetry algebra, which is related to
the mass and entropy of the black hole.

$\displaystyle\mathcal{Z}(Q)\;=\;\frac{Q^{T}\,\Sigma\,\Pi}{\sqrt{-\text{i}\Pi^{\dagger}\,\Sigma\,\Pi}}~{},$
(1.12)

The intersection matrix $\Sigma$ appearing here is given in our chosen basis
by

$\displaystyle\Sigma\;=\;\begin{pmatrix}\phantom{-}0&\mathbbm{1}\\\
-\mathbbm{1}&0\end{pmatrix}.$

The dynamics of the scalars $z^{a}$ and the function $U$ can be written in
terms of a new coordinate $\rho=1/r$:

$\displaystyle\frac{dU(\rho)}{d\rho}$
$\displaystyle\;=\;-e^{U(\rho)}|\mathcal{Z}|~{},\qquad\frac{dz^{a}(\rho)}{d\rho}\;=\;-2e^{U(\rho)}g^{a\,\bar{b}}\partial_{\bar{b}}|\mathcal{Z}|~{},$
(1.13)

where $g_{a\,\bar{b}}$ is the special geometry metric on the moduli space
$\mathcal{M}_{\mathbbl{C}}$ and $z^{a}$ are the scalars of the
$\mathcal{N}{\;=\;}2$ vector multiplets, which are viewed as coordinates on
the moduli space $\mathcal{M}_{\mathbbl{C}}$. These equations describe flows
of scalar field values from $\rho=0$ at spatial infinity to $\rho=\infty$ at
the black hole horizon. The terminology “attractor mechanism” owes to the fact
that the values of $z^{a}$ are fixed at the black hole horizon at $r=0$
($\rho=\infty$) by the black hole charges, and so are independent of the
values they take at $r=\infty$ ($\rho=0$), at least for sufficiently small
variations of boundary values.

The second equation, (1.13), can be used to find a gradient flow equation:

$\frac{d|\mathcal{Z}|}{d\rho}\;=\;-4e^{U(\rho)}g^{a\bar{b}}\partial_{a}|\mathcal{Z}|\partial_{\bar{b}}|\mathcal{Z}|~{},$

revealing that the end points of attractor flows are critical values of
$|\mathcal{Z}|$, since the metric $g_{a\bar{b}}$ is positive definite. One can
also see that $|\mathcal{Z}|$ decreases along attractor flows.

It was pointed out early on by Strominger [36] that at an attractor point
where the central charge is not vanishing, the period vector of the Calabi-Yau
manifold satisfies a vector relation that we shall refer to as the alignment
equations

$\displaystyle\text{Im}\left[C\Pi\right]$ $\displaystyle\;=\;Q~{},$ (1.14)

where $\Pi$ is the period vector of the Calabi-Yau manifold, $Q$ is the charge
vector of the corresponding black hole, and

$\displaystyle
C\;=\;2\text{e}^{-K/2}\overline{\mathcal{Z}},\qquad\text{where}\qquad\text{e}^{-K}\;\buildrel\rm
def\over{=}\;\text{i}\int_{X}\Omega\wedge\overline{\Omega}~{}.$

In the following, we often choose a gauge in which $z^{0}=1$. The equation
(1.14) determines the relation between the charges and the prepotential.

If, on the other hand, the central charge vanishes, then the corresponding
relation is

$\displaystyle Q^{T}\,\Sigma\,\Pi\;=\;0~{},$ (1.15)

which we will refer to as the orthogonality equation. This determines the
prepotential in such a case.

The one-parameter attractor equations for a certain classes of Calabi-Yau
manifolds were studied for example in [37, 38, 39].

#### The alignment equations

It is convenient to write $C=a+\text{i}b$ and express the alignment equation
(1.14) in the form

$\displaystyle a\,\text{Im}\Pi+b\,\text{Re}\Pi\;=\;Q~{}.$ (1.16)

This equation can also be understood in terms of cohomology, and amounts to
the condition that

$\displaystyle\Gamma\in H^{(3,0)}(X,\mathbbl{Z})\oplus
H^{(0,3)}(X,\mathbbl{Z})~{}.$

We can think of the period vector $\Pi$ as defining a 2-plane over the reals,
spanned by $\text{Re}\Pi$ and $\text{Im}\Pi$. This two plane varies as we move
in the moduli space giving a family of 2-planes in $\mathbbl{R}^{4}$,
parametrised by the moduli-space coordinate $t$.

In order to satisfy the alignment equations, the plane

$\displaystyle V\;=\;\text{Span}(\text{Re}\Pi,\text{Im}\Pi)$

must contain a non-zero point of the charge lattice
$H^{3}(X,\mathbbl{Z})\simeq\mathbbl{Z}^{4}$, and so a lattice line consisting
of integer multiples of smallest such point. If the intersection is a lattice
line, the attractor is said to be of rank one. Exceptionally, the plane $V$
will intersect the charge lattice in two linearly independent points, and so
in a lattice plane. Then the attractor point is said to be of rank two. If
this is so, the alignment equations are satisfied for two linearly independent
vectors $Q$.

#### The orthogonality equation

In addition to the attractor points solving the alignment equations, there are
attractor points where the central charge vanishes, since these points are
necessarily local minima of the nonnegative quantity $|\mathcal{Z}(Q)|$. In
this case

$\displaystyle\Gamma\in H^{(2,1)}(X,\mathbbl{Z})\oplus
H^{(1,2)}(X,\mathbbl{Z}).$

There is again an interpretation of this equation in terms of planes and
lattices. The charge vector $\Gamma$ lies in the plane $V^{\perp}$ that is
orthogonal to $V$ under the symplectic inner product, corresponding to
$\Sigma$. For there to be an attractor point, the plane $V^{\perp}$ must also
intersect the charge lattice $H^{3}(X,\mathbbl{Z})$, and this intersection can
also be of rank one, or exceptionally of rank two.

Solutions to the alignment equations and the orthogonality equation are
related. The fact that $V$ is to some extent rational with respect to the
charge lattice has consequences for the rationality of the orthogonal plane
$V^{\perp}$. In particular, if $V$ intersects the lattice in a lattice plane,
then $V^{\perp}$ does so also. To see this, note that in this case
$V{\;=\;}\text{Span}(Q_{1},Q_{2})$ for some independent lattice vectors
$Q_{1}$ and $Q_{2}$. Then the space $V^{\perp}$ that is orthogonal to $V$
under the symplectic inner product, can also be taken to be spanned by two
lattice vectors $V{\;=\;}\text{Span}(Q_{3},Q_{4})$. The alignment equations at
rank two attractor points can be then written as

$\displaystyle\text{Span}(\text{Re}\Pi,\text{Im}\Pi)\;=\;\text{Span}(Q_{1},Q_{2})^{\phantom{\perp}}=\;\text{Span}(Q_{3},Q_{4})^{\perp}~{}.$

Likewise, the orthogonality equation can be written as

$\displaystyle\text{Span}(\text{Re}\Pi,\text{Im}\Pi)\;=\;\text{Span}(Q_{1},Q_{2})^{\perp}=\;\text{Span}(Q_{3},Q_{4})^{\phantom{\perp}}~{}.$

Thus alignment equations can be, in this case, written as an orthogonality
equation and vice versa. In the case that $V$ intersects the lattice in a
lattice line, the situation is more complicated, and is explored in more
detail in §6.

We wish to be clear about an issue of terminology. The term “attractor point”
is often used to refer exclusively with regards to massive black holes, and so
to solutions of the alignment equations, ignoring possible solutions to the
orthogonality equation. In the following, we will understand an attractor
point to be a point that is either a solution of the alignment equations or a
solution of the orthogonality equation. By a rank two attractor point, we will
continue to mean that there are two independent solutions to either the
alignment or the orthogonality equation, and hence that either $V$ or
$V^{\perp}$ intersects the charge lattice in a plane.

$Q_{1}$

$Q_{2}$

$\text{Im}\Pi$

$\text{Re}\Pi$

Figure 2: A schematic figure of the $H^{3}(X,\mathbbl{R})$, which we identify
with $\mathbbl{R}^{4}$. Here we show the real and imaginary parts of $\Pi$ as
vectors in this space, along with the charge vectors $Q_{1}$, $Q_{2}$. As the
moduli space coordinate $t$ varies, the vectors $\text{Re}\,\Pi$ and
$\text{Im}\,\Pi$, as well as the vectors orthogonal to these, move. When we
have a solution to the alignment equations, at least one of the vectors
$Q_{1}$ and $Q_{2}$ will align with the plane generated by $\text{Re}\,\Pi$
and $\text{Im}\,\Pi$.

#### Interpretation

The solutions to the alignment and orthogonality equations can be given very
different physical interpretations, owing to their relation to the BPS state
existence conjecture and the nature of the corresponding attractor flows. The
BPS state existence conjecture [4, 40, 41, 3, 2] asserts that, in the full
quantum theory of gravity, the existence of BPS states with certain charge
vectors is determined by the nature of the attractor point where the flow
ends. If we leave aside the possibility of a limiting cycle, there are three
different types of flows:

  1. 1.

The attractor flow ends at a point where $|\mathcal{Z}|{\,\neq\,}0$.

  2. 2.

The attractor flow ends at a point where $|\mathcal{Z}|{\,=\,}0$, and the
point is a regular point on the moduli space of the Calabi-Yau manifold.

  3. 3.

The attractor flow ends at a point where $|\mathcal{Z}|{\,=\,}0$, and the
point is a singular point on the moduli space of the Calabi-Yau manifold.

The BPS state existence conjecture asserts that BPS states always exist in the
first case, and never in the second case, while in the third case BPS states
may or may not exist, and this question cannot be determined from the
attractor flow data alone [3, 2]. Thus the conjecture asserts that the
solutions to the alignment equations should have an associated BPS state in
the full theory, while the solutions to the orthogonality equation do not
correspond to such states.

Even though the BPS state existence conjecture asserts that solutions to the
orthogonality equation do not correspond to BPS states, this does not mean
that the solutions are not significant. First of all, as we have seen, the
alignment and orthogonality equations are intimately connected: in some cases,
solutions to the alignment equations can be found from solutions to the
orthogonality equation. In addition, the orthogonality equation has a relation
to flux compactifications [42, 43] (see also section 4 of [15]). Finally, it
has been suggested [15] that, at least at rank two attractor points, the
existence of solutions to the orthogonality equation may indicate
supersymmetry enhancement.

#### The central charge and entropy

In the case of Calabi-Yau compactifications, the attractor equations not only
make it possible to find the complex or complexified Kähler structure of the
internal manifold, but also can be used to compute the central charge (1.12)
of the supergravity theory, which in turn is related to the Bekenstein-Hawking
entropy of the black hole associated to the attractor point.

$\displaystyle S_{\mbox{\tiny BH}}\;=\;\frac{A}{4}\;=\;\pi|\mathcal{Z}|^{2}.$

Solutions for the attractor equations in the approximation where the instanton
corrections to the prepotential are ignored, but the genus 1 perturbative
contribution is included, were found in [9, 22] and used to give an
approximate formula for the Bekenstein-Hawking entropy in any Calabi-Yau
compactification, where the internal manifold has a large complex structure.
For black holes for which $p^{0}=0$ but the other charges are large,

$\displaystyle S_{\mbox{\tiny
BH}}=\pi\sqrt{-\frac{2}{3}Y(\widetilde{p}^{3}+c_{2}\widetilde{p})\left(\widetilde{q}_{0}-\frac{1}{2\widetilde{p}Y}\widetilde{q}^{2}\right)}~{}.$
(1.17)

Here $\widetilde{p}$, $\widetilde{q}$ and $\widetilde{q}_{0}$ are the
microscopic charges, that is numbers of D4-, D2-, and D0-branes. The relation
to the macroscopic charges $p$, $q$ and $q_{0}$ can be found in appendix C.

In the calculation of this formula $Y_{000}$ and $Y_{110}$ cancel out, and
$Y_{100}$ and $Y_{111}$ appear via $c_{2}$ and $Y$, respectively. The
existence of such a result, which bears a clear resemblance to Cardy growth
[44], raises the question whether there is a modular form related to the
microstate counting, whose asymptotics reproduce the above formula in a manner
similar to the celebrated results of Strominger and Vafa [16]. This question
was first tackled in [17], where the authors showed that this entropy can be
reproduced, under some mild assumptions, by counting massless degrees of
freedom in an M-theory scenario with fivebranes wrapping a four-cycle on $X$
and the M-theory circle, together with momentum $q_{0}$ around the circle.
This setup gives rise to a two-dimensional $\mathcal{N}=(4,0)$ conformal field
theory on the fivebrane world-volume. It is possible to define a _modified
elliptic genus_ , which is a supersymmetric index that it is invariant under
continuous deformations of the theory, at least for small enough deformations.
This modified elliptic genus can be expressed in terms of a vector-valued
modular form. The coefficients of this modular form contain information about
the number of microstates. The asymptotic growth of the number of microstates
is then given by the Cardy formula [44] with the central charges

$\displaystyle\begin{split}c_{R}&=Y\left(\widetilde{p}^{3}+c_{2}\widetilde{p}\right),\qquad
c_{L}=Y\left(\widetilde{p}^{3}+\frac{1}{2}c_{2}\widetilde{p}\right).\end{split}$

In the large charge limit $\widetilde{p}\to\infty$, this correctly reproduces
(1.17).

For cases where the charge $\widetilde{p}$ is small, the corresponding
modified elliptic genera have been found explicitly, for some simple but
interesting Calabi-Yau manifolds in [45, 46]. To do this, one can consider the
geometry of divisors, curves, and points on the manifold. The fivebranes are
understood as divisors, and the lower-dimensional branes as curves or points
on these. The coefficients in the modular form that appears in the modified
elliptic genus can then be found by studying the moduli spaces of various such
configurations of branes and computing their Witten indices.

There is, however, an alternative method, which uses _split attractor flows_
[40, 47, 41] to compute elliptic genera. This was used in [48] to repeat the
calculation for the Calabi-Yau manifolds referred to above. This technique is
based on the observation that at least some of the coefficients count
microscopic degrees of freedom in situations where the black hole can be
viewed as a bound state of a D6 and an anti-D6 brane [4]. Such microstate
indices can be expressed in terms of Donaldson-Thomas invariants [49, 50],
which in turn can be expressed in terms of the Gromov-Witten invariants.
Interestingly, these invariants also appear in our macroscopic expressions,
such as (1.6). Unfortunately, the modified elliptic genus is not currently
known for the manifold AESZ34, which we use as the main example of a one-
parameter Calabi-Yau manifold in this paper owing to that fact that it has
rank two attractor points that are known explicitly. We hope to revisit this
issue elsewhere.

It can also be shown, as we will review in §7.2, that the central charge can
be viewed as the leading-order contribution to the Wald entropy [51] of the
black hole in the large charge limit [19]. The Wald entropy can be obtained
from an analogue of the attractor equations, known as the stabilisation
equations, where the period vector $\Pi$ is replaced by a more general vector
that includes the higher-genus prepotentials. However, as we will see in §7.2,
in the large charge limit the stabilisation equations reduce to the alignment
equations (1.14).

One of the aims of the present work is to extend these results by including
the contributions of instantons that have not been included in the
approximation used in [9, 24, 17, 8]. We set up an iterative procedure that
generates the $N$-expansions, which have a universal form that depends on the
manifold through the perturbative coefficients of the prepotential together
with the instanton numbers. A simple example of such an expansion has been
given in (1.6).

The black hole charge determines the location of the attractor point, and so
also the period vector. The value of the central charge therefore depends in a
detailed way on the particular brane system under consideration. Nevertheless,
we close this subsection with some general remarks, which we will later refine
in order to obtain explicit expressions for different brane systems.

From the alignment equations it follows that at an attractor point the charge
vector can be expressed as a linear combination of real and imaginary parts of
the period vector. Substituting (1.16) into the definition of the central
charge, the square of the central charge can be simplified

$\displaystyle|\mathcal{Z}|^{2}$
$\displaystyle\;=\;\frac{a^{2}+b^{2}}{2}\left|\text{Re}\Pi^{T}\,\Sigma\;\text{Im}\Pi\right|.$

Without yet specialising to any particular case, we can expand this near the
large complex structure limit, so for large $y$, in the form

$\displaystyle\frac{|\mathcal{Z}|^{2}}{a^{2}+b^{2}}$
$\displaystyle\;=\;\frac{1}{3}y^{3}Y+\frac{1}{6}Y\sigma+\sum_{k=1}^{\infty}\frac{(2\pi
ky+1)\cos(2\pi kx)}{(2\pi k)^{3}}N^{\text{GW}}_{k}\text{e}^{-2\pi ky}~{}.$

This formula, however, is deceptively simple. It depends on the location of
the attractor point $x+\text{i}y$, which in turn depends on the charge vector
$Q$. Our aim is to solve this dependence explicitly and so find a formula for
the central charge that depends only on the black hole charges and the
internal geometry of the manifold. An interesting aspect of this expression is
the appearance of the Gromov-Witten invariants, or equivalently the instanton
numbers, since these hint at counting of microstate contributions. Since $x$
and $y$ also have expansions in terms of the Gromov-Witten invariants, this
feature is preserved when we eliminate $x$ and $y$ from the expression.

#### Rank two attractor points

A particularly interesting examples of attractor points are provided by rank
two attractor points where two linearly independent charges solve the
alignment equation (1.14) at the same point in the moduli space. These are
much rarer than rank one attractor points, as the period vector has to satisfy
much more stringent conditions. However, precisely due to these restrictions,
we also have more control over various quantities associated to these points.

As first emphasised by Moore [3, 2], rank two attractor point also have
interesting connections to number theory. The explicit examples of rank two
attractor points found for the manifold AESZ34 in [15] display some intriguing
connections between attractor point coordinates, central charge and
L-functions. For example, it was conjectured (supported by extensive numerical
evidence) that there is a rank two attractor point at $\varphi=33-8\sqrt{17}$
whose location in the complexified Kähler moduli space is

$\displaystyle t\;=\;\text{i}\frac{5}{16\cdot
17}(9+\sqrt{17})\frac{\pi\lambda_{4}(1)}{\lambda_{4}(2)}~{},$

where $\lambda_{4}(s)$ are the real parts of an L-function $L(s)$ for the
group $\Gamma_{1}(34)$ with the LMFDB designation 34.4.b.a [52]. In this
paper, we will be able to rederive some of these results using a complementary
point of view that emphasises the relation of the attractor point coordinate
and the charge vector, whereas the method utilised in [15] uses zeta-function
factorisation to find the location attractor point in terms of the mirror
coordinate $\varphi$. As a result of our different approach, we are able to
derive some interesting identities between the L-function values
$\lambda_{4}(1)$, $\lambda_{4}(2)$ and Gromov-Witten invariants. Identities
like this were already anticipated by Moore [3].

###  1.2 Outline of the paper

In this paper, our aim is to solve the attractor equations for an arbitrary
one-parameter family of Calabi-Yau manifolds by developing $N$-expansions for
the attractor point coordinate $t$. Although we consider an arbitrary
manifold, we specialise the charge vector for simplicity and concentrate on
two important special cases that correspond to the D6-D2-D0 and D4-D2-D0 brane
systems. The main reason for this choice is that restricting to a case where
one of the charges vanishes makes the resulting equations much more tractable.
In addition, these systems are of particular interest with microstate counting
in mind, as their microstate indices can be viewed either as Donaldson-Thomas
invariants or as Witten indices of divisor moduli spaces [48, 45, 46].

In §2, prior to studying the two special cases separately, we discuss the
general method that will be used to derive the $N$-expansions. We use this
section to set notation and find expressions that we will later specialise.

In §3, we review some basic facts concerning partitions. While this material
is standard, we take the opportunity to set the notation that is used
extensively in the rest of the paper. We also define operations on partitions
that are useful for expressing various functions as perturbative expansions.

Following these preliminaries, in §5 and §4 we are able to calculate fully
instanton-corrected solutions to the attractor equations for the general
D6-D2-D0 and D4-D2-D0 configurations. The alignment and orthogonality
equations are considered separately. The solutions are separated into a
perturbative part and instanton corrections.

Our strategy is to express the instanton corrections as series in
$q_{y}=\text{e}^{-2\pi y_{0}}$. These series have coefficients which are
polynomials in the perturbative coordinates $x_{0}$, $y_{0}$, as well as the
related quantities $\sin 2\pi x_{0}$ and $\cos 2\pi x_{0}$. The polynomials
are most conveniently expressed as solutions to recurrence relations. These
relations can be quite complicated, but in §6 we see that in some important
cases there exist closed-form expressions. In particular, the polynomials
appearing in the solutions to the D4-D2-D0 orthogonality equation and a
special case of the D6-D2-D0 alignment equations can be written in terms of
modified spherical Bessel functions.

This special case of the D6-D2-D0 alignment equations is of particular
interest since the rank two attractor points of AESZ34, found in [15], are
included in this case. The highly constrained nature of the rank two attractor
points allows one, for example, to write the entropy in the particularly
simple form (1.8). Combining the solutions obtained here to the formulae in
[15], where the entropy at rank two attractor points was related to L-function
values, we are able to derive an intriguing formula relating L-function values
to Gromov-Witten invariants and Bessel functions.

In §7 we discuss some applications of our expressions. First the attractor
points in the large complex structure region are divided according to whether
or not the black hole central charge vanishes: an important distinction for
the physical interpretation of the solutions. We are able, based on
consideration of the black hole charge only, to determine whether solutions to
the alignment or orthogonality equations exist near the large complex
structure point.

To conclude this section, we consider our entropy formulae from the
perspective of microstate counting and verify that in leading order our
expressions agree with those in [17] and with modular form asymptotics. We
are, however, considering a limit that is different from that in [17] and so
differences appear in subleading orders. Building on the work of [22, 23, 24,
18, 19, 20, 21] (for a review, see [6]), we show that the proper
interpretation of our entropy formula is that it gives the leading-order Wald
entropy of the black hole in the large charge limit. In this sense, our
results extend those in [9, 17, 18].

Matters that would otherwise disrupt the narrative are relegated to four
appendices. In appendix A, we discuss the action on our formulae of a
$\mathbbl{Z}^{3}$ subgroup of $\text{Sp}(4,\mathbbl{Z})$, and identify certain
combinations of the Yukawa couplings and the charges which are invariant under
this group. In appendix B, we examine the role of the monodromy
transformations around the large complex structure limit, showing that these
can either be interpreted as a symmetry of the solutions or used to extend our
solutions to cases where the D4 brane charge divides the D6 charge. Appendix C
sets out the relation of the charge vector $Q$ to the various D-brane charges.
In appendices D and E, we list a set of polynomials which can be used to
express the perturbative solutions to the D6-D2-D0 attractor equations.
Finally, in appendix F, we discuss the asymptotics and convergence of
solutions we find.

###  1.3 Notation

We introduce notation in an attempt to streamline otherwise complicated
equations, and collect the notation which is used throughout this paper, in
the tables below.

$Y_{111}=Y$ | $\text{Im}\,Y_{000}=Y\sigma$ | $\zeta=\frac{\sigma}{y_{0}^{3}}$ | $\widehat{\sigma}=\frac{\sigma}{\beta^{3}}$
---|---|---|---
$\alpha=-\frac{Y_{110}+\Upsilon}{\widetilde{\beta}Y}$ |  $\beta=\sqrt{\frac{Y_{100}+2\Upsilon}{Y}}$ |  $\widetilde{\beta}=\sqrt{\frac{Y_{100}+2\Lambda}{Y}}$ |  $\gamma=-\frac{3\Lambda}{\beta^{3}Y}$

Table 1: Shorthand for $Y_{ijk}$\- and charge-related quantities. We take
$\beta$ and $\widetilde{\beta}$ to have argument $0$ or $\frac{\pi}{2}$,
depending on the argument of the square root.

In the main body of this paper, we choose a symplectic basis for
$H^{3}(X,\mathbbl{Z})$ in which $Y_{000}$ has no real part. When attention is
restricted to a system with no D6 charge, the $\mathbbl{Z}^{3}$ symmetry
discussed in appendix A leaves the combinations $\alpha$ and
$\widetilde{\beta}$ invariant. In the setup where the D4 charge vanishes, this
symmetry leaves $\beta$ and the quantity $\beta^{3}\gamma
Y-\text{Re}\,Y_{000}$ invariant.

Table 2: Quantities that are used throughout the paper with references to
where they are defined.

Notation |  Definition |  Description |  Ref
---|---|---|---
$\Pi$ |  $(\partial_{0}\mathcal{F},\partial_{1}\mathcal{F},z^{0},z^{1})^{T}$ |  period vector of $X$ |  (1.9)
$Q$ |  $(q_{0},q,p_{0},p)^{T}$ |  charge vector |  (1.10)
$\mathcal{F}$ |  $-\frac{1}{3!}Y_{abc}\frac{z^{a}z^{b}z^{c}}{z^{0}}-(z^{0})^{2}\mathcal{I}\left(\frac{z^{1}}{z^{0}}\right)$ |  Calabi-Yau prepotential |  (1.2)
$t=x+\text{i}y$ |  $z^{1}/z^{0}$ |  attractor point coordinate |  §1.1
$t_{0},x_{0},y_{0}$ |  $t_{0}=x_{0}+\text{i}y_{0}$ |  perturbative coordinates |  §2.3
$x_{00},y_{00}$ |  $x_{0}|_{\sigma=0}$, $y_{0}|_{\sigma=0}$ |  coordinates at $\sigma=0$ |  §2.3
$\mathcal{I}$ |  $\frac{1}{(2\pi\text{i})^{3}}\sum_{k=1}^{\infty}n_{k}\text{Li}_{3}(\text{e}^{2\pi\text{i}kt})$ |  instanton sum |  (1.4)
$\mathcal{I}_{1},\,\mathcal{I}_{2}$ |  $\text{Re}[\mathcal{I}],\,\text{Im}[\mathcal{I}]$ |  real and imaginary parts |  §2.2
$Y_{ijk}$ |  $\int_{\widetilde{X}}e_{i}\wedge e_{j}\wedge e_{k}$ |  $Y_{ijk}$ coefficients |  (1.3)
$\mathcal{Z}$ |  $\frac{Q^{T}\Sigma\Pi}{\sqrt{-\text{i}\Pi^{\dagger}\Sigma\Pi}}$ |  central charge |  (1.12)
$n_{k}$ |  |  instanton numbers |  (1.4)
$N^{\mbox{\tiny GW}}_{k}$ |  |  Gromov-Witten invariants |  (1.5)
$N_{k}$ |  $\sum_{d|k}d^{3}n_{d}=k^{3}N^{GW}_{k}$ |  scaled GW invariants |  (1.5)
$q_{t}$ |  $\text{e}^{2\pi\text{i}t_{0}}$ |  small expansion parameter |  (2.7)
$q_{x},q_{y}$ |  $q_{x}=\text{e}^{2\pi\text{i}x_{0}},\,q_{y}=\text{e}^{-2\pi y_{0}}$ |  phase and modulus of $q_{t}$ |  (2.7)
$t_{i}$ |  $t=\sum_{j=0}^{\infty}t_{j}q_{t}^{j}$ |  $q_{t}$-expansion of $t$ |  (2.7)

Table 2 continued.
---
Notation |  Definition |  Description |  Ref
$x_{i},y_{i}$ |  $x=\sum_{j=0}^{\infty}x_{j}q_{y}^{j}$, $y=\sum_{j=0}^{\infty}y_{j}q_{y}^{j}$ |  real and imaginary parts |  (2.1)
$R^{t}_{\mathfrak{p}}$ |  $t_{j}=\sum_{\mathfrak{p}\in\text{pt}(j)}N_{\mathfrak{p}}R^{(t)}_{\mathfrak{p}}$ |  $N_{k}$-expansion of $t_{j}$ |  (2.2)
$R^{x}_{\mathfrak{p}},\,R^{y}_{\mathfrak{p}}$ |  $\text{Re}\left[R^{t}_{\mathfrak{p}}\right],\,\text{Im}\left[R_{\mathfrak{p}}^{t}\right]$ |  real and imaginary parts |  (2.2)
$\mathfrak{p},\mathfrak{q}$ |  $\sum_{j}\mu_{j}j,\;\sum_{j}\nu_{j}j$ |  partitions |  §3
$c_{\mathfrak{p}}$ |  $\frac{\left(\sum_{j=1}^{\infty}\mu_{j}\right)!}{\prod_{j=1}^{\infty}\mu_{j}!};\qquad\mathfrak{p}=\sum_{j}\mu_{j}j$ |  multinomial coefficient |  (3.5)
$a_{\mathfrak{p}}$ |  $\prod_{j=1}^{\infty}\frac{1}{j^{2\mu_{j}}\mu_{j}!},\,\text{when}\,\,\mathfrak{p}=\sum_{j=1}^{\infty}\mu_{j}j$ |  coeffs. in the solutions in §6 |  (6.3)
$(F\hskip 1.0pt{*}\hskip 1.0ptG)_{\mathfrak{p}}$ |  $\sum_{\mathfrak{q}\subseteq\mathfrak{p}}F_{\mathfrak{p}-\mathfrak{q}}\,G_{\mathfrak{q}}$ |  natural convolution-like product |  (3.3)
$(F{\circ}G)_{\mathfrak{p}}$ |  $\sum_{\mathfrak{q}\subset\mathfrak{p}}F_{\mathfrak{p}-\mathfrak{q}}\,G_{\mathfrak{q}}$ |  as above, empty partitions excluded |  (3.3)
$\Lambda,\Upsilon$ |  varies according to context |  charge ratios |  (2.3)
$|\mathfrak{p}|$ |  the integer with partition $\mathfrak{p}$ |  |  §3
$\bm{\ell}(\mathfrak{p})$ |  number of parts of a partition $\mathfrak{p}$ |  length of $\mathfrak{p}$ |  §3
$\text{pt}(k)$ |  $\left\\{\,\mathfrak{p}~{}\big{|}~{}|\mathfrak{p}|=k\right\\}$ |  set of all partitions of $k$ |  (3.1)
$\text{pt}_{s}(k)$ |  $\\{\,\mathfrak{p}~{}\big{|}~{}|\mathfrak{p}|=k~{}~{}\text{and}~{}~{}\bm{\ell}(\mathfrak{p})=s\\}$ |  all partitions of length $s$ |  (3.2)
$\eta_{k}$ |  $\sum_{j=0}^{k}\frac{(-1)^{j}}{y_{0}^{j+1}}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}c_{\mathfrak{p}}y_{\mathfrak{p}}$ |  coeffs. in the $q_{y}$-expansion of $1/y$ |  (3.6)
$\bm{\eta}_{\mathfrak{p}}$ |  $\frac{1}{y_{0}}\sum_{j=0}^{l(\mathfrak{p})}\frac{(-1)^{j}}{y_{0}^{j}}r^{y_{\,}}_{(j,j),\mathfrak{p}}$ |  coeffs. in the expansion of $1/y$ |  (3.8)
$\text{e}^{y}(\alpha)$ |  $\sum_{j=0}^{k}\frac{(2\pi m)^{j}}{j!}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}c_{\mathfrak{p}}y_{\mathfrak{p}}$ |  coeffs. in the $q_{y}$-expansion of $\text{e}^{2\pi y}$ |  (3.6)
$\text{{\boldmath e}}^{y}_{\mathfrak{p}}$ |  $\sum_{j=0}^{l(\mathfrak{p})}\frac{(2\pi\alpha)^{j}}{j!}r^{y_{\,}}_{(j,j),\mathfrak{p}}$ |  coeffs. in the expansion of $\text{e}^{2\pi y}$ |  (3.8)
$r^{x}_{(m,n),\mathfrak{p}}$ |  $\displaystyle\sum_{\begin{subarray}{c}\sum_{j=1}^{n}\mathfrak{p}_{j}=\mathfrak{p}\\\ \mathfrak{p}_{1},...,\mathfrak{p}_{m}\neq\emptyset\\\\[3.0pt] \end{subarray}}c\big{(}\hskip-2.0pt\left\\{\mathfrak{p}_{j}\right\\}_{j=1}^{i}\big{)}\prod_{j=1}^{i}R^{x}_{\mathfrak{p}_{j}}$ |  combinations of functions $R^{x}$ |  (3.7)
$\bm{k}_{n}(z)$ |  $\sqrt{\frac{2}{\pi z}}K_{n+\frac{1}{2}}(z)$ |  spherical $k$ Bessel functions |  (6.4)
$f_{n}(y)$ |  $\text{e}^{y}y^{n+1}\bm{k}_{n}(y)$ |  polynomials associated to $\bm{k}_{n}$ |  (6.4)
$\mathcal{F}^{(g)}(X)$ |  |  genus $g$ Calabi-Yau prepotential |  §7.2
$\mathpzc F(X,W^{2})$ |  $\sum_{g=0}^{\infty}\mathcal{F}^{(g)}(X)W^{2g}$ |  all-genus CY prepotential |  (7.3)

##  2 Solving the Attractor Equations: a General Overview

Our strategy for solving the attractor equations relies on using the
expression (1.9) for the period vector $\Pi$ to express both the alignment
equations (1.16) and the orthogonality equation (1.15) explicitly as a set of
equalities involving geometric data (the topological quantities $Y_{abc}$ and
Gromov-Witten invariants of the Calabi-Yau manifold) and the black hole
charges.

In the following sections §4 and §5, we solve these equations perturbatively
in the instanton corrections. We first find the zeroth-order solution
corresponding to what we shall refer to as the perturbative part of the
prepotential, meaning the $Y_{abc}$-terms, corresponding to ignoring the
contributions of the instanton sums. This gives us a pair of polynomial
equations of degree at most four. Explicit solutions of such polynomial
equations are of course known since the Renaissance, but we dedicate some
discussion to giving the most concise solutions. We show that the solutions
can, in most cases, be given by simple trigonometric expressions.

Near the large complex structure point the instanton corrections are
exponentially small and we expand in terms of the quantity
$q_{y}{\;=\;}\text{e}^{-2\pi y_{0}}$, where $y_{0}$ is the perturbative value
of the attractor coordinate, found by using the perturbative equations as
outlined above. We find that the instanton corrections take the form of a
series in $q_{y}$,

$\displaystyle x\;=\;\sum_{j=0}^{\infty}x_{j}q_{y}^{j}~{},\qquad
y\;=\;\sum_{j=0}^{\infty}y_{j}q_{y}^{j}~{},$ (2.1)

with coefficients $x_{j}$ and $y_{j}$ that are rational functions of $x_{0}$,
$y_{0}$, $\sin 2\pi x_{0}$, and $\cos 2\pi x_{0}$.

$\displaystyle x_{j}$
$\displaystyle\;=\;\sum_{\mathfrak{p}\in\text{pt}(j)}\hskip-5.0ptN_{\mathfrak{p}}R^{x_{\,}}_{\mathfrak{p}}(x_{0},y_{0}),\qquad
y_{j}\;=\;\sum_{\mathfrak{p}\in\text{pt}(j)}\hskip-5.0ptN_{\mathfrak{p}}R^{y_{\,}}_{\mathfrak{p}}(x_{0},y_{0}),$
(2.2)

where $\mathfrak{p}$ is a partition of $j{\,\in\,}\mathbbl{Z}^{+}$,
$R^{x_{\,}}_{\mathfrak{p}}(x_{0},y_{0})$ and
$R^{y_{\,}}_{\mathfrak{p}}(x_{0},y_{0})$ are the rational functions of
$x_{0}$, $y_{0}$, $\sin(2\pi x_{0})$ and $\cos(2\pi x_{0})$ and are
independent of the Gromov-Witten invariants. The $N_{\mathfrak{p}}$ are
combinations of the scaled Gromov-Witten invariants as in (1.7). We give
recurrence relations for the $R_{\mathfrak{p}}^{x}$ and
$R_{\mathfrak{p}}^{y}$, which are amenable to rapid computer calculation.
Since the individual terms in the sum are exponentially small the first few
terms suffice for many practical purposes.

A pleasant surprise is that even though the recurrence relations are somewhat
complicated in the general case, we are able to give simple closed-form
solutions in important special cases. In particular, for the D4-D2-D0
orthogonality equation and a special case of D6-D2-D0 alignment equations, we
will find in §6 interesting expressions for the location of the attractor
point which curiously involve spherical Bessel functions.

###  2.1 Particular brane systems

We study extremal charged black holes that, in the string theory picture,
arise from wrappings of D-branes on cycles on the internal Calabi-Yau
manifold. We restrict our attention to arrangements containing either D6- or
D4-branes but not both. We attach to these some D2- and D0-branes. Recalling
the form of a generic charge vector (1.10), this means that we consider charge
vectors of the form

$Q_{\text{D4}}=\kappa\left(\begin{matrix}\Lambda\\\ \Upsilon\\\ 0\\\
1\end{matrix}\right),\hskip
56.9055ptQ_{\text{D6}}=\kappa\left(\begin{matrix}\Lambda\\\ \Upsilon\\\ 1\\\
0\end{matrix}\right).$ (2.3)

The physical charge vectors have integral components, so $\Lambda$ and
$\Upsilon$ are rational numbers, and $\kappa$ is a suitable integer. The
charge vectors are written in this way partly to simplify later expressions
since the locations of attractor points depend only on the charge ratios
$\Upsilon$ and $\Lambda$. Furthermore, this decomposition allows us to make an
important distinction between what we call the large charge limit, where
${|\kappa|\to\infty}$ and the large charge ratio limits, where either
$|\Lambda|$ or $|\Upsilon|$ is large.

In [15], where rank two attractor points for one-parameter families were first
explicitly identified, the associated charge vectors were of this type, and
charge vectors of this form have received much attention in the literature.
Microscopic state counting has been performed both for black holes of
$Q_{\text{D4}}$ charge using elliptic genera [45, 46] and D6-D0 bound states
using Donaldson-Thomas invariants [4, 48].

By utilising the monodromy around the large complex structure point we can
extend our solutions to any brane system where the D6-charge divides the
D4-charge, as we will show in detail in appendix B.

###  2.2 General comments concerning the alignment equations

The alignment equations (1.14) depend on the quantity $\text{Im}[C\Pi]$, which
is manifestly not holomorphic in $t$, so it becomes useful to decompose the
coordinate $t$ into its real and imaginary parts, $t=x+\text{i}y$. Similarly,
the constant $C$ is written as $a+\text{i}b$. All of the $Y_{abc}$ can be
chosen to be real, apart from $Y_{000}=\text{i}Y\sigma$ which becomes pure
imaginary. The instanton sum $\mathcal{I}(t)$ and its derivative
$\mathcal{I}^{\prime}(t)$ can be decomposed as
$\mathcal{I}(t)=\mathcal{I}_{1}(x,y)+\text{i}\mathcal{I}_{2}(x,y)$ and
$\mathcal{I}^{\prime}(t)=\mathcal{I}_{1}^{\prime}(x,y)+\text{i}\mathcal{I}_{2}^{\prime}(x,y)$.

In terms of the components of the period vector (1.9) the alignment equations
(1.16) are

$\begin{split}&Q\;=\;\\\ &a\hskip-3.0pt\begin{bmatrix}\vrule
height=13.0pt,width=0.0pt\hskip-2.0pt\frac{1}{6}Y\hskip-2.0pty\hskip-2.0pt\left(3x^{2}{-}y^{2}\right){-}\frac{1}{2}Y_{100}y{-}\frac{1}{3}Y\hskip-2.0pt\sigma{+}y\mathcal{I}_{1}^{\prime}{+}x\mathcal{I}_{2}^{\prime}{-}2\mathcal{I}_{2}\\!{}\\\\[3.0pt]
-Yxy{-}Y_{110}y{-}\mathcal{I}_{2}^{\prime}\\\\[3.0pt] 0\\\\[3.0pt]
y\end{bmatrix}\hskip-2.0pt+b\hskip-3.0pt\begin{bmatrix}\vrule
height=13.0pt,width=0.0pt\hskip-2.0pt\frac{1}{6}Y\hskip-2.0ptx\hskip-2.0pt\left(x^{2}{-}3y^{2}\right){-}\frac{1}{2}Y_{100}x{+}x\mathcal{I}_{1}^{\prime}{-}y\mathcal{I}_{2}^{\prime}{-}2\mathcal{I}_{1}\\!{}\\\\[5.0pt]
\frac{1}{2}Y(y^{2}{-}x^{2}){-}Y_{110}x{-}\frac{1}{2}Y_{100}{-}\mathcal{I}_{1}^{\prime}\\\\[3.0pt]
1\\\\[3.0pt] x\end{bmatrix}\end{split}$ (2.4)

The lower two components of this vector equation determine $a$ and $b$

$(a,b)=\begin{cases}\frac{\kappa}{y}\left(-x\,,y\right),&\text{D6 system}\\\
&\\\ \frac{\kappa}{y}\left(\phantom{-}1\,,0\right),&\text{D4
system.}\end{cases}$ (2.5)

Substituting these two values into the upper two components of equation (2.4)
leaves two equations for $x$ and $y$. This offers another perspective on the
ubiquity of rank one attractor points and rarity of those of rank two: the
alignment equations written in the form (2.4) can be solved very often, but
satisfying this equation for two linearly independent charge vectors involves
four equations and only two unknowns. This latter system is overdetermined and
usually insoluble.

After substituting in (2.5), there is an overall factor of $\kappa$ on both
sides of (2.4) which cancels. This leaves the equations with which we begin
the discussion of the alignment equations in §4 and §5.

###  2.3 Setting up the iteration

To set up the perturbative problem, we will separate, in the period vector,
those terms that do not contain instanton corrections from those that do.

$\begin{gathered}\Pi=\Pi_{0}+\Delta_{\mathcal{I}}\Pi,\\\\[15.0pt]
\Pi_{0}=\left(\begin{matrix}\phantom{-}\frac{1}{6}Yt^{3}-\frac{1}{2}Y_{100}t-\frac{1}{3}\text{i}Y\sigma\\\\[3.0pt]
-\frac{1}{2}Yt^{2}-Y_{110}t-\frac{1}{2}Y_{100}\\\ 1\\\
t\end{matrix}\right),\quad\quad\Delta_{\mathcal{I}}\Pi=\left(\begin{matrix}t\mathcal{I}^{\prime}-2\mathcal{I}\\\
-\mathcal{I}^{\prime}\\\ 0\\\ 0\end{matrix}\right).\end{gathered}$

As stated above, we wish to treat the instanton terms as small corrections,
and so in all cases we first solve the zeroth-order perturbative equations.
These are obtained from the full alignment or orthogonality equations by
simply replacing the full period vector $\Pi$ by $\Pi_{0}$,

$\displaystyle\text{Im}[C\Pi_{0}]\;=\;Q~{},\qquad\qquad\Pi_{0}^{T}\Sigma
Q\;=\;0~{}.$ (2.6)

We denote the solutions to the perturbative alignment equations by
$t_{0}=x_{0}+\text{i}y_{0}$, and largely work directly with the real and
imaginary parts. We can find a closed-form solution for $x_{0}$ and $y_{0}$,
which depends on $\sigma$. This solution is usually most conveniently
expressed in terms of trigonometric functions. We can also give $x_{0}$ and
$y_{0}$ as a power series in $\sigma$. Thus, in addition to the perturbative
coordinates $x_{0}$ and $y_{0}$, we define quantities $x_{00}$ and $y_{00}$ as
the $\sigma=0$ limits of these.

$\displaystyle x_{00}\;=\;x_{0}\big{|}_{\sigma=0}~{},\qquad
y_{00}\;=\;y_{0}\big{|}_{\sigma=0}~{}.$

The utility of this expansion will be seen in §7, where it is a natural
expansion to use for the large charge ratio limits, as well as for comparing
supergravity results to microscopic formulae.

Solving the orthogonality equation proceeds in a very similar manner. The
principal difference is that, as the orthogonality equation (1.15) depends on
$t$ holomorphically, it is most convenient to work directly with $t$ itself
instead of splitting it into real and imaginary parts. Similarly, we do not
split the instanton sums $\mathcal{I}$ and $\mathcal{I}^{\prime}$ into their
real and imaginary parts. We denote the solution to the perturbative equations
(2.6) by $t_{0}$ and express this both as a closed-form expression involving
trigonometric functions, as well as a Taylor series expansion around
$\sigma=0$. To include the instanton corrections we make the ansatz

$\displaystyle t\;=\;\sum_{j=0}^{\infty}t_{j}q_{t}^{j}~{}.$ (2.7)

The small parameter we use here is $q_{t}=\text{e}^{2\pi\text{i}t_{0}}$. Note
however that $|q_{t}|=q_{y}$, so we can expect similar convergence. As before,
we are able to derive recurrence relations for the coefficients $t_{j}$. In
(2.7) the $t_{j}$ are also expressed in terms of rational functions and
Gromov-Witten invariants in the form

$\displaystyle
t_{j}\;=\;\sum_{\mathfrak{p}\in\text{pt}(j)}\hskip-5.0ptN_{\mathfrak{p}}R^{t}_{\mathfrak{p}}(t_{0})~{}.$
(2.8)

##  3 Partitions

Since many of the formulae we will come across while solving the attractor
equations involve partitions, we find it useful to introduce some terminology
and notation from the domain of enumerative combinatorics and provide a
rudimentary introduction to the theory of partitions. This serves to fix our
notation and give some of the basic results that will be extensively used in
the following. A more thorough treatment can be found in, for example, [53],
whose notation we largely follow.

###  3.1 Basic definitions and notation

A _partition_ $\mathfrak{p}$ of a positive integer $k\in\mathbbl{Z}_{>0}$ is a
sequence of positive integers

$0<p_{1}\leqslant p_{2}\leqslant\ldots\leqslant p_{m}\quad\text{such
that}\quad\sum_{i=1}^{m}p_{i}\;=\;k~{}.$

If $\mathfrak{p}$ is a partition of $k$, we write $|\mathfrak{p}|{\;=\;}k$ and
refer to the terms $p_{i}$ as the _parts_ of $\mathfrak{p}$. The number of
parts is called the _length_ of the partition, which is denoted by
$\bm{\ell}(\mathfrak{p})$. If $\mu_{i}$ denotes the number of parts equal to
$i$, we can write partitions by means of a multiset notation
$\mathfrak{p}{\;=\;}\\{1^{\mu_{1}},2^{\mu_{2}},...\\}$, where only a finite
number of $\mu_{i}$ are non-zero.

The partitions of an integer $k$ can be also viewed as the different ways of
expressing $k$ as a sum of positive integers without distinguishing the order
of the summands. In virtue of this, partitions are often also written using a
summation notation:

$\mathfrak{p}\;=\;\sum_{i=1}^{\infty}\mu_{i}i~{}.$

Every integer $k$ is a partition of itself, so we can regard integers as
special cases of partitions.

Although partitions are defined as sequences of positive numbers, it is
sometimes convenient to generalise the notion of partition to include
_partitions with zeros_. A partition with zeros $\mathfrak{p}$ of
$k{\,\in\,}\mathbbl{Z}_{>0}$ is a sequence of non-negative integers

$0\leqslant p_{1}\leqslant p_{2}\leqslant...\leqslant p_{m}\quad\text{such
that}\quad\sum_{i=1}^{m}p_{i}\;=\;k~{}.$

The definitions of $\bm{\ell}(\mathfrak{p})$ and $|\mathfrak{p}|$ are as
before.

There is an important special case where the partition corresponds to an empty
multiset $\mathfrak{p}=\emptyset$. This is the _empty partition_ , and we take
$\bm{\ell}(\emptyset)=0$ and $|\emptyset|=0$. It is important to distinguish
the empty partition $\emptyset$ from the partition with zeros $\\{0^{n}\\}$.
For example, we have $\bm{\ell}(\emptyset)=0$, while
$\bm{\ell}(\\{0^{n}\\})=n$.

###  3.2 Operations on partitions

In this subsection

$\displaystyle\mathfrak{p}\;=\;\sum_{i=0}^{\infty}\mu_{i}i\;=\;\\{0^{\mu_{0}},1^{\mu_{1}},2^{\mu_{2}},...\\}\qquad\text{and}\qquad\mathfrak{q}\;=\;\sum_{i=0}^{\infty}\nu_{i}i\;=\;\\{0^{\nu_{0}},1^{\nu_{1}},2^{\nu_{2}},...\\}$

are either partitions or partitions with zeros, for which we use the multiset
or sum notation, as convenient.

We write $\mathfrak{p}\subseteq\mathfrak{q}$ if, as multisets,
$\mathfrak{p}\subseteq\mathfrak{q}$, which means that
$\mu_{i}\leqslant\nu_{i}$ for all $i$. We use $\subset$ when
$\mu_{i}<\nu_{i}$.

We can always define a sum of two partitions as a union of the partitions:
$\mathfrak{p}+\mathfrak{q}{\;=\;}\mathfrak{p}\cup\mathfrak{q}$, where in
writing $\mathfrak{p}\cup\mathfrak{q}$ we understand $\mathfrak{p}$ and
$\mathfrak{q}$ as multisets. Alternatively, in the sum notation, we can write

$\mathfrak{p}+\mathfrak{q}\;=\;\sum_{i=0}^{\infty}(\mu_{i}+\nu_{i})\,i~{}.$

It is clear that this is always a partition, and that
$|\mathfrak{p}+\mathfrak{q}|{\;=\;}|\mathfrak{p}|+|\mathfrak{q}|$, and
$\bm{\ell}(\mathfrak{p}+\mathfrak{q}){\;=\;}\bm{\ell}(\mathfrak{p})+\bm{\ell}(\mathfrak{q})$.

When $\mathfrak{p}\subseteq\mathfrak{q}$, we can define also the difference of
these partitions as
$\mathfrak{q}-\mathfrak{p}{\;=\;}\mathfrak{q}\setminus\mathfrak{p}$ or,
slightly informally, using the sum notation, as

$\mathfrak{q}-\mathfrak{p}\;=\;\sum_{i=0}^{\infty}(\nu_{i}-\mu_{i})\,i~{}.$

Note that there is a potential source of confusion here with the treatment of
zeros. We emphasise that this sum is just an informal way of writing the
difference, and the unambiguous definition is given via the multiset
difference. Similarly to the sum, we have that
$|\mathfrak{q}-\mathfrak{p}|{\;=\;}|\mathfrak{q}|-|\mathfrak{p}|$ and
$\bm{\ell}(\mathfrak{q}-\mathfrak{p}){\;=\;}\bm{\ell}(\mathfrak{q})-\bm{\ell}(\mathfrak{p})$.
In the special case $\mathfrak{p}=\mathfrak{q}$, we have

$\displaystyle\mathfrak{p}-\mathfrak{p}\;=\;\emptyset,$

which follows unambiguously by taking the multiset difference.

It is useful also to set notation for some important sets of partitions. We
denote the set of all partitions of $k$, of any length, by

$\text{pt}(k)\;\buildrel\rm
def\over{=}\;\left\\{\,\mathfrak{p}~{}\big{|}~{}|\mathfrak{p}|=k\right\\}~{}.$
(3.1)

The set of all partitions of $k$ with length $s$ is denoted by

$\text{pt}_{s}(k)\;\buildrel\rm
def\over{=}\;\\{\,\mathfrak{p}~{}\big{|}~{}|\mathfrak{p}|=k~{}~{}\text{and}~{}~{}\bm{\ell}(\mathfrak{p})=s\\}~{}.$
(3.2)

In addition, we define some unions of the above sets:

$\text{pt}_{{\scriptscriptstyle\leqslant}\,s}(k)\;\buildrel\rm
def\over{=}\;\bigcup_{r=1}^{s}\text{pt}_{r}(k)\,,~{}~{}\text{and}~{}~{}\text{pt}_{{\scriptscriptstyle<}\,s}(k)\;\buildrel\rm
def\over{=}\;\bigcup_{r=1}^{s-1}\text{pt}_{r}(k)~{},$

with the sets $\text{pt}_{{\scriptscriptstyle\geqslant}\,s}(k)$ and
$\text{pt}_{{\scriptscriptstyle>}\,s}(k)$ defined similarly.

The set of partitions of $k$ with zeros is denoted by $\text{pt}^{0}(k)$, and
the definitions of the variants such as
$\text{pt}^{0}_{{\scriptscriptstyle\leqslant}\,s}(k)$ are given
straightforwardly, as in the above. To avoid any confusion arising from the
somewhat subtle issue of the empty partition, we take
$\text{pt}(0)=\\{\emptyset\\}$, and $\emptyset\notin\text{pt}^{0}_{i}(k)$ for
any $i$ or $k$.

It is sometimes convenient to use a partition $\mathfrak{p}$ as a multi-index,
specifying a product. This is understood exclusively for the following five
quantities:

$x_{\mathfrak{p}}\;=\;\prod_{j=0}^{\infty}x_{j}^{\mu_{j}}~{},\qquad
y_{\mathfrak{p}}\;=\;\prod_{j=0}^{\infty}y_{j}^{\mu_{j}}~{},\qquad
t_{\mathfrak{p}}\;=\;\prod_{j=0}^{\infty}t_{j}^{\mu_{j}}~{},\qquad
N_{\mathfrak{p}}\;=\;\prod_{j=0}^{\infty}N_{j}^{\mu_{j}}~{},\qquad\bm{m}_{\mathfrak{p}}\;=\;\prod_{j=0}^{\infty}\bm{m}_{j}^{\mu_{j}}~{}.$

In all other cases where $\mathfrak{p}$ appears in a subscript, such as the
later $\mathcal{E}_{j,\mathfrak{p}}$, it is meant as an index that associates
a quantity to a partition in some other way specific to that symbol. In other
words, said symbol is a different function of $\mathfrak{p}$.

Functions arise that depend on partitions and also Gromov-Witten invariants.
We often expand these functions as power series in the rescaled Gromov-Witten
invariants $N_{k}$. Using the partition notation, the resulting series can be
written as

$F\;=\;\sum_{\mathfrak{p}}N_{\mathfrak{p}}F_{\mathfrak{p}}~{}.$

We shall refer to such an expansion as an _$N$ -expansion_.

We can think of the coefficients $F_{\mathfrak{p}}$ as functions of the
partitions $\mathfrak{p}$. For two such functions $F_{\mathfrak{p}}$ and
$G_{\mathfrak{p}}$, there is a natural convolution-like product

$(F\hskip 1.0pt{*}\hskip 1.0ptG)_{\mathfrak{p}}\,\buildrel\rm
def\over{=}\,\sum_{\mathfrak{q}\subseteq\mathfrak{p}}F_{\mathfrak{p}-\mathfrak{q}}\,G_{\mathfrak{q}}~{}.$
(3.3)

This product is clearly commutative and associative, so we write

$F\hskip 1.0pt{*}\hskip 1.0pt(G\hskip 1.0pt{*}\hskip 1.0ptH)\;=\;(F\hskip
1.0pt{*}\hskip 1.0ptG)\hskip 1.0pt{*}\hskip 1.0ptH\;=\;F\hskip 1.0pt{*}\hskip
1.0ptG\hskip 1.0pt{*}\hskip 1.0ptH~{}.$

The main utility of this product is that if

$\displaystyle
F(j)\;=\;\hskip-3.0pt\sum_{\mathfrak{p}\in\text{pt}(j)}\hskip-5.0ptF_{\mathfrak{p}}\,N_{\mathfrak{p}}~{},\qquad\text{and}\qquad
G(j)\;=\;\hskip-3.0pt\sum_{\mathfrak{p}\in\text{pt}(j)}\hskip-5.0ptG_{\mathfrak{p}}\,N_{\mathfrak{p}}~{},$

then we have the following identity

$\sum_{j=0}^{k}F(j)\,G(k-j)\;=\sum_{\mathfrak{p}\in\text{pt}(k)}(F\hskip
1.0pt{*}\hskip 1.0ptG)_{\mathfrak{p}}\,N_{\mathfrak{p}}~{}.$

We will find frequent use for this identity in formulating different
recurrence relations.

The sum in the convolution product includes the cases
$\mathfrak{q}{\;=\;}\mathfrak{p}$ and $\mathfrak{q}{\;=\;}\emptyset$, which
both correspond to the empty partition. It is sometimes useful to consider a
convolution product which does not include these two terms. We denote this
product by $\circ$.

$(F{\circ}G)_{\mathfrak{p}}\,\buildrel\rm
def\over{=}\hskip-3.0pt\sum_{\begin{subarray}{c}\mathfrak{q}\subset\mathfrak{p},\mathfrak{q}\neq\emptyset\end{subarray}}F_{\mathfrak{p}-\mathfrak{q}}\,G_{\mathfrak{q}}\;=\;(F\hskip
1.0pt{*}\hskip
1.0ptG)_{\mathfrak{p}}-F_{\emptyset}G_{\mathfrak{p}}-F_{\mathfrak{p}}G_{\emptyset}~{}.$

###  3.3 Series expansions

The main utility of the theory of partitions to the discussion of the
attractor equations is that the generating functions of many series that arise
may be expressed in terms of the quantities defined above. The prototypical
quantities where sums or series can be written with the help of partitions are
the powers of sums. For example

$\left(\sum_{j=1}^{\infty}x_{j}q^{j}\right)^{n}\;=\;\sum_{k=1}^{\infty}q^{k}\sum_{\mathfrak{p}\in\text{pt}_{n}(k)}\hskip-5.0ptc_{\mathfrak{p}}x_{\mathfrak{p}}~{},$
(3.4)

where $c_{\mathfrak{p}}$ is the multinomial coefficient associated to the
partition $\mathfrak{p}{\;=\;}\sum_{j=1}^{\infty}\mu_{j}j$

$c_{\mathfrak{p}}\;=\;\frac{\left(\sum_{j=1}^{\infty}\mu_{j}\right)!}{\prod_{j=1}^{\infty}\mu_{j}!}~{}.$
(3.5)

Expressions like this will be appear repeatedly in what follows. In this way,
we are led to series expansions for

$\displaystyle x\;=\;\sum_{i=0}^{\infty}x_{i}\,q_{y}^{i}~{},\quad
y\;=\;\sum_{i=0}^{\infty}y_{i}\,q_{y}^{i}~{},\quad
t\;=\;\sum_{i=0}^{\infty}t_{i}\,q_{t}^{i}~{},$

and the following closely related quantities. These expressions follow
straightforwardly from (3.4).

$\displaystyle x^{n}$
$\displaystyle\;=\;\sum_{k=0}^{\infty}q_{y}^{k}\sum_{\mathfrak{p}\in\text{pt}^{0}_{n}(k)}\hskip-5.0ptc_{\mathfrak{p}}x_{\mathfrak{p}}~{},\qquad
n>0~{},$ (3.6) $\displaystyle\frac{1}{y}$
$\displaystyle\;=\;\sum_{k=0}^{\infty}\eta_{k}\,q_{y}^{k}~{},\qquad\qquad\qquad\text{where}\;\qquad\qquad\eta_{k}\;\buildrel\rm
def\over{=}\;\sum_{j=0}^{k}\frac{(-1)^{j}}{y_{0}^{j+1}}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}\hskip-5.0ptc_{\mathfrak{p}}y_{\mathfrak{p}}~{},$
$\displaystyle\text{e}^{-2\pi my}$
$\displaystyle\;=\;\sum_{k=0}^{\infty}\text{e}_{k}^{y_{\,}}(-m)\,q_{y}^{k+m}~{},\;\,\qquad\text{where}\qquad\;\,\text{e}_{k}^{y_{\,}}(m)\;\buildrel\rm
def\over{=}\;\sum_{j=0}^{k}\frac{(2\pi
m)^{j}}{j!}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}\hskip-5.0ptc_{\mathfrak{p}}y_{\mathfrak{p}}~{},$
$\displaystyle\text{e}^{2\pi\text{i}mx}$
$\displaystyle\;=\;\sum_{k=0}^{\infty}\text{e}_{k}^{x_{\,}}(\text{i}m)\,q_{y}^{k}q_{x}^{m}~{},\;\;\;\,\qquad\text{where}\qquad\;\,\text{e}_{k}^{x_{\,}}(m)\;\buildrel\rm
def\over{=}\;\sum_{j=0}^{k}\frac{(2\pi
m)^{j}}{j!}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}\hskip-5.0ptc_{\mathfrak{p}}x_{\mathfrak{p}}~{},$
$\displaystyle\text{e}^{2\pi\text{i}mt}$
$\displaystyle\;=\;\sum_{k=0}^{\infty}\text{e}_{k}^{t_{\,}}(\text{i}m)\,q_{t}^{k+m}~{},\;\;\;\,\qquad\text{where}\qquad\;\,\text{e}_{k}^{t_{\,}}(m)\;\buildrel\rm
def\over{=}\;\sum_{j=0}^{k}\frac{(2\pi
m)^{j}}{j!}\sum_{\mathfrak{p}\in\text{pt}_{j}(k)}\hskip-5.0ptc_{\mathfrak{p}}t_{\mathfrak{p}}~{}.$

It is also useful to express these in terms of the quantities in (2.2) and
(2.8). Substituting these for $x$ and $y$, we find a set of expressions listed
below. In the intermediate expressions, the following quantity appears often:

$\displaystyle r^{x_{\,}}_{(m,n),\mathfrak{p}}$ $\displaystyle\;\buildrel\rm
def\over{=}\sum_{\begin{subarray}{c}\sum_{j=1}^{n}\mathfrak{p}_{j}=\mathfrak{p}\\\
\mathfrak{p}_{1},...,\mathfrak{p}_{m}\neq\emptyset\end{subarray}}c\Bigl{(}\hskip-2.0pt\left\\{\mathfrak{p}_{j}\right\\}_{j=1}^{n}\Bigr{)}\prod_{j=1}^{n}R^{x_{\,}}_{\mathfrak{p}_{j}}~{}.$
(3.7)

The sum here is over all partitions such that their sum is equal to
$\mathfrak{p}$, with the condition that the partitions
$\mathfrak{p}_{1},...,\mathfrak{p}_{m}$ are non-empty (equivalently, there are
between 0 and $n-m$ empty partitions). The coefficients
$c(\\{\mathfrak{p}_{j}\\})$ are given by $c_{\mathfrak{q}}$, where
$\mathfrak{q}$ is the partition $\sum_{j=1}^{n}q_{j}$ formed out of the
multiplicities $q_{j}$ of distinct partitions in the set
$\\{\mathfrak{p}_{j}\\}_{j=1}^{n}$. We define functions
$r^{y}_{(m,n),\mathfrak{p}}$ and $r^{t}_{(m,n),\mathfrak{p}}$ similarly, with
the only difference being that $R^{x_{\,}}$ in replaced by $R^{y_{\,}}$ or
$R^{t_{\,}}$.

With the help of these quantities, we can express various quantities appearing
in (3.6), in a form that is convenient for later computations, for example:

$\displaystyle\eta_{k}$
$\displaystyle\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}\hskip-3.0ptN_{\mathfrak{p}}\sum_{j=1}^{l(\mathfrak{p})}\frac{(-1)^{j}}{y_{0}^{j+1}}r^{y_{\,}}_{(j,j),\mathfrak{p}}\,$
$\displaystyle\buildrel\rm
def\over{=}\,\sum_{\mathfrak{p}\in\text{pt}(k)}\bm{\eta}_{\mathfrak{p}}N_{\mathfrak{p}}~{},$
$\displaystyle\eta^{\prime}_{k}$
$\displaystyle\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}\hskip-3.0ptN_{\mathfrak{p}}\sum_{j=2}^{l(\mathfrak{p})}\frac{(-1)^{j}}{y_{0}^{j+1}}r^{y_{\,}}_{(j,j),\mathfrak{p}}\,$
$\displaystyle\buildrel\rm
def\over{=}\,\sum_{\mathfrak{p}\in\text{pt}(k)}\bm{\eta^{\prime}}_{\mathfrak{p}}N_{\mathfrak{p}}~{},$
(3.8) $\displaystyle\text{e}_{k}^{x_{\,}}(\alpha)$
$\displaystyle\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}\hskip-3.0ptN_{\mathfrak{p}}\sum_{j=0}^{l(\mathfrak{p})}\frac{(2\pi\alpha)^{j}}{j!}r^{x_{\,}}_{(j,j),\mathfrak{p}}\,$
$\displaystyle\buildrel\rm
def\over{=}\,\sum_{\mathfrak{p}\in\text{pt}(k)}\text{{\boldmath
e}}_{\mathfrak{p}}^{x_{\,}}(\alpha)\,N_{\mathfrak{p}}~{},$

with similar expressions for $\text{e}^{y}_{k}(\alpha)$ and
$\text{e}^{t}_{k}(\alpha)$. When $\mathfrak{p}=\emptyset$, it is convenient to
define exceptional values

$\displaystyle\bm{\eta}_{\emptyset}\;=\;\frac{1}{y_{0}}~{},\qquad\bm{\eta}^{\prime}_{\emptyset}\;=\;1~{}.$
(3.9)

We give some combinations of these functions their own names due to the fact
that these will appear in the instanton sums.

$\displaystyle\begin{split}\mathcal{C}_{j}(k)&\;=\;\hskip-3.0pt\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}\,\mathcal{C}_{j,\mathfrak{p}}~{},\qquad\mathcal{S}_{j}(k)\;=\;\hskip-3.0pt\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}\mathcal{S}_{j,\mathfrak{p}}~{},\qquad\mathcal{E}_{j}(k)\;=\;\hskip-3.0pt\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}\mathcal{E}_{j,\mathfrak{p}}~{},\\\\[5.0pt]
\mathcal{C}_{j,\mathfrak{p}}&\;=\;\frac{1}{Y(2\pi)^{j}}\sum_{m\in\mathfrak{p}}\frac{1}{2m^{j}}\Big{(}q_{x}^{m}\big{(}\text{{\boldmath
e}}^{x_{\,}}(\text{i}m)\hskip 1.0pt{*}\hskip 1.0pt\text{{\boldmath
e}}^{y_{\,}}(-m)\big{)}_{\mathfrak{p}-m}+q_{x}^{-m}\big{(}\text{{\boldmath
e}}^{x_{\,}}(-\text{i}m)\hskip 1.0pt{*}\hskip 1.0pt\text{{\boldmath
e}}^{y_{\,}}(-m)\big{)}_{\mathfrak{p}-m}\Big{)}~{},\\\\[5.0pt]
\mathcal{S}_{j,\mathfrak{p}}&\;=\;\frac{1}{Y(2\pi)^{j}}\sum_{m\in\mathfrak{p}}\frac{1}{2\text{i}m^{j}}\Big{(}q_{x}^{m}\big{(}\text{{\boldmath
e}}^{x_{\,}}(\text{i}m)\hskip 1.0pt{*}\hskip 1.0pt\text{{\boldmath
e}}^{y_{\,}}(-m)\big{)}_{\mathfrak{p}-m}-q_{x}^{-m}\big{(}\text{{\boldmath
e}}^{x_{\,}}(-\text{i}m)\hskip 1.0pt{*}\hskip 1.0pt\text{{\boldmath
e}}^{y_{\,}}(-m)\big{)}_{\mathfrak{p}-m}\Big{)}~{},\\\\[5.0pt]
\mathcal{E}_{j,\mathfrak{p}}&\;=\;\frac{1}{Y(2\pi\text{i})^{j}}\sum_{m\in\mathfrak{p}}\frac{1}{m^{j}}\text{{\boldmath
e}}^{t_{\,}}(\text{i}m)_{\mathfrak{p}-m}~{},\end{split}$ (3.10)

where the sums over $m\in\mathfrak{p}$ denote the sum where every $i$ with
$\mu_{i}\neq 0$ is counted once. Now we can write the instanton sums and their
derivatives compactly as

$\displaystyle\begin{split}\mathcal{I}_{1}&\;=\;-Y\sum_{k=1}^{\infty}q_{y}^{k}\,\mathcal{S}_{3}(k)~{},\qquad\mathcal{I}_{2}\;=\;\phantom{-}Y\sum_{k=1}^{\infty}q_{y}^{k}\,\mathcal{C}_{3}(k)~{},\\\\[5.0pt]
\mathcal{I}^{\prime}_{1}&\;=\;-Y\sum_{k=1}^{\infty}q_{y}^{k}\,\mathcal{C}_{2}(k)~{},\qquad\mathcal{I}^{\prime}_{2}\;=\;-Y\sum_{k=1}^{\infty}q_{y}^{k}\,\mathcal{S}_{2}(k)~{},\\\\[5.0pt]
\mathcal{I}&\;=\;\phantom{-}Y\sum_{k=1}^{\infty}q_{t}^{k}\mathcal{E}_{3}(k)~{},\qquad\;\mathcal{I}^{\prime}\;=\;\phantom{-}Y\sum_{k=1}^{\infty}q_{t}^{k}\mathcal{E}_{2}(k)~{}.\end{split}$

##  4 The D4-D2-D0 System

In this and the following section, we use the methods outlined previously to
find the perturbative solutions to the zeroth-order attractor equations and
the recurrence relations for the instanton corrections. We will consider the
orthogonality and alignment equations separately.

We start with the case where the charge vector takes the following form,
corresponding to the D4-D2-D0 system:

$\displaystyle Q_{\text{D4}}=\kappa\begin{pmatrix}\Lambda\\\ \Upsilon\\\ 0\\\
1\end{pmatrix}.$

The orthogonality equation for a charge vector of this form turns out to be
the simplest among the four different cases we study in this paper.
Consequently, this is one of the cases where we are able to find a closed-form
solution for the instanton corrections, as will be shown in detail in §6.

With regards to microstate counting, the alignment equations in this case are
particularly interesting. For example in [45, 46] the microstates of black
holes with these charges are counted by the coefficients of modular forms.
Using these results, it is possible to express the supergravity entropy in
terms of the perturbative corrections corresponding to $\sigma$ as well as the
instanton corrections. This can be used to compare the macroscopic expression
to the asymptotics of modular forms, a subject to which we intend to return
elsewhere. We discuss this in a little more detail in §7.2, where we give the
relation of our results to the Wald entropy of the black hole associated to
the attractor point we find here. We show that the leading-order asymptotics
of the coefficients of the modular forms agree with the formulae in this
section.

###  4.1 The orthogonality equation

The orthogonality equation depends only on the complex period ${\Pi}$ and not
on the real and imaginary parts separately. Therefore it is most natural to
write the orthogonality equation as a condition on the complex coordinate $t$
instead of separating this into the real and imaginary parts $x$ and $y$ as we
are obliged to do for the non-holomorphic alignment equations later.

The orthogonality equation $\Pi^{T}\,\Sigma\,Q_{\text{D4}}=0$ can be written
as

$\displaystyle
t^{2}-2t\alpha\widetilde{\beta}+\widetilde{\beta}^{2}+\frac{2}{Y}\,\mathcal{I}^{\prime}\;=\;0~{}.$
(4.1)

#### The perturbative equations

Ignoring the instanton corrections, the equation is just a quadratic.
Moreover, it is independent of $\sigma$. Of the two solutions, the solution
that can, for suitable values of $\Lambda$ and $\Upsilon$, lie near the large
complex structure point is

$\displaystyle
t_{0}\;=\;\widetilde{\beta}\left(\alpha+\sqrt{\alpha^{2}-1}\right).$ (4.2)

For this to correspond to a solution that lies in the large complex structure
region, $t_{0}$ needs to have a large imaginary part. Noting that
$\alpha\widetilde{\beta}=-\frac{Y_{110}+\Upsilon}{Y}$ is always real and
$\widetilde{\beta}$ has a phase $0$ or $\frac{\pi}{2}$, we have that the pair
$\alpha$ and $\widetilde{\beta}$ are either both real or both purely
imaginary. In fact, in the large complex structure region they cannot both be
imaginary, for then $t_{0}$ would be real. Thus we require that
$\widetilde{\beta}>0$ and $-1<\alpha<1$, in order for $t_{0}$ to have a
positive imaginary part. We will discuss the conditions for existence of
perturbative solutions near the large complex structure point in more detail
in §7.1. This said, the real and imaginary parts $x_{0}$ and $y_{0}$ can be
written as

$\displaystyle x_{0}\;=\;\widetilde{\beta}\alpha~{},\qquad
y_{0}\;=\;\widetilde{\beta}\sqrt{1-\alpha^{2}}~{}.$ (4.3)

#### Instanton corrections

The instanton corrections to the solutions of the orthogonality equation are
included as a series in $q_{t}$ by writing $t=\sum_{j}t_{j}q_{t}^{j}$. To
solve the attractor equations, we require that the coefficient of every power
of $q_{t}$ vanishes. This requirement can be written as

$\displaystyle 0$
$\displaystyle\;=\;\frac{1}{2}\sum_{\mathfrak{p}\in\text{pt}^{0}_{2}(k)}\hskip-6.0ptc_{\mathfrak{p}}t_{\mathfrak{p}}-\alpha\widetilde{\beta}t_{k}+\mathcal{E}_{2}(k)~{}.$

It is simple to solve this equation for $t_{k}$ in terms of $t_{i}$ with
$i<k$. As $\mathcal{E}_{2}(k)$ is independent of $t_{k}$, the above equation
contains $t_{k}$ multiplied by the factor $(t_{0}-\alpha\widetilde{\beta})$.
Thus $t_{k}$ is given by

$\displaystyle t_{k}$
$\displaystyle\;=\;\frac{1}{\alpha\widetilde{\beta}-t_{0}}\left(\frac{1}{2}\sum_{\mathfrak{p}\in\text{pt}_{2}(k)}\hskip-6.0ptc_{\mathfrak{p}}t_{\mathfrak{p}}+\mathcal{E}_{2}(k)\right).$

We now express $t_{k}$ in terms of the Gromov-Witten invariants in the form
(2.8). The rational functions $R^{t}_{\mathfrak{p}}$ are chosen precisely to
be themselves independent of the scaled Gromov-Witten invariants $N_{k}$. This
leads to a recurrence relation

$\displaystyle R^{t}_{\mathfrak{p}}(t_{0})$
$\displaystyle\;=\;\frac{1}{\alpha\widetilde{\beta}-t_{0}}\left(\frac{1}{2}\left(R^{t}{\circ}R^{t}\right)_{\mathfrak{p}}+\mathcal{E}_{2,\mathfrak{p}}\right).$
(4.4)

From this we see that the $R^{t}_{\mathfrak{p}}$ are rational functions of
$t_{0}$. The denominators of these rational functions are due to the overall
prefactor $1/(\alpha\widetilde{\beta}-t_{0})$, and can be written explicitly.
We can concentrate on the numerators by introducing polynomials
$P^{t}_{\mathfrak{p}}$

$\displaystyle
P^{t}_{\mathfrak{p}}(w)\;=\;-\frac{\text{i}\pi\mathbbl{\Delta}(w)^{\bm{\ell}(\mathfrak{p})}}{w}R^{t}_{\mathfrak{p}}\left(\alpha\widetilde{\beta}+\text{i}\frac{w}{\pi}\right),$

where in this subsection we set

$\displaystyle\mathbbl{\Delta}(w)$ $\displaystyle\;=\;-4Yw^{2}~{}.$

This formula differs somewhat from the relation between the polynomials and
rational functions in the other cases considered later, since here the
argument of $R^{t}_{\mathfrak{p}}$ is shifted by $\alpha\widetilde{\beta}$,
while in the other cases we just multiply the argument $v$ by an overall
factor. The reason for the shift in this case, as will soon become apparent,
is that with this definition we can later substitute $v=y_{0}$ to re-obtain
the rational functions $R_{\mathfrak{p}}^{t}(t_{0})$ satisfying the recurrence
relation (4.4). This is the consequence of the fact that the instanton
corrections depend essentially on $y_{0}$ only.

It is easy to write down the first few polynomials. These have a simple form,
and will later play an important role in finding solutions to the attractor
equations in closed form.

$\displaystyle\begin{split}P_{\\{1\\}}^{t}(w)&\;=\;\phantom{-}\hskip
2.0pt1~{},\\\\[3.0pt] P_{\\{2\\}}^{t}(w)&\;=\;\phantom{-}\hskip
2.0pt\frac{1}{4}~{},\\\\[3.0pt] P_{\\{1^{2}\\}}^{t}(w)&\;=\;-\hskip
2.0pt\frac{1}{2}\hskip 3.0pt\left(1+4w\right),\\\\[3.0pt]
P_{\\{1,2\\}}^{t}(w)&\;=\;-\hskip 2.0pt\frac{1}{4}\hskip
3.0pt\left(1+6w\right),\\\\[3.0pt]
P_{\\{1^{3}\\}}^{t}(w)&\;=\;\phantom{-}\hskip 2.0pt\frac{1}{2}\hskip
3.0pt\left(1+6w+12w^{2}\right),\\\\[3.0pt]
P_{\\{1^{2},2\\}}^{t}(w)&\;=\;\phantom{-}\hskip 2.0pt\frac{1}{8}\hskip
3.0pt\left(3+24w+64w^{2}\right),\\\\[3.0pt]
P_{\\{1^{4}\\}}^{t}(w)&\;=\;-\frac{1}{24}\left(15+120w+384w^{2}+512w^{3}\right).\end{split}$

Combining these observations, and recalling from (4.3) that
$\alpha\widetilde{\beta}-t_{0}=-\text{i}y_{0}$, we are able to write the
coordinate $t$ in the form

$\displaystyle\begin{split}t&\;=\;t_{0}+\text{i}y_{0}\sum_{k=1}^{\infty}\text{e}^{2\pi\text{i}kt_{0}}\\!\sum_{\mathfrak{p}\in\text{pt}(k)}\frac{N_{\mathfrak{p}}}{\mathbbl{\Delta}\left(\pi
y_{0}\right)^{\bm{\ell}(\mathfrak{p})}}P^{t}_{\mathfrak{p}}\left(\pi
y_{0}\right).\end{split}$

Let us also consider the real and imaginary parts of $t$. The polynomials
$P^{t}_{\mathfrak{p}}(v)$ are real for real $v$, as can be seen from the
recurrence (4.4), so the real and imaginary parts of $t$ are given by

$\displaystyle\begin{split}x\;=\;x_{0}-y_{0}\sum_{k=0}^{\infty}\text{e}^{-2\pi
ky_{0}}\sin(2\pi
kx_{0})\sum_{\mathfrak{p}\in\text{pt}(k)}\frac{N_{\mathfrak{p}}}{\mathbbl{\Delta}\left(\pi
y_{0}\right)^{\bm{\ell}(\mathfrak{p})}}P^{t}_{\mathfrak{p}}\left(\pi
y_{0}\right),\\\\[5.0pt] y\;=\;y_{0}+y_{0}\sum_{k=0}^{\infty}\text{e}^{-2\pi
ky_{0}}\cos(2\pi
kx_{0})\sum_{\mathfrak{p}\in\text{pt}(k)}\frac{N_{\mathfrak{p}}}{\mathbbl{\Delta}\left(\pi
y_{0}\right)^{\bm{\ell}(\mathfrak{p})}}P^{t}_{\mathfrak{p}}\left(\pi
y_{0}\right).\end{split}$ (4.5)

###  4.2 The alignment equations

The alignment equations are most conveniently expressed in terms of the real
coordinates $x$ and $y$. After replacing $a$ and $b$ in the equations (2.4)
via (2.5), we are left with two nontrivial conditions:

$\displaystyle\begin{split}(3x^{2}-y^{2})y-2\sigma+\frac{6}{Y}\left(x\mathcal{I}_{2}^{\prime}-2\mathcal{I}_{2}+y\mathcal{I}_{1}^{\prime}\right)-3\widetilde{\beta}^{2}y&\;=\;0~{},\\\\[10.0pt]
x+\frac{1}{Yy}\mathcal{I}_{2}^{\prime}-\alpha\widetilde{\beta}&\;=\;0~{}.\end{split}$
(4.6)

#### The perturbative equations

Neglecting the instanton contributions, the attractor equations become

$\displaystyle\begin{split}\left(3x_{0}^{2}-y_{0}^{2}\right)y_{0}-2\sigma-3\widetilde{\beta}^{2}y_{0}&\;=\;0~{},\qquad
x_{0}-\alpha\widetilde{\beta}\;=\;0~{}.\end{split}$ (4.7)

The second of these equations is solved trivially, and substituting this
solution into the first equation yields a cubic for $y_{0}$:

$\displaystyle
y_{0}^{3}-3\widetilde{\beta}^{2}(\alpha^{2}-1)y_{0}+2\sigma\;=\;0~{}.$

This makes it possible to give a simple solution for $y_{0}$ in terms of
trigonometric functions. When $\widetilde{\beta}\neq 0$ and $|\alpha|\neq 1$,
we make the substitution

$\displaystyle y_{0}\;=\;2\widetilde{\beta}\sqrt{\alpha^{2}-1}\sin\theta~{},$
(4.8)

which brings the equation to the form

$\displaystyle-4\sin^{3}\theta+3\sin\theta\;=\;\frac{\sigma}{\widetilde{\beta}^{3}(\alpha^{2}-1)^{3/2}}~{},$

and we recognise the left-hand side as $\sin 3\theta$, with $\theta$ defined
by

$\displaystyle\sin
3\theta\;=\;\frac{\sigma}{\widetilde{\beta}^{3}(\alpha^{2}-1)^{3/2}}~{}.$
(4.9)

A cubic has three solutions, corresponding here to the freedom to replace
$\theta$ by $\theta+\frac{2\pi}{3}$ or $\theta+\frac{4\pi}{3}$. We want to
choose a branch so that $y_{0}$ is real and $x_{0}+\text{i}y_{0}$ lies in the
large complex structure region. This choice depends on the values of the
charges and Yukawa couplings, and so on $\alpha,\widetilde{\beta},$ and
$\sigma$, and on the size of the large complex structure region. To better
describe these various possibilities we tabulate different relevant cases in
Table 3 below.

Charge data | Positive solutions for $y_{0}$ | Definition of $\theta$
---|---|---
$0{\leqslant}\sigma{\leqslant}\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}$ |  $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\theta$, $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\left(\theta{+}\frac{2\pi}{3}\right)\\!$ | $\theta=\frac{1}{3}\sin^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}{\leqslant}\sigma{\leqslant}0$ | $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\left(\theta{+}\frac{4\pi}{3}\right)$ | $\theta=\frac{1}{3}\sin^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$-\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}{\leqslant}\sigma{\leqslant}0$ | $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\left(\theta{+}\frac{2\pi}{3}\right)$ | $\theta=\frac{1}{3}\sin^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$0{\leqslant}\sigma{\leqslant}-\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}$ |  $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\theta$, $2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sin\left(\theta{+}\frac{4\pi}{3}\right)\\!$ | $\theta=\frac{1}{3}\sin^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$\sigma{<}\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}{<}0$ | $-2\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\cosh\nu$ |  $\theta{=}\frac{3\pi}{2}{+}\text{i}\nu$, $\nu{=}\frac{1}{3}\cosh^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$\sigma{<\\!}-\widetilde{\beta}^{3}(\alpha^{2}{-}1)^{3/2}{<}0$ | $2\widetilde{\beta}\sqrt{\alpha^{2}-1}\sin\theta$ |  $\theta=\frac{5\pi}{6}{+}\text{i}\nu$ $\nu{=}\frac{1}{3}\cosh^{-1}\frac{\sigma}{(\widetilde{\beta}\sqrt{\alpha^{2}-1})^{3}}$
$\sigma{<}0$, $\text{i}\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\in\mathbbl{R}$ | $2|\widetilde{\beta}\sqrt{\alpha^{2}{-}1}\sinh\nu|$ |  $\theta{=}\text{i}\nu$, $\nu{=}\frac{1}{3}\sinh^{-1}\frac{\sigma}{|\widetilde{\beta}\sqrt{\alpha^{2}-1}|^{3}}$

Table 3: Solutions to the perturbative attractor equations (4.7). Whenever an
inverse trigonometric function appears in this list, we take the principal
branch value. We have only included here the solutions for which $y_{0}>0$, as
these are the only solutions that can possibly lie within the large complex
structure region. However, even if a solution is included here, it may not lie
near the large complex structure point for all Calabi-Yau manifolds.

As we do not have a general formula for the radius of the large complex
structure region, we will include all solutions for which a positive solution
for $y_{0}$ exists.

In addition to the cases listed in Table 3, there are degenerate cases with
$\widetilde{\beta}=0$ or $\alpha^{2}=1$. In the case $\widetilde{\beta}=0$, we
have that $\alpha\to\infty$ in such a way that

$\displaystyle\alpha\widetilde{\beta}\;=\;-\frac{Y_{110}+\Upsilon}{Y_{111}}~{},\qquad\text{and}\qquad\widetilde{\beta}\sqrt{\alpha^{2}-1}\,\to\,\left|\frac{Y_{110}+\Upsilon}{Y_{111}}\right|.$

These expressions resolve the ambiguities in Table 3.

In the two cases where either $\alpha^{2}{\;=\;}1$ or
$\widetilde{\beta}{\;=\;}\widetilde{\beta}\alpha{\;=\;}0$, both with
$\sigma<0$, the only possibly admissible solution is

$\displaystyle x_{0}\;=\;0~{},\qquad y_{0}\;=\;|\sigma|^{1/3}.$

The values of charges that are not included in the Table 3, or among the two
degenerate cases, do not have a solution which lies near the large complex
structure point. We provide some intuition for the situations where multiple
$y_{0}$ are possible for a set of charges in Figure 3.

$1$ $y$ $z=f(y)$ $-y^{3}+4y$
$\sigma\widetilde{\beta}^{-3}(\alpha^{2}-1)^{-3/2}$ Figure 3: In the case
where $\beta\in\mathbbl{R}$ and $|\alpha|>1$ the number of positive solutions
for $y_{0}$ varies, subject to the value of $\sigma$. The red line moves up
and down as $\sigma$ is varied, with our solution being the value of $y$ such
that the red line intersects the blue curve. As can be seen from the figure,
when the red line lies in the region $0<z<1$ there are two positive solutions,
but when $z>1$ there are no positive solutions. A similar plot can be borne in
mind for other charge ranges.

It is also useful to expand $y_{0}$ as a series in $\sigma$. This gives a
perturbative expansion for $y_{0}$ which is convergent when
$\sigma/(\widetilde{\beta}\sqrt{\alpha^{2}-1})$ is small. This ratio is
suitably small in the large charge ratio limits, for example.

If $\sigma{\;=\;}0$, then there are solutions to (4.7) with positive $y_{0}$
only if $\widetilde{\beta}^{2}\left(\alpha^{2}-1\right)>0$. These solutions
with $\sigma{\;=\;}0$ are

$x_{00}\;=\;\widetilde{\beta}\alpha~{},\qquad
y_{00}\;=\;\sqrt{3}\left|\widetilde{\beta}\sqrt{\alpha^{2}-1}\right|.$

This $y_{00}$ is the zeroth-order term in the $\sigma$ expansion, with the
full series given by

$y_{0}\;=\;y_{00}\left[1-\frac{1}{12\pi}\sum_{n=1}^{\infty}\frac{\Gamma(\frac{n}{2}-\frac{1}{6})\,\Gamma(\frac{n}{2}+\frac{1}{6})}{n!}\left(\frac{2\,\sqrt{27}\,\sigma}{y_{\text{00}}^{3}}\right)^{n}\;\right].$
(4.10)

By separating the sum into odd and even terms, we find an expression in terms
of hypergeometric functions

$\frac{y_{0}}{y_{00}}\;=\;{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{0}{\frac{1}{6},\,-\frac{1}{6}}{\frac{1}{2}};~{}\frac{27\sigma^{2}}{y_{00}^{6}}\right)-\frac{\sigma}{y_{00}^{3}}\,{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{0}{\frac{1}{3},\,\frac{2}{3}}{\frac{3}{2}};~{}\frac{27\sigma^{2}}{y_{00}^{6}}\right).$

#### Instanton corrections

Having written the alignment equations in terms of $x$ and $y$, it turns out
that the most convenient way of writing the instanton corrections is as a
series in $q_{y}$. In other aspects the procedure for solving the alignment
equations is nearly identical to that used for the orthogonality equation.

Using the $q_{y}$ expansions (2.1) for $x$ and $y$, the second instanton-
corrected equation (4.6) becomes

$\displaystyle\sum_{k=1}^{\infty}q_{y}^{k}\left(x_{k}-\sum_{j=1}^{k}\eta_{k-j}\,\mathcal{S}_{2}(j)\right)\;=\;0~{}.$

Demanding that the coefficient of $q_{y}^{k}$ vanishes gives that

$\displaystyle x_{k}\;=\;\sum_{j=1}^{k}\eta_{k-j}\,\mathcal{S}_{2}(j)~{}.$

This lends itself to an iterative solution as the left-hand side only involves
$x_{j}$ and $y_{j}$ with $j<k$.

Next we use the $N$-expansions (2.2), (3.8), and (3.10) to give $x_{j}$,
$\eta_{j}$, and $\mathcal{S}_{2}(j)$ in terms of the rational functions
$R^{x_{\,}}_{\mathfrak{p}}$ and $R^{y_{\,}}_{\mathfrak{p}}$:

$\displaystyle\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}R^{x_{\,}}_{\mathfrak{p}}\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}(\bm{\eta}\hskip
1.0pt{*}\hskip 1.0pt\mathcal{S}_{2})_{\mathfrak{p}}~{}.$

The convolution $(\bm{\eta}\hskip 1.0pt{*}\hskip
1.0pt\mathcal{S}_{2})_{\mathfrak{p}}$ involves $R^{x_{\,}}_{\mathfrak{q}}$ and
$R^{y_{\,}}_{\mathfrak{q}}$ with $|\mathfrak{q}|<k$, so $R^{x}_{\mathfrak{p}}$
with $|\mathfrak{p}|=k$ appears only on the right-hand side. By construction,
the $R^{x_{\,}}_{\mathfrak{p}}$ are themselves independent of the Gromov-
Witten invariants, so the coefficients of different $N_{\mathfrak{p}}$ satisfy
the equation independently. Thus we have

$\displaystyle R^{x_{\,}}_{\mathfrak{p}}$ $\displaystyle\;=\;(\bm{\eta}\hskip
1.0pt{*}\hskip 1.0pt\mathcal{S}_{2})_{\mathfrak{p}}~{},$ (4.11)

which determines the $R^{x_{\,}}_{\mathfrak{p}}$ recursively.

The requirement that the coefficient of $q_{y}^{k}$ vanishes in the first
instanton-corrected equation (4.6) can be written as

$\displaystyle\begin{split}0&\;=\;\frac{1}{6}\sum_{\mathfrak{p}\in\text{pt}^{0}_{2}(k)}\hskip-8.0ptc_{\mathfrak{p}}(3x_{\mathfrak{p}}-y_{\mathfrak{p}})-\frac{\sigma}{3}\eta_{k}-2\sum_{i=1}^{k}\eta_{k-i}\,\mathcal{C}_{3}(i)-\mathcal{C}_{2}(k)-\hskip-12.0pt\sum_{0\leq
j\leq i\leq
k-1}\hskip-12.0ptx_{i-j}\,\eta_{j}\,\mathcal{S}_{2}(k-i)~{}.\end{split}$

Here $y_{k}$ appears only in $\eta_{k}$ and in the first sum when
$\mathfrak{p}$ takes the value $\mathfrak{p}=k+0$. Thus $y_{k}$ appears
multiplied by the factor

$\frac{\sigma}{3y_{0}^{2}}-\frac{1}{3}y_{0}\;=\;-\frac{1}{3}y_{0}(1-\zeta)~{}.$

We rearrange the equation to express $y_{k}$ in terms of $x_{i}$ with
$i\leqslant k$ and $y_{i}$ with $k<i$.

$y_{k}{=}\frac{3}{y_{0}(1{-}\zeta)}\hskip-2.0pt\left(\hskip-2.0pt\frac{1}{6}\sum_{\mathfrak{p}\in\text{pt}_{2}(k)}\hskip-8.0ptc_{\mathfrak{p}}(3x_{\mathfrak{p}}{-}y_{\mathfrak{p}}){+}x_{0}x_{k}{-}\frac{\sigma}{3}\hskip
1.0pt\eta^{\prime}_{k}-2\sum_{i=1}^{k}\eta_{k-i}\,\mathcal{C}_{3}(i)-\mathcal{C}_{2}(k)-\hskip-12.0pt\sum_{0\leqslant
j\leqslant i\leqslant
k-1}\hskip-12.0ptx_{i-j}\,\eta_{j}\,\mathcal{S}_{2}(k{-}i)\hskip-2.0pt\right).$

We have solved for $x_{k}$ in terms of $x_{i},y_{i}$ with $i<k$ above. This is
therefore a well-defined recurrence relation for $y_{k}$. To find the
recurrence for $R^{y_{\,}}_{\mathfrak{p}}$, we use the expressions (2.2) to
obtain

$\displaystyle\begin{split}R^{y_{\,}}_{\mathfrak{p}}(x_{0},y_{0})&{=}\frac{3}{y_{0}(1{-}\zeta)}\left(\frac{1}{2}r^{x_{\,}}_{(2,2),\mathfrak{p}}{-}\frac{1}{6}r^{y_{\,}}_{(2,2),\mathfrak{p}}{+}x_{0}R^{x_{\,}}_{\mathfrak{p}}{-}\frac{y_{0}^{3}\zeta}{3}\,\bm{\eta}^{\prime}_{\mathfrak{p}}{-}2\left(\mathcal{C}_{3}\hskip
1.0pt{*}\hskip
1.0pt\bm{\eta}\right)_{\mathfrak{p}}\right.\hskip-2.0pt-\mathcal{C}_{2\mathfrak{p}}{-}\left(\mathcal{S}_{2}\hskip
1.0pt{*}\hskip 1.0ptR^{x_{\,}}\\!\\!\hskip 1.0pt{*}\hskip
1.0pt\bm{\eta}\right)_{\mathfrak{p}}\hskip-2.0pt\biggr{)}.\end{split}$ (4.12)

The denominators are again simple, given essentially by the prefactor
appearing in the expression above. We therefore extract the numerators to give
us polynomials.

$\displaystyle P^{x_{\,}}_{\mathfrak{p}}(u,v)$
$\displaystyle\;=\;\phantom{-}Y\pi
vR^{x_{\,}}_{\mathfrak{p}}\left(\frac{u}{\pi},\frac{v}{\pi}\right),\ \ $
$\displaystyle l(\mathfrak{p})=1~{},$ (4.13) $\displaystyle
P^{x_{\,}}_{\mathfrak{p}}(u,v)$
$\displaystyle\;=\;-\frac{\pi\mathbbl{\Delta}(v)^{\bm{\ell}(\mathfrak{p})}}{v^{2}(1-\zeta)^{3}}\,R^{x_{\,}}_{\mathfrak{p}}\left(\frac{u}{\pi},\frac{v}{\pi}\right),\
\ $ $\displaystyle l(\mathfrak{p})>1~{},$ $\displaystyle
P^{y_{\,}}_{\mathfrak{p}}(u,v)$
$\displaystyle\;=\;-\frac{\pi\mathbbl{\Delta}(v)^{\bm{\ell}(\mathfrak{p})}}{v(1-\zeta)}\,R^{y_{\,}}_{\mathfrak{p}}\left(\frac{u}{\pi},\frac{v}{\pi}\right),$

where, in this subsection,

$\displaystyle\mathbbl{\Delta}(v)$ $\displaystyle\;=\;Yv^{3}(1-\zeta)^{2}.$

We record the first few polynomials

$\displaystyle\begin{split}P_{\\{\hskip 2.0pt1\hskip
2.0pt\\}}^{x_{\,}}(u,v)&\;=\phantom{-}\;\frac{1}{4}\sin(2u)~{},\\\\[5.0pt]
P_{\\{\hskip 2.0pt2\hskip
2.0pt\\}}^{x_{\,}}(u,v)&\;=\phantom{-}\frac{1}{16}\sin(4u)~{},\\\\[5.0pt]
P_{\\{1^{2}\\}}^{x_{\,}}(u,v)&\;=-\frac{1}{32}\Big{(}3+9v+2(4-\zeta)v^{2}\Big{)}\sin(4u)~{},\\\\[5.0pt]
P_{\\{\hskip 2.0pt1\hskip
2.0pt\\}}^{y_{\,}}(u,v)&\;=\phantom{-}\;\frac{3}{4}(1+v)\cos(2u)~{},\\\\[5.0pt]
P_{\\{\hskip 2.0pt2\hskip
2.0pt\\}}^{y_{\,}}(u,v)&\;=\phantom{-}\frac{3}{32}(1+2v)\cos(4u)~{},\\\\[5.0pt]
P_{\\{1^{2}\\}}^{y_{\,}}(u,v)&\;=\phantom{-}\frac{3}{64}\Big{(}9+6(4-\zeta)v+3(8-4\zeta-\zeta^{2})v^{2}+4(2-\zeta-\zeta^{2})v^{3}\\\\[5.0pt]
&\phantom{-}\phantom{-}\phantom{-}\hskip
3.0pt+\left(9+6(4-\zeta)v+3(10-8\zeta+\zeta^{2})v^{2}+4(4-5\zeta+\zeta^{2})v^{3}\right)\cos(4u)\hskip-1.0pt\Big{)}~{}.\end{split}$

In summary, the full solution is

$\displaystyle\begin{split}x&\;=\;x_{0}-\sum_{k=0}^{\infty}\text{e}^{-2\pi
ky_{0}}\left(\sum_{\mathfrak{p}\in\text{pt}_{>1}(k)}\hskip-5.0ptN_{\mathfrak{p}}\frac{\pi
y_{0}^{2}(1-\zeta)^{3}}{\mathbbl{\Delta}(\pi
y_{0})^{\bm{\ell}(\mathfrak{p})}}P^{x_{\,}}_{\mathfrak{p}}(\pi x_{0},\pi
y_{0})-\frac{N_{k}P^{x_{\,}}_{k}(\pi x_{0},\pi
y_{0})}{Y\pi^{2}y_{0}}\right),\\\\[5.0pt]
y&\;=\;y_{0}-\sum_{k=0}^{\infty}\text{e}^{-2\pi
ky_{0}}\hskip-5.0pt\sum_{\mathfrak{p}\in\text{pt}(k)}\hskip-5.0ptN_{\mathfrak{p}}\frac{y_{0}(1-\zeta)}{\mathbbl{\Delta}(\pi
y_{0})^{\bm{\ell}(\mathfrak{p})}}P^{y_{\,}}_{\mathfrak{p}}(\pi x_{0},\pi
y_{0})~{},\end{split}$

where $x_{0}$ and $y_{0}$ are the perturbative solutions to (4.7). The
polynomials $P^{x_{\,}}_{\mathfrak{p}}$ and $P^{y_{\,}}_{\mathfrak{p}}$ are
given in terms of the rational functions $R^{x_{\,}}_{\mathfrak{p}}$ and
$R^{y_{\,}}_{\mathfrak{p}}$ in (4.13). These, in turn, are given by the
recurrence relations (4.11) and (4.12).

##  5 The D6-D2-D0 System

In this section, we consider a charge vector that corresponds to a black hole
consisting of some D6 branes bound to a set of D2 and D0 branes. In other
words, we will take the charge vector to be given by

$\displaystyle Q_{D6}\;=\;\kappa\begin{pmatrix}\Lambda\\\ \Upsilon\\\ 1\\\
0\end{pmatrix}.$

Although for generic charges of this form the alignment equations are the most
complicated of the four cases studied here, these will simplify considerably
when we specialise to the important special case $\Lambda=0$, corresponding to
attractor points lying on the imaginary axis of the moduli space. This special
case allows us to present a closed-form solution to the alignment equations,
as we will discuss in detail in §6. The equations in this section are of
interest for studying rank two attractor points as the explicit example found
recently in [15] has a charge vector of this form with $\Lambda=0$.

###  5.1 The orthogonality equation

The condition that the central charge vanishes is again best written as a
single complex equation

$\displaystyle
t^{3}-3\beta^{2}t-2\text{i}\sigma+2\beta^{3}\gamma-\frac{12}{Y}\,\mathcal{I}+\frac{6}{Y}\,t\mathcal{I}^{\prime}\;=\;0~{}.$
(5.1)

#### The perturbative equations

Neglecting the instanton contributions from this equation leaves a cubic
equation, which is already of a form which, when $\beta\neq 0$, we can solve
using a trigonometric substitution as in §4. We write the equation as

$\displaystyle-4\left(\frac{t_{0}}{2\beta}\right)^{3}+3\frac{t_{0}}{2\beta}\;=\;\gamma-\text{i}\frac{\sigma}{\beta^{3}}~{},$
(5.2)

and the solution is given by

$t_{0}\;=\;2\beta\sin\theta~{},\qquad\text{where}\qquad\sin
3\theta\;=\;\gamma-\text{i}\frac{\sigma}{\beta^{3}}~{}.$

Again, the three solutions correspond to $\theta$, $\theta+\frac{2\pi}{3}$ and
$\theta+\frac{4\pi}{3}$. In addition, we shall only be concerned with the
values of $\theta$ that give a $t_{0}$ potentially lying in the large complex
structure region, that is with $\text{Im}\,t_{0}>0$. Whether such a solution
actually lies within the large complex structure region depends on the exact
value of $t_{0}$ and the radius of the large complex structure region. It is
useful to also write down the real and imaginary parts of the coordinate
$t_{0}$ explicitly. In order to do this we will write

$\theta\;=\;\theta_{1}+\text{i}\theta_{2}~{},$

with $\theta_{1}$ and $\theta_{2}$ real. The range of values $\theta_{1}$ and
$\theta_{2}$ take depends on the solution that is chosen. The real coordinates
$x_{0}$ and $y_{0}$ are then given by Table 4.

Phase of $\beta$ | $x_{0}$ | $y_{0}$
---|---|---
$~{}\beta\in\mathbbl{R}^{+}$ | $\phantom{-}2\beta\sin\theta_{1}\cosh\theta_{2}$ | $2\beta\cos\theta_{1}\sinh\theta_{2}$
$\beta\in\text{i}\mathbbl{R}^{+}$ | $-2|\beta|\cos\theta_{1}\sinh\theta_{1}$ | $2|\beta|\sin\theta_{1}\cosh\theta_{2}$

Table 4: The real and imaginary parts of $t_{0}$. We refrain in this case from
giving as detailed a table as in the section on the D4 alignment equations.
The expressions are conceptually straightforward but typographically lengthy.

One should take a suitable branch of
$\arcsin\left(\gamma-\text{i}\frac{\sigma}{\beta^{3}}\right)$ such that
$y_{0}$ is positive. For all possible $\gamma$, $\sigma$, and nonzero $\beta$
there is at least one suitable choice, and at most two.

The analysis runs slightly differently when $\beta$ vanishes. When this
occurs, $\gamma$ necessarily tends to infinity in such a way that
$\beta^{3}\gamma\to-\frac{3\Lambda}{Y}$, which is a finite constant. In this
case the perturbative equations reduce to

$t_{0}^{3}\;\buildrel\rm
def\over{=}\;r_{0}^{3}\,\text{e}^{3\text{i}\phi_{0}}=-2\beta^{3}\gamma+2\text{i}\sigma~{},\qquad\text{where
}\,r_{0}>0\,\text{ and }\,0\leqslant\phi_{0}<\frac{2}{3}\pi~{}.$

The solutions with positive imaginary part are then obtained by choosing the
correct branch of the cube root, and are given by the table.

Charge Data | $t_{0}$
---|---
$\beta^{3}\gamma<0$ and $\sigma=0$ | $r_{0}\,\text{e}^{\text{i}\left(\phi_{0}+\frac{2}{3}\pi\right)}$
$\sigma>0$ | $r_{0}\,\text{e}^{\text{i}\phi_{0}},\;\;r_{0}\,\text{e}^{\text{i}\left(\phi_{0}+\frac{2}{3}\pi\right)}$
otherwise | $r_{0}\,\text{e}^{\text{i}\phi_{0}}$

Table 5: Solutions with positive imaginary part when $\beta$ vanishes.

We turn now to the discussion of solving the equations (5.1) for $y_{0}$ as a
power series in $\sigma$. There are a number of cases that require separate
consideration. These are listed in Table 6.

Charge data | $x_{00}$ | $y_{00}$ | Definition of $\nu$
---|---|---|---
$~{}\beta\in\mathbbl{R}^{+}$, $\gamma>1$ | $\phantom{-}\beta\cosh\nu$ | $\sqrt{3}\beta\sinh\nu$ | $\hskip-5.0pt\nu=\frac{1}{3}\cosh^{-1}\gamma$
$\beta\in\mathbbl{R}^{+}$, $\gamma<-1$ | $-\beta\cosh\nu$ | $\sqrt{3}\beta\sinh\nu$ | $\phantom{-}\hskip-5.0pt\nu=\frac{1}{3}\cosh^{-1}-\gamma$
$\beta=\text{i}\mathbbl{R}^{+}$ | $\phantom{-}|\beta|\sinh\nu$ | $\sqrt{3}|\beta|\cosh\nu$ | $\phantom{-}\nu=\frac{1}{3}\sinh^{-1}\text{Im}\,\gamma$
$\beta=0$ | $\frac{\nu}{2}\,|2\beta^{3}\gamma|^{1/3}$ | $\frac{\sqrt{3}}{2}\,|2\beta^{3}\gamma|^{1/3}$ | $\hskip-17.0pt\nu=\text{sign}\,\beta^{3}\gamma$

Table 6: Table of solutions for $x_{00}$ and $y_{00}$ with $y_{00}>0$. All the
inverse trigonometric functions appearing in the table are understood to have
their values in the principal branch. The case $|\gamma|<1$ does not appear in
the table as there are then no solutions with $\text{Im}\,t_{0}>0$.

We now give the $\sigma$-expansions around these zeroth-order solutions. In
this context, let us write $P_{n}(z)$ for the Jacobi polynomial
$P_{n}^{(\frac{1}{3},-\frac{1}{3})}(z)$. Additionally, we introduce the
quantity

$\text{T}(k)\;=\;2\frac{d^{k}}{dz^{k}}\sin(z)\bigg{|}_{z=\frac{2\pi}{3}}\;=\;\begin{cases}\phantom{-}\sqrt{3}~{},&k\;=\;0\text{
mod }4\\\ \hskip 9.0pt-1~{},&k\;=\;1\text{ mod }4\\\
-\sqrt{3}~{},&k\;=\;2\text{ mod }4\\\ \hskip
10.0pt\phantom{-}1~{},&k\;=\;3\text{ mod }4.\end{cases}$

It is shown in appendix D that when $\beta$ is real we can write

$\frac{y_{0}}{\beta}\;=\;\sqrt{3}\sinh\nu+\sum_{n=1}^{\infty}\frac{\text{T}(n)}{6n}\bigg{[}\text{e}^{\nu}P_{n-1}\big{(}{-}\coth
3\nu\big{)}{-}\text{e}^{-\nu}P_{n-1}\big{(}\coth
3\nu\big{)}\bigg{]}\left(\frac{\sigma}{\beta^{3}\sinh 3\nu}\right)^{n}.$ (5.3)

In the case where $\beta\in\text{i}\mathbbl{R}^{+}$, we have instead that

$\frac{y_{0}}{|\beta|}\;=\;\sqrt{3}\cosh\nu+\sum_{n=1}^{\infty}\frac{\text{T}(n)}{6n}\bigg{[}\text{e}^{\nu}P_{n-1}\big{(}{-}\tanh
3\nu\big{)}{+}\text{e}^{-\nu}P_{n-1}\big{(}\tanh
3\nu\big{)}\bigg{]}\left(\frac{\sigma}{|\beta|^{3}\cosh 3\nu}\right)^{n}.$
(5.4)

In the $\beta$=0 case, we have

$t_{00}\;=\;\exp\left[\left(1-\frac{\nu}{3}\right)\frac{\pi\text{i}}{2}\right]\left(\frac{6|\Lambda|}{Y}\right)^{1/3},\;\;\;\;t_{0}\;=\;t_{00}\left(1+\frac{\text{i}Y\sigma}{3\Lambda}\right)^{1/3}.$

This can be expanded in $\sigma$. Picking out the imaginary part, we can write

$y_{0}\;=\;\left(\frac{3|\Lambda|}{4Y}\right)^{1/3}\sum_{n=0}^{\infty}\text{T}(n)\frac{\left(-\frac{1}{3}\right)_{n}}{n!}\left(-\frac{\sigma
Y}{3|\Lambda|}\right)^{n}.$

The polynomials $P_{n}$ satisfy useful identities that follow from the general
properties of Jacobi polynomials, and which we briefly recall.

The $P_{n}$ satisfy the differential equation:

$\displaystyle(1-z^{2})\frac{d^{2}}{dz^{2}}P_{n}(z)-\left(2z+\frac{2}{3}\right)\frac{d}{dz}P_{n}(z)+n(n+1)P_{n}(z)\;=\;0~{}.$

In addition, these are easily generated by means of a recurrence relation

$\displaystyle
9n^{2}(n-1)P_{n}(z)+9n(n-1)(1-2n)zP_{n-1}(z)+n(3n-2)(3n-4)P_{n-2}(z)\;=\;0~{},$

with initial values

$\displaystyle P_{0}(z)\;=\;1~{},\qquad\text{and}\qquad
P_{1}(z)\;=\;\frac{1}{3}+z.$

After solving for $y_{0}$, the $\sigma$-expansions for $x_{0}$ can be easily
obtained. After separating the equation (5.2) into its real and imaginary
parts, one of these will be a quadratic equation for $x_{0}$. Precisely which
one is quadratic depends on the phase of $\beta$. Solving this gives

$\displaystyle
x_{0}\;=\;\pm\sqrt{\beta^{2}+\frac{y_{0}^{3}+2\sigma}{3y_{0}}}~{},$

where the $\pm$ sign is chosen so that consistency with Table 6 is maintained
in the $\sigma\rightarrow 0$ limit.

#### Instanton Corrections

As before, we require that the coefficient of every power vanishes in the
$q_{t}$ expansion and this leads to recurrence relations for
$R^{t}_{\mathfrak{p}}$. Explicitly, the condition that the coefficient of
$q_{t}^{k}$ vanishes can be written compactly as

$\displaystyle\frac{1}{6}\sum_{\mathfrak{p}\in\text{pt}_{3}^{0}(k)}\hskip-6.0ptc_{\mathfrak{p}}t_{\mathfrak{p}}-\frac{1}{2}\beta^{2}t_{k}+\sum_{i=0}^{k}t_{k-i}\mathcal{E}_{2}(i)-2\mathcal{E}_{3}(k)\;=\;0~{}.$

Here $t_{k}$ appears in the first term when
$\mathfrak{p}{\;=\;}0\,{+}\,0\,{+}\,k$, with the coefficient
$c_{0{+}0{+}k}=3$, and also in the second term, so overall it is multiplied by
$\frac{1}{2}(t_{0}^{2}-\beta^{2})$. Using this, it is easy to write down the
recurrence relation for $t_{k}$:

$\displaystyle t_{k}$
$\displaystyle\;=\;\frac{2}{\beta^{2}-t_{0}^{2}}\left(\frac{1}{6}\sum_{\mathfrak{p}\in\text{pt}_{3}(k)}\hskip-6.0ptc_{\mathfrak{p}}t_{\mathfrak{p}}+\frac{1}{2}t_{0}\hskip-5.0pt\sum_{\mathfrak{p}\in\text{pt}_{2}(k)}\hskip-6.0ptt_{\mathfrak{p}}+\sum_{i=0}^{k}t_{k-i}\mathcal{E}_{2}(i)-2\mathcal{E}_{3}(k)\right).$

We can use this to derive the recurrence relation for the functions
$R^{t}_{\mathfrak{p}}$,

$\displaystyle R^{t}_{\mathfrak{p}}$
$\displaystyle\;=\;\frac{2}{\beta^{2}-t_{0}^{2}}\left(\frac{1}{6}r^{t}_{(2,3),\mathfrak{p}}+\left(\mathcal{E}_{2}\hskip
1.0pt{*}\hskip
1.0ptR^{t}\right)_{\mathfrak{p}}-2\mathcal{E}_{3,\mathfrak{p}}\right)~{}.$
(5.5)

Once again we can define functions $P^{t}_{\mathfrak{p}}$, which are
polynomials in $t_{0}$, that correspond to the numerators of
$R^{t}_{\mathfrak{p}}$. The denominators of $R^{t}_{\mathfrak{p}}$ with
$t_{0}$ dependence come from the overall prefactor in (5.5), so we define

$\displaystyle P^{t}_{\mathfrak{p}}(w)$
$\displaystyle\;=\;\frac{\text{i}\pi\,\mathbbl{\Delta}(\text{i}w)^{\bm{\ell}(\mathfrak{p})}}{\pi^{2}\beta^{2}+w^{2}}R^{t}_{\mathfrak{p}}\left(\text{i}\frac{w}{\pi}\right),$

where in this section

$\displaystyle\mathbbl{\Delta}(w)$
$\displaystyle\;=\;Y\left(\pi^{2}\beta^{2}-w^{2}\right)^{2}.$

These polynomials can be then computed from the recurrence relation (5.5),
together with the above relation between the rational functions
$R^{t}_{\mathfrak{p}}$ and the polynomials $P^{t}_{\mathfrak{p}}$. We list
here the first few polynomials:

$\displaystyle\begin{split}P^{t}_{\\{1\\}}(w)&\;=\;\hskip
4.0pt\frac{1}{2}\hskip 3.0pt\left(1+w\right),\\\\[5.0pt]
P^{t}_{\\{2\\}}(w)&\;=\;\hskip 2.0pt\frac{1}{16}\hskip
1.0pt\left(1+2w\right),\\\\[5.0pt]
P^{t}_{\\{1^{2}\\}}(w)&\;=\;\hphantom{1}\frac{1}{4}\hskip
3.0pt\left(1+w\right)\Big{(}w+2w^{2}+2w^{3}+\pi^{2}\beta^{2}\left(1+2w\right)\Big{)},\\\\[5.0pt]
P^{t}_{\\{1,2\\}}(w)&\;=\;\;\frac{1}{32}\left(2w+9w^{2}+18w^{3}+12w^{4}+\pi^{2}\beta^{2}\left(3+14w+12w^{2}\right)\vphantom{\left(w^{2}\right)^{2}}\hskip-3.0pt\right).\end{split}$

The location of the attractor point $t$ can be written in terms of these
polynomials, giving us a relatively simple solution to the attractor
equations.

$\displaystyle\begin{split}t&\;=\;t_{0}-\text{i}\pi\left(\beta^{2}-t_{0}^{2}\right)\sum_{k=1}^{\infty}\text{e}^{2\pi\text{i}kt_{0}}\sum_{\mathfrak{p}\in\text{pt}(k)}\frac{N_{\mathfrak{p}}}{\mathbbl{\Delta}(\pi
t_{0})^{\bm{\ell}(\mathfrak{p})}}P_{\mathfrak{p}}^{t}(-\text{i}\pi
t_{0})~{}.\end{split}$

###  5.2 The alignment equations

As discussed in §2 we use (2.5) to reduce the alignment equations to a pair:

$\displaystyle\begin{split}x(x^{2}+y^{2})-\frac{\sigma
x}{y}+\frac{3}{Y}\left(2\mathcal{I}_{1}-\frac{2x}{y}\mathcal{I}_{2}+\frac{x^{2}+y^{2}}{y}\mathcal{I}_{2}^{\prime}\right)-\beta^{3}\gamma&\;=\;0~{},\\\\[10.0pt]
\phantom{-}x^{2}+y^{2}+\frac{2}{Y}\left(\frac{x}{y}\mathcal{I}_{2}^{\prime}-\mathcal{I}_{1}^{\prime}\right)-\beta^{2}\phantom{\gamma}&\;=\;0~{}.\end{split}$
(5.6)

#### The perturbative equations

In this case, solving the perturbative equations reduces to finding the roots
of a quartic recorded in (5.10), which allows for a closed-form solution in
terms of radicals. However, the resulting expression is not particularly
illuminating. We will first solve the equations in the special case $\sigma=0$
and use the solutions $x_{00}$, $y_{00}$ to give us the explicit solution in
terms of radicals, as well as an expansion in $\sigma$. Each of these
formulations of the solution is useful for slightly different purposes
although ultimately both solutions are of course equivalent.

Neglecting instantons, the equations for $x_{0}$ and $y_{0}$ read

$\displaystyle x_{0}\,y_{0}$
$\displaystyle\;=\;\beta\gamma\,y_{0}+\frac{\sigma}{\beta^{2}}\,x_{0}~{},$
$\displaystyle x_{0}^{2}+y_{0}^{2}$ $\displaystyle\;=\;\beta^{2}~{}.$ In the
special case $\sigma=0$, these are solved by $\displaystyle x_{0}$
$\displaystyle\;=\;x_{00}\;=\;\beta\gamma~{},$ $\displaystyle y_{0}$
$\displaystyle\;=\;y_{00}\;=\;\beta\sqrt{1-\gamma^{2}}~{}.$ (5.7)

We also note for later discussion that the second equation in (5.2) tells us
that an attractor point only exists for this brane configuration if $\beta$ is
real, and therefore so is $\gamma$. The modulus $|t_{0}|{\;=\;}\beta$ is
independent of $\sigma$. For $\sigma{\;=\;}0$, we are also forced to have
${|\gamma|<1}$, and so we introduce a real angle $\phi_{00}\;{\in}\;(0,\pi)$
by

$\gamma\;=\;\cos\phi_{00}~{},\qquad\text{ so that }\qquad
x_{00}\;=\;\beta\cos\phi_{00}\;\;\text{ and
}\;\;y_{00}\;=\;\beta\sin\phi_{00}~{}.$

It is optimal to work with the rescaled parameter
$\widehat{\sigma}=\frac{\sigma}{\beta^{3}}$. We define also the rescaled
coordinates

$\widehat{x}\;=\;\frac{x_{0}}{\beta}~{},\hskip
30.0pt\widehat{y}\;=\;\frac{y_{0}}{\beta}~{}.$

In the rescaled variables the alignment equations are

$\widehat{x}\widehat{y}-\gamma\widehat{y}-\widehat{\sigma}\widehat{x}\;=\;0~{},\hskip
30.0pt\widehat{x}^{2}+\widehat{y}^{2}\;=\;1~{}.$ (5.8)

The first equation above provides an expression for $\widehat{x}$ in terms of
$\widehat{y}$:

$\widehat{x}\;=\;\frac{\gamma\widehat{y}}{\widehat{y}-\widehat{\sigma}}~{}.$
(5.9)

On substituting (5.9) into the second of equations (5.8) and clearing the
denominators, we obtain a quartic equation for $\widehat{y}$:

$(\widehat{y}^{2}-1)\left(\widehat{y}-\widehat{\sigma}\right)^{2}+\widehat{y}^{2}\gamma^{2}\;=\;0~{}.$
(5.10)

Some general remarks about this equation are in order. The highest term is
$\widehat{y}^{4}$, and the constant term is $-\widehat{\sigma}^{2}$, so the
product of the four roots of this equation is $-\widehat{\sigma}^{2}\leqslant
0$.

We are interested in solutions of (5.10) with positive positive imaginary
part, which is necessary for them to lie in the large complex structure
region, and how the number of these solutions varies as a function of
$\widehat{\sigma}$. To any quartic polynomial with real coefficients is
associated a discriminant $D_{4}$, which for our polynomial is

$D_{4}\;=\;16\,\gamma^{2}\widehat{\sigma}^{2}\left(\left(1-\gamma^{2}-\widehat{\sigma}^{2}\right)^{3}-27\gamma^{2}\widehat{\sigma}^{2}\right).$

When $D_{4}$ is positive, the quartic has either four or no real roots, while
when $D_{4}$ is negative the quartic has precisely two real solutions.

When $|\gamma|\geqslant 1$, this discriminant is negative for all
$\widehat{\sigma}$. So when $|\gamma|\geqslant 1$ there are two real solutions
to (5.10), and by our earlier observation that the product of the roots is
$-\widehat{\sigma}^{2}$, precisely one of these is positive. So in the
following we will restrict our attention to the case
$|\gamma|=\cos\phi_{00}<1$. When $\widehat{\sigma}=0$, the quartic has four
real roots. These are $\pm\sin\phi_{00}$ and $0$, the latter with multiplicity
two. When $\widehat{\sigma}$ is small but non-vanishing, the solutions
$\pm\sin\phi_{00}$ are perturbed slightly and both of the previously vanishing
solutions become nonzero.

Since a root can only vanish when $\widehat{\sigma}=0$, these new nonzero
roots cannot change sign as $|\widehat{\sigma}|$ increases. Furthermore, the
four roots multiply together to give a negative number. Since there is always
one positive and one negative root, this forces both newly nonzero roots to
have the same sign. Solving to leading order in $\widehat{\sigma}$, these
solutions are

$\widehat{y}\;=\;\frac{\widehat{\sigma}}{1-\gamma}+\mathcal{O}\left(\widehat{\sigma}^{2}\right)~{},\
\ \ \ \
\widehat{y}\;=\;\frac{\widehat{\sigma}}{1+\gamma}+\mathcal{O}\left(\widehat{\sigma}^{2}\right)~{}.$

So, irrespective of the value of $\phi_{00}$, the newly nonzero solutions are
both of the same sign as $\widehat{\sigma}$.

The sign of $D_{4}$ is determined by the cubic

$d(\phi_{00},\xi)\;\buildrel\rm
def\over{=}\;\left.\frac{D_{4}}{16\gamma^{2}\widehat{\sigma}^{2}}\right|_{\widehat{\sigma}^{2}=\xi}\;=\;\left(\sin^{2}\phi_{00}-\xi\right)^{3}-27\cos^{2}\phi_{00}\,\xi~{}.$

For all $\phi_{00}\,{\in}\,(0,\pi)$, this quantity $d(\phi_{00},\xi)$ is
monotonically decreasing as a function of $\xi$. Such a monotonic cubic has
one real root, which here is located at

$\left(1-|\gamma|^{2/3}\right)^{3}\;\buildrel\rm
def\over{=}\;\Xi(\gamma)^{2}.$

The sign of $D_{4}$ changes when $\widehat{\sigma}^{2}=\Xi(\gamma)^{2}$.

When $\widehat{\sigma}\,{=}\,\left(1-|\gamma|^{2/3}\right)^{3/2}$, the
solution flowing from $\sin\phi_{00}$ meets the larger solution that vanishes
at $\widehat{\sigma}=0$. For larger $\widehat{\sigma}$ they both become
nonreal, and so lose relevance for us. The solution flowing from
$\sin\phi_{00}$ remains a solution for every negative value of
$\widehat{\sigma}$.

These observations are summarised in the following Table 7, where we also
include equation numbers for the closed-form solutions that we find later.
This phenomenon of the splitting of attractors as $\sigma$ varies was first
noted in [38].

Case |  Topological/charge data |  Solutions with $y_{0}>0$
---|---|---
i |  $|\gamma|\geqslant 1$, $\,\,\,\,\phantom{0<}\widehat{\sigma}=0$ | none
ii |  $|\gamma|\geqslant 1$, $\,\,\,\,\phantom{0<}\widehat{\sigma}\neq 0$ | $y_{0}^{*}$
iii |  $|\gamma|<1$, $\,\,\,\,\phantom{0<}\widehat{\sigma}\leqslant 0$ | $y_{0}$
iv |  $|\gamma|<1$, $\,\,0<\widehat{\sigma}<\Xi(\gamma)$ |  $y_{0}$, $y_{0}^{\pm}$
v |  $|\gamma|<1$, $\,\,\,\,\phantom{0<}\widehat{\sigma}=\Xi(\gamma)$ | $y_{0}^{\pm}$
vi |  $|\gamma|<1$, $\,\,\,\,\phantom{0<}\widehat{\sigma}>\Xi(\gamma)$ |  $y_{0}^{+}$ or $y_{0}^{-}$

Table 7: Solutions to the alignment equations in different cases. The
quantities $y_{0}$, $y_{0}^{\pm}$ and $y_{0}^{*}$ are defined in (5.12),
(5.13), and (5.14).

We can illustrate the above cases with some plots that perhaps clarify the
changes in number of solutions. The locus of points defined by the second of
the equations (5.8) is a circle, while the locus corresponding to the first
equation is a rectangular hyperbola with asymptotes
$\widehat{x}=\widehat{\sigma}$ and $\widehat{y}=\gamma$. The following Figure
4 illustrates the behaviour described in Table 7. Note the progression
iv$\rightarrow$v$\rightarrow$vi for increasing positive $\widehat{\sigma}$.

(a) i (b) ii

(c) iii

(d) iv (e) v

(f) vi

Figure 4: Illustrations of the conic intersections. The black lines are the
coordinate axes $\widehat{x}=0$ and $\widehat{y}=0$. Our interest is in the
solutions with positive $\widehat{y}$, that is the intersections above the
$\widehat{x}$-axis.

When $\widehat{\sigma}=0$ and $|\gamma|<1$, the hyperbola is degenerate and
intersects the circle once in the upper half plane. Turning $\widehat{\sigma}$
on will perturb this solution. For a small positive $\widehat{\sigma}$ this
perturbed solution is the uppermost point of intersection in the Figure iv
above. We are able to give explicit expressions for $y_{0}$ as a power series
in $\widehat{\sigma}$ about $y_{00}$, in terms of hypergeometric functions of
argument $\gamma^{2}$. The details of this can be found in Appendix E.

$\begin{split}y_{0}\;=\;y_{00}-\frac{\beta\gamma^{2}}{2}\,\sum_{k=1}^{\infty}(k+1)\,{}_{3}F_{2}\left(\genfrac{}{}{0.0pt}{0}{\frac{k+1}{2},\,\frac{k+2}{2},\,\frac{k+3}{2}}{\frac{3}{2},~{}2};~{}\gamma^{2}\right)\widehat{\sigma}^{k}~{}.\end{split}$
(5.11)

It is also possible to solve the equations (5.2) in closed form in terms of
radicals. The simplest expression for the root $y_{0}$ that becomes $y_{00}$
when $\sigma=0$ is

$\displaystyle\frac{2y_{0}}{\beta}$
$\displaystyle\;=\;\widehat{\sigma}+\sqrt{Q_{2}+\widehat{\sigma}^{2}}+\sqrt{2\sin^{2}\phi_{00}-Q_{2}-\frac{2\widehat{\sigma}\left(2-\sin^{2}\phi_{00}\right)}{\sqrt{Q_{2}+\widehat{\sigma}^{2}}}}~{},$
(5.12)

where we have introduced auxiliary quantities $Q_{1}$ and $Q_{2}$ with

$\displaystyle\begin{split}Q_{1}&\;=\;\frac{3}{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}\left(\,\widehat{\sigma}\cos\phi_{00}+\text{i}\sqrt{\left(\frac{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}{3}\right)^{3}-\widehat{\sigma}^{2}\cos^{2}\phi_{00}}\;\right)^{2/3},\\\\[10.0pt]
Q_{2}&\;=\;\frac{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}{3}\left(Q_{1}+2+\frac{1}{Q_{1}}\right).\end{split}$

We are using the branch of the function $z^{1/3}$ that is
$|z|^{1/3}\text{e}^{\text{i}\arg(z)/3}$, with $-\pi<\arg(z)\leqslant\pi$.

The two additional solutions in case iv are given by

$\frac{2y_{0}^{\pm}}{\beta}\;=\;\widehat{\sigma}-\sqrt{Q_{2}^{\pm}+\widehat{\sigma}^{2}}+\sqrt{2\sin^{2}\phi_{00}-Q_{2}^{\pm}+\frac{2\widehat{\sigma}\left(2-\sin^{2}\phi_{00}\right)}{\sqrt{Q_{2}^{\pm}+\widehat{\sigma}^{2}}}}~{},$
(5.13)

with

$\displaystyle\begin{split}Q_{1}^{\pm}&\;=\;\frac{3}{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}\left(\,\pm\widehat{\sigma}\cos\phi_{00}+\text{i}\sqrt{\left|\frac{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}{3}\right|^{3}-\widehat{\sigma}^{2}\cos^{2}\phi_{00}}\;\right)^{2/3},\\\\[10.0pt]
Q_{2}^{\pm}&\;=\;\frac{\sin^{2}\phi_{00}-\widehat{\sigma}^{2}}{3}\left(Q_{1}^{\pm}+2+\frac{1}{Q_{1}^{\pm}}\right).\end{split}$

At the critical value $\widehat{\sigma}=\Xi(\gamma)$ the
larger222$y_{0}^{-}>y_{0}^{+}$ if and only if $\cos\phi_{00}>0$. of these
becomes equal to solution (5.12), which is the case v in the Table 7. For
$\widehat{\sigma}>\Xi(\gamma)$ one of these is no longer real and we enter the
case vi with the real member of this pair giving the only solution.

In the case ii where $|\gamma|{\,\geqslant\,}1$ and
$\widehat{\sigma}{\,\neq\,}0$ there is always a single relevant solution. This
solution is

$\frac{2y_{0}^{*}}{\beta}\;=\;\widehat{\sigma}-\text{sign}(\sigma)\sqrt{Q^{*}_{2}+\widehat{\sigma}^{2}}+\sqrt{\frac{2|\widehat{\sigma}|(2+\sinh^{2}\varphi)}{\sqrt{Q_{2}^{*}+\widehat{\sigma}^{2}}}-2\sinh^{2}\varphi-
Q_{2}^{*}}~{},$ (5.14)

where

$\displaystyle\begin{split}\gamma&\;=\;\pm\cosh\varphi,\\\\[10.0pt]
Q_{1}^{*}&\;=\;\frac{3}{\sinh^{2}\varphi+\widehat{\sigma}^{2}}\left(-\widehat{\sigma}\cosh\varphi+\sqrt{\left(\frac{\sinh^{2}\varphi+\widehat{\sigma}^{2}}{3}\right)^{3}+\widehat{\sigma}^{2}\cosh^{2}\varphi}\;\right)^{2/3},\\\\[10.0pt]
Q_{2}^{*}&\;=\;\frac{\sinh^{2}\varphi+\widehat{\sigma}^{2}}{3}\left(Q_{1}^{*}-2+\frac{1}{Q_{1}^{*}}\right).\end{split}$

In any nondegenerate case with $\widehat{\sigma}\neq 0$, a value for $y_{0}$
singles out one point on the relevant hyperbola in Figure 4, which uniquely
specifies $x_{0}$:

$x_{0}\;=\;\frac{\beta\gamma y_{0}}{y_{0}-\beta\widehat{\sigma}}~{}.$ (5.15)

#### Instanton corrections

The instanton corrections to the alignment equations take the form of a series
in $q_{y}$. To solve these equations perturbatively, we again require that the
coefficient of every power of $q_{y}$ vanishes. This leads to two conditions

$\displaystyle\begin{split}0&\;=\;-\frac{1}{3}\sum_{0\leqslant j\leqslant
i\leqslant
k}\hskip-5.0pt(x_{k-i}x_{i-j}x_{j}+x_{k-i}y_{i-j}y_{j})+\frac{\sigma}{3}\sum_{i=0}^{k}\eta_{k-i}x_{i}+2\mathcal{S}_{3}(k)\\\
&\hskip 56.9055pt\phantom{-}+2\hskip-9.0pt\sum_{0\leqslant j\leqslant
i\leqslant
k}\hskip-9.0pt\mathcal{C}_{3}(k-i)\eta_{i-j}x_{j}+\hskip-15.0pt\sum_{0\leqslant
l\leqslant j\leqslant i\leqslant
k}\hskip-13.0ptx_{k-i}x_{i-j}\eta_{j-l}\mathcal{S}_{2}(l)+\sum_{i=0}^{k}y_{k-i}\mathcal{S}_{2}(i)~{},\\\\[10.0pt]
0&\;=\;\frac{1}{2}\sum_{i=0}^{k}(x_{k-i}x_{i}+y_{k-i}y_{i})+\mathcal{C}_{2}(k)-\hskip-10.0pt\sum_{0\leqslant
j\leqslant i\leqslant
k}\hskip-5.0ptx_{k-i}\eta_{i-j}\mathcal{S}_{2}(j)~{}.\end{split}$

In the first equation, $x_{k}$ and $y_{k}$ appear in the first three terms,
while other terms contain only $x_{i}$ and $y_{i}$ with $i<k$. The prefactors
of $x_{k}$ and $y_{k}$ are

$\displaystyle\frac{1}{3}\Big{(}(1-\zeta)y_{0}^{2}-3x_{0}^{2}\Big{)},\qquad\text{and}\qquad-\frac{1}{3}(2+\zeta)x_{0}y_{0}~{}.$

In the second equation the coefficients are $x_{0}$ and $y_{0}$, respectively.
We can then solve the two equations above for $x_{k}$ and $y_{k}$. In this
way, we get a recurrence relation for $x_{k}$ and $y_{k}$ in terms of $y_{i}$
and $x_{i}$ with $i<k$, which is of the form

$\displaystyle\begin{split}x_{k}\;=\;\frac{A(k)+(2+\zeta)B(k)x_{0}}{(1-\zeta)\beta^{2}}~{},\quad\qquad
y_{k}\;=\;-\frac{A(k)x_{0}+B(k)\big{(}3x_{0}^{2}+(1-\zeta)y_{0}^{2}\big{)}}{y_{0}(1-\zeta)\beta^{2}}~{},\end{split}$

where $A(k)$ and $B(k)$ can be written as $N$-expansions

$\displaystyle A(k)$
$\displaystyle\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}A_{\mathfrak{p}}~{},\qquad
B(k)\;=\;\sum_{\mathfrak{p}\in\text{pt}(k)}N_{\mathfrak{p}}B_{\mathfrak{p}}~{},$