$\left(\begin{matrix}-K&0\\\\[5.0pt]
0&g_{\alpha\bar{\beta}}\end{matrix}\right)=-2y\bar{y}\begin{pmatrix}\bar{Z}^{I}\\\\[5.0pt]
\nabla_{\alpha}Z^{I}\end{pmatrix}\text{{I}}\left(Z^{I}\,\,\overline{\nabla}_{\bar{\beta}}\bar{Z}^{I}z\right).$
(4.58)

A comment on the layout of this equation is in order. The matrix on the LHS is
read in the obvious way. In the leftmost matrix on the RHS $\bar{Z}^{I}$ is
understood as a row vector and $\nabla_{\alpha}Z^{I}$ is an $n\times(n+1)$
matrix. $\text{{I}}=\text{Im}[\mathcal{N}]$ is read in the obvious way. In the
rightmost matrix on the LHS $Z^{I}$ is a column vector and
$\bar{\nabla}_{\bar{\beta}}\bar{Z}^{I}$ is an $(n+1)\times n$ matrix.

The point of this involved digression is to arrive at a useful formula in the
derivation of the stabilisation equations. One can rearrange (4.58) to reach

$\text{{I}}^{-1}=-2y\bar{y}\left(\begin{matrix}Z^{I}&\overline{\nabla}_{\bar{\beta}}\bar{Z}^{I}\end{matrix}\right)\left(\begin{matrix}-K^{-1}&0\\\\[5.0pt]
0&g^{\alpha\bar{\beta}}\end{matrix}\right)\left(\begin{matrix}\bar{Z}^{I}\\\\[5.0pt]
\nabla_{\alpha}Z^{I}\end{matrix}\right).$ (4.59)

The matrix multiplication can be carried out to get an equation for the
components of $I^{-1}$.

$\left(\text{{I}}^{-1}\right)^{IJ}=\frac{2}{K}\bar{X}^{I}X^{J}-2g^{\alpha\bar{\beta}}\left(\nabla_{\alpha}X^{I}\right)\left(\overline{\nabla}_{\bar{\beta}}\bar{X}^{J}\right).$
(4.60)

It is now possible to extremise the central charge. First, substitute the
third identity from (4.56) in (4.55) and use the fact that $\mathcal{N}_{IJ}$
is symmetric to obtain

$\left(\nabla_{\alpha}X^{I}\right)q_{I}-\bar{\mathcal{N}}_{IJ}\left(\nabla_{\alpha}X^{I}\right)p^{J}=0.$
(4.61)

Next contract with
$g^{\alpha\bar{\beta}}\left(\overline{\nabla}_{\bar{\beta}}\bar{X}^{K}\right)$
and use formula (4.60) to replace
$g^{\alpha\bar{\beta}}\left(\nabla_{\alpha}X^{I}\right)\left(\overline{\nabla}_{\bar{\beta}}\bar{X}^{K}\right)$.

$\left(\frac{1}{K}\bar{X}^{I}X^{K}-\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}\right)\left(q_{I}-\bar{\mathcal{N}}_{IJ}p^{J}\right)=0.$
(4.62)

One then rearranges this equation,

$\begin{aligned}
0&=\frac{1}{K}\left(\bar{X}^{I}q_{I}-\bar{\mathcal{N}}_{IJ}\bar{X}^{I}p^{J}\right)X^{K}-\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}q_{I}+\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}\bar{\mathcal{N}}_{IJ}p^{J}\\\\[5.0pt]
&=\frac{1}{K}\left(\bar{X}^{I}q_{I}-\bar{F}_{J}p^{J}\right)X^{K}-\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}q_{I}+\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}\left(\text{{R}}_{IJ}-\text{i}\text{{I}}_{IJ}\right)p^{J}\\\\[5.0pt]
&=\frac{\kappa^{2}}{2K}\bar{\mathcal{Z}}X^{K}-\frac{1}{2}\left(\text{{I}}^{-1}\right)^{IK}\left(q_{I}-\text{{R}}_{IJ}p^{J}\right)-\frac{\text{i}}{2}p^{K}.\end{aligned}.$
(4.63)

Passing from (4.62) to the first line of (4.63) involves only moving terms
around. The passage from the first to the second line makes use of the second
lowering identity in (4.56) and breaks $\bar{\mathcal{N}}$ into its real and
imaginary parts. Passing from the second to the third line necessitates
formula (4.54) and multiplying some matrices together.

Since $\kappa^{2}$, I, R, $K$, $p^{I}$ and $q_{I}$ are all real, one can
extract the imaginary part of the last equation:

$p^{K}=\text{Im}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}X^{K}\right].$
(4.64)

The real part of (4.63), contracted with $\text{{I}}_{JK}$, is

$\displaystyle q_{J}$
$\displaystyle=\text{{R}}_{JK}p^{K}+\text{Re}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}\text{{I}}_{JK}X^{K}\right]$
(4.65)
$\displaystyle=\text{Im}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}\text{{R}}_{JK}X^{K}\right]+\text{Im}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}\text{i}\text{{I}}_{JK}X^{K}\right]$
$\displaystyle=\text{Im}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}\mathcal{N}_{JK}X^{K}\right]$
$\displaystyle=\text{Im}\left[\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}F_{J}\right].$

The quantity $\frac{\kappa^{2}}{K}\bar{\mathcal{Z}}$ depends on the dilatation
gauge and the charges. Denoting this quantity’s value at an attractor point as
$C$, we have reached the stabilisation equations that relate the scalar values
at an attractor point to the electromagnetic charges when the central charge
does not vanish:

$\begin{pmatrix}p^{I}\\\
q_{I}\end{pmatrix}=\text{Im}\left[C\begin{pmatrix}X^{I}\\\
F_{I}\end{pmatrix}\right].$ (4.66)

These are $2n+2$ real equations for $2n+4$ real variables — the real and
imaginary parts of $C$ and the $2n+2$ attractor coordinates on the rigid
manifold. Fixing a choice of gauge constrains two of these variables, and so
the number of equations equals the number of physical variables to solve for.

##### Multi-centred solutions

In [96] the attractor mechanism was extended to multi-centered black holes in
$\mathcal{N}=2$ supergravity. It was shown that in the presence of $N$ sources
located at $\vec{x}_{i}$, $i=1,...,N$, each of charge $\Gamma_{i}$, the vector
multiplet scalars take attractor values associated to the charge vector
$\Gamma_{i}$ as $\vec{x}_{i}$ is approached.

###  4.3 Calabi-Yau attractors in IIB

In a IIB compactification on a Calabi-Yau threefold $X$, a charged black hole
has a string-theoretic description as a bound state of D3-branes wrapping
special lagrangian cycles in the middle cohomology of $X$. The charge vector
$Q$ then is associated to a three-cycle $\gamma$ in $H_{3}(X,\mathbbl{Z})$.
This is dual in cohomology to a threeform $\Gamma$, which we will expand in
the same integral symplectic basis as used in (4.10), whose components we
collect in the vector $Q$.

$\Gamma=p^{I}\alpha_{I}-q_{I}\beta^{I}~{},\qquad Q=\begin{pmatrix}q_{I}\\\
p^{I}\end{pmatrix}~{}.$ (4.67)

The stabilisation equations (4.66) give the following constraint on the
holomorphic three-form $\Omega$:

$\Gamma=\text{Im}\left[C\,\Omega\right]~{},\qquad\text{or in components
}\,\,Q=\text{Im}\left[C\,\Pi\right]~{}.$ (4.68)

Note that the ordering of the electric and magnetic charges is reversed
compared to that in §4.2, which is purely a matter of changing convention and
not conceptually significant. If we fix the charge vector, then the above
equation is a constraint on the complex structure moduli of $X$. A Calabi-Yau
threefold whose moduli satisfy the above equation for some integral charge
vector is said to be an attractor variety.

If we let the complex constant $C$ have real part $a$ and imaginary part $b$,
then the above equation reads

$Q=a\,\text{Im}\left[\Pi\right]+b\,\text{Re}\left[\Pi\right]~{}.$ (4.69)

In the Dolbeault decomoposition of $H^{3}(X,\mathbbl{Z})$, the holomorphic
three form generates the $(3,0)$ part. The real and imaginary part of $\Omega$
both belong to $H^{(3,0)}\oplus H^{(0,3)}$, and so the above equation tells us
that inside $H^{(3,0)}\oplus H^{(0,3)}$ lives the span of the integral vector
$Q$. The statement is that the $(3,0)+(0,3)$ part of $X$’s Dolbeault
cohomology contains a 1-dimensional integral lattice.

In some cases, there may be a point $\varphi_{*}$ in the complex structure
moduli space of $X$ so that, at this point, the equation (4.68) is solved for
two independent charge vectors $Q_{1},\,Q_{2}$ (with different values of $C$).
“Independence” here means that the symplectic inner product $Q_{1}\Sigma
Q_{2}$ does not vanish, and so $Q_{1}$ and $Q_{2}$ are necessarily linearly
independent.

For such moduli values $\varphi_{*}$, $X$’s cohomology has the property that
$H^{(3,0)}\oplus H^{(0,3)}$ is the complexification of a rank-two integral
lattice in $H^{3}(X,\mathbbl{Z})$ [14]. This gives a splitting of the Hodge
structure. We see this at the level of deRham cohomology, however
conjecturally this persists into a suitable étale cohomology and so restricts
the form of the zeta function. We shall look at this more closely in §4.10.

###  4.4 One-parameter Calabi-Yau attractors in IIA

We will only be concerned with IIA compactifications on manifolds $Y$ with
$h^{1,1}=1$. If the period vector $\Pi$ solves the attractor equations for two
independent charge vectors $Q_{1},\,Q_{2}$, then there must exist independent
integral vectors $Q_{3},\,Q_{4}$ such that

$\Pi\Sigma Q_{3}=\Pi\Sigma Q_{4}=0~{}.$ (4.70)

This must be so, because the attractor equations imply

$\text{Span}\left(\text{Re}[\Pi]~{},\,\text{Im}[\Pi]\right)=\text{Span}\left(Q_{1}~{},\,Q_{2}\right)~{}.$
(4.71)

The symplectic complement of
$\text{Span}\left(Q_{1}~{},\,Q_{2}\right)\subset\mathbbl{Z}^{4}$ is two-
dimensional, with generators some $Q_{3}~{},\,Q_{4}$, which are by the above
relation symplectic-orthogonal to $\text{Re}[\Pi]$ and $\text{Im}[\Pi]$. We
shall in what follows make reference to the _orthogonality equation_

$Q\Sigma\Pi=0~{},$ (4.72)

in contrast to the attractor equations (4.68) .

This equivalence will be useful when we discuss a set of summation identities
based on solutions to the IIA attractor equations, which we now set up. In a
one-parameter compactification the IIA prepotential (4.6) reads

$\displaystyle\mathcal{F}$
$\displaystyle=-\left(z^{0}\right)^{2}\left[\frac{1}{6}Y_{111}t^{3}+\frac{1}{2}Y_{110}t^{2}+\frac{1}{2}Y_{100}t+\frac{1}{6}Y_{000}+\mathcal{I}\left(t\right)\right]~{},$
(4.73) $\displaystyle\mathcal{I}$
$\displaystyle=\frac{1}{\left(2\pi\text{i}\right)^{3}}\sum_{k=1}^{\infty}n^{(0)}_{k}\text{Li}_{3}\left(\text{e}^{2\pi\text{i}\,kt}\right)=\frac{1}{\left(2\pi\text{i}\right)^{3}}\sum_{k=1}^{\infty}\frac{N_{k}}{k^{3}}\text{e}^{2\pi\text{i}\,kt}~{}.$

We have written $t=\frac{z^{1}}{z^{0}}$, and in the above expression for the
instanton sum $\mathcal{I}$ we have repackaged the genus-0 instanton numbers
into _scaled Gromov-Witten invariants_ $N_{k}$. These are computed from the
$n^{(0)}_{k}$ and related to the usual Gromov-Witten invariants $N^{GW}_{k}$
[97] by

$N_{k}=\sum_{d|k}d^{3}n^{(0)}_{d}~{},\qquad N_{k}=k^{3}N^{GW}_{k}~{}.$ (4.74)

The four-component period vector $\Pi$ is

$\Pi(t)=\begin{pmatrix}\frac{\partial}{\partial z^{0}}\mathcal{F}\\\
\frac{\partial}{\partial z^{1}}\mathcal{F}\\\ z^{0}\\\
z^{1}\end{pmatrix}=z^{0}\begin{pmatrix}\frac{1}{6}Yt^{3}-\frac{1}{2}Y_{100}t-\frac{1}{3}Y_{000}-2\mathcal{I}(t)+t\mathcal{I}^{\prime}(t)\\\
-\frac{1}{2}Yt^{2}-Y_{110}t-\frac{1}{2}Y_{100}-\mathcal{I}^{\prime}(t)\\\ 1\\\
t\end{pmatrix}~{},\qquad Y=Y_{111}~{}.$ (4.75)

We will now fix a gauge444Note that this is done only after taking the above
derivatives $\frac{\partial}{\partial z^{0}}\mathcal{F}$, $z^{0}=1$ . The
period vector is seen to be a sum of a quantity $\Pi_{0}$ that is corrected by
instantons:

$\begin{gathered}\Pi=\Pi_{0}+\Delta_{\mathcal{I}}\Pi~{},\\\\[10.0pt]
\Pi_{0}=\begin{pmatrix}\frac{1}{6}Yt^{3}-\frac{1}{2}Y_{100}t-\frac{1}{3}Y_{000}\\\
-\frac{1}{2}Yt^{2}-Y_{110}t-\frac{1}{2}Y_{100}\\\ 1\\\
t\end{pmatrix}~{},\qquad\Delta_{\mathcal{I}}\Pi=\begin{pmatrix}-2\mathcal{I}(t)+t\mathcal{I}^{\prime}(t)\\\
-\mathcal{I}^{\prime}(t)\\\ 0\\\ 0\end{pmatrix}~{}.\end{gathered}$ (4.76)

In [4] a method of solving the attractor and orthogonality equations for the
above form of period vector was displayed. We provide a brief summary of the
solution method. First, the uncorrected equations

$Q=\text{Im}\left[C\Pi_{0}\right]\qquad\text{or}\qquad Q\Sigma\Pi_{0}=0$
(4.77)

are solved, to find a ‘perturbative’ solution $t_{0}$, with real and imaginary
part $x_{0}$ and $y_{0}$, so

$t_{0}=x_{0}+\text{i}\,y_{0}~{}.$ (4.78)

Notice that either of (4.77) gives algebraic equations for $x_{0}$ and
$y_{0}$. In [98] the attractor equations for such uncorrected prepotentials
were considered. A full ‘instanton-corrected’ solution $t$ is found by
performing perturbation theory in the small parameter

$\text{e}^{-2\pi y_{0}}=|\text{e}^{2\pi\text{i}t_{0}}|~{}.$ (4.79)

The precise form of the solution depends on the charge vector $Q$. The most
general form, with electric charges $q_{0},q$ and magnetic charges $p^{0},p$,
is

$Q=\begin{pmatrix}q_{0}\\\ q\\\ p^{0}\\\
p\end{pmatrix}=\begin{pmatrix}q_{\text{D0}}\\\ q_{\text{D2}}\\\
q_{\text{D6}}\\\ q_{\text{D4}}\end{pmatrix}~{},$ (4.80)

where in the second equality we interpret the integral charges as giving the
wrapping number of even-dimensional D-branes. We adopt a convention that does
not include charges induced on each brane by world-volume curvature coupling.

We will consider only charge vectors that take one of the two following forms:

$Q_{\text{D4}}=\kappa\begin{pmatrix}\Lambda\\\ \Upsilon\\\ 0\\\
1\end{pmatrix}~{},\qquad Q_{\text{D6}}=\kappa\begin{pmatrix}\Lambda\\\
\Upsilon\\\ 1\\\ 0\end{pmatrix}~{}.$ (4.81)

The integer $\kappa$ gives the total D4 or D6 charge, while $\Lambda$ and
$\Upsilon$ are integer multiples of $\frac{1}{\kappa}$ .

Our attention will be fixed to solutions of the ‘D4-D2-D0 orthogonality
equations’:

$Q_{\text{D4}}\Sigma\Pi=0~{}.$ (4.82)

To explain this choice, we note first that if $t$ is a rank-two attractor,
then an integral vector of the form $Q_{\text{D4}}$ can be found so that the
above equation (4.82) is satisfied. We work with the equations (4.82) and not
the attractor equations themselves because, at present, the solution to (4.82)
takes a much simpler form where the terms in the perturbative series can be
given in closed form (rather than solely by recurrences).

Neglecting instantons, the equation $Q_{\text{D4}}\Sigma\Pi_{0}=0$ is a simple
quadratic,

$Y_{111}t_{0}^{2}+2\left(Y_{110}+\Upsilon\right)t_{0}+Y_{100}+2\Lambda=0~{}.$
(4.83)

We will assume that
$\left(Y_{110}+\Upsilon\right)^{2}-Y_{111}\left(Y_{100}+2\Lambda\right)<0$. In
that case, the solution to (4.83) is $t_{0}=x_{0}+\text{i}y_{0}$ with real and
imaginary parts

$x_{0}=\frac{-Y_{110}-\Upsilon}{Y_{111}}~{},\qquad
y_{0}=\frac{\sqrt{Y_{111}\left(Y_{100}+2\Lambda\right)-\left(Y_{110}+\Upsilon\right)^{2}}}{Y_{111}}~{}.$
(4.84)

To reiterate, this algebraic number $x_{0}+\text{i}y_{0}$ does not solve the
attractor or orthogonality equations, but is termed the _perturbative_
solution of the orthogonality equation as it solves the second of the
equations (4.77). Then, the full solution of (4.82), as conjectured555The
conjectural part of this analysis is in identifying the combinatoric functions
$a_{\mathfrak{p}}$ and modified Bessel functions in the series coefficients.
It is a theorem that some series solution with recursively defined
coefficients exists when $y_{0}$ is sufficiently small, see [4]. in [4], is

$t\;=\;t_{0}-\frac{\text{i}}{\sqrt{2\pi^{3}Y_{111}}}\sum_{j=1}^{\infty}\text{e}^{2\pi\text{i}x_{0}j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N_{\mathfrak{p}}\left(\frac{j}{2\pi
y_{0}Y_{111}}\right)^{\ell(\mathfrak{p})-1/2}K_{\ell(p)-1/2}\left(2\pi
jy_{0}\right)~{}.$ (4.85)

$K_{\nu}(z)$ is the modified Bessel function of the second kind.
$\text{pt}(j)$ is the set of partitions of the integer $j$. For such a
partition $\mathfrak{p}$, $j$ is partitioned into a set of integers as
$j=\sum_{k=1}^{\infty}\mu_{k}k$. That is to say, $\mu_{k}$ is the multiplicity
of the integer $k$ in the partition $\mathfrak{p}$ of $j$.

That said, $a_{\mathfrak{p}}$ is a combinatorial factor given by

$a_{\mathfrak{p}}\;=\;\prod_{k=1}^{\infty}\frac{1}{k^{2\mu_{k}}\mu_{k}!}$
(4.86)

and $N_{\mathfrak{p}}$ is the following product of the enumerative invariants
$N_{k}$:

$N_{\mathfrak{p}}\;=\;\prod_{k=1}^{\infty}N_{k}^{\mu_{k}}~{}.$ (4.87)

Finally, $\ell(\mathfrak{p})$ is the length of the partition $\mathfrak{p}$,

$\ell(\mathfrak{p})=\sum_{k=0}^{\infty}\mu_{k}~{}.$ (4.88)

###  4.5 Flux compactifications

Flux compactifications are string theory compactifications with nontrivial
background values for the $(p+1)$-form field strengths. Our account of this
topic follows [99] and [12]. The first supersymmetric configurations were
given in [100]. The prospect of realising deSitter vacua was addressed in
[101].

Type IIB supergravity’s massless bosonic field content includes the even
$p$-form Ramond-Ramond fields $C_{0}$, $C_{2}$, and $C_{4}$ which have field
strengths

$F_{i}=\text{d}C_{i-1}~{}.$ (4.89)

Additionally there is the Kalb-Ramond two-form $B_{2}$ which has field
strength

$H_{3}=\text{d}B_{2}~{}.$ (4.90)

There is also the dilaton $\phi$, which is packaged along with the axion
$C_{0}$ into a single complex scalar, the axiodilaton

$\tau=C_{0}+\text{i}\text{e}^{-\phi}~{}.$ (4.91)

Another notational change introduces the three-form field $G_{3}$, defined as

$G_{3}=F_{3}-\tau H_{3}~{}.$ (4.92)

The utility of these two redefinitions is that it simplifies the action of the
$\text{SL}(2,\mathbbl{Z})$ symmetry of the IIB theory. The axiodilaton $\tau$
and complexified field strength $G_{3}$ transform via

$\tau\mapsto\frac{a\tau+b}{c\tau+d}~{},\qquad
G_{3}\mapsto\frac{1}{c\tau+d}G_{3}~{},\qquad\text{with }\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\text{SL}(2,\mathbbl{Z})~{}.$ (4.93)

In a flux compactification, we dimensionally reduce on a Calabi-Yau threefold
$X$, or an orientifold thereof, but do so with nonzero values given to $F_{3}$
and $H_{3}$ supported on the cohomology of $X$. Expanding in an integral
symplectic basis, we write the components of $F_{3}$ and $H_{3}$ as

$F_{3}=(2\pi)^{2}\alpha^{\prime}\left(f^{a}\alpha_{a}-f_{b}\beta^{b}\right)~{},\qquad
H_{3}=(2\pi)^{2}\alpha^{\prime}\left(h^{a}\alpha_{a}-h_{b}\beta^{b}\right)~{},$
(4.94)

where $\alpha^{\prime}$ is the string coupling and
$f_{a},\,f^{b},\,h_{a},\,h^{b}$ are all integers. We collect these into the
component vectors

$F=\begin{pmatrix}f^{a}\\\ f_{b}\end{pmatrix}~{},\qquad
H=\begin{pmatrix}h^{a}\\\ h_{b}\end{pmatrix}~{}.$ (4.95)

When compactifying with such fluxes turned on, more supersymmetry is broken
and the resulting four-dimensional supergravity has $\mathcal{N}=1$
supersymmetry. The geometric moduli of $X$ (both complex structure and Kähler)
and the axiodilaton are scalars in $\mathcal{N}=1$ chiral multiplets [99].
These scalars are coupled together by a potential term $V$ in the action,
which in accordance with $\mathcal{N}=1$ supersymmetry is constructed in a
standard manner from a superpotential $W$. Incorporating or neglecting
nonperturbative string theory contributions, like D-brane instantons, leads to
different superpotentials $W$. We shall work with the uncorrected, classical
superpotential which only depends on the complex structure moduli of $X$ and
the axiodilaton, reading

$W=\int_{X}G_{3}\wedge\Omega=(2\pi)^{2}\alpha^{\prime}\left(F-\tau
H\right)^{T}\Sigma\Pi~{},\qquad\Sigma=\begin{pmatrix}\phantom{-}0&\mathbb{I}\\\
-\mathbb{I}&0\end{pmatrix}~{}.$ (4.96)

There are two constraints on the choices of flux vectors $F$ and $H$, which
bound $F^{T}\Sigma H$ from above and below.

$0<F^{T}\Sigma H\leq\frac{\chi\left(X_{4}\right)}{24}~{}.$ (4.97)

The lower bound is necessary to get nontrivial supersymmetric solutions to the
equations of motion, while the upper bound is the D3 tadpole condition, which
is a consistency condition coming from F-theory. Namely, the different sources
of D3 charge must locally cancel out one another. Flux compactifications on an
orientifold of $X$ can be lifted to F-theory on an elliptically fibred Calabi-
Yau fourfold $X_{4}$ [102], and there is a condition

$\frac{1}{2\kappa_{10}^{2}T_{3}}\int_{X}H_{3}\wedge
F_{3}+Q^{\text{D3}}=\frac{\chi\left(X_{4}\right)}{24}~{}.$ (4.98)

$Q^{\text{D3}}$ is the total charge of any present D3 branes. We can add D3
branes (but not anti-D3 branes) without breaking supersymmetry, from which the
upper bound follows.

###  4.6 F-theory lifts

Sen demonstrated in [102] that F-theory on an elliptically fibred Calabi-Yau
fourfold reduces at weak coupling to an Orientifold compactification of type
IIB string theory. We shall briefly review his construction, informed also by
[12, 11].

The modulus $\tau$ of the elliptic fibration is identified with the
axiodilaton of the IIB theory. This gives a geometric interpretation to the
$\text{SL}(2,\mathbbl{Z})$ symmetry of IIB. The base of the fibration admits a
double covering by a Calabi-Yau threefold, as depicted in figure 4.1.

${\mathcal{E}_{\tau}}$${X_{4}}$${X}$${\mathcal{B}}$$\scriptstyle{2:1}$

Figure 4.1: The F-theory fourfold is an elliptic fibration over the base
$\mathcal{B}$. This base is in turn doubly covered by the Calabi-Yau threefold
$X$.

A method for constructing these fourfolds as intersections in toric varieties,
given $X$ as such an intersection, was described in [103, 104] .

Sen’s description gave an explicit relation between the elliptic fibration and
the locations of D7 branes and O7 planes in the IIB limit. The most general
elliptic fibration over a base $\mathcal{B}$ has a Weierstrass form

$y^{2}=x^{3}+f(u)x+g(u)~{},$ (4.99)

with $u$ being coordinates on the base $\mathcal{B}$. As $u$ varies so does
the above elliptic curve. For certain values of $u$, given by the vanishing of
the discriminant

$\Delta(u)=4f^{3}+27g(u)^{2}~{},$ (4.100)

the elliptic curve becomes singular. Sen computed monodromies of $\tau$ about
this singular locus, and recognised the same monodromies as one expects upon
circling D7 branes or O7 planes. In so doing, the singular locus $\Delta(u)=0$
on the base $\mathcal{B}$ is recognised as giving the intersections of
spacetime filling 7-dimensional extended objects with the internal orientifold
geometry.

To delineate between the D7 and O7 objects, it is convenient to rewrite $f$
and $g$ in terms of functions $h$, $\eta$, and $\chi$, and a constant $C$
(which is not the $C$ of the previous section):

$\displaystyle f(u)$ $\displaystyle=C\eta(u)-3h(u)^{2}~{},\qquad
g(u)=h(u)\left[C\eta(u)-2h(u)^{2}\right]+C^{2}\chi(u)~{}.$ (4.101)

Having done this, the discriminant is

$\Delta(u)=C^{2}\left[\eta^{2}\left(4C\eta-9h^{2}\right)+54h\left(C\eta-2h^{2}\right)\chi+27C^{2}\chi^{2}\right]~{}.$
(4.102)

The $j$-invariant $j_{\mathcal{E}}$ of an elliptic curve is, in terms of the
Weierstrass data $f,g$,

$j_{\mathcal{E}}(u)=\frac{6912f^{3}}{4f^{3}+27g^{2}}~{}.$ (4.103)

Our prefactor 6912 differs to Sen’s convention. Since the elliptic fibration’s
complex structure parameter $\tau$ (itself some function of the base
coordinate $u$) is to be identified with the axiodilaton $\tau$, we should
have

$j_{\mathcal{E}}(u)=j\left(\tau(u)\right)~{},$ (4.104)

where on the right hand side we have Klein’s $j$-invariant

$j(\tau)=e^{-2\pi\text{i}\tau}+744+196884\text{e}^{2\pi\text{i}\tau}+21493760\text{e}^{4\pi\text{i}\tau}+864299970\text{e}^{6\pi\text{i}\tau}+\,...~{}.$
(4.105)

Note that this differs from the function $J(\tau)=\frac{1}{1728}j$, the former
being implemented in Mathematica as KleinInvariantJ.

In terms of the functions $f,\,h,\,\chi$, this $j$-invariant is

$j\left(\tau(u)\right)=\frac{6912\left(C\eta-3h\right)^{2}}{C^{2}\left[\eta^{2}\left(4C\eta-9h^{2}\right)+54h\left(C\eta-2h^{2}\right)\chi+27C^{2}\chi^{2}\right]}~{}.$
(4.106)

In the limit $C\mapsto 0$ the above $j$-invariant diverges like $C^{-2}$. As
discussed in [102] (wherein their $\lambda$ is our $\tau$), up to
$\text{SL}(2,\mathbbl{Z})$ transformations this limit corresponds to sending
the axiodilaton $\tau$ to $\text{i}\infty$ via
$\tau=\frac{1}{4\pi\text{i}}\log(C)+O(C)$ .

In this large $\tau$ limit, taken with constant $h,\,\eta,\,\chi$ and
$C\mapsto 0$, the discriminant (4.102) becomes

$\Delta(u)=-C^{2}h^{2}\left(\eta^{2}+12h\chi\right)~{}.$ (4.107)

This vanishes on the loci

$h=0~{},\qquad\text{ and }\qquad\eta=\pm\sqrt{-12h\chi}~{}.$ (4.108)

Sen computed the monodromies of the axiodilaton around these loci, and
interpreted these as being due to charged objects sourcing the axiodilaton
being present. Seven-dimensional extended objects fill the macroscopic four-
dimensional spacetime and intersect the base $\mathcal{B}$ on the loci
$\Delta(u)=0$. The locus $h=0$ gives the position of an $O7$ plane, while the
loci $\eta=\pm\sqrt{-12h\chi}$ give D-brane positions.

In F-theory, a supersymmetric configuration will be one where the D7 and O7s
coincide. By making a choice $\chi=0$ and $h^{2}=\eta$, both objects will be
positioned at $\eta=0$.

At the IIB level a maximally supersymmetric configuration should have a
spatially constant axiodilaton profile. The sought profile is obtained from
Sen’s fibration, and the choices made for $\eta,\,h,\,\chi$ give the following
$f$ and $g$ as in (4.101):

$f(u)=(C-3)h(u)^{2}~{},\qquad g(u)=(C-2)h(u)^{3}~{}.$ (4.109)

This gives an elliptic fibration over $\mathcal{B}$ with Weierstrass model

$y^{2}=x^{3}+(C-3)h(u)^{2}+(C-2)h(u)^{3}~{}.$ (4.110)

The singular locus of this curve is $C^{2}(4C-9)h(u)^{6}$. Away from the locus
$h(u)=0$ we can make a rescaling of coordinates $y\rightarrow
h^{3/2}y~{},\,x\rightarrow hx$ so that our model is constant over the base,
away from the charged objects.

$y^{2}=x^{3}+(C-3)x+C-2~{}.$ (4.111)

The $j$-invariant of this curve is

$j_{C}=\frac{6912(C-3)^{3}}{C^{2}(4C-9)}~{}.$ (4.112)

From this follows a cubic equation for $C$, which can be solved in terms of
$j(\tau)$. This choice of a $C$ then gives the correct elliptic fibre for the
F-theory uplift of the IIB supersymmetric flux vacuum with axiodilaton $\tau$.

###  4.7 The scalar potential

Let us temporarily forget the fact that we have an internal Calabi-Yau
manifold, and discuss some general features of these supergravity theories
following [92]. In any four-dimensional $\mathcal{N}\,{=}\,1$ supergravity
coupled to $N$ chiral multiplets, the scalars in those multiplets are
coordinates on a projective Kähler manifold. The theory is specified by the
choice of two functions of the moduli; the Kähler potential $\mathcal{K}$ for
the scalar manifold and a superpotential $W$. From these two functions, one
calculates the following potential:

$V\buildrel\rm
def\over{=}\text{e}^{\mathcal{K}}\left(G^{\alpha\bar{\beta}}D_{\alpha}W\overline{D_{\beta}W}-3|W|^{2}\right).$
(4.113)

The sum in the above expression runs over all moduli, so in the case of a
Calabi-Yau compactification on $X$,
$\alpha,\beta=1,\,...\,,h^{1,1}(X)+h^{1,2}(X)+1$. The metric
$G_{\alpha\bar{\beta}}$ is the metric on the scalar manifold, and, by virtue
of the Kähler condition,
$G_{\alpha\bar{\beta}}=\partial_{\alpha}\partial_{\bar{\beta}}\,\mathcal{K}$.
As in [93, 24], the quantity $D_{\alpha}W$ is the Kähler covariant derivative
of $W$, which has Kähler weight (1,0), so

$D_{\alpha}W\;=\;\partial_{\alpha}W+\left(\partial_{\alpha}\mathcal{K}\right)W~{}.$

In the case of a type IIB Calabi-Yau compactification, the total scalar
manifold is the product of the upper half plane (in which the axiodilaton is
valued) and the moduli space of the Calabi-Yau manifold $X$, which factorises,
at least locally, into the moduli spaces of complex structures and
complexified Kähler structures of $X$. The Kähler potential for the total
moduli space is then a sum of the Kähler potentials for the axiodilaton,
complex structure, and Kähler structure factors of the moduli space:

$\mathcal{K}\;=\;K^{\text{AD}}+K^{\mathbbl{C}\text{S}}+K^{\mathbbl{C}\text{K}}~{}.$

These depend on the moduli as follows:

$\displaystyle
K^{\text{AD}}=-\log\big{(}{-}\text{i}\left(\tau-\overline{\tau}\right)\big{)}~{},\quad
K^{\mathbbl{C}\text{S}}=-\log\left(-\text{i}\,\Pi^{\dagger}\,\Sigma\,\Pi\right)~{},\quad
K^{\mathbbl{C}\text{K}}=-\log\left(-\text{i}\,\amalg^{\dagger}\Sigma\amalg\right)~{},$
(4.114)

where $\amalg$ is the period vector for the mirror manifold, so a function of
the Kähler moduli of $X$. The Kähler potentials for the axiodilaton and
complex structure moduli are exact at tree level, while the potential for
complexified Kähler structures, $K^{{\mathbbl{C}\text{K}}}$, is corrected by
fundamental string instantons and $\alpha^{\prime}$ corrections (this
corrected $K^{{\mathbbl{C}\text{K}}}$ is the uncorrected
$\widetilde{K}^{{\mathbbl{C}\text{S}}}$ of the mirror manifold). Additionally,
nonperturbative effects can modify $W$ in such a way as to give it a
dependence on the Kähler moduli of $X$. If these latter corrections are
neglected, then derivatives of $W$ with respect to Kähler moduli vanish and
the potential (4.113) reduces to

$V_{(W\text{
classical})}\;=\;\text{e}^{\mathcal{K}}\bigg{(}\text{Im}[\tau]^{2}\,|D_{\tau}W|^{2}+g^{i\bar{\jmath}}D_{i}W\overline{D_{j}W}+\left(\widetilde{g}^{\,r\bar{s}}K^{{\mathbbl{C}\text{K}}}_{r}K^{{\mathbbl{C}\text{K}}}_{\bar{s}}-3\right)|W|^{2}\bigg{)}~{}.$
(4.115)

In this expression $g_{i\bar{\jmath}}$ is the metric on the space of complex
structures and $\widetilde{g}_{r\bar{s}}$ is the metric on the space of
complexified Kähler structures. The indices $i,j$ run from 1 to $h^{2,1}(X)$,
and $r,s$ run from 1 to $h^{1,1}(X)$.

In supersymmetric configurations the superpotential $W$ vanishes, which
greatly simplifies the above expression for the scalar potential $V$, which
must vanish666Note that $V$ does not automatically vanish on a locus in moduli
space where $W$ vanishes, because $V$ depends on derivatives of $W$ in each
moduli coordinate. in a vacuum configuration.

$V\;=\;\text{e}^{\mathcal{K}}\bigg{(}\text{Im}[\tau]^{2}\,|\partial_{\tau}W|^{2}+g^{i\bar{\jmath}}\,\partial_{\varphi^{i}}W\,\overline{\partial_{\varphi^{j}}W}\bigg{)}~{}.$
(4.116)

This potential is manifestly positive semidefinite, and thus the equations
defining a supersymmetric flux vacuum, requiring that both $W$ and $V$ vanish,
read:

$W\;=\;0~{},\qquad\partial_{\tau}W\;=\;0~{},\qquad\partial_{\varphi^{i}}W\;=\;0~{}.$
(4.117)

Depending on the vacuum expectation values of the field strengths $F_{3}$ and
$H_{3}$, these equations might be solved for the complex structure moduli
$\varphi^{i}$ as well as the value of the axiodilaton field. The Kähler moduli
are unconstrained, although this ceases to be true when instanton corrections
to the action are incorporated [105].

###  4.8 Corrections to the potential and a digression on moduli stabilisation

In what follows, we only consider the case where the superpotential is given
by the classical formula (4.96), that is, we work in the approximation where
we can safely ignore the non-perturbative stringy corrections. Incorporating
these corrections to $W$ could alter the space of supersymmetric flux vacua.
In particular, the condition of vanishing $W$, which has a cohomological
interpretation central to the modularity that we will discuss, only holds when
the classical expression (4.96) for $W$ is used. Additionally, these
corrections give $W$ a dependence on the Kähler moduli, and so additional
derivative terms will appear in the potential.

The nonperturbative corrections to $W$ that we are referring to are the same
as those considered in [105], where it was argued that precisely these
corrections allowed for a realisation of metastable de Sitter vacua. One
possible source of these corrections is Euclidean D3-instantons, first
discussed in [106]. Additionally, in some setups gluino condensation could
occur on spacetime-filling D7-branes which gives a contribution to $W$.
Neither of these will be discussed here.

In attempting to build realistic cosmological models from flux
compactifications, one can expect to meet the moduli problem. Namely, string
theory constructions typically come with a large number of massless scalars.
This is phenomenologically undesirable because there is no evidence for the
existence such massless scalars. In fact, since these scalars will be coupled
to gravity, they would give rise to unobserved fifth-force effects [107, 108]
that would, among other things, affect planetary orbits. Hopes of getting
around this problem lie in finding suitable mechanisms of moduli
stabilisation. This involves recognising nonperturbative corrections to the
scalar potential, so that it acquires a steep local minimum in which the
scalars will settle. When the scalars have taken this minimising value, the
scalar potential term in the action becomes an effective cosmological
constant. To realise a deSitter spacetime necessitates that this minimum be
positive. There then remains the problem of giving a natural explanation for
the cosmological constant’s value being of the order $10^{-122}$ in Planck
units.

We will not have anything to say about this problem. However, we will make a
short digression now to discuss how the topology of the internal geometry $X$
can affect the large volume behaviour of the potential for the Kähler moduli.
We proceed from the scalar potential (4.115) that follows from the
superpotential (4.96), which crucially does not depend on the Kähler moduli.

The square-bracketed quantities in the first line of (4.115) are positive
definite, so $V$ can be minimised by setting them to zero. This gives
$h^{2,1}+1$ equations for the $h^{2,1}+1$ quantities of the axiodilaton $\tau$
and the full set of complex structure moduli $\bm{\varphi}$. Let us assume
that these equations all provide independent constraints, so that all complex
structure moduli and the axioldilaton are fixed. Let these respectively be
fixed to values $\tau_{*}$ and $\bm{\varphi}_{*}$. What remains is

$V(\tau_{*},\bm{\varphi_{*}})\;=\;\frac{\text{e}^{C(\bm{\varphi}_{*})}}{\text{Im}[\tau_{*}]}|W_{*}|^{2}\cdot\text{e}^{K}\left(K^{i\bar{j}}K_{i}K_{\bar{j}}-3\right)~{},$
(4.118)

which depends on the as-yet-unfixed Kähler moduli $\bm{t}$ through
$K(\bm{t})$.

We want to study the contraction

$\mathcal{C}\;=\;K^{i\bar{j}}K_{i}K_{j}~{}.$ (4.119)

Recall that $K$ can be computed from the prepotential $\mathcal{F}$ via

$\text{e}^{-K}=-\text{i}\left(z^{a}\bar{\mathcal{F}}_{a}-\bar{z}^{a}\mathcal{F}_{a}\right)~{}.$
(4.120)

$\mathcal{F}$ depends on the projectivised Kähler moduli by

$\mathcal{F}=-\frac{1}{3!}Y_{abc}\frac{z^{a}z^{b}z^{c}}{z^{0}}-\left(z^{0}\right)^{2}\mathcal{I}\left(\frac{\bm{z}}{z^{0}}\right)~{},\qquad\mathcal{I}(\bm{t})\;=\;\frac{1}{(2\pi\text{i})^{3}}\sum_{\bm{k}\in
H_{2}(X,\mathbb{Z})}n_{\bm{k}}\text{Li}_{3}\left(\text{e}^{2\pi\text{i}\bm{k}\cdot\bm{t}}\right)~{}.$
(4.121)

with $a,b,c$ running from 0 to $h^{1,1}$, the above sum includes the
perturbative worldsheet corrections to $X$’s quantum volume, in the terms with
a zero index. Including or excluding the $Y_{0ij}$ and $Y_{00i}$ terms do not
lead to any change in $\mathcal{C}$, a fact already well-appreciated in the
supergravity literature (ultimately because $Y_{0ij}$ and $Y_{00i}$ are real).
However the $Y_{000}$ term does play heavily in our discussion, so we take a
moment to fix a notation.

$Y_{000}\equiv-3\frac{\zeta(3)}{(2\pi\text{i})^{3}}\chi(X)\;=\;3\frac{\zeta(3)\text{i}}{8\pi^{3}}\chi\;=\;6\text{i}\zeta~{},\qquad\text{where}\qquad\zeta=\frac{\zeta(3)}{16\pi^{3}}\chi~{}.$
(4.122)

A salient point, $\zeta$ has the same sign as the Euler characteristic $\chi$.

We shall write the real and imaginary parts of $\bm{t}$ as $\bm{\mu}$ and
$\bm{\rho}$,

$\bm{t}=\bm{\mu}+\text{i}\bm{\rho}~{}.$ (4.123)

We will denote $\partial_{t^{i}}\mathcal{I}$ and
$\partial_{t^{i}}\partial_{t^{j}}\mathcal{I}$ by $\mathcal{I}_{i}$ and
$\mathcal{I}_{ij}$. A computation gives

$\text{e}^{-K}=4\left|z^{0}\right|^{2}\bigg{[}\frac{1}{3}Y_{ijk}\rho^{i}\rho^{j}\rho^{k}-\rho^{i}\,\text{Re}\left[\mathcal{I}_{i}\right]+\text{Im}\left[\mathcal{I}\right]+\zeta\bigg{]}~{}.$
(4.124)

Differentiating this equation yields

$\displaystyle\text{e}^{-K}K_{i}$
$\displaystyle\;=\;\phantom{-}2\left|z^{0}\right|^{2}\bigg{[}\text{i}Y_{imn}\rho^{m}\rho^{n}+\rho^{m}\mathcal{I}_{im}-\text{Im}\left[\mathcal{I}_{i}\right]\bigg{]}~{},$
(4.125) $\displaystyle\text{e}^{-K}K_{\bar{j}}$
$\displaystyle\;=\;-2\left|z^{0}\right|^{2}\bigg{[}\text{i}Y_{imn}\rho^{m}\rho^{n}-\rho^{m}\mathcal{I}_{im}+\text{Im}\left[\mathcal{I}_{i}\right]\bigg{]}~{},$
$\displaystyle\text{e}^{-K}K_{i\bar{j}}-\text{e}^{-K}K_{i}K_{\bar{j}}$
$\displaystyle\;=\;-2\left|z^{0}\right|^{2}\bigg{[}Y_{ijn}\rho^{n}+\text{Im}\left[\mathcal{I}_{ij}\right]\bigg{]}~{}.$

By introducing the matrix M with components

$\text{M}_{ij}=Y_{ijn}\rho^{n}-\text{i}\,\mathcal{I}_{ij}$ (4.126)

we can write

$K_{i\bar{j}}=-\text{e}^{K}\left(\text{M}+\overline{\text{M}}\right)_{ij}+4\text{e}^{2K}\bigg{[}\text{M}_{in}\rho^{n}+\text{i}\,\text{Im}\left[\mathcal{I}_{i}\right]\bigg{]}\bigg{[}\overline{\text{M}}_{jm}\rho^{m}-\text{i}\,\text{Im}\left[\mathcal{I}_{j}\right]\bigg{]}~{}.$
(4.127)

Computing $\mathcal{C}$ necessitates inverting this matrix, so now is a good
time to note an elementary formula. For an invertible symmetric matrix A and
vectors $\bm{a}$, $\bm{b}$, the matrix B with components

$\text{B}_{ij}\;=\;\text{A}_{ij}+a_{i}b_{j}$ (4.128)

has inverse

$\text{B}^{ij}\;=\;\text{A}^{ij}-\frac{1}{1+\bm{a}^{T}\text{A}^{-1}\bm{b}}\text{A}^{im}\text{A}^{jn}a_{m}b_{n}~{}.$
(4.129)

Let us introduce yet more notation,

$\text{P}_{ij}=\text{Re}\left[\text{M}_{ij}\right]~{},\qquad
v_{i}\;=\;\text{M}_{in}\rho^{n}+\text{i}\,\text{Im}\left[\mathcal{I}_{i}\right].$
(4.130)

We find

$\mathcal{C}\;=\;\frac{2\bm{v}^{T}\text{P}^{-1}\bar{\bm{v}}+4\text{e}^{K}\bigg{(}\left[\bm{v}^{T}\text{P}^{-1}\bm{v}\right]\left[\bar{\bm{v}}^{T}\text{P}^{-1}\bar{\bm{v}}\right]-\left[\bm{v}^{T}\text{P}^{-1}\bar{\bm{v}}\right]^{2}\bigg{)}}{2\bm{v}^{T}\text{P}^{-1}\bar{\bm{v}}-\text{e}^{-K}}~{}.$
(4.131)

If all terms in the instanton expansion are set to zero, then
$\text{e}^{-K}=\frac{4}{3}Y_{ijk}\rho^{i}\rho^{j}\rho^{k}+4\zeta$ and
$\bm{v}^{T}\text{P}^{-1}\bar{\bm{v}}=\bar{\bm{v}}^{T}\text{P}^{-1}\bar{\bm{v}}=\bm{v}^{T}\text{P}^{-1}\bar{\bm{v}}=Y_{ijk}\rho^{i}\rho^{j}\rho^{k}$.
(4.131) then reduces to

$\mathcal{C}\;=\;\frac{3Y_{ijk}\rho^{i}\rho^{j}\rho^{k}}{Y_{ijk}\rho^{i}\rho^{j}\rho^{k}-6\zeta}~{}.$
(4.132)

If the Euler characteristic of the manifold is $0$, then the above expression
takes the constant value 3. Then (4.118) vanishes, recovering the standard no-
scale supergravity potential, for instance appearing in [105].

However, if the Euler characteristic is not zero, then $\mathcal{C}$ is not
constant and $V$ is less trivial. In fact, $V$ exhibits typical Dine-Seiberg
behaviour [109]. The classical value of the manifold’s volume is

$\upsilon=Y_{ijk}\rho^{i}\rho^{j}\rho^{k}~{},$ (4.133)

and for large $\upsilon$, $\mathcal{C}$ asymptotes to the value 3. The
difference $\mathcal{C}-3$ falls off like $\frac{1}{\upsilon}$. The asymptote
is from above if $\chi>0$, and from below if $\chi<0$. In the first case,
$\chi>0$, a barrier prevents $\mathcal{C}$ from becoming smaller than
$\upsilon$ (in dimensionless variables) and the potential runs away to
infinity. For $\chi<0$, there is a runaway towards $\upsilon=0$. Since this
latter point is at small volume, we can expect that mechanisms not
incorporated in our analysis become important and so we cannot say much at all
about $\mathcal{C}$ near $\upsilon=0$.

The denouement of this section is that equation (4.131) allows for one to
study how instantons affect the previous paragraph’s content. For $\chi\neq 0$
there is no substantial change, but the $\chi=0$ scenario is altered.

If $\chi=0$ but instantons are not neglected, then at large $\rho$ the
contraction $\mathcal{C}$ will asymptote to 3 from above or below depending on
the sign of the instanton corrections. What is more, the asymptotic approach
will be exponential, rather than the powerlike decay discussed before. This is
because, if $\chi=0$, the first nonclassical term in (4.121) that contributes
to (4.131) is the first instanton contribution, which falls off exponentially
as the Kähler parameters are increased.

None of that has direct relevance to any useful string models, because (among
other problems) we have not stabilised any moduli. We have demonstrated how
the values of $\chi$ and the leading instanton number can affect the large
volume behaviour of this unrealistic model.

###  4.9 Modularity of supersymmetric vacua in flux compactifications

The central equations of this section are those giving the simultaneous
vanishing of the potential $V$ and superpotential $W$ in a IIB flux
compactification on a CY threefold $X$. We work with the classical
superpotential, which has no dependence on the Kähler moduli. The potential
$V$ then has the form (4.116) which we repeat here.

$W=\int_{X}G_{3}\wedge\Omega=(2\pi)^{2}\alpha^{\prime}(F-\tau
H)\Sigma\Pi~{}~{},\qquad
V\;=\;\text{e}^{\mathcal{K}}\left(\text{Im}[\tau]^{2}\,|\partial_{\tau}W|^{2}+g^{i\bar{\jmath}}\,\partial_{\varphi^{i}}W\,\overline{\partial_{\varphi^{j}}W}\right)~{}.$
(4.134)

The equations (4.117) satisfied by supersymmetric vacuum configurations,
together with the consistency condition $F^{T}\Sigma H\neq 0$, can be recast
as the SFV equations

$F^{T}\Sigma\Pi=0~{},\qquad H^{T}\Sigma\Pi=0~{},\qquad F^{T}\Sigma H\neq
0~{},\qquad\left(F-\tau
H\right)\Sigma\left(\partial_{\varphi^{i}}\Pi\right)=0~{}.$ (4.135)

It is unknown if, for a given manifold $X$, there is always a pair of flux
vectors $F$ and $H$ so that within $X$’s moduli space exists a region where
the above equations are satisfied. In [12] it was conjectured by Kachru,
Nally, and Yang that such manifolds, with their moduli on the SFV locus and
valued in an algebraic number field, are weight-two modular. They were able to
test this conjecture by appealing to two sources of information. One of these
was a solution method produced by deWolfe [110] for solving the SFV equations,
valid for $X$ the mirror of a hypersurface in weighted projective space. The
other key source of information was the thesis [51] of Kadir, in which tables
of zeta functions were given, with various choices of moduli, for the mirror
of the octic hypersurface in $\mathbbl{W}\mathbbl{P}_{1,1,2,2,2}$. Kachru,
Nally, and Yang supported their conjecture by reporting that on deWolfe’s
solution locus, the tables of Kadir demonstrated weight-two modularity.

We seek to make further tests of this conjecture. This requires more solutions
to the SFV equations and more computations of zeta functions. Concerning the
first requirement, we determine a method of solving the SFV equations that
works on any manifold $X$ whose period vector possesses a $\mathbbl{Z}_{2}$
symmetry. In the thesis [111], the method of [50] was extended to
multiparameter models which, using the solution method that we now describe,
allowed for the flux modularity conjecture to be extensively tested, with
additional tests carried out in [3].

####  4.9.1 Solving the SFV equations via permutation symmetry

It is required that the period vector $\Pi$ has two pairs of entries such that
each pair is permuted when two of the moduli are swapped. That is to say, we
should be able to identify a pair of moduli $\varphi^{I},\varphi^{J}$ so that

$\displaystyle\Pi^{1+a}(\varphi^{I},\varphi^{J})$
$\displaystyle=\Pi^{1+b}(\varphi^{J},\varphi^{I})~{},$ (4.136)
$\displaystyle\Pi^{2+h^{2,1}+a}(\varphi^{I},\varphi^{J})$
$\displaystyle=\Pi^{2+h^{2,1}+b}(\varphi^{J},\varphi^{I})~{},$

where we have suppressed $\Pi$’s dependence on the other moduli. The
superscripts indicate components of the vector $\Pi$, with $1\leq a,b\leq
h^{2,1}$.

The dependence of the other components of $\Pi$ on any remaining moduli is
unimportant. If $\Pi$ has this property, then on the locus
$\varphi^{I}=\varphi^{J}$ flux vectors $F$ and $H$ satisfying the first three
equations in (4.135) are

$F=e_{2+h^{2,1}+I}-e_{2+h^{2,1}+J}~{},\qquad H=e_{1+I}-e_{1+J}~{},$ (4.137)

where the $e_{k}$ are standard orthonormal basis vectors for
$\mathbbl{R}^{2+2h^{2,1}}$, with all components 0 except for the $k^{th}$
component being 1.

With the flux vectors and this restriction on the moduli space
$\varphi^{I}=\varphi^{J}$ specified, all bar two of the SFV equations are
satisfied for any value of the axiodilaton. The remaining two equations are
equivalent and fix the axiodilaton:

$\displaystyle\tau(\varphi)$
$\displaystyle=-\frac{F^{T}\Sigma\left(\partial_{\varphi^{I}}-\partial_{\varphi^{J}}\right)\Pi}{H^{T}\Sigma\left(\partial_{\varphi^{I}}-\partial_{\varphi^{J}}\right)\Pi}\bigg{|}_{\varphi^{I}=\varphi^{J}}$
(4.138)
$\displaystyle=\frac{i}{2\pi}\cdot\frac{\partial_{\varphi^{I}}\left(\varpi_{2,I}-\varpi_{2,J}\right)}{\partial_{\varphi^{I}}\left(\varpi_{1,I}-\varpi_{1,J}\right)}\bigg{|}_{\varphi^{I}=\varphi^{J}}+Y_{0IJ}-Y_{0II}~{}.$

It is perhaps interesting to note that for real moduli in the large complex
structure region, the real part of the axiodilaton is 0 or $1/2$ depending on
the value of $Y_{0IJ}-Y_{0II}$.

We will now give a select few worked examples. Many more are possible to find
than those given here.

####  4.9.2 Examples of supersymmetric flux vacua

##### The Hulek-Verrill Manifold

The first example of such a suitable symmetric manifold is the Hulek-Verrill
manifold, which is mirror to the CICY

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 1&1\\\ 1&1\\\ 1&1\end{matrix}\right]^{(5,45)}_{\chi\,=-80~{}}$ (4.139)

and has the following topological quantities:

$Y_{ijk}\;=\;\begin{cases}2\quad i,j,k\text{ distinct.}\\\
0\quad\text{otherwise,}\end{cases}\quad Y_{ij0}\;=\;0~{},\quad
Y_{i00}\;=\;-2~{},\quad
Y_{000}\;=\;240\frac{\zeta(3)}{(2\pi\text{i})^{3}}~{}.$ (4.140)

Analytic expressions for the periods are available in terms of integrals of
Bessel functions, as detailed in [2]. Subject to the condition

$\displaystyle\text{Re}\left[\sum_{i=1}^{5}\sqrt{\varphi^{i}}\right]<1~{},$
(4.141)

we can give the relevant periods as

$\displaystyle\varpi_{0}(\bm{\varphi})$
$\displaystyle\;=\;\int_{0}^{\infty}\text{d}z\;z\,\text{K}_{0}(z)\prod_{i=1}^{5}\text{I}_{0}\left(\sqrt{\varphi^{i}}z\right)~{},$
(4.142) $\displaystyle\varpi_{1,j}(\bm{\varphi})$
$\displaystyle\;=\;-2\int_{0}^{\infty}\text{d}z\;z\,\text{K}_{0}(z)\text{K}_{0}\left(\sqrt{\varphi^{j}}z\right)\prod_{i\neq
j}\text{I}_{0}\left(\sqrt{\varphi^{i}}z\right)~{},$
$\displaystyle\varpi_{2,j}(\bm{\varphi})$
$\displaystyle\;=\;8\sum_{\begin{subarray}{c}m<n\\\ m,n\neq
j\end{subarray}}\int_{0}^{\infty}\text{d}z\;z\,\text{K}_{0}(z)\text{K}_{0}\left(\sqrt{\varphi^{m}}z\right)\text{K}_{0}\left(\sqrt{\varphi^{n}}z\right)\prod_{i\neq
m,n}\text{I}_{0}\left(\sqrt{\varphi^{i}}z\right)-4\pi^{2}\varpi_{0}(\bm{\varphi})~{}.$

The five-parameter Hulek-Verrill manifolds support supersymmetric flux vacua
on the loci where any two complex structure moduli $\varphi^{i}$ are equal.
For definiteness, let us set two of the $\varphi^{1}$ and $\varphi^{2}$ equal
to a value $\varphi$, and relabel the three remaining $\varphi^{i}$ as
$\psi^{1},\,\psi^{2},\,\psi^{3}$. Use of standard Bessel function identities
reveals that in the case of Hulek-Verrill manifolds, the axiodilaton given in
(4.138) is

$\tau\left(\psi^{1},\,\psi^{2},\,\psi^{3}\right)\;=\;\frac{2\text{i}}{\pi}\,\frac{\int_{0}^{\infty}\text{d}z\;z\,\text{K}_{0}(z)\left[\text{K}_{0}\left(\sqrt{\psi^{1}}z\right)\,\text{I}_{0}\left(\sqrt{\psi^{2}}z\right)\,\text{I}_{0}\left(\sqrt{\psi^{3}}z\right)\phantom{\bigg{|}}+\text{cyclic}\right]}{\phantom{\bigg{|}}\int_{0}^{\infty}\text{d}z\;z\,\text{K}_{0}(z)\,\text{I}_{0}\left(\sqrt{\psi^{1}}z\right)\,\text{I}_{0}\left(\sqrt{\psi^{2}}z\right)\,\text{I}_{0}\left(\sqrt{\psi^{3}}z\right)}~{}.$
(4.143)

The dependence of the $j$-invariant can be found numerically by computing the
value of $j(\tau)$ on numerous points on the moduli space, and fitting the
points on a rational function. It turns out that the $j$-function takes a
remarkably simple form

$j\left(\tau\left(\psi^{1},\,\psi^{2},\,\psi^{3}\right)\right)\;=\;\frac{\left(\Delta_{F}+16\psi^{1}\psi^{2}\psi^{3}\right)^{3}}{\Delta_{F}\left(\psi^{1}\psi^{2}\psi^{3}\right)^{2}},$
(4.144)

where the polynomial $\Delta_{F}$, which is related to the discriminant
(3.25), is given by

$\displaystyle\Delta_{F}$ $\displaystyle\;=\;\prod_{\epsilon_{i}=\pm
1}\left(1+\epsilon_{1}\sqrt{\psi^{1}}+\epsilon_{2}\sqrt{\psi^{2}}+\epsilon_{3}\sqrt{\psi^{3}}\right)$
(4.145)
$\displaystyle\;=\;\left(\left(1-\psi^{1}-\psi^{2}-\psi^{3}\right)^{2}-4\left(\psi^{1}\psi^{2}+\psi^{2}\psi^{3}+\psi^{3}\psi^{1}\right)\right)^{2}-64\;\psi^{1}\psi^{2}\psi^{3}~{}.$

Note that to derive (4.143) and then subsequently arrive at (4.144), we have
used the expressions (4.142) for the periods, which are valid only in the
region (4.141). However, since the expression (4.144) for the $j$-function is
well-defined everywhere outside of the discriminant locus, this is the unique
analytic continuation of the left-hand side into this region and the
expression (4.144) is correct throughout moduli space.

It is interesting to note that (4.143) and (4.144) do not involve the complex
structure coordinate $\varphi$, and only depend on the three coordinates
$\psi^{1},\,\psi^{2},\,\psi^{3}$. In other examples, as we will see in
imminent subsections, $\tau$ does depend on the value to which we set two of
the coordinates in our solution method. For this particular example at hand,
the elimination of $\varphi$ from these formula was explained geometrically in
[3] in terms of the fibred product structure of the Hulek-Verrill manifold.
Understanding why this $\varphi$ independence does or does not occur for other
examples is an open problem.

##### The mirror bicubic

This manifold is mirror to the CICY

$\begin{matrix}\mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}3\\\
3\end{matrix}\right]_{\chi\,=-162~{},}$ (4.146)

with the constants $Y_{abc}$ given by

$Y_{ijk}\;=\;\begin{cases}0\quad i=j=k~{},\\\
3\quad\text{otherwise,}\end{cases}\quad
Y_{ij0}\;=\;\begin{cases}0&i\;=\;j~{},\\\
\frac{1}{2}&i\;\neq\;j~{},\end{cases}\,\quad Y_{i00}\;=\;-3~{},\quad
Y_{000}\;=\;486\frac{\zeta(3)}{(2\pi\text{i})^{3}}~{}.$ (4.147)

The periods can be written in terms of integrals of hypergeometric and Meijer
G-functions:

$\displaystyle\varpi_{0}$
$\displaystyle\;=\;\phantom{-}\int_{0}^{\infty}\text{d}u\;\text{e}^{-u}\,{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi^{1}\right)\,{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi^{2}\right)~{},$
(4.148) $\displaystyle\varpi_{1,i}$
$\displaystyle\;=\;-\int_{0}^{\infty}\text{d}u\;\text{e}^{-u}\,G^{2,0}_{0,3}\left(0,0,0|u^{3}\varphi^{i}\right)\,{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi^{j}\right)~{},\qquad
j\neq i~{},$ $\displaystyle\varpi_{2,i}$
$\displaystyle\;=\;3\int_{0}^{\infty}\text{d}u\,\text{e}^{-u}\Big{[}G^{2,0}_{0,3}\left(0,0,0|u^{3}\varphi^{i}\right)\,G^{2,0}_{0,3}\left(0,0,0|u^{3}\varphi^{j}\right)$
$\displaystyle\hskip
80.0pt+{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi^{i}\right)\left(G^{3,0}_{0,3}\left(0,0,0|-u^{3}\varphi^{j}\right)+\text{i}\pi
G^{2,0}_{0,3}\left(0,0,0|u^{3}\varphi^{j}\right)\right)\Big{]}$
$\displaystyle\hskip 120.0pt-2\pi^{2}\varpi_{0}~{},\hskip 110.0ptj\neq i~{},$

whence it follows that the axiodilaton profile on the $\mathbbl{Z}_{2}$
symmetric locus $\varphi^{1}=\varphi^{2}$ is given by

$\tau(\varphi){=}\frac{3\text{i}}{2\pi}\,\frac{\int_{0}^{\infty}\text{d}u\;\text{e}^{-u}\,\left[{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi\right)G^{3,0}_{0,3}(0,0,1|{-}u^{3}\varphi)-{}_{0}F_{2}\left(2,\,2;~{}u^{3}\varphi\right)G^{3,0}_{0,3}(1,1,1|{-}u^{3}\varphi)\right]}{\int_{0}^{\infty}\text{d}u\;\text{e}^{-u}\,\left[{}_{0}F_{2}\left(1,\,1;~{}u^{3}\varphi\right)G^{2,0}_{0,3}(0,1,0|u^{3}\varphi)+{}_{0}F_{2}\left(2,\,2;~{}u^{3}\varphi\right)G^{2,0}_{0,3}(1,1,1|u^{3}\varphi)\right]}{-}1~{}.$

Even though it is not immediately obvious, this expression has, for real
$\varphi$, a real part equal to $\frac{1}{2}$. One should bear in mind that we
have further simplified the ratio of integrals to arrive at the above
expression.

Numerical methods strongly suggest that the $j$-invariant $j(\tau(\varphi))$
is given by the following rational function of the moduli:

$\displaystyle
j(\tau(\varphi))\;=\;-\frac{(1+24\varphi)^{3}}{\varphi^{3}(1+27\varphi)}~{}.$
(4.149)

##### The mirror maximally split quintic

This manifold is the mirror of the maximal split of the quintic hypersurface
in $\mathbbl{P}^{5}$, given by the configuration described by the CICY matrix

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}\;1\;1\;1\;1\;1\;\\\
\;1\;1\;1\;1\;1\;\end{matrix}\right]_{\chi=-100}.$ (4.150)

The quantities $Y_{abc}$ of the split quintic are

$Y_{ijk}\;=\;\begin{cases}5\quad\;\;i=j=k~{},\\\ 10\quad\text{otherwise,}\
\end{cases}\quad Y_{0ij}\;=\;\begin{cases}\frac{1}{2}&i\;=\;j~{},\\\
0&i\;\neq\;j~{},\end{cases}\qquad Y_{i00}\;=\;-\frac{25}{6}~{},\quad
Y_{000}\;=\;\frac{300\zeta(3)}{(2\pi\text{i})^{3}}~{}.$ (4.151)

Denoting the Meijer G-functions as
$G^{a,b}_{c,d}(0,0,0,0,0|x)=G^{a,b}_{c,d}(x)$, and the hypergeometric
functions as ${}_{a}F_{b}(1,1,1,1;x)={}_{a}F_{b}(x)$, the periods relevant to
our analysis can be written as

$\displaystyle\varpi_{0}$
$\displaystyle\;=\;\phantom{-}\int_{0}^{\infty}\text{d}u\;G^{5,0}_{0,5}(u)\;{}_{0}F_{4}(u\varphi^{1})\;{}_{0}F_{4}(u\varphi^{2})~{},$
(4.152) $\displaystyle\varpi_{1,i}$
$\displaystyle\;=\;-\int_{0}^{\infty}\text{d}u\;G^{5,0}_{0,5}(u)\;G^{2,0}_{0,5}(u\varphi^{i})\;{}_{0}F_{4}(u\varphi^{j})~{},\hskip
156.0pti\neq j~{},$ $\displaystyle\varpi_{2,i}$
$\displaystyle\;=\;5\int_{0}^{\infty}\text{d}u\;G^{5,0}_{0,5}(u)\left[2G^{2,0}_{0,5}(u\varphi^{i})\;G^{2,0}_{0,5}(u\varphi^{j})\right.+\left(G^{3,0}_{0,5}(-u\varphi^{i})+\text{i}\pi
G^{2,0}_{0,5}(u\varphi^{i})\right){}_{0}F_{4}(u\varphi^{2})$
$\displaystyle+2\left(G^{3,0}_{0,5}(-u\varphi^{j})+\text{i}\pi
G^{2,0}_{0,5}(u\varphi^{j})\right){}_{0}F_{4}(u\varphi^{i})\left.-\;\frac{5\pi^{2}}{3}\;{}_{0}F_{4}(u\varphi^{i}){}_{0}F_{4}(u\varphi^{j})\right]~{},\qquad
i\neq j~{},$

and the axiodilaton is given on the $\mathbbl{Z}_{2}$ symmetric locus
$\varphi^{1}=\varphi^{2}=\varphi$ by

$\displaystyle\tau(\varphi)$
$\displaystyle{=}\frac{5\text{i}}{2\pi}\frac{\int_{0}^{\infty}\text{d}u\;G^{5,0}_{0,5}(u)\left[G^{3,0}_{0,5}(0,0,1,0,0|-u\varphi){}_{0}F_{4}(u\varphi){-}G^{3,0}_{0,5}(1,1,1,1,1|-u\varphi){}_{0}F_{4}\left(2,\,2,\,2,\,2;~{}u\varphi\right)\right]}{\int_{0}^{\infty}\text{d}u\;G^{5,0}_{0,5}(u)\left[G^{2,0}_{0,5}(0,1,0,0,0|u\varphi){}_{0}F_{4}(u\varphi){-}G^{2,0}_{0,5}(1,1,1,1,1|u\varphi){}_{0}F_{4}\left(2,\,2,\,2,\,2;~{}u\varphi\right)\right]}$
(4.153) $\displaystyle\phantom{-}\phantom{-}\phantom{-}\phantom{-}+2~{}.$

For real $\varphi$, this expression has real part equal to $-\frac{1}{2}$.
Attempts to numerically integrate the above combinations of Meijer G functions
are met with problems in Mathematica, so instead it is best to expand each of
the hypergeometric functions ${}_{0}F_{4}$ as a power series in $u$ up to some
large order (we used 360). After interchanging the order of summation and
integration, each integral in the resulting sum can be evaluated in
Mathematica exactly as a single Meijer G function, yielding a series
expression amenable to fast evaluation.

The numerical evidence strongly suggests that the $j$-invariant $j(\tau)$ is
again a rational function of the complex structure parameter $\varphi$,

$\displaystyle
j(\tau(\varphi))\;=\;-\frac{\left(1+12\varphi+14\varphi^{2}-12\varphi^{3}+\varphi^{4}\right)^{3}}{\varphi^{5}(1+11\varphi-\varphi^{2})}~{}.$
(4.154)

####  4.9.3 Speculation on elliptic curves inside the F-theory fourfold

Consider again the F-theory fourfold, as discussed in §4.6. This is an
elliptic fibration, and for the supersymmetric vacuum configurations which we
have described, the fibre is the elliptic curve $E_{\tau}$ with $\tau$ the
axiodilaton (4.138) of our solution to the supersymmetric flux vacuum
equations.

The base of this fibration, $\mathcal{B}$, is doubly covered by the threefold
$X$, which we have argued is modular of weight two. In line with the
conjectures of Kachru, Nally, and Yang in [12, 11], the weight-two modular
form associated to $X$ is itself associated to the elliptic curve $E_{\tau}$
in the sense of the modularity theorem for elliptic curves.

For the case of the Hulek-Verrill family, it was explained in [3] that the
threefold $X$ contained a ruled surface $E_{\tau}\times\mathbbl{P}^{1}$, which
was nontrivial in homology. This gave an account of why the threefold’s zeta
function numerator contained a factor corresponding to the elliptic curve
$E_{\tau}$.

For this Hulek-Verrill example then, the F-theory fourfold that appears in the
uplift of the supersymmetric flux vacuum that we describe contains $E_{\tau}$
in two different ways: once as the fibre and again in (a double cover) of the
base. It could be interesting to understand whether this happens for every
example, and more speculatively, to see if any string duality explains or
utilises this twofold appearance of $E_{\tau}$ in the F-theory fourfold.

###  4.10 New weight four modular manifolds

We have seen in §4.2 that the attractor mechanism fixes the values taken by
vector multiplet scalar fields on the horizon of an extremal black hole in
$\mathcal{N}=2$ supergravity. In IIB Calabi-Yau compactifications those
scalars give the complex structure moduli of the Calabi-Yau $X$, and the fixed
points of attractor flows are the solutions of Strominger’s equations

$Q=\text{Im}\left[C\Pi\right]~{}.$ (4.155)

In this section we will describe a method, first worked through in [14] and
later fully developed in [50], for finding rank-two attractors. These are
complex structure moduli $\varphi_{*}$ that solve the above equation (4.155)
for two independent charge vectors $Q_{1},\,Q_{2}$. Tables that display the
main piece of information that we use, factorisation counts, are collected in
§B. Our search returns a number of known examples, but also some new weight-
four manifolds.

We will search among a subset of the one-parameter manifolds. These possess
fourth-order Picard Fuchs operators which are of Calabi-Yau type [112, 113].
Such operators are tabulated in the database [65], which is attached to the
paper [112]. We shall refer to geometries by the AESZ label of their operator.
Note that in the one-parameter setting, rank-two attractors are also solutions
to the supersymmetric flux vacuum equations, and indeed all examples of
weight-four modular one-parameter manifolds are also weight-two modular.

#### The search process

If a rational number $\varphi_{*}$ is such that $X_{\varphi_{*}}$ is weight-
four modular, then the numerator of the zeta function
$\zeta_{p}\left(X_{\varphi_{*}};T\right)$ should factorise for each prime $p$.
Turning this on its head, we will search for rank-two attractors by first
tabulating the zeta function for $\varphi^{p}\in\\{0,...,p-1\\}$, for a number
of primes $p$. Fixing a prime $p$, we will count the number of times the zeta
function factorises for $\varphi^{p}\in\\{0,...,p-1\\}$. If this number is
‘usually’ greater than zero (with exceptions expected to correspond to a small
number of bad primes), then we inspect the zeta function numerators for the
$\varphi^{p}_{*}$ that have factorised numerators and search the LMFDB
database [40] for a weight-four modular form whose $p^{th}$ Fourier
coefficient is that read off of the zeta function numerator, to the extent of
our tables.

If such a modular form can be found, then we seek a rational $\varphi_{*}$
that reduces modulo $p$ to $\varphi^{p}_{*}$. Having found such a value
$\varphi_{*}$, we can numerically check to see if a pair of rational charge
vectors $Q_{1},\,Q_{2}$ can be found so that (4.155) is solved by
$\Pi(\varphi_{*})$.

We perform this check for 61 of the operators on the AESZ list. In Appendix §B
we give bar charts showing the number of factorisations for each prime $5\leq
p\leq 131$, for a total of 30 primes. To spot a rational rank-two attractor,
we scroll down this list and stop where the bars all have heights greater than
0. In this way we recover a number of known examples: AESZ34 [14], and the
pair AESZ4 and AESZ11 [13], as well as a number of examples identified
previously in [47]. We will highlight those examples in Appendix §B, and also
comment on this choice of 61 operators.

Moreover, we are also able to identify two hitherto-undiscovered attractors.
These are in the moduli spaces of AESZ17 and AESZ22, and both at
$\varphi_{*}=-1$.

The operators AESZ17 and AESZ22 possess a property not shared by any other
operators for which rank-two attractors are known: 17 and 22 both possess two
MUM points, at 0 and infinity. By a change of variables that exchanges the MUM
points, the operators AESZ17 and AESZ22 can be respectively transformed to
AESZ118 and AESZ290. Indeed, the charts in §B for the pair (22,118) are
identical. We give the associated modular forms in the following table.

(AESZ no. | Attractor | weight 2 form | weight 4 form
---|---|---|---
17 | $-1\phantom{{}^{-5}}$ | 14.2.a.a | 14.4.a.b
22 | $-1\phantom{{}^{-5}}$ | 11.2.a.a | 33.4.a.b
Table 4.1: New rank two attractors.

One should bear in mind that these are only conjectured to be modular
varieties, and that conjecture is supported by tabulating the zeta function
for finitely many primes.

We will say more about these geometries in §4.11, and wrap up this section
with the first few terms of the above modular form’s $q$-expansions, with
$q=\text{e}^{2\pi\text{i}\tau}$:

$\displaystyle f_{14.2.a.a}(\tau)$
$\displaystyle=q-q^{2}-2q^{3}+q^{4}+2q^{6}+q^{7}-q^{8}+q^{9}+\,...~{},$
(4.156) $\displaystyle f_{14.4.a.b}(\tau)$
$\displaystyle=q+2q^{2}-2q^{3}+4q^{4}-12q^{5}-4q^{6}+7q^{7}+8q^{8}-23q^{9}+\,...~{},$
$\displaystyle f_{11.2.a.a}(\tau)$
$\displaystyle=q-2q^{2}-q^{3}+2q^{4}+q^{5}+2q^{6}-2q^{7}-2q^{9}+\,...~{},$
$\displaystyle f_{33.4.a.b}(\tau)$
$\displaystyle=q-q^{2}-3q^{3}-7q^{4}-4q^{5}+3q^{6}-26q^{7}+15q^{8}+9q^{9}+\,...~{}.$

LMFDB [40] provides the following eta-function expressions for the weight-two
forms:

$f_{14.2.a.a}(\tau)=\eta(\tau)\eta(2\tau)\eta(7\tau)\eta(14\tau)~{},\qquad
f_{11.2.a.a}(\tau)=\eta(\tau)^{2}\eta(11\tau)^{2}~{},$ (4.157)

with $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^{n}\right)~{}.$

###  4.11 Summation identities: L-value ratios from GW invariants

Note that all summation identities in this section are at the time of writing
conjectures, supported to the various degrees of numerical precision that we
give.

In this section, we will specialise the series solution (4.85) to the
orthogonality equations to a selection of rank-two attractors. Following [14,
13], we can express the Frobenius periods as linear combinations of the two
critical L-values associated to a weight-four modular form. Therefore, the
mirror coordinate $t$ is a rational function of those L-values. This value $t$
is a solution to the instanton-corrected IIA attractor equations, and we
explained in §4.4 how such a $t$ necessarily solved the orthogonality
equations. By expressing $t$ via the solution (4.85), we arrive at a series
identity. Certain combinations of L-values can be expressed as infinite sums
whose terms are built out of Gromov-Witten invariants.

In practice, we evaluate no more than the first 75 terms of the series,
because the $j^{th}$ term contains a sum over partitions of $j$. Beyond $j=75$
there are too many partitions to work with on a laptop.

These sums converge for small enough values of $y_{0}$, and in those cases we
get an honest numerical equality. Shanks transformations [114] can be applied
to the series in order to improve the number of figures to which our
identities hold. However, for a number of rank-two attractors the appropriate
value of $y_{0}$ is too large and the sums formally diverge.

Building on the work of [4], where such divergences were not further
addressed, we identify Padé resummation as the appropriate scheme for summing
these series for larger values of $y_{0}$. We shall write

$\text{LHS}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\sum_{j=1}\text{e}^{2\pi\text{i}x_{0}j}c\left(j,y_{0}\right)$
(4.158)

to indicate that while the series on the right hand side diverges, the Padé
approximants about 0 in the variable $X$ to the series
$\sum_{j=1}^{\infty}X^{j}c\left(j,y_{0}\right)$ approach (as the order of the
approximant is increased) the value on the LHS when the substitution
$X=\text{e}^{2\pi\text{i}x_{0}}$ is made.

Our examples will feature rank two attractors found in [14, 13] for the
operators AESZ34, AESZ4, and AESZ11, as well as the new attractors found in
§4.10 for AESZ22 and AESZ17.

Note that when we give a $\chi$ among a set of topological data at the start
of a subsection, we do so for the B-model geometry.

#### AESZ4, The mirror of $\mathbbl{P}^{5}[3,3]$

The Picard-Fuchs equation for this geometry is the following hypergeometric
equation:

$\left[\theta^{4}-729\,z\left(\theta+\frac{1}{3}\right)^{2}\left(\theta+\frac{2}{3}\right)^{2}\right]F(z)\;=\;0~{}.$
(4.159)

The topological data is

$Y_{111}\;=\;9~{},\qquad Y_{110}\;=\;\frac{1}{2}~{},\qquad
Y_{100}\;=\;-\frac{9}{2}~{},\qquad\chi\;=\;144~{}.$ (4.160)

It was shown in [13] that this geometry has a rank two attractor at
$z=-2^{-3}\cdot 3^{-6}$. The Frobenius periods were shown to be given by the
following combinations of critical L-values associated to the weight-4 modular
form with LMFDB label 54.4.a.c :

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-18\,\frac{L(2)}{(2\pi\text{i})^{2}}~{},$ (4.161)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-9\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{9}{4}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;\phantom{-}135\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{81}{8}\,\frac{L(1)}{(2\pi\text{i})}~{},$
$\displaystyle\widehat{\varpi}_{3}$
$\displaystyle\;=\;\phantom{-}\frac{297}{4}\,\frac{L(2)}{(2\pi\text{i})^{2}}+\frac{81}{16}\,\frac{L(1)}{2\pi\text{i}}+\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}~{}.$

These critical L-values have decimal expansions

$\displaystyle L(1)$ $\displaystyle\;=\;3.84405587339769322173023\,...~{},$
(4.162) $\displaystyle L(2)$
$\displaystyle\;=\;2.18020008917513916885513\,...~{}.$

In the $t-$plane, the attractor is

$t\;=\;\frac{1}{2}+\frac{\text{i}\pi}{4}\frac{L(1)}{L(2)}~{}.$ (4.163)

The attractor equations are solved by $z=-2^{-3}\cdot 3^{-6}$ for the
following charge vectors:

$Q_{1}\;=\;(0,-5,0,1)^{T},\qquad Q_{2}\;=\;(-3,12,1,0)^{T}~{}.$ (4.164)

It follows that the integral basis period vector $\Pi$ is symplectic-
orthogonal to the charge vectors

$Q_{3}\;=\;(-3,0,1,0)^{T},\qquad Q_{4}\;=\;(12,-5,0,1)^{T}~{}.$ (4.165)

$Q_{4}$ is of ‘D4-type’, as the entry corresponding to the D6 charge is zero.
This means that we can form an identity as in [4]. The perturbative solution
of the orthogonality equation is given by

$t_{0}=\frac{1}{2}+\frac{\text{i}\sqrt{69}}{6}~{}.$ (4.166)

We arrive at

$\frac{3\pi}{2}\cdot\frac{L(1)}{L(2)}\;=\;\sqrt{69}-\sqrt{\frac{2}{\pi^{3}}}\,\sum_{j=1}^{\infty}(-1)^{j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{j}{3\pi\sqrt{69}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{j\,\pi\sqrt{69}}{3}\right)~{}.$
(4.167)

This sum converges, and there is no need for Padé resummation. The equality
should be understood as usual. Using 60 terms of the series, we observe
agreement to 73 figures. By performing a Shanks transformation up to 15 times,
we observe improvement to 105 figures. Applying further Shanks transformations
does not yield further improvement.

### AESZ11, The mirror of $\mathbbl{W}\mathbbl{P}^{5}_{1,1,1,1,1,2}[4,3]$

This geometry also has a hypergeometric Picard-Fuchs equation:

$\left[\theta^{4}-1728\,z\left(\theta+\frac{1}{4}\right)\left(\theta+\frac{1}{3}\right)\left(\theta+\frac{2}{3}\right)\left(\theta+\frac{3}{4}\right)\right]F(z)\;=\;0~{}.$
(4.168)

The topological data is

$Y_{111}\;=\;6~{},\qquad Y_{110}\;=\;0~{},\qquad
Y_{100}\;=\;-4~{},\qquad\chi\;=\;156~{}.$ (4.169)

It was shown in [13] that this geometry has a rank two attractor at
$z=-2^{-4}\cdot 3^{-3}$. The Frobenius periods were again shown to be given by
combinations of critical L-values , this time associated to the weight-4
modular form with LMFDB label 180.4.a.e :

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-18\,\frac{L(2)}{(2\pi\text{i})^{2}}~{},$ (4.170)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-9\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{6}{10}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;\phantom{-}45\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{9}{5}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{3}$
$\displaystyle\;=\;\phantom{-}27\,\frac{L(2)}{(2\pi\text{i})^{2}}+\frac{3}{5}\,\frac{L(1)}{2\pi\text{i}}+\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}~{}.$

These critical L-values have decimal expansions

$\displaystyle L(1)$ $\displaystyle\;=\;10.01558944150908412493925\,...~{},$
(4.171) $\displaystyle L(2)$
$\displaystyle\;=\;\phantom{0}1.99897335900796011552100\,...~{}.$

In the $t-$plane, the attractor this time is

$t\;=\;\frac{1}{2}+\frac{\text{i}\pi}{15}\cdot\frac{L(1)}{L(2)}~{}.$ (4.172)

The value $z=-2^{-4}\cdot 3^{-3}$ solves the attractor equations for the
following charge vectors:

$Q_{1}\;=\;(-1,6,1,0)^{T},\qquad Q_{2}\;=\;(1,-3,0,1)^{T}~{}.$ (4.173)

One computes that the integral basis period vector $\Pi$ is symplectic-
orthogonal to the charge vectors

$Q_{3}\;=\;(-1,1,1,0)^{T},\qquad Q_{4}\;=\;(6,-3,0,1)^{T}~{}.$ (4.174)

$Q_{4}$ is of D4-type, so we can form an identity. The perturbative solution
is given by

$t_{0}=\frac{1}{2}+\frac{\text{i}\sqrt{39}}{6}~{}.$ (4.175)

We thereby find

$\frac{2\pi}{5}\cdot\frac{L(1)}{L(2)}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\sqrt{39}-\sqrt{\frac{3}{\pi^{3}}}\,\sum_{j=1}^{\infty}(-1)^{j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{j}{2\pi\sqrt{39}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{j\,\pi\sqrt{39}}{3}\right)~{}.$
(4.176)

This sum does not converge. As discussed in the overview, the identity is
supported numerically by Padé resummation. A diagonal Padé approximant of
order 30 gives agreement to 38 figures.

### AESZ34, A quotient of the Hulek-Verrill manifold

Three rank two attractor points were found for this geometry in [14]. One of
these, at $z=33-8\sqrt{17}$, lies inside the LCS region and was a central
example in [4], where the first sum of the form considered here was given. We
discuss now the other two attractor points, at $z=-7^{-1}$ and
$z=33+8\sqrt{17}$, which lie outside of the LCS region and so require Padé
resummation.

The Picard-Fuchs equation is not hypergeometric, and is

$\displaystyle\left[(1-z)(1-9z)(1-25z)\,\theta^{4}+2z\left(675z^{2}-518z+35\right)\theta^{3}+z\left(2925z^{2}-1580z+63\right)\theta^{2}\right.$
(4.177)
$\displaystyle\left.+4z\left(675z^{2}-272z+7\right)\theta+5z\left(180z^{2}-57z+1\right)\right]F(z)\;=\;0~{}.$

The topological data is

$Y_{111}\;=\;24~{},\qquad Y_{110}\;=\;0~{},\qquad
Y_{100}\;=\;-2~{},\qquad\chi\;=\;16~{}.$ (4.178)

We will divide the remainder of this section into subsections discussing each
rank two attractor separately. We choose to work with the ‘$\kappa=2$’
geometry of [14], which has triple intersection number $Y_{111}=24$. The
choice does not affect the summation identities, as all factors of $\kappa$
drop out.

#### $z\;=\;-\frac{1}{7}$

The periods were conjectured (with 1000 digits of numerical precision
supporting) to be given in terms of critical L-values for the modular form
14.4.a.a .

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-28\,\frac{L(2)}{(2\pi\text{i})^{2}}~{},$ (4.179)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-14\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{5}{2}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;-28\frac{L(2)}{(2\pi\text{i})^{2}}-30\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{3}$
$\displaystyle\;=\;\phantom{-}14\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{11}{2}\frac{L(1)}{2\pi\text{i}}+\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}~{}.$

The decimal expansions of these L-values begin

$\displaystyle L(1)$ $\displaystyle\;=\;0.67496319716994177129270\,...~{},$
(4.180) $\displaystyle L(2)$
$\displaystyle\;=\;0.91930674266912115653914\,...~{}.$

The attractor is, in the $t$-plane,

$t\;=\;\frac{1}{2}+\frac{5\pi\text{i}}{28}\cdot\frac{L(1)}{L(2)}~{}.$ (4.181)

Independent charge vectors for which the attractor equations are solved are

$Q_{1}\;=\;(-\frac{8}{5},6,1,0)^{T}~{},\qquad
Q_{2}\;=\;(\frac{16}{5},-12,0,1)^{T}~{}.$ (4.182)

The integral symplectic period vector is then orthogonal to

$Q_{3}\;=\;(-\frac{8}{5},\frac{16}{5},1,0)^{T}~{},\qquad
Q_{4}\;=\;(6,-12,0,1)^{T}~{}.$ (4.183)

We shall form our identity using $Q_{4}$. The perturbative solution is

$t_{0}\;=\;\frac{1}{2}+\frac{\text{i}}{\sqrt{6}}~{}.$ (4.184)

From these considerations, we can find the identity

$\frac{15\pi}{14}\cdot\frac{L(1)}{L(2)}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\sqrt{6}-\frac{1}{2}\cdot\sqrt{\frac{3}{\pi^{3}}}\,\sum_{j=1}^{\infty}(-1)^{j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{j}{8\pi\sqrt{6}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{\pi
j\sqrt{6}}{3}\right)~{}.$ (4.185)

A diagonal Padé approximant of order 30 gives agreement to 34 figures.

#### $z\;=\;33+8\sqrt{17}$

This constitutes perhaps our strangest example. The real parts of the critical
L-values associated to 34.4.b.a have decimal expansions beginning

$\displaystyle\lambda(1)$
$\displaystyle\;=\;0.61300748403501690756896\,...~{},$ (4.186)
$\displaystyle\lambda(2)$
$\displaystyle\;=\;0.72053904959503349611019\,...~{}.$

The complete L-values are

$\displaystyle L(1)$
$\displaystyle\;=\;\lambda(1)\left(1+\text{i}\left(\frac{1+\sqrt{17}}{4}\right)^{3}\right)~{},$
(4.187) $\displaystyle L(2)$
$\displaystyle\;=\;\lambda(2)\left(1-\text{i}\,\frac{1-\sqrt{17}}{4}\right)~{}.$

Introduce symbols for the following numbers:

$\epsilon_{-}\;=\;4-\sqrt{17}~{},\qquad\delta_{+}\;=\;\frac{3+\sqrt{17}}{2}~{},\qquad\delta_{-}\;=\;\frac{3-\sqrt{17}}{2}~{}.$
(4.188)

In terms of the real parts $\lambda(1),\,\lambda(2)$ of the L-values, the
periods as computed in [14] are (again, conjecturally with 1000 digits of
precision found numerically by those authors)

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-\frac{7\sqrt{17}\epsilon_{-}^{2}\delta_{+}}{2^{3}\pi^{2}}\lambda(2)-\text{i}\,\frac{3\cdot
5\delta_{-}^{3}}{2^{4}\sqrt{17}\pi}\lambda(1)~{},$ (4.189)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-\frac{\sqrt{17}\epsilon_{-}^{2}\delta_{+}}{2\pi^{2}}\lambda(2)-\text{i}\,\frac{5\delta_{-}^{3}}{2^{5}\sqrt{17}\pi}\lambda(1)~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;-\frac{5^{2}\sqrt{17}\epsilon_{-}^{2}\delta_{+}}{2^{3}\pi^{2}}\lambda(2)+\text{i}\,\frac{3\cdot
5\delta_{-}^{3}}{2^{4}\sqrt{17}\pi}\lambda(1)~{},$
$\displaystyle\widehat{\varpi}_{3}$
$\displaystyle\;=\;-\frac{\sqrt{17}\epsilon_{-}^{2}\delta_{+}}{2\pi^{2}}\lambda(2)+\text{i}\,\frac{13\delta_{-}^{3}}{2^{5}\sqrt{17}\pi}\lambda(1)+\frac{\chi\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}~{}.$

In the $t$-plane, the attractor is

$t\;=\;\frac{1}{6}+\frac{1156L(2)}{2856L(2)-45\pi\text{i}\left(9+\sqrt{17}\right)L(1)}~{}.$
(4.190)

The attractor equations are solved for the charge vectors

$Q_{1}\;=\;(-\frac{72}{85},\frac{66}{17},1,0)^{T}~{},\qquad
Q_{2}\;=\;(\frac{296}{85},-\frac{192}{17},0,1)^{T}~{}.$ (4.191)

The period vector is orthogonal to the charge vectors

$Q_{3}\;=\;(-\frac{72}{85},\frac{296}{85},1,0)^{T}~{},\qquad
Q_{4}\;=\;(\frac{66}{17},-\frac{192}{17},0,1)^{T}~{}.$ (4.192)

$Q_{4}$ shall be used to obtain an identity. The pertubative solution is

$t_{0}\;=\;\frac{8}{17}+\frac{\text{i}}{34}\sqrt{\frac{65}{3}}~{}.$ (4.193)

We arrive at

$\displaystyle\frac{2^{3}17^{3}\lambda(2)}{2856\lambda(2)-45\pi\text{i}\left(9+\sqrt{17}\right)\lambda(1)}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\frac{31}{3}+\text{i}\sqrt{\frac{65}{3}}$
(4.194) $\displaystyle\hskip
30.0pt-\frac{17\text{i}}{2\sqrt{3\pi^{3}}}\sum_{j=1}^{\infty}\text{e}^{\frac{16\pi\text{i}}{17}\cdot
j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{17j}{8\pi\sqrt{195}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{\pi
j}{17}\sqrt{\frac{65}{3}}\right)~{}.$

This identity is much trickier to verify numerically. Using a diagonal Padé
approximant of order 30, agreement can be found to six figures. Our
conjectural summation identity for this example then is not very well
supported. In fact, agreement to six figures can be found using diagonal Padé
approximants of orders 23 through 30.

In case our series (4.194) does not hold, and in case the other (better
evidenced) conjectural series identities from this section should prove true,
it would be interesting to understand what sets this example apart.

### The pair AESZ22 and AESZ118

The following two operators are mapped to one another by a change of variables
and scaling transformation

$z\;=\;\frac{1}{32x}~{},\qquad F_{22}(z)=\frac{1}{32x}F_{118}(x)~{}.$ (4.195)
AESZ22: $\displaystyle\hskip
10.0pt\left[(7-4z)^{2}\left(1+11z-z^{2}\right)\theta^{4}\right.$ (4.196)
$\displaystyle\hskip
15.0pt\left.-2z(7-4z)\left(143+4942z-2084z^{2}+256z^{3}\right)\theta^{3}\right.$
$\displaystyle\hskip
15.0pt\left.-z\left(1638+102261z-72568z^{2}+23024z^{3}-3072z^{4}\right)\theta^{2}\right.$
$\displaystyle\hskip
15.0pt\left.-z\left(637+66094z-30072z^{2}+12896z^{3}-2048z^{4}\right)\theta\right.$
$\displaystyle\hskip
15.0pt\left.-2z\left(49+7868z-1904z^{2}+1472z^{3}-256z^{4}\right)\right]F_{22}(z)\;=\;0~{},$
AESZ118: $\displaystyle\hskip
10.0pt\left[(1-x)(1-56x)^{2}\left(1-352x-1024x^{2}\right)\theta^{4}\right.$
$\displaystyle\hskip
15.0pt-2x(1-56x)(9-64x)\left(33+384x-1792x^{2}\right)\theta^{3}$
$\displaystyle\hskip
15.0pt-x\left(431+15136x-335424x^{2}+4386816x^{3}-19267584x^{4}\right)\theta^{2}$
$\displaystyle\hskip
15.0pt-2x\left(67+7072x-41088x^{2}+996532x^{3}-6422528x^{4}\right)\theta$
$\displaystyle\hskip
15.0pt\left.-16x\left(1+176x-144x^{2}+23296x^{3}-20070x\right)\right]F_{118}(x)\;=\;0~{}.$

In the first of the above equations, $\theta$ denotes the operator
$z\frac{\text{d}}{\text{d}z}$, while in the second $\theta$ denotes
$x\frac{\text{d}}{\text{d}x}$.

Every operator in the AESZ list is ‘adapted’ to a MUM point, in that the point
$0\in\mathbbl{C}$ is a point of maximal unipotent monodromy for any operator
and the exponents for a basis of solutions about this point are
$\\{0,0,0,0\\}$.

However, as demonstrated by these examples, Calabi-Yau operators can have
multiple MUM points. AESZ22 has, in addition to the origin, a MUM point at
infinity with exponents $\\{1,1,1,1\\}$. Under the above change of variables,
this point is mapped to 0 while 0 is mapped to AESZ118’s secondary MUM point
at infinity. The scaling transformation $F(z)\mapsto xF(x)$ is performed so
that the exponents about the origin are $\\{0,0,0,0\\}$.

The $q$-expansion of either the Yukawa coupling or the prepotential about
$z=0$ returns integer invariants $n_{d}^{\text{RC}}$ that are interpreted as
counts of genus 0 curves with degree $d$ on a mirror geometry, the Reye
Congruence. This is a $\mathbb{Z}_{2}$ quotient of the complete intersection

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}\;1\;1\;1\;1\;1\;\\\
\;1\;1\;1\;1\;1\;\end{matrix}\right]_{\chi=-100.}$ (4.197)

by the freely acting $\mathbb{Z}_{2}$ quotient that exchanges the two
$\mathbbl{P}^{4}$ factors. The first few $n_{d}^{\text{RC}}$ are

$n_{1}^{\text{RC}}\;=\;50~{},\,n_{2}^{\text{RC}}\;=\;325~{},\,n_{3}^{\text{RC}}\;=\;1475~{},\,n_{4}^{\text{RC}}\;=\;15325~{},\,n_{5}^{\text{RC}}\;=\;148575~{},\,n_{6}^{\text{RC}}\;=\;1885575~{}.$
(4.198)

On the other hand, $q$-expanding about $x=0$ returns integers
$n_{d}^{\text{QS}}$ that count genus 0 degree $d$ curves on the orthogonal
linear section of the double quintic symmetroid [68].

The first few of these numbers are

$n_{1}^{\text{QS}}\;=\;550~{},\,n_{2}^{\text{QS}}\;=\;19150~{},\,n_{3}^{\text{QS}}\;=\;1165550~{},\,n_{4}^{\text{QS}}\;=\;106612400~{},\,n_{5}^{\text{QS}}\;=\;12279982850~{}.$
(4.199)

The formal statement, around which we avoid working, is that these two spaces
RC and QS are derived equivalent, having the same derived category of coherent
sheaves [115]. They are not birational, possessing different topological
invariants (such as their nonequal triple intersection numbers), but have the
same Hodge numbers and the mirror symmetry computation of their curve counts
takes place in a common complex structure moduli space for which the Picard-
Fuchs operator can be taken to be AESZ22 or AESZ118 by suitable choices of
coordinates.

#### A rational rank-two attractor

We have identified the point $z=-1$ as a rank-two attractor. Equivalently,
this is $x=-1/32$. The zeta function numerator factorises and the ensuing
weight-four modular form has LMFDB label 33.4.a.b .

Upon a Mellin transformation, this modular form gives an L-function with
special values

$\displaystyle L(1)$ $\displaystyle\;=\;-1.0538249565444346000240\,...~{},$
(4.200) $\displaystyle L(2)$ $\displaystyle\;=\;\phantom{-}0~{}.$

About either $z=0$ or $x=0$ a Frobenius basis of periods can be computed. It
is possible to express the imaginary parts of the period vectors
$\widehat{\varpi}$ as rational multiples of $\frac{L(1)}{2\pi}$ (except that
$\widehat{\varpi}_{3}$ involves $\frac{\zeta(3)}{(2\pi\text{i})^{3}}$). The
real parts involve another nonzero number $\frac{\Phi}{(2\pi)^{2}}$, with

$\Phi\;=\;16.671069275646809663\,...\,~{}.$ (4.201)

Since $L(2)$ is zero, it is not useful to this end. It would be interesting to
approach this issue from the perspective of Beilinson’s conjecture and give
$\Phi$ in terms of the nonzero value $L^{\prime}(2)$, a problem that would
involve calculating a suitable Beilinson regulator [116]. However, it is
possible to express $\Phi$ in terms of the L-values for a different modular
form related to 33.4.a.b by a twist. In particular, $\Phi$ can be expressed in
terms of a critical value of the Mellin transform of the form with LMFDB label
528.4.a.h:

$\Phi\;=\;\frac{\pi}{6}L_{\textbf{528.a.h}}(1)~{},$ (4.202)

##### The Reye Congruence

Pertaining to the mirror geometry RC, the topological invariants are

$Y_{111}\;=\;35~{},\qquad Y_{110}\;=\;\frac{1}{2}~{},\qquad
Y_{100}\;=\;-\frac{25}{6}~{},\qquad\chi\;=\;50~{}.$ (4.203)

at $z=-1$, the Frobenius periods (about $z=0$) evaluate as follows:

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-\frac{\Phi}{(2\pi\text{i})^{2}}-\frac{5}{3}\,\frac{L(1)}{2\pi\text{i}}~{},$
(4.204) $\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-\frac{1}{2}\,\frac{\Phi}{(2\pi\text{i})^{2}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;-\frac{10}{3}\,\frac{\Phi}{(2\pi\text{i})^{2}}+\frac{175}{36}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{3}-\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}$
$\displaystyle\;=\;-\frac{5}{24}\,\frac{\Phi}{(2\pi\text{i})^{2}}+\frac{5}{6}\,\frac{L(1)}{2\pi\text{i}}~{}.$

The attractor equations are solved for charge vectors given in the integral
basis by

$Q_{1}\;=\;(-\frac{1}{2},5,1,0)^{T}~{},\qquad
Q_{2}\;=\;(\frac{7}{2},-13,0,1)~{}.$ (4.205)

The orthogonality equations are solved by

$Q_{3}\;=\;(5,-13,0,1)^{T}~{},\qquad
Q_{4}\;=\;(-\frac{1}{2},\frac{7}{2},1,0)~{}.$ (4.206)

$Q_{3}$ has zero $D6$ charge. One can determine the perturbative solution to
the orthogonality equation to be

$t_{0}\;=\;\frac{15}{42}+\frac{\sqrt{69}}{42}\text{i}~{}.$ (4.207)

These considerations lead to a sum

$-\frac{3-25\frac{\pi L(1)}{\Phi}\text{i}}{3+10\frac{\pi
L(1)}{\Phi}\text{i}}\text{i}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\frac{\sqrt{69}}{6}-\sqrt{\frac{7}{10\pi^{3}}}\sum_{j=1}^{\infty}\text{e}^{\frac{15\pi\text{i}}{21}j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{3j}{5\pi\sqrt{69}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{\pi
j\sqrt{69}}{21}\right)~{},$ (4.208)

in which the $N_{k}$ are the scaled Gromov-Witten invariants constructed from
the invariants $n_{d}^{\text{RC}}$. We use a diagonal Padé approximant of
order 30, and find agreement to six figures. This example is also only weakly
supported, and our conjectural sum here is not well-evidenced.

##### The Quintic Symmetroid

This time, for the mirror geometry QS, the topological invariants are

$Y_{111}\;=\;10~{},\qquad Y_{110}\;=\;0~{},\qquad
Y_{100}\;=\;-\frac{10}{3}~{},\qquad\chi\;=\;50~{}.$ (4.209)

at $x=-1/32$, the Frobenius periods (about $x=0$) evaluate to:

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-2\,\frac{\Phi}{(2\pi\text{i})^{2}}~{},$ (4.210)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;\phantom{2}-\frac{\Phi}{(2\pi\text{i})^{2}}+\frac{10}{3}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;\phantom{-}\frac{5}{3}\,\frac{\Phi}{(2\pi\text{i})^{2}}+\frac{50}{3}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{3}-\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}$
$\displaystyle\;=\;\phantom{-}\frac{5}{3}\,\frac{\Phi}{(2\pi\text{i})^{2}}+\frac{10}{9}\frac{L(1)}{2\pi\text{i}}~{}.$

The attractor equations are solved for the charge vectors

$Q_{1}\;=\;(-1,5,1,0)^{T}~{},\qquad Q_{2}\;=\;(2,-5,0,1)^{T}~{}.$ (4.211)

The orthogonality equations are in turn solved by

$Q_{3}\;=\;(-1,2,1,0)^{T}~{},\qquad Q_{4}\;=\;(5,-5,0,1)^{T}~{}.$ (4.212)

From $Q_{4}$, one can determine the perturbative solution

$t_{0}\;=\;\frac{1}{2}+\frac{\text{i}\sqrt{15}}{6}~{}.$ (4.213)

Collecting the various pieces, one can reach the sum

$-20\frac{\pi
L(1)}{\Phi}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\sqrt{15}-\frac{1}{\sqrt{10\pi^{3}}}\sum_{j=1}^{\infty}(-1)^{j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{j}{60\pi\sqrt{15}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\pi
j\sqrt{\frac{5}{3}}\right)~{},$ (4.214)

in which the scaled Gromov-Witten invariants are constructed from the
instanton numbers $n_{d}^{\text{QS}}$. We use a diagonal Padé approximant of
order 36, and find agreement to 29 figures.

### The pair AESZ17 and AESZ290

These are another pair of operators with two MUM points that are exchanged by
a coordinate transformation and scaling

$z\;=\;z\mapsto-\frac{1}{3^{6}x}~{},\qquad
F_{17}(z)=-\frac{1}{3^{6}x}F_{290}(x)~{}.$ (4.215) AESZ17:
$\displaystyle\hskip
10.0pt\big{[}(5-9z)^{2}(1-27z)\left(1+27z^{2}\right)\theta^{4}$ (4.216)
$\displaystyle\hskip
15.0pt-36z(5-9z)\left(7-15z+621z^{2}-729z^{3}\right)\theta^{3}$
$\displaystyle\hskip
15.0pt-6z\left(59049z^{4}-91935z^{3}+39591z^{2}-541z+180\right)\theta^{2}$
$\displaystyle\hskip
15.0pt-6z\left(39366z^{4}-64233z^{3}+34155z^{2}-155z+75\right)\theta$
$\displaystyle\hskip
15.0pt-3z\left(19683z^{4}-32562z^{3}+21060z^{2}-30z+25\right)\big{]}F_{17}(z)\;=\;0~{},$
AESZ290: $\displaystyle\hskip
10.0pt\big{[}(1+27x)(1+405x)^{2}\left(1+19683x^{2}\right)\theta^{4}$
$\displaystyle\hskip
15.0pt-108x(1+405x)\left(7-729x-177147x^{2}-7971615x^{3}\right)\theta^{3}$
$\displaystyle\hskip
15.0pt-6x\left(80-37017x-8155323x^{2}-1506635235x^{3}-87169610025x^{4}\right)\theta^{2}$
$\displaystyle\hskip
15.0pt-6x\left(17-17415x-3720087x^{2}-789189885x^{3}-58113073350x^{4}\right)\theta$
$\displaystyle\hskip
15.0pt-9x\left(1-1998x-454896x^{2}-111602610x^{3}-9685512225x^{4}\right)\big{]}F_{290}(x)\;=\;0~{}.$

The geometric picture is markedly different to the pair AESZ22/118, in that
$q$-expansions about $x=0$ do not return the BPS invariants of a manifold
because there is no mirror manifold. Indeed, we will see that any putative
mirror should have $Y_{111}=-30/13$, which defies an interpretation as a
triple intersection number.

The MUM point $z=0$ is mirror the the large volume point of a one-parameter
family of threefolds constructed as the $\mathbbl{Z}_{3}$ quotient of the
split bicubic:

$\begin{matrix}\mathbbl{P}^{2}\\\ \mathbbl{P}^{2}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}\;1\;1\;1\;\;\\\
\;1\;1\;1\;\;\\\ \;1\;1\;1\;\;\end{matrix}\right]_{/\mathbbl{Z}_{3}}$ (4.217)

The above quotient geometry has Euler characteristic $\chi=-30$. Since
$h^{1,1}=1$, we have that $h^{2,1}=16$ for the above space. AESZ17 is the
Picard Fuchs operator for a mirror $X$ to (4.217). From the Hodge diamond
reflection, we have that $h^{1,1}(X)=16$ and $=h^{2,1}(X)=1$. If the mum point
$x=0$ (which is $z=\infty$) in the complex structure moduli space of $X$ has a
mirror $Y$, then $Y$’s Hodge numbers must read
$h^{1,1}(Y)=1~{},h^{2,1}(Y)=16$. That is to say, $Y$ should have the same
Hodge numbers as (4.217). As a consequence, $Y$ and (4.217) must have the same
Euler characteristic.

Now we consider monodromies [117] for the operator AESZ290. This has conifold
singularities at $x=-1/27,\,x=\pm\frac{\sqrt{3}\text{i}}{343}$. As usual,
since the operator’s local exponents at $x=0$ are $(0,0,0,0)$ we can construct
a Frobenius basis

$\displaystyle\varpi_{0}(x)$ $\displaystyle=f_{0}(x)~{},$ (4.218)
$\displaystyle\varpi_{1}(x)$
$\displaystyle=\frac{1}{2\pi\text{i}}\left(f_{0}(x)\log(x)+f_{1}(x)\right)~{},$
$\displaystyle\varpi_{2}(x)$
$\displaystyle=\frac{Y_{111}}{2(2\pi\text{i})^{2}}\left(f_{0}(x)\log(x)^{2}+2f_{1}(x)\log(x)+f_{2}(x)\right)~{},$
$\displaystyle\varpi_{3}(x)$
$\displaystyle=\frac{Y_{111}}{6(2\pi\text{i})^{3}}\left(f_{0}(x)\log(x)^{3}+3f_{1}(x)\log(x)^{2}+3f_{2}(x)\log(x)+f_{3}(x)\right)~{},$

where all of the $f_{i}$ are power series, with $f_{0}(0)=1$ and
$f_{1}(0)=f_{2}(0)=f_{3}(0)=0$. Now we attempt to construct an integral
symplectic period vector

$\Pi=\begin{pmatrix}-\frac{\chi(Y)\zeta(3)}{(2\pi\text{i})^{3}}&-\frac{1}{2}Y_{100}&0&1\\\
-\frac{1}{2}Y_{100}&-Y_{110}&-1&0\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}\varpi~{}.$ (4.219)

The task at hand then is to find real values of $Y_{111},\,Y_{100}$ and
$Y_{110}$ so that the monodromy matrices about each conifold are integral.
Note that we have fixed the value $\chi(Y)=30$ by our earlier argument. In
practice we compute monodromy matrices for a modified version of the Frobenius
vector that does not include the factors of $Y_{111}$, and then restore these
symbolically with a diagonal change of basis.

It is instructive to check the monodromy matrix that we compute at
$x=\frac{\text{i}\sqrt{3}}{343}$ for the above $\Pi$. The $(1,1)$ component is

$-\frac{7}{8}+\frac{9Y_{100}}{Y_{111}}+\frac{9\text{i}\left(30+13Y_{111}\right)\zeta(3)}{4Y_{111}\pi^{3}}~{}.$
(4.220)

In order for this to be real, we must choose $Y_{111}$ so that the imaginary
part vanishes. So we learn that the triple intersection number of our manifold
$Y$ should be

$Y_{111}=-\frac{30}{13}~{}.$ (4.221)

On these grounds, we dispense with the idea that there is a mirror manifold at
$x=0$. We cannot by any choice of $Y_{100}$ and $Y_{110}$ render the conifold
matrix integral, so we press on without fixing any value for these.

#### A rational rank-two attractor

Here we are able to identify a rank-two attractor at $z=-1$, which is
$x=\frac{1}{3^{6}}$, with the corresponding weight-four modular form having
LMFDB label 14.4.a.b . Mellin transforming this gives the critical L-values

$\displaystyle L(1)$ $\displaystyle\;=\;0.81476235013261396706625\,...~{},$
(4.222) $\displaystyle L(2)$
$\displaystyle\;=\;1.13620338571911095621858\,...~{}.$

##### AESZ17

The topological invariants are

$Y_{111}\;=\;30~{},\qquad Y_{110}\;=\;0~{},\qquad
Y_{100}\;=\;-3~{},\qquad\chi\;=\;30~{}.$ (4.223)

The Frobenius periods about $z=0$ can be evaluated at $z=-1$, giving

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;-14\,\frac{L(2)}{(2\pi\text{i})^{2}}~{},$ (4.224)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;-7\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{3}{4}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;-42\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{45}{4}\,\frac{L(1)}{2\pi\text{i}}~{},$
$\displaystyle\widehat{\varpi}_{3}-\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}$
$\displaystyle\;=\;-\frac{7}{2}\,\frac{L(2)}{(2\pi\text{i})^{2}}-\frac{19}{8}\frac{L(1)}{2\pi\text{i}}~{}.$

Independent charge vectors for which the attractor equations are solved are

$Q_{1}\;=\;\left(-\frac{4}{3},6,1,0\right)^{T}~{},\qquad
Q_{2}\;=\;\left(\frac{14}{3},-15,0,1\right)~{},$ (4.225)

while the orthogonality equations are solved for

$Q_{3}\;=\;\left(6,-15,0,1\right)^{T}~{},\qquad
Q_{4}\;=\;\left(-\frac{4}{3},\frac{14}{3},1,0\right)~{}.$ (4.226)

Using $Q_{3}$ one can find the perturbative solution

$t_{0}\;=\;\frac{1}{2}+\frac{\text{i}\sqrt{5}}{10}~{}.$ (4.227)

We obtain the sum

$\frac{3\pi}{14}\frac{L(1)}{L(2)}\;\stackrel{{\scriptstyle\mathclap{\mbox{P}}}}{{=}}\;\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{15\pi^{3}}}\sum_{j=1}^{\infty}(-1)^{j}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}\left(\frac{j}{6\pi\sqrt{5}}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\frac{\pi
j}{\sqrt{5}}\right)~{}.$ (4.228)

A diagonal Padé approximant of order 36 provides agreement to 10 figures.

##### AESZ290

We lack a geometric interpretation for a mirror at $x=0$, but proceed with the
computation using the topological invariants

$Y_{111}\;=\;-\frac{30}{13}~{},\qquad Y_{110}\;=\;Y_{110}~{},\qquad
Y_{100}\;=\;Y_{100}~{},\qquad\chi\;=\;30~{}.$ (4.229)

The rank two attractor $z=-1$ maps to $x=3^{-6}$, and we find that at this
point the Frobenius periods for AESZ290 evaluate to

$\displaystyle\widehat{\varpi}_{0}$
$\displaystyle\;=\;\frac{9\sqrt{3}}{2}\frac{L(1)}{(2\pi)}~{},$ (4.230)
$\displaystyle\widehat{\varpi}_{1}$
$\displaystyle\;=\;21\text{i}\sqrt{3}\frac{L(2)}{(2\pi)^{2}}~{},$
$\displaystyle\widehat{\varpi}_{2}$
$\displaystyle\;=\;\frac{45\sqrt{3}}{8}\frac{L(1)}{2\pi}~{},$
$\displaystyle\widehat{\varpi}_{3}-\frac{\chi\,\zeta(3)}{(2\pi\text{i})^{3}}\widehat{\varpi}_{0}$
$\displaystyle\;=\;\frac{315\text{i}\sqrt{3}}{52}\frac{L(2)}{(2\pi)^{2}}~{}.$

The attractor equations are solved for the charge vectors

$Q_{1}\;=\;\left(\frac{15}{52}-\frac{Y_{100}}{2},\,-Y_{110},\,0,\,1\right)^{T}~{},\qquad
Q_{2}\;=\;\left(0,\,-\frac{5}{4}-\frac{Y_{100}}{2},\,1,\,0\right)^{T}~{}.$
(4.231)

The orthogonality equations are solved for charges

$Q_{3}\;=\;\left(0,\,\frac{15}{52}-\frac{Y_{100}}{2},\,1,\,0\right)^{T}~{},\qquad
Q_{4}\;=\;\left(-\frac{5}{4}-\frac{Y_{100}}{2},\,-Y_{110},\,0,1\right)^{T}~{}.$
(4.232)

The orthogonality equation for the charge vector $Q_{4}$ is solved at the
perturbative level by

$t_{0}=\frac{\text{i}}{2}\sqrt{\frac{13}{3}}~{}.$ (4.233)

From this information, we construct the sum

$\frac{14}{3\pi}\frac{L(2)}{L(1)}\;=\;\sqrt{\frac{13}{3}}-\sqrt{\frac{13}{15\pi^{3}}}\sum_{j=1}^{\infty}\sum_{\mathfrak{p}\in\text{pt}(j)}a_{\mathfrak{p}}N^{\mathfrak{p}}(-1)^{l(\mathfrak{p})}\left(\frac{\sqrt{\frac{13}{3}}j}{10\pi}\right)^{l(\mathfrak{p})-1/2}K_{l(\mathfrak{p})-1/2}\left(\pi
j\sqrt{\frac{13}{3}}\right)~{}.$ (4.234)

There is no Padé resummation here, as this sum converges (by the same
arguments as appeared in Appendix F of [4]). By computing 60 terms in the
series, we find agreement to 50 figures. By taking two Shanks transformations
[114], this agreement can be improved to 55 figures. Three Shanks
transformations also yields 55 figures of agreement, and further Shanks
transformations worsen the agreement.

##  Appendix A Coordinates on complex structure moduli space

> Smith : Because of you, I’m no longer an Agent of this system. Because of
> you, I’ve
> Smith :  changed. I’m unplugged. A new man, so to speak. Like you,
> apparently, free.
> Neo -- : Congratulations.
> Smith : Thank you. But, as you well know, appearances can be deceiving,
> which brings
> Smith :  me back to the reason why we’re here.
>
> Lana and Lilly Wachowski, The Matrix Reloaded

In this appendix, we explain our choice of coordinates for the complex
structure moduli spaces of the mirrors of complete intersections. We follow
the presentations of [27, 118], whose choice of coordinates we use. This
involves a combinatoric problem in toric geometry, a subject that we do not go
into detail on, instead referring to the textbook [119]. The construction of
mirror manifolds via operations on polyhedrons that encode intersections in
toric varieties goes back to work of Batyrev, Borisov, and Nill [120, 121,
122, 123, 124].

The presentation that we offer is solely given to define a choice of
coordinates. We will not discuss the proofs underlying these methods.

As in the main body of the thesis, $Y$ will be a family of complete
intersections. For this appendix we shall allow the ambient space to be a
product of weighted projective spaces, which is a mild generalisation of the
CICYs discussed in §2.2. So the data of $Y$ is

$\begin{matrix}\mathbbl{W}\mathbbl{P}^{n_{1}}_{\left(w_{1}^{(1)}~{},\,...~{},\,w_{n_{1}+1}^{(1)}\right)}\\\
\\\ \vdots\\\ \\\
\mathbbl{W}\mathbbl{P}^{n_{K}}_{\left(w_{1}^{(K)}~{},\,...~{},\,w_{n_{K}+1}^{(K)}\right)}\end{matrix}\\!\left[\begin{matrix}d^{\,(1)}_{1}~{},\,...~{},\,d^{\,(1)}_{C}\\\
\\\ \vdots\\\ \\\
d^{\,(K)}_{1}~{},\,...~{},\,d^{\,(K)}_{C}\end{matrix}\right]~{}.$ (A.1)

We assume, following [27], that the intersection $Y$ does not intersect with
the singularities of the ambient variety
$\mathcal{A}=\mathbbl{W}\mathbbl{P}^{n_{1}}_{\left(w_{1}^{(1)}~{},\,...~{},\,w_{n_{1}+1}^{(1)}\right)}\times\,...\,\times\mathbbl{W}\mathbbl{P}^{n_{K}}_{\left(w_{1}^{(K)}~{},\,...~{},\,w_{n_{K}+1}^{(K)}\right)}$,
and without loss of generality take $w_{n_{i}+1}^{(i)}=1$ for each $i$. We
will also assume a favourability condition: $h^{1,1}=K$. We will display here
formulae relevant for such intersections in products of weighted projective
spaces. Results for CICYs in products of (unweighted) projective spaces, as in
all our main lines of discussion in this thesis, can be obtained by setting
all $w^{(k)}_{i}$ equal to 1.

Each weighted projective space $\mathbbl{W}\mathbbl{P}^{n_{i}}_{\bm{w}^{(i)}}$
is a toric variety, associated to a reflexive simplicial polyhedron
$\Delta_{i}$. $\Delta_{i}$ has integer vertices in $\mathbbl{R}^{n_{i}}$. The
ambient space $\mathcal{A}$ is also a toric variety, and the associated
polyhedron is

$\Delta=\Delta_{1}\times\,...\,\times\Delta_{K}\subset\mathbbl{R}^{n_{1}}\times\,...\,\times\mathbbl{R}^{n_{K}}~{}.$
(A.2)

The Batyrev-Borisov procedure produces a mirror $X$ to an intersection in a
toric variety, like $Y$ above, from the data of the polyhedron $\Delta^{*}$,
which is the polar dual of $\Delta$. We intentionally gloss over the specifics
of $\Delta$ and its association to $\mathcal{A}$, and skip to the presentation
of the vertices $\nu_{i,j}^{*}$ of $\Delta^{*}$. For each fixed $i$, there are
vertices $\nu_{i,j}^{*}$ that have components $0$ in each factor
$\mathbbl{R}^{n_{j}}$ of
$\mathbbl{R}^{n_{1}}\times\,...\,\times\mathbbl{R}^{n_{K}}$ with $j\neq i$.
The components of $\nu_{i,j}^{*}$ in the factor $\mathbbl{R}^{n_{i}}$ are

$\begin{gathered}\nu_{i,1}^{*}=(1,0,\,...\,,0)~{},\qquad...~{},\qquad\nu_{i,n_{i}}^{*}=(0,\,...\,,0,1)~{},\\\
\nu_{i,n_{i}+1}^{*}=\left(-w_{1}^{(1)}~{},\,...~{},\,-w_{n_{i}}^{(i)}\right)~{}.\end{gathered}$
(A.3)

Denote the set of all the vertices by $E$, and partition $E$ into distinct
sets $E_{m}$, $1\leq m\leq C$. For each fixed $i$, each set $E_{m}$ contains
$d^{\,(i)}_{m}$ vertices from the set $\nu^{*}_{i,j}$.

Now for each fixed $m$, each vertex $\nu_{i,j}^{*}\in E_{m}$ is extended to

$\overline{\nu}_{i,j}^{*}=\left(e^{(m)},\nu_{i,j}^{*}\right)\in\mathbbl{R}^{C}\times\mathbbl{R}^{n_{1}}\times\,...\,\times\mathbbl{R}^{n_{K}}~{}.$
(A.4)

By $e^{(m)}$ we mean the unit vector in the $m^{th}$ direction of
$\mathbbl{R}^{C}$. We then form additional vectors

$\overline{\nu}_{0,p}^{*}=\left(e^{(p)}~{},\bm{0}\right)\in\mathbbl{R}^{C}\times\mathbbl{R}^{n_{1}}\times\,...\,\times\mathbbl{R}^{n_{K}}~{}.$
(A.5)

Importantly, it turns out that there are $K$ independent linear relations
between the vectors $\overline{\nu}_{i,j}^{*}$,

$\sum_{i,j}l^{(s)}_{i,j}\,\overline{\nu}_{i,j}^{*}=\bm{0}~{},\qquad 1\leq
s\leq K~{}.$ (A.6)

The components of $l^{(s)}_{i,j}$ are

$\displaystyle l^{(s)}_{0,j}$ $\displaystyle=-d_{j}^{\,(s)}~{},\quad$
$\displaystyle 1\leq j\leq C~{},$ (A.7) $\displaystyle l^{(s)}_{i,j}$
$\displaystyle=\delta_{i}^{\,(s)}w_{j}^{(s)}~{},\quad$ $\displaystyle 1\leq
i\leq K~{},$ $\displaystyle\quad 1\leq j\leq n^{(s)}_{n_{s}+1}~{},$
$\displaystyle\delta_{i}^{(s)}$ $\displaystyle=\begin{cases}0&i\neq s~{},\\\
1&i=s~{}.\end{cases}$

The choice of overall sign, with minus signs in the components of
$l^{(s)}_{0,j}$, is one of convention. We fix this choice in our work, so that
MUM points are located at the origin. We remark that for the more general case
of intersections in toric varieties, relations of the form (A.6) hold. It is
only in our simplified case of complete intersections in weighted projective
spaces that the particular form (A.7), with the Kronecker delta, hold.

Let $\Delta_{i}^{*}$ denote the convex hull of $E_{i}$ and the origin. The
mirror family of $Y$ consists of varieties $X$ birational to the vanishing
loci of $C$ equations $P_{r}=0$ in the toric variety
$\mathbbl{P}_{\Delta_{1}^{*}+\,...\,+\Delta_{C}^{*}}$, where $+$ denotes the
Minkowski sum of polyhedra. The ambient space coordinates are $X_{m,n}$, with
$1\leq m\leq K$ and $1\leq n\leq n_{m}$. Those equations read

$P_{r}\equiv a_{0r}-\sum_{\nu_{i,j}^{*}\in
E_{r}}a_{i,j}X^{\nu_{i,j}^{*}}=0~{}.$ (A.8)

Let us clarify the notation here: $X^{\nu_{i,j}^{*}}$ means
$\prod_{j=1}^{n_{i}}X_{i,j}^{\nu_{i,j}^{*}}$.

The set of $a_{ij}$ that appear in (A.8) outnumber the set of complex
structure moduli of $X$. Coordinate redefinitions remove the redundancies, and
these are encoded in the relations $l^{s}_{i,j}$. We work with coordinates

$\varphi^{s}=\frac{\prod_{j=1}^{n_{s}+1}a_{s,j}}{\prod_{j=1}^{C}a_{0,j}^{d_{j}^{\,(s)}}}\equiv
a^{l^{(s)}}~{}.$ (A.9)

Inspecting this process, one sees that there is some superficial ambiguity in
the choice of sets $E_{m}$ which can lead to different polynomials $P_{m}$.
The set of coordinates obtained from (A.9) does not depend on this choice. The
apparent ambiguity in the polynomials is removed when these varieties are
desingularised, but we do not discuss that in this appendix.

##  Appendix B Zeta function factorisation counts

> We’re not banging rocks together here.
>
> Cave Johnson

This appendix contains bar charts that display for each labelled operator,
with $p$ on the $x$-axis, the number of times that the zeta function
$\zeta_{p}(\varphi;T)$ factorises as $\varphi$ runs over $\mathbbl{F}_{p}$.
Our tables are for primes $5\leq p\leq 131$.

It should be understood that this list is not exhaustive: other operators have
been studied elsewhere (see [14, 13, 47]), and many more remain
uninvestigated. We depend on the digital database [65]. The computation that
obtains these factorisation counts needs the topological numbers $Y_{111}$ and
$\chi$, which are not given for every operator in [65]. Further, a number of
the operators listed in [65] are typeset in such a way that posed a problem
for our code that recovered the operators from the website. Rather than fix
these surmountable problems, we omit those examples. We also did not make an
effort to work through the whole set for which there was sufficient data
available for our purposes. This choice of 61 is a matter of simplistic
practicality and does not belie anything interesting.

These tables were generated using the methods of [50], and are available in
electronic form [1].

AESZ4 was studied in [13, 47] and is known to possess a rational rank two
attractor.
AESZ11 was studied in [13, 47] and is known to possess a rational rank two
attractor.
The existence of a rational rank two attractor for AESZ17 is a new result of
this thesis.
The existence of a rational rank two attractor for AESZ22 is a new result of
this thesis.
AESZ34 was studied in [14]. It is known to possess one rational and two
quadratic rank two attractors.
AESZ36 was studied in [47] and is known to possess a rational rank two
attractor.
AESZ55 was studied in [47] and is known to possess a rational rank two
attractor.
AESZ118 is equivalent to AESZ22 after a change of variables (see the AESZ22
plot).

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