The House or College of Scholars of Merton
in the University of Oxford
and
Hertford College
Doctor of Philosophy Trinity 2023

# Investigations in Calabi-Yau
Modularity and Mirror Symmetry

Joseph McGovern

###### Abstract

This thesis lays out a number of different projects, linked by the common
thread of Calabi-Yau manifolds.

We produce tables of instanton numbers for various multiparameter Calabi-Yau
manifolds. Studying those tables reveals, in some cases, a Coxeter group of
symmetries that act on sets of instanton numbers. Instanton numbers are
constant over the orbits of group actions on the curve classes, and these
groups can be infinite.

We investigate to what extent this symmetry can be used to constrain the
holomorphic ambiguity that arises in higher genus topological string free
energy computations. In the example we study, at genus 2, combining this
Coxeter constraint with the set of vanishing numbers fixes the holomorphic
ambiguity.

A new class of solutions is provided to the supersymmetric flux vacuum
equations, which have been conjectured elsewhere to give weight-two modular
manifolds.

Another search, complementary to those that have already been carried out, is
made for rank-two attractors in the AESZ list. Two novel examples are found,
both of which belong to moduli spaces with two points of maximal unipotent
monodromy each. For one of these operators, the additional MUM point
corresponds to another derived equivalent geometry. For the other operator, we
compute _nonintegral_ values of the triple intersection number and genus 0
instanton numbers, and so a geometric interpretation is lacking for the second
MUM point.

We provide several instances of summation identities that express a ratio of
critical L-values as an infinite sum, whose terms contain the Gromov-Witten
invariants of a Calabi-Yau manifold $Y$. In one example there is no manifold
$Y$, only an operator, but a set of invariants can nonetheless be computed for
this and a summation identity is found. The L-functions are associated to a
modular manifold $X$ which is mirror to $Y$. These sums can be divergent, but
Padé resummation cures this.

In addition to these results, we briefly review informing aspects of Calabi-
Yau geometry, black holes in 4d $\mathcal{N}=2$ supergravity, flux
compactifications, topological string theory, and number theory.

The contents of the tables of this thesis are available to download at [1].

Dedicated to my
sister and brother,
Charlotte and Stewart.

Parts of this thesis are based on joint work with Philip Candelas, Xenia de la
Ossa, and Pyry Kuusela.

Chapters One and Two serve as an introduction and review of relevant
background material, neither contains new results.

Chapter Three describes ongoing work, building on ideas that appeared in [2]
where the appearance of the Coxeter groups was noted.

Chapter Four also describes ongoing work. Parts of the discussion of flux
vacua appear in [3]. The series solution method for the IIA attractor
equations was first given in [4].

I will always be grateful to my supervisor, Philip Candelas, for affording me
the opportunity to do this degree. He has been patient, kind, generous with
his time, and shared many fascinating things with me. Xenia de la Ossa, who
began cosupervising me in my second year, has also been of immense help and I
feel very lucky to have worked with her, sharing in the beautiful world of her
work. It has been very interesting to work with Pyry Kuusela, my collaborator
and former fellow graduate student. Our projects together were accelerated by
his technical wizardry and fearsome work ethic. I thank him for his
willingness to share and discuss his ideas, and his friendship. Mohamed Elmi,
as a former student of this working group, has been very helpful to me, in
Oxford during my early days and later on during his postdoc. We have benefited
from interesting conversations with Duco van Straten.

My course through life greatly benefited from the supervision of Kolya Gromov
during my undergraduate degree in KCL. I was extremely lucky to find a mentor
in him, and working with him was a very happy period. For our meetings and
their help during those times, I thank Andreá Cavaglia, Fedya Levkovich-
Maslyuk, and Julius. Also from KCL, I thank Nadav Drukker, Jürgen Berndt,
Sameer Murthy, Alexander Pushnitski, Paul Cook, and Dionysios Anninos, for the
varied help that they gave me at different times.

I would like here to belatedly thank the anonymous JHEP referee of my first
paper on Wilson loop insertions. In my naivety I neglected to do so in the
acknowledgements section of that paper, in spite of their helpful comments.

I am grateful to James Sparks and Fernando Alday for giving their time to
critically assess my work at stages of this degree. For this I also thank Erik
Panzer, as well as for some additional inspirational conversations. I am also
very glad that Kilian Bönisch was able to visit, and offer me some helpful
advice on technical matters which greatly helped in §4.11. I also thank my
college advisor Dawid Kielak for some helpful conversations.

I expect that regularly seeing Sujay Nair in our office will be the part of
Oxford that I most come to miss. I feel that our time together has shaped me,
and I look forward to discussing graphene with him one day. I know that he
will do very well, and probably without making a big deal of it.

I will miss Carmen Jorge-Diaz, who was an excellent colleague, housemate, and
friend. By transitivity I will also miss Lockheed Martin Purkiss. I could not
legally submit this thesis without first thanking Doctor Mega Ultra Mateo
$G_{2}$ Galdeaño Feliz “Solans” Huesca. We had many spooky adventures. Even
after he left he was still being helpful, because I then had use of his desk,
desktop, and sword. Thank you Diego Berdeja Suarez, first for introducing me
to the group as my DPhil buddy, and then for being my friend. James Hefford
has been a wonderful person to know, he gave me a lot more food than I gave
him. Also from Hertford, I thank Anastasia Sigutina for her sense of humour
and the heartfelt effort she puts into her friendships. I appreciated Manuel
Schmigdall’s approach to 214b.

From my time living in Mansfield, I am happy to have known Tom McAuliffe,
David Rytz, Thomas Han, and the chief Sigma males Armands Strikis and Lily
Mott (Armands is the favourite of those two). Thank you Alex Cliffe for not
only being my friend, but for organising happy hours.

Living in Norham Gardens was an unexpected blessing, even more so for the
characters within. Rick Longley, you are hilarious and thank you for your
sincerity. Thank you Esraa Shaban, Tom Anglim Lagones, Ryan O’Sullivan,
Facundo Herrero, Katie Hearne, Christine Arndt, Janko Ondras, Ben Ta, Insaif
Bakeer Markar, Pamela, Philip, Sarah, and everyone from the Labyrinth.

My final year in Oxford was enhanced by the chance to work in Merton, which I
enjoyed greatly. I am grateful for their decision to employ me, and I thank
the many interesting people I met there, and those who I worked with. I thank
all of those who I had the opportunity to teach, particularly those at Merton
in this last year, who were a most impressive collection.

My year spent playing chess in the City Club was memorable, so thank you to
them, in particular Phil Hayward and Will van Noordt.

My friends from KCL changed my view of what was possible, thank you Jack
Heimrath and Mary Czarnecka for among so much else hosting us in Poland,
Marcello Harle-Cowan, Louis Yudowitz, Yusuf Broujerdi, Sil Linskens, Maite
Arcos-Enriquez, Andreas Sandbakken, Vlad Petrica, Enzo Desiage. My time in
Falmouth Road with Jonathan Gorard and Jan Stanczuk was special.

I thank Keir Crawley and the Music Academy, for making my life so much richer.
Time spent with Chris Williams, Michael Papasavva, Warren Alexander, Moeed
Majeed, Terry Murphy, Kyriacos Papasavva, Shiv Sharma, and Alex Kempson has
been wonderful.

I pay my respects to the brave warriors in the Tiger Mafia and associated
entities, for 2 min.

I am grateful to the organisers of the GLSMs@30 conference in the SCGP, Long
Island, for facilitating interesting conversations. I also thank the
organisers of the Elliptics and Beyond 2023 conference, in ETH Zurich, for the
opportunity to talk about my work and for their hospitality.

I now proceed to thank my favourite members of the vast number of MP DPhil
students/postdocs (and affine extensions). Thank you Horia Magureanu for your
wisdom. I thank Marieke and Pieter Tunstall van Beest, and Pietro Ferrero,
most recently for driving me around Long Island. Our group was extremely
fortunate to have so often as hosts Andrea Boido, Enrico Marchetto, and Palash
Singh, their generosity has made for many enjoyable evenings. I thank Jay Swar
for the variety of conversations, and also just the variety. I thank Federico
Zerbini for his help. I formally rethank Julius Eckhard. Max Hübner is pretty
good. Spending time with Maria Nocchi and Adam Kmec has been entertaining and
joyful. I thank Giulia Albonico for her kindness, Guillermo Arias Tamargo for
his Smith-Morra, Christopher Raymond for his explanation of loose units.
Filipo Revello, Jean Garret, Johan Henriksson, Murat Kologlu, Connor Behan,
Matteo Sacchi, Atul Sharma, Giuseppe Bogna, Lea Bottini, Dario Ascari, the
mysterious Dewi Gould, Izar Alonso Lorenzo, Evyatar Sabag, Chris Couzens,
Zhenghao Zhong, Dylan Butson, and everyone else who I forgot.

None of the good things in my life would be possible without the love and
provision of my mother Helen Clark. Thank you for all that you have done for
me all through my life, for working so hard and giving me all that you have.

I also thank my grandfather Kenneth Clark for his support, my stepfather
Richard Starnes, and Midna, Barney, Martin, Charlotte and Stewart.

My father Vincent McGovern has done so much for me. Thank you for your
support, the adventures across Ireland, and your belief in me. Your loving
presence in my life means a great deal to me.

I thank also Patrick McGovern, Katrina McLoughlin, my cousins Malachy, Emer
and Brian McGovern, my grandparents Annie and Joe McGovern, Tyke McGovern, and
Sunita Ramasawmy.

I finish by thanking Stephen Wallman, Ehsan Pedram, Alison Wilding, Robert
Cowen, Alan Godfrey, Despo Speel, and Thore Lindahl Itturiate.

I apologise to the people that I have forgotten in writing this.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Thesis overview
  2. 2 Some essentials
    1. 2.1 Periods and the triple intersection form
    2. 2.2 CICYs
    3. 2.3 Splittings and the conifold transition
    4. 2.4 Calabi-Yau modularity
      1. 2.4.1 The local zeta function
      2. 2.4.2 Modular forms and modularity of elliptic curves
      3. 2.4.3 Modularity of Calabi-Yau threefolds
  3. 3 Reflections in the mirror
    1. 3.1 The genus 0 prepotential
    2. 3.2 Genus 1 mirror symmetry: Counting elliptic curves
    3. 3.3 Discriminant loci and Yukawa couplings
    4. 3.4 Contractions and instanton summations
    5. 3.5 Coxeter groups
    6. 3.6 Higher genus mirror symmetry
    7. 3.7 A worked example at genus 2
  4. 4 Supergravity Compactifications
    1. 4.1 4d $\mathcal{N}=2$ matter coupled supergravities
    2. 4.2 The attractor mechanism
    3. 4.3 Calabi-Yau attractors in IIB
    4. 4.4 One-parameter Calabi-Yau attractors in IIA
    5. 4.5 Flux compactifications
    6. 4.6 F-theory lifts
    7. 4.7 The scalar potential
    8. 4.8 Corrections to the potential and a digression on moduli stabilisation
    9. 4.9 Modularity of supersymmetric vacua in flux compactifications
      1. 4.9.1 Solving the SFV equations via permutation symmetry
      2. 4.9.2 Examples of supersymmetric flux vacua
      3. 4.9.3 Speculation on elliptic curves inside the F-theory fourfold
    10. 4.10 New weight four modular manifolds
    11. 4.11 Summation identities: L-value ratios from GW invariants
  5. A Coordinates on complex structure moduli space
  6. B Zeta function factorisation counts

##  Chapter 1 Introduction

> I must go down to the seas again, for the call of the running tide
> Is a wild call and a clear call that may not be denied;
>
> John Masefield, Sea Fever

###  1.1 Motivation

It is difficult to overstate the wealth of interesting mathematics attached to
Calabi-Yau manifolds as they appear in physics. This thesis documents an
attempt to look into these geometries, specifically compact threefolds, to see
what can be found. This approach is informed by two broad ideas, both of which
have seen major recent advancement on which we rely. These are _modularity_
and _mirror symmetry_.

Modularity has a good modern physics pedigree, appearing for instance as a
highly constraining symmetry in CFT partition functions. The modular group
also appears as a symmetry of type IIB string theory, which is vital for the
geometric uplift to F-theory by way of elliptic fibrations. The “modular” here
refers to the modular group $\Gamma=\text{PSL}(2,\mathbbl{Z})$.

This group $\Gamma$ has a deep significance to the arithmetic of elliptic
curves. Arithmetic, by which we mean the study of varieties over finite
fields, is perhaps less familiar to physicists of some schools than the
aforementioned role played by $\Gamma$. It is a fact that the set of point
counts for an elliptic curve, considered as a variety over a finite field, are
collected into a generating function that transforms in a highly constrained
way under a subgroup of $\Gamma$. Namely, this function is a weight-two
modular newform for a congruence subgroup of $\Gamma$. This is the modularity
theorem for elliptic curves [5, 6, 7]. We will better explain that claim in
§2.4, but for now press on to make the point that this can, rarely, happen for
Calabi-Yau threefolds [8].

Whether or not this happens can change as the Calabi-Yau is subject to a
complex structure deformation, so that the question of “Which Calabi-Yau
threefolds are modular?” becomes the question ”Which families contain modular
members, and where in the moduli space are they?”. The reason that the
physicist should care about this question lies in two putative answers, each
reliant on some conjectures [9, 10, 11, 12]:

  * •

Rank-two attractors, in the context of four dimensional $\mathcal{N}=2$
supergravity, are weight four modular.

  * •

Supersymmetric vacua in IIB flux compactifications are weight-two modular.

At the most extreme end of optimism then, one could hope that a better
understanding of arithmetic modularity could shed light on black hole physics
and string model building. This would see a well-developed area of mathematics
being brought to bear on some persistent problems in physics. We come far
short of that in this thesis, but these aspirations, and other recent works
realising these possibilities, serve as the ultimate motivation. For example,
the works [13, 14] were able to express D-brane masses and semiclassical black
hole entropies using Mellin transforms of the modular forms associated to
modular geometries.

The other raison d’être of this thesis is mirror symmetry, specifically two-
dimensional mirror symmetry. This is a duality between certain quantum field
theories, specifically twisted $\mathcal{N}=(2,2)$ nonlinear sigma models with
Calabi-Yau target space. Any duality between two quantum field theories should
be squeezed for all that can be learnt from it, and this duality comes with a
particularly nice geometric picture. It is possible to obtain, suitably
defined, counts of holomorphic embeddings of genus-$g$ curves of various
degrees into a Calabi-Yau threefolds $Y$ by studying the complex-structure
periods of a mirror geometry $X$.

There is a captivating problem in carrying out this computation at higher
genera. There is a procedure for doing this, BCOV recursion [15],[16], which
at each genus suffers from a holomorphic ambiguity. This is a rational
function that must be fixed, up to which the complete B-model free energy at
genus $g$ is determined by the lower genus results. This ambiguity can be
fixed by a number of considerations, for instance [17], but these do not
suffice to reach arbitrarily high genus. A full solution to this stubborn
problem could involve a strong handle on patterns in the sets of instanton
numbers, which motivates our work along these lines.

Additionally, the mirror symmetry computation is understood to give higher
derivative stringy correction terms to the supergravity action [18, 19]. This
duality then, is offering one way to approach the problem of formulating
string theory nonperturbatively. Topological string theory is a simplified
version of string theory, and so it is a natural expectation that a full
understanding of the latter will be helped by first understanding the
topological theory, as discussed recently in [20].

Mirror symmetry gives a duality between supergravity compactifications. These
are low energy limits of string theories, with there conjecturally being a
duality between two full string compactifications that exchanges the
nonperturbative contents of both theories. This supergravity duality gives one
a means of studying the interplay between mirror symmetry and the modularity
that can be present in supergravity compactifications. One can consider
extremal black holes from the IIB or IIA perspectives. It is in IIB that we
will first consider rank two attractors. In the IIA description the
enumerative invariants, computed by mirror symmetry techniques, enter as
quantum corrections to the supergravity action. The upshot is that the mirror
map provides identities relating the enumerative geometry of one space to the
modularity of another, and these are naturally encountered in the course of
investigating black hole solutions.

###  1.2 Thesis overview

We have throughout this thesis strived to denote Calabi-Yau manifolds by $X$
and $Y$, and always have it so that B-model computations are performed on $X$
while A-model computations are reserved for $Y$. We will consider IIB on $X$
and IIA on $Y$. Where $Y$ is a CICY, $X$ is a CICY mirror. We will always
refer to instanton numbers of $Y$.

##### Chapter §2

This is a brief collection of some preliminary results and terminology. We
define Calabi-Yaus, and introduce the favourable CICY set. These will recur
throughout the thesis as a sufficiently varied set of examples for our
purposes. We go on to set up the period computation that we will use
frequently. We explain how, for CICY mirrors $X$, the CICY matrix defines the
topological quantities and periods of $X$. We also review the splitting
process for CICYs, and all of these will bear relation to discussion in §3.

We go on to give a very basic account of arithmetic modularity, sufficient for
the purposes of searching for rational rank-two attractors later on in §4. We
will meet the zeta function, which provides the means of testing for
modularity. The zeta function is first defined by suitably collecting point
counts into a power series, but remarkably this is resummed into a rational
function. For modular manifolds, the numerator of this function factorises. It
is from these factors that the relevant modular form is read off.

##### Chapter §3

Here we detail our studies in mirror symmetry. We compute genus 0 numbers for
manifolds with $h^{1,1}$ as high as 7. We compute genus one numbers for some
geometries with lower $h^{1,1}$, but cannot do this for the 6 and 7 parameter
models. The genus 1 computations requires a piece of data not strictly
necessary at genus 0, the manifold’s discriminant locus. We compute a number
of examples of these, and comment on some patterns in this data set, in the
hopes of eventually returning to the 6 and 7 parameter models.

It is already known that the process of splitting a Calabi-Yau partitions its
instanton numbers, so obtaining the instanton numbers of the new manifold. We
showcase some examples of this, for a wider class of models than simply CICYs.

The data from our tables reveals a surprising fact, that sometimes sets of
instanton numbers can be symmetric under the action of a Coxeter group on the
manifold’s second homology. These are groups that are generated by
reflections, and are encoded by diagrams similar to Dynkin diagrams. The Weyl
group associated to a Dynkin diagram is also a Coxeter group, but Coxeter
groups are a much larger set with a different classification theory. We go on
to tabulate some Coxeter groups that appear in our data, giving their diagrams
and their representation on the curve classes of the Calabi-Yaus to which they
are associated.

Armed with Coxeter symmetry, we address the higher genus computation. We
review the construction of the propagators out of which higher genus B-model
free energies can be built, and cover BCOV recursion. We work an example at
genus 2, presented in such a way that we hope is valuable to anybody hoping to
carry out this computation. When it comes time to fix the genus-2 holomorphic
ambiguity, we apply the Coxeter symmetry and observe that this does not fully
fix the ambiguity. We discuss what other information is required.

##### Chapter §4

This chapter opens by recalling some textbook information on Calabi-Yau
supergravity compactifications. Then, we undertake a substantial review of the
attractor mechanism’s derivation, including the recasting of the attractor
equations into Strominger’s form for the fixed points of the flow. With the
attractor mechanism explained for general 4d $\mathcal{N}=2$ supergravities,
we then discuss its details where Calabi-Yau compactifications are concerned.

Next, we depart from black holes and instead look at flux compactifications.
We review both the IIB construction and its lift into F-theory á la Sen, to
see how precisely the IIB axiodilaton profile is realised as the modulus for
the elliptic fibre in a Calabi-Yau fourfold.

At this point in the thesis, we make a small digression. We discuss possible
sources of corrections to the potential in a Calabi-Yau flux compactification,
and explain how the no-scale relation is violated by a nonzero Euler
characteristic $\chi$, or by worldsheet instantons if $\chi=0$.

Then, we return to flux compactifications and explain our solution method for
the equations defining supersymmetric vacua. We provide a number of examples,
the crux of these being their axiodilaton’s dependence on the complex
structure moduli. This allows for the F-theory curve to be identified.

Next we turn our sights back to black holes, and use zeta function
computations to identify two new rank-two attractors. There exists a way,
given a rank-two attractor, to construct a summation identity relating L value
ratios to Gromov-Witten invariants. We display a number of these. Not all of
these sums converge, but the divergent ones can be treated with Padé
resummation.

The identities so given bring together critical L-values and instanton
numbers, in some examples distinct sets of instanton numbers, for different
spaces that are derived equivalent. We discuss some of the geometric aspects
of these derived equivalent partners.

##  Chapter 2 Some essentials

> This is where the fun begins.
>
> Anakin Skywalker

A manifold is Calabi-Yau if it is complex, Kähler, and has vanishing first
Chern class. In this thesis we will be concerned with compact Calabi-Yau
manifolds. They possess a twofold utility in string theory. One side of this
is that the two-dimensional $\mathcal{N}=(2,2)$ nonlinear sigma model is
conformal exactly when the target space is Calabi-Yau. The other side is that
an equivalent characterisation of a Calabi-Yau $n$-fold is that the manifold
has $\text{SU}(n)$ special holonomy, and so possesses a covariantly constant
spinor. This allows for supersymmetry to be preserved in compactifications of
supergravity or string theory on these manifolds. Specifically, Calabi-Yau
compactifications preserve one half of the supercharges found in the ten-
dimensional theory. They saw introduction to the theoretical physics
literature in [21].

In complex dimension 1, the Calabi-Yau condition is met only by elliptic
curves. These can be realised as quotients of the complex plane $\mathbbl{C}$
by a lattice $\Lambda_{\tau}$, with complex structure parameter the lattice
parameter $\tau$. The choices $\tau$ and $-\tau$ give isomorphic elliptic
curves. Two lattices $\Lambda_{\tau}$, $\Lambda_{\tau^{\prime}}$ are equal if
$\tau$ and $\tau^{\prime}$ are related by an $\text{SL}(2,\mathbbl{Z})$
transformation, and so the elliptic curves obtained as quotients
$\mathbbl{C}/\Lambda_{\tau}$ and $\mathbbl{C}/\Lambda_{\tau^{\prime}}$ are
also equal. Therefore, the moduli space of one-dimensional Calabi-Yau
manifolds is given by the upper half plane modulo $\text{SL}(2,\mathbbl{Z})$
transformations.

In complex dimension two, the set of complex Kähler manifolds with vanishing
first Chern class consists of products of elliptic curves, Abelian surfaces,
and K3 surfaces (see [22, 23]). Only K3 surfaces have holonomy $SU(2)$, and
they all lie in one complex analytic, but not algebraic, family.

The situation in three dimensions and higher is starkly different, and does
not currently possess a full characterisation. There are distinct families,
unrelated by complex structure deformation, and it is not known if the number
of such families in any given dimension is finite. Our studies will concern
the three-dimensional manifolds. It was shown in [24] that the moduli space of
a Calabi-Yau manifold $X$ factorises locally, into a product of the space of
Kähler structures on $X$, $\mathcal{M}_{K}(X)$, with the space of complex
structures on $X$, $\mathcal{M}_{\mathbbl{C}S}(X)$. The first of these spaces
can be thought of as the coefficients of $X$’s Kähler form when expanded in a
basis of $H^{2}(X,\mathbbl{Z})$ while the second parametrises complex
structure deformations of $X$, which in the simplest cases are variations of
the coefficients of the polynomials defining $X$ as a complete intersection.
Recall that the spaces of the $n^{th}$ cohomology of a complex manifold admit
a Hodge decomposition into $(p,q)$-forms, with $p+q=n$, with each of these
spaces having dimension equal to the Hodge number $h^{p,q}$.

We will only consider examples with full $SU(3)$ holonomy and not a subgroup.
This is the case for manifolds with $h^{1,0}=h^{2,0}=0$, because it can be
shown that the existence of a holomorphic $(1,0)$ or $(2,0)$ form reduces the
holonomy from $SU(3)$ to a subgroup [25]. In so doing we exclude the product
of three elliptic curves, and the product of an elliptic curve with a K3
surface. All Hodge numbers of $X$ are fixed once $h^{1,1}$ and $h^{1,2}$ are
specified. This is because complex conjugation equates $h^{p,q}$ to $h^{q,p}$,
while Hodge duality equates $h^{p,q}$ to $h^{3-q,3-p}$. We have stated that
$\text{Dim }\mathcal{M}_{K}(X)=\text{Dim }H^{2}(X,\mathbbl{Z})$, which in turn
equals $h^{1,1}$ because $h^{2,0}=h^{0,2}=0$. It can also be shown that
$\text{Dim }\mathcal{M}_{\mathbbl{C}S}(X)=h^{2,1}(X)$ [24].

###  2.1 Periods and the triple intersection form

Up to scale, Calabi-Yau manifolds possess a unique holomorphic $(3,0)$ form
denoted $\Omega$. After expanding $\Omega$ in a basis of
$H^{3}(X,\mathbbl{Z})$, one obtains the period vector of $X$. The
$b_{3}=2h^{1,2}+2$ components of this vector vary as the complex structure of
$X$ is modified, and satisfy the Picard-Fuchs system of $X$. For manifolds
with a one-dimensional space of complex structures (so $h^{2,1}=1$), the
Picard-Fuchs equation is a fourth order Fuchsian differential equation (that
is, a fourth order ODE, with coefficients that are rational functions in the
dependent variable, with regular singular points). There are a number of
choices of the basis in which one expands $\Omega$, but the two that we will
most frequently see are the Frobenius basis and integral symplectic basis. The
Frobenius period vector, denoted $\mathbf{\varpi}$, arises as the first set of
solutions obtained by the method of Frobenius. The integral symplectic basis
sees the most use in physical applications. Recall that linear systems of
differential equations possess monodromies, whereby continuing a vector of
solutions analytically around a singularity effects a linear transformation on
the solution vector. Importantly, the moduli space
$\mathcal{M}_{\mathbbl{C}S}$ possesses a point of maximal unipotent monodromy,
or MUM point, around which the monodromy matrix M obeys
$(\text{M}-\mathbbm{1})^{b_{3}}=0$ but $(\text{M}-\mathbbm{1})^{b_{3}-1}\neq
0$.

We shall write $\varphi^{i}$, $i=1,\,...\,,\,h^{1,2}$ for the complex
structure moduli coordinates of $X$, or $\bm{\varphi}$ to denote the list of
all coordinates. The Picard-Fuchs system admits as a solution a power series
$\varpi_{0}$ about a MUM point, which we shall take to be at
$\bm{\varphi}=\bm{0}$, with radius of convergence given by the distance from
the MUM point to the nearest singularity of the system. By normalising this
function $\varpi_{0}$ to equal 1 at the MUM point we uniquely specify it. Once
an analytic expression for the coefficients in the power series defining
$\varpi_{0}$ in an expansion about $\bm{\varphi}=\bm{0}$ is known (as a ratio
of factorials), along with some topological data that we will explain, it is
possible to obtain the other periods using the method of [26, 27] as follows.

First, introduce dependence on formal parameters $\epsilon_{i}$ into
$\varpi_{0}$ as follows:

$\varpi_{0}(\bm{\varphi})=\sum_{\bm{m}\geq
0}c(\bm{m})\bm{\varphi}^{\bm{m}}\qquad\to\qquad\bm{\varpi}^{\bm{\epsilon}}=\sum_{\bm{m}\geq
0}\frac{c(\bm{m}+\bm{\epsilon})}{c(\bm{\epsilon})}\bm{\varphi}^{\bm{m}+\bm{\epsilon}}~{}.$
(2.1)

Our choice of coordinates $\bm{\varphi}$ is discussed in Appendix §A. The
function $c(\bm{m})$ is extended to a function of non-integer argument above
by replacing factorials with Gamma functions, $n!\mapsto\Gamma(1+n)$.
$h^{2,1}$ periods with logarithmic singularities at the MUM point can then be
found by taking $\epsilon$-derivatives:

$\displaystyle\varpi_{1,i}$
$\displaystyle=\partial_{\epsilon_{i}}\varpi^{\bm{\epsilon}}\bigg{|}_{\bm{\epsilon}=0}~{}.$
(2.2) We will shortly introduce and define a set of constants $Y_{ijk}$. This
can be used to construct a further $h^{2,1}$ solutions with logarithm-squared
singularities at the MUM point like so: $\displaystyle\varpi_{2,i}$
$\displaystyle=\frac{1}{2}Y_{ijk}\partial_{\epsilon_{j}}\partial_{\epsilon_{k}}\varpi^{\bm{\epsilon}}\bigg{|}_{\bm{\epsilon}=0}~{}.$
(2.3) The last period, which has logarithm-cubed behaviour near
$\bm{\varphi}=0$, is obtained via $\displaystyle\varpi_{3}$
$\displaystyle=\frac{1}{6}Y_{ijk}\partial_{\epsilon_{i}}\partial_{\epsilon_{j}}\partial_{\epsilon_{k}}\varpi^{\bm{\epsilon}}\bigg{|}_{\bm{\epsilon}=0}~{}.$
(2.4)

These functions are then collected into the Frobenius period vector,

$\varpi=\begin{pmatrix}\varpi_{0}\\\ \varpi_{1,i}\\\ \varpi_{2,i}\\\
\varpi_{3}\end{pmatrix}~{}.$ (2.5)

Let us draw attention to a notational point. Where one-parameter manifolds
have been studied in the literature, it has sometimes been common to use the
symbols $\varpi_{2}$ and $\varpi_{3}$ to denote the above functions divided by
$Y_{111}$, i.e. it has been common to have $\varpi_{2}$ and $\varpi_{3}$ with
expansions beginning respectively with $\log(\varphi)^{2}$ and
$\log(\varphi)^{3}$. In this thesis we will consider one-parameter manifolds
on par with the multiparameter geometries, and so will always use the above
contraction with $Y_{ijk}$.

The quantities $Y_{ijk}$ are topological quantities not on $X$, but on a
_mirror manifold_ to $X$. If $Y$ denotes this mirror family, then $Y_{ijk}$
gives the triple intersection form of two-cycles on $Y$. That is, if $e_{i}$
form a basis of the second cohomology of $Y$,

$Y_{ijk}=\int_{Y}e_{i}\wedge e_{j}\wedge e_{k}~{}.$ (2.6)

This quantity is an integer, equalling the intersection number of the four
cycles that are Poincaré dual to each $e_{i}$.

Having explained how to form a Frobenius basis of solutions from the data of
$\varpi_{0}$ and $Y_{ijk}$, it remains to determine analytic expressions for
the coefficients in $\varpi_{0}$’s series expansion. This can be done by
analysing the Picard-Fuchs equations directly, but then one must first write
these down, which is not always straightforward. $\varpi_{0}$ can be
determined by certain residue integrals, as in [28]. The method that we will
adopt (outside of the one-parameter cases where we have the Picard-Fuchs
operator), is to use Liebgober’s formula that relates the sought coefficient
$c(\bm{m})$ to the total Chern class of $Y$, when $X$ and $Y$ are
intersections in toric varieties [29]. The total Chern class can be formally
written as a rational function of generators of the second cohomology $e_{i}$,
with a numerator and denominator that factorise into linear factors. By
effecting for each such factor the replacement

$\left(1+\sum_{i=1}^{h^{2,1}(X)}a_{i}e_{i}\right)\qquad\to\qquad\Gamma\left(1+\sum_{i=1}^{h^{2,1}(X)}a_{i}m_{i}\right)^{-1}~{},$
(2.7)

we obtain $c(\bm{m})$ as a ratio of Gamma functions.

When we turn to studying string-theory compactifications, we shall find a need
to express period vectors in an integral symplectic basis, as we now explain.
Consider a path in moduli space that begins and ends at some value
$\varphi_{*}$, and goes around a singularity. Encircling singularities in the
complex structure moduli space effects monodromy transformations of the period
vector. Since such an encircling returns us to the same point $\varphi_{*}$ in
moduli space that we started at, these monodromy transformations must be a
symmetry of the theory. By choosing to expand the period vector in a basis
such that monodromy transformations are integral symplectic matrices, we can
pair the monodromy transformations of the period vector with symplectic
transformations of the charge vector, to have a manifest monodromy invariance
of quantities like D-brane masses or black hole central charges (these are the
symplectic inner products of charge vectors with period vectors). Indeed,
$\mathcal{N}=2$ Maxwell theory possesses a symplectic duality symmetry. We
explain here how to make the change of basis from Frobenius to integral
symplectic.

First, one divides each Frobenius period $\varpi$ by an appropriate power of
$2\pi\text{i}$, since upon circling $0$ anticlockwise we have
$\frac{1}{2\pi\text{i}}\log(\varphi)\mapsto\frac{1}{2\pi\text{i}}\log(\varphi)+1$:

$\widehat{\varpi}_{0}=\varpi_{0}~{},\qquad\widehat{\varpi}_{1,i}=\frac{1}{2\pi\text{i}}\varpi_{1,i},\qquad\widehat{\varpi}_{2,i}=\frac{1}{(2\pi\text{i})^{2}}\varpi_{2,i}~{},\qquad\widehat{\varpi}_{3}=\frac{1}{(2\pi\text{i})^{3}}\varpi_{3}~{}.$
(2.8)

The integral symplectic period vector $\Pi$ is then computed from

$\Pi=\rho\,\widehat{\varpi}~{},\qquad\rho=\begin{pmatrix}-\frac{1}{3}Y_{000}&-\frac{1}{2}\mathbf{Y}_{00}^{T}&\phantom{-}\mathbf{0}^{T}&\phantom{-}1\\\\[3.0pt]
-\frac{1}{2}\mathbf{Y}_{00}&-\mathbbl{Y}_{0}&-\mathbbm{1}\phantom{-}&\phantom{-}\mathbf{0}\\\\[3.0pt]
1&\phantom{-}\mathbf{0}^{T}&\phantom{-}\mathbf{0}^{T}&\phantom{-}0\\\\[3.0pt]
\mathbf{0}&\mathbbm{1}&\,\hbox{{\elevenrmfromseventeenrm 0}\kern-3.5pt\vrule
height=6.9pt,depth=-0.2pt,width=0.35pt\kern 1.0pt\vrule
height=6.9pt,depth=-0.2pt,width=0.35pt\kern
2.0pt}&\phantom{-}\bm{0}\end{pmatrix}~{}.$ (2.9)

We have already seen the triple intersection numbers $Y_{ijk}$, with $i$
running from 1 to $h^{2,1}(X)$. it will turn out to be natural to consider
these as members of a larger set $Y_{abc}$ with indices running from 0 to
$h^{2,1}$. The vector $\mathbf{Y}_{00}$ has components $Y_{00i}$, which are
defined in terms of the mirror manifold $Y$’s second Chern class $c_{2}$ as

$Y_{00i}=-\frac{1}{12}\int_{Y}c_{2}\wedge e_{i}~{}.$ (2.10)

The quantity $Y_{000}$ is

$Y_{000}=\frac{3\zeta(3)}{(2\pi\text{i})^{3}}\chi(X)=-\frac{3\zeta(3)}{(2\pi\text{i})^{3}}\chi(Y)~{},$
(2.11)

where in the last equality we have used that the mirror pairing exchanges the
complex structure and complexified Kähler structure of $X$ and $Y$, and so
their Hodge numbers $h^{1,1}$ and $h^{2,1}$ are interchanged, and thereby $X$
and $Y$ have Euler characteristics that differ by a sign.

The quantities $Y_{0ij}$, which we collect into the matrix $\mathbbl{Y}_{0}$,
can be taken to have values $0$ or $1/2$. Symplectic transformations allow for
shifts by integer values, so we make the choice to take values in
$\\{0,1/2\\}$. As was explained in [27], exactly which number each $Y_{0ij}$
equals is determined by demanding that the monodromy matrix that acts on $\Pi$
after encircling the MUM point at $\bm{\varphi}=0$ is integral and symplectic.
This condition relates the $Y_{0ij}$ to the $Y_{ijk}$, and one can compute in
the $1,\,2,\,3$ modulus cases that

$Y_{0ij}=\frac{Y_{ijj}\text{ Mod }2}{2}~{}.$ (2.12)

###  2.2 CICYs

A set that admits a particularly simple characterisation are the _Complete
Intersection Calabi-Yaus_ , or CICYs [30]. These are the intersection of
hypersurfaces in products of projective space. Such a manifold $Y$ can be
represented by a configuration matrix, a rectangular array where for each
projective space factor in the ambient space there is a row. Each column
represents one of the defining equations. The $(i,j)$-entry of the array gives
the degree of the $j^{th}$ equation in the $i^{th}$ projective space. Three
illustrative examples are

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
3\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}3&3\end{matrix}\right]~{},\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1\\\
4&1\end{matrix}\right]~{}.$ (2.13)

The first of these represents a hypersurface (the vanishing set of a single
equation) in ${\mathbbl{P}^{1}\times\mathbbl{P}^{1}\times\mathbbl{P}^{2}}$
defined by a polynomial with degrees $(2,2,3)$ in the homogeneous coordinates
on each projective space. The second represents the intersection of two
degree-3 hypersurfaces in $\mathbbl{P}^{5}$. The third represents the
intersection of two hypersurfaces in the space
$\mathbbl{P}^{1}\times\mathbbl{P}^{4}$. The general configuration matrix,
which we display to fix notation in this section, is

$\begin{matrix}\mathbbl{P}^{n_{1}}\\\ \vdots\\\
\mathbbl{P}^{n_{K}}\end{matrix}\\!\left[\begin{matrix}d^{\,(1)}_{1}&...&d^{\,(1)}_{C}\\\
\vdots&...&\vdots\\\ d^{\,(K)}_{1}&...&d^{\,(K)}_{C}\end{matrix}\right]~{}.$
(2.14)

For the purposes of this section, $K$ will be the number of projective space
factors in the ambient space and $C$ will be the number of equations defining
the intersection.

Not any configuration matrix yields a Calabi-Yau. One must verify that the
first Chern class $c_{1}(TY)$ vanishes. This is an exercise in classical
algebraic geometry: one can form the total Chern class from the entries of the
configuration matrix, which is Taylor expanded to get the Chern classes
$c_{1}$, $c_{2}$ and $c_{3}$. The first of these needs to vanish, while the
third can be used to compute the Euler characteristic $\chi(Y)$. The space $Y$
is an intersection in an ambient space
$\mathcal{A}=\mathbbl{P}^{n_{i}}\times\,...\,\times\mathbbl{P}^{n_{K}}$, and
there is a split exact sequence

$0\to TY\to T\mathcal{A}\to NY\to 0~{},$ (2.15)

where the tangent bundle $TY$ of $Y$ injects into the tangent bundle of
$\mathcal{A}$, which in turn projects onto the normal bundle $NY$ of $Y$.
Stated in more pedestrian terms, the following decomposition (which is non-
canonical, depending on the choice of the embedding of $NY$ as a subbundle of
$T\mathcal{A}$) holds at points on $Y$:

$T\mathcal{A}=TY\oplus NY~{}.$ (2.16)

The Chern class of the vector sum of two spaces is the product of their
individual Chern classes, and so

$\text{Ch}(T\mathcal{A})=\text{Ch}(TY)\cdot\text{Ch}(NY)~{},\qquad\text{whereby}\qquad\text{Ch}(TY)=\frac{\text{Ch}(T\mathcal{A})}{\text{Ch}(NY)}~{}.$
(2.17)

The total Chern class of $\mathcal{A}$ is the product of the Chern classes of
the factors $\mathbbl{P}^{n_{i}}$. Since
$\text{Ch}(\mathbbl{P}^{n_{i}})=(1+E_{i})^{n_{i}+1}$, where $E_{i}$ generates
the second cohomology of $\mathbbl{P}^{n_{i}}$, we get

$\text{Ch}(T\mathcal{A})=\prod_{i=1}^{K}\left(1+E_{i}\right)^{n_{i}+1}~{}.$
(2.18)

It is through the total Chern class of $NY$ that the entries of the
configuration matrix enter. One has that

$\text{Ch}(NY)=\prod_{j=1}^{C}\left(1+\sum_{i=1}^{K}d^{\,(i)}_{j}E_{i}\right)~{}.$
(2.19)

The total Chern class then is

$\text{Ch}(TY)=\frac{\prod_{i=1}^{K}\left(1+E_{i}\right)^{n_{i}+1}}{\prod_{j=1}^{C}\left(1+\sum_{i=1}^{K}d^{\,(i)}_{j}E_{i}\right)}~{},$
(2.20)

which we expand to linear order in the cohomology generators $E_{i}$ to find
the first Chern class $c_{1}$. This gives

$c_{1}(TY)=\sum_{i=1}^{K}\left(1+n_{i}-\sum_{j=1}^{C}d^{\,(i)}_{j}\right)E_{i}~{},$
(2.21)

whereupon we learn that the Calabi-Yau condition is met precisely when the sum
of the configuration matrix’s $i^{th}$ row equals $n_{i}+1$. Since we are
interested in threefolds, we must have $c+3=\sum_{i=1}^{K}n_{i}$.

There is a computational problem in enumerating all possible rectangular
matrices with nonnegative entries that satisfy the above two conditions.
However, there are a number of possible redundancies which mean that a
straightforward listing of all possible configuration matrices overcounts the
number of CICY threefolds. For example, for $m$-vectors
$\mathbf{a},\,\mathbf{b}$, an $n\times m$ matrix M, and a ‘vector’ $P$ of $n$
projective spaces, there is the identity

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
P\end{matrix}\\!\left[\begin{matrix}1&\mathbf{a}^{T}\\\ 1&\mathbf{b}^{T}\\\
\mathbf{0}&\text{M}\end{matrix}\right]=\begin{matrix}\mathbbl{P}^{1}\\\
P\end{matrix}\\!\left[\begin{matrix}\mathbf{a}^{T}+\mathbf{b}^{T}\\\
\text{M}\end{matrix}\right]~{},\qquad\text{for
example}\qquad\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 0&2\\\ 0&3\end{matrix}\right]=\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{2}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
3\end{matrix}\right]~{}.$ (2.22)

This follows from the fact that

$\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1\\\
1\end{matrix}\right]\quad=\quad\mathbbl{P}^{1}~{}.$ (2.23)

Upon removing such redundant descriptions as above, one arrives at a naive
number of 7890 CICY threefolds. This is still not the full story, as
additional nontrivial redundancies exist beyond those used to find this
number. This list is held in many places, we have in this work made use of
Lukas’s website [31], from which one can for instance find the redundancy

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 1&1\\\ 1&1\\\
1&1\end{matrix}\right]\;=\;\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}0&2\\\ 1&1\\\ 1&1\\\ 1&1\\\
1&1\end{matrix}\right]~{}.$ (2.24)

This is a family to which we shall give some attention in later sections,
preferring to use the more symmetric description on the left-hand side.

More generally, one could consider intersections in toric varieties (of which
CICYs are but a special case). Hypersurfaces in toric varieties are in
bijection with reflexive four-dimensional polytopes, which were tabulated in
[32]. The enumeration of more general intersections of hypersurfaces in toric
varieties of dimension greater than 4 has not been completed. Using gauge
theory constructions, intersections in toric varieties are realised as phases
of an abelian gauged linear sigma model. By dropping this abelian condition,
one could reach a wider-still set of manifolds. Examples of this construction
include the Rødland manifold [33] and the two-parameter model of Knapp and
Hori [34].

As we turn to the study of particular problems, we shall lean on the CICY
threefolds as our examples. However, we stress that a number of the properties
that we will observe can be expected to hold in more general examples.

There is an important partition of the CICY manifolds into two sets: those
that are _favourable_ and those that are not. A complete intersection $Y$ is
favourable if its second cohomology is spanned by the pullbacks to $Y$ of the
generators for each projective space’s second cohomology. For example,

$\begin{matrix}\mathbbl{P}^{5}\end{matrix}\\!\left[\begin{matrix}3,3\end{matrix}\right]\qquad\text{and}\qquad\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&4\end{matrix}\right]$ (2.25)

respectively have one- and two-dimensional second cohomologies. In the first
example this is spanned by the pullback of the Kähler form for the ambient
$\mathbbl{P}^{5}$, while in the second the pullbacks of the Kähler forms for
the ambient $\mathbbl{P}^{1}$ and $\mathbbl{P}^{4}$ span the manifold’s second
cohomology. Note that a favourable manifold necessarily has $h^{1,1}=K$. This
is to be contrasted with non-favourable CICYs, for which $h^{1,1}>K$ (we will
always take $K$ to be the number of projective space factors in the
representative among redundant configurations with the smallest number of
rows). An example of a CICY with this feature is

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}0&0&2&2&1\\\
2&2&0&0&1\end{matrix}\right]~{},$ (2.26)

which has $h^{1,1}=12$. The pullbacks of the Kähler form for each
$\mathbbl{P}^{4}$ factor each appear in the manifold’s second cohomology, but
there are ten other generators. To see where these come from, we should recall
that the (non-Calabi-Yau) intersection
$\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}2&2\end{matrix}\right]$
is the Segre surface, isomorphic to the projective plane blown up in 5 points.
In addition to the hyperplane class, this space has an additional 5 generators
in the second cohomology dual to cycles wrapping each blown up point. So by
viewing the above CICY manifold as an intersection in the product of two Segre
surfaces, we can account for $h^{1,1}=12$ by pulling back to $X$ the six
independent generators in each Segre surface’s cohomology. More generally, the
$h^{1,1}$ of a non-favourable manifold can be explained in terms of the
cohomology classes on del Pezzo surfaces that arise from the configuration
matrix in a similar manner as above.

To compute $h^{2,1}$, it suffices to compute $h^{1,1}$ as in the previous
paragraph and then use the total Chern class the compute $\chi=\int_{Y}c_{3}$.
Recall that the Euler characteristic of $Y$ is the alternating sum of Betti
numbers, $\chi(Y)=\sum_{i=0}^{6}(-1)^{i}b_{i}$. Now, $b_{0}=b_{6}=1$ as our
manifolds each have a single connected component. Further, $b_{1}=b_{5}=0$ (in
light of full $\text{SU}(3)$ holonomy). We have that $b_{2}=b_{4}=h^{1,1}$ and
$b_{3}=2+2h^{1,2}$, so that

$\chi=2\left(h^{1,1}-h^{1,2}\right)~{}.$ (2.27)

We see that $h^{2,1}$ can be determined via $h^{2,1}=h^{1,1}-\frac{1}{2}\chi$,
with $\chi$ determined from the configuration matrix via the total Chern class
formula. Note that for a CICY $Y$, $\chi(Y)\leq 0$.

Following [30], the triple intersection numbers of a favourable CICY $Y$ can
be computed by noting that each two-form $e_{i}$ on $Y$ is the pullback, with
respect to the inclusion map $\iota:Y\mapsto\mathcal{A}$, of the Kähler forms
$E_{i}$ of each $\mathbbl{P}^{n_{i}}$. That is,

$e_{i}=\iota^{*}(E_{i}).$ (2.28)

As a result of this

$Y_{ijk}=\int_{Y}e_{i}e_{j}e_{k}=\int_{\mathcal{A}}E_{i}E_{j}E_{k}\prod_{j=1}^{C}\left(1+\sum_{i=1}^{K}d^{\,(i)}_{j}E_{i}\right)~{},$
(2.29)

where the product inserted in the right-hand integral is the Poincaré dual to
$Y$ in the cohomology of $\mathcal{A}$. The above integral is performed by
singling out the coefficient of the volume form on $\mathcal{A}$, which is the
product $\prod_{i=1}^{K}E_{i}^{n_{i}}$.

We have now provided the total Chern class and triple intersection numbers of
a favourable CICY $Y$. We can now use (2.20) and (2.29) to compute the periods
of mirror manifolds $X$ as in (2.1).

###  2.3 Splittings and the conifold transition

The data of a CICY matrix gives only the degrees of the polynomials defining
the Calabi-Yau variety, and says nothing about the coefficients of these
polynomials. Generic choices of these polynomials, related by variation of
complex structure, will be smooth and be of the same topological type (in
particular they will have the same Hodge numbers and triple intersection
form). Nonetheless, for nongeneric choices of these coefficients the geometry
will acquire singularities. Deformation (varying the polynomial coefficients
again) will return a smooth manifold in the generic family, but one could also
resolve these singularities via a blowup to obtain some smooth member of a
different family of manifolds, with different topological properties. The
latter family obtained from the singular space by resolution is said to be a
‘split’ of the former family obtained from the singular space via deformation.

Let us be more concrete, and explain the splittings via operations on CICY
matrices. Let us begin with the family $Y$, with configuration matrix

$Y:\qquad\begin{matrix}\mathbf{P}\end{matrix}\\!\left[\begin{matrix}\mathbf{v}&\text{M}\end{matrix}\right]~{},$
(2.30)

with $\mathbf{P}$ denoting a list of projective spaces, $\mathbf{v}$ being a
vector of nonnegative integers, and M being a matrix. Now it may be the case
that for some nongeneric choice of coefficients, the first equation of the
above configuration can be expressed as a determinant. That is to say, the
equation read off from $\mathbf{P}[\mathbf{v}]$ is

$\text{Det}(\text{C}^{a}_{\phantom{-}b})\;=\;0~{},$ (2.31)

where the square matrix C, of some dimension $(n+1)\times(n+1)$, has entries
that are polynomials in the coordinates of $\mathbf{P}$. The set of such
equations as the polynomials are varied may have members defining singular
spaces, in which case the splitting we go on to describe is said to be
_effective_ and yields a manifold of a new topological type. Alternatively we
may be describing an _ineffective_ splitting if the family of vanishing loci
of $\text{det}(\text{C})$ has no singularities, so that we are not changing
the topology of the family considered, instead just giving a different
redundant description.

Vanishing of C’s determinant implies the existence of a nullvector for C, a
nonzero $n+1$ vector $x^{b}$ such that

$\text{C}^{a}_{\phantom{-}b}\,x^{b}\;=\;0~{}.$ (2.32)

There are $n+1$ such equations for each choice of the index $a$. Promoting
$x^{b}$ to homogeneous coordinates on $\mathbbl{P}^{n}$, The above equation
then is some element of the set of polynomials read off from

$\begin{matrix}\mathbbl{P}^{n}\\\
\mathbf{P}\end{matrix}\\!\left[\begin{matrix}1&1&...&1\\\
\mathbf{u}_{1}&\mathbf{u}_{2}&...&\mathbf{u}_{n+1}\end{matrix}\right]~{},$
(2.33)

where $\mathbf{v}=\sum_{i=1}^{n+1}\mathbf{u}_{i}$ . We learn that the family
$X$ can be split to a family $X^{\prime}$ with configuration

$Y^{\prime}:\qquad\begin{matrix}\mathbbl{P}^{n}\\\
\mathbf{P}\end{matrix}\\!\left[\begin{matrix}1&1&...&1&\mathbf{0}^{T}\\\
\mathbf{u}_{1}&\mathbf{u}_{2}&...&\mathbf{u}_{n+1}&\text{M}\end{matrix}\right]~{},$
(2.34)

which is guaranteed to be Calabi-Yau by the fact that $Y$ was Calabi-Yau (one
can check that the rows sum to appropriate values). Some examples of pairs
$(A,B)$ where $B$ splits $A$ are

$\left(\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}5\end{matrix}\right]\;,\;\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&4\end{matrix}\right]\right)~{},\qquad\left(\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}5\end{matrix}\right]\;,\;\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]\right)~{},\qquad\left(\begin{matrix}\mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\ 2\\\
2\end{matrix}\right]\;,\;\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\ 1&1\\\ 1&1\\\ 1&1\\\
1&1\end{matrix}\right]\right)~{}.$ (2.35)

Note that a manifold can have distinct splits, as exemplified by the Quintic
$\begin{matrix}\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}5\end{matrix}\right]$
above.

###  2.4 Calabi-Yau modularity

We turn now to explaining what is meant by a _modular_ Calabi-Yau manifold.
Having done this, we will go on to see how Calabi-Yau modularity is related to
supersymmetric physics.

The conventional physicist’s view of a Calabi-Yau manifold is that it is an
algebraic variety over the field of complex numbers $\mathbbl{C}$. From a
wider perspective, one is free to consider Calabi-Yau manifolds over any
field. Of particular relevance to modularity are the finite fields
$\mathbbl{F}_{p^{n}}$, where in the subscript $p$ is a prime and $n$ a
positive integer. This number $p^{n}$ is the order of the finite field, which
is necessarily the power of a prime.

Given a polynomial with rational coefficients, we can multiply through by the
least common multiple of the coefficient’s denominators in order to get a
polynomial with integer coefficients. This in turn can be reduced modulo a
prime.

Alternatively, suppose now that a Calabi-Yau $X$’s complex structure moduli
take rational values $m/n$. $X$ is then an algebraic variety over
$\mathbbl{Q}$, for which one writes $X/\mathbbl{Q}$. The integers $m$, $n$
have representatives in $\mathbbl{F}_{p}$, simply found as $m\text{ Mod }p$
and $n\text{ Mod }p$. Now, $\mathbbl{F}_{p}$ is a field and so for each
integer $n$ such that $p\nmid n$, there is an element $1/n$. So the product
$m\cdot\frac{1}{n}$ can be found in $\mathbbl{F}_{p}$. This allows for the
equations defining $X$ as an algebraic variety over $\mathbbl{C}$, assumed to
have rational coefficients, to be replaced with equations with coefficients in
$\mathbbl{F}_{p}$. All of that is to say, we can consider the variety $X$ over
the field $\mathbbl{F}_{p}$, denoted $X/\mathbbl{F}_{p}$.

The variety $X/\mathbbl{F}_{p}$ is the set of solutions to these equations,
where the indeterminates (coordinates on the ambient space containing $X$)
take values in $\mathbbl{F}_{p}$. So, this variety $X/\mathbbl{F}_{p}$ is the
union of a finite number of points in the ambient space.

The finite fields $\mathbbl{F}_{p^{n}}$ can be realised as the field extension
of $\mathbbl{F}_{p}$ by the root of an irreducible polynomial with order $n$.
That is to say, we can realise $\mathbbl{F}_{p^{n}}$ by taking

$\mathbbl{F}_{p^{n}}\cong\mathbbl{F}_{p}(\alpha)~{},\qquad\text{with $\alpha$
a root of an irreducible polynomial $Q$ with coefficients in
$\mathbbl{F}_{p}$.}$ (2.36)

There is a standard field embedding

$\mathbbl{F}_{p}\hookrightarrow\mathbbl{F}_{p^{n}}$ (2.37)

which simply sends the field element $r\in\mathbbl{F}_{p}$ to
$r\in\mathbbl{F}_{p}(\alpha)\cong\mathbbl{F}_{p^{n}}$.

Since $\mathbbl{F}_{p}$ embeds this way into $\mathbbl{F}_{p^{n}}$, we can
also take the equations defining the variety $X/\mathbbl{F}_{p}$ to define
another variety $X/\mathbbl{F}_{p^{n}}$, which consists of another union of a
finite number of coordinate vectors with elements in $\mathbbl{F}_{p^{n}}$ so
that these equations are solved.

Crucially, the information contained in each of these finite sets ‘fits
together’ in a way that we shall now describe.

####  2.4.1 The local zeta function

Fix a prime $p$, and a set of algebraic moduli $\varphi$ for the variety
$X_{\varphi}$, which we shall subscript with $\varphi$ in this subsection so
as to make dependencies clear. We will write $N_{p^{n}}(\varphi)$ for the
number of points constituting the variety $X_{\varphi}/\mathbbl{F}_{p^{n}}$.

Now we introduce a formal variable $T$ and collect these point counts into the
following generating function:

$\zeta_{p}\left(X_{\varphi};T\right)=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\,N_{p^{n}}\,(\varphi)T^{n}\right)~{}.$
(2.38)

This is the local zeta function of $X_{\varphi}/\mathbbl{F}_{p}$ . The three
Weil conjectures [35], proved by Dwork [36], Grothendieck [37], and Deligne
[38] , describe the behaviour of this function. We shall state them on the
assumption that $X_{\varphi}/\mathbbl{C}$ is a $d$-complex dimensional
projective variety, and not necessarily a Calabi-Yau threefold.

##### (1) Rationality

The zeta function is a rational function of $T$, with the numerator and
denominator factorising over $\mathbbl{Z}$ into the form

$\zeta_{p}\left(X_{\varphi};T\right)=\frac{P_{1}\left(X_{\varphi};T\right)P_{3}\left(X_{\varphi};T\right)\,...\,P_{2d-1}\left(X_{\varphi};T\right)}{P_{0}\left(X_{\varphi};T\right)P_{2}\left(X_{\varphi};T\right)\,...\,P_{2d}\left(X_{\varphi};T\right)}~{},$
(2.39)

where each polynomial $P_{i}\left(X_{\varphi};T\right)$ has degree equal to
the Betti number $b_{i}$ . In addition,

$P_{0}=1-T\qquad\text{ and }\qquad P_{2d}=1-p^{d}T~{}.$ (2.40)

##### (2) Functional Equation

This function obeys the following transformation of argument property:

$\zeta_{p}\left(X_{\varphi};\frac{1}{p^{d}T}\right)=\pm
p^{\frac{d\chi}{2}}T^{\chi}\zeta_{p}\left(X_{\varphi};T\right)~{},$ (2.41)

with $\chi$ the Euler characteristic of $X_{\varphi}$.

##### (3) The Riemann hypothesis

Each polynomial $P_{i}$ factorises over $\mathbbl{C}$,

$P_{i}\left(X_{\varphi};T\right)=\prod_{j=1}^{b_{i}}\left(1-\lambda_{ij}(X_{\varphi})T\right)~{},$
(2.42)

and each $\lambda_{ij}$ is an algebraic integer. That is, each $\lambda_{ij}$
is a root in $\mathbbl{C}$ of a monic polynomial with integer coefficients.
Further, each $\lambda_{ij}$ has absolute value $p^{i/2}$.

Note that the rational function (2.39) is further simplified when
$X_{\varphi}$ is a Calabi-Yau threefold. Then $d=3$, and the Betti numbers are
$b_{0}=b_{6}=1$, $b_{2}=b_{4}=h^{1,1}$, $b_{1}=b_{5}=0$, and
$b_{3}=2h^{2,1}+2$. The form of (2.39) is further constrained by the
functional equation (2.41), so that for a Calabi-Yau threefold we get

$\zeta_{p}\left(X_{\varphi};T\right)=\frac{R_{p}\left(X_{\varphi};T\right)}{\left(1-T\right)\left(1-pT\right)^{h^{1,1}}\left(1-p^{2}T\right)^{h^{1,1}}\left(1-p^{3}T\right)}~{},$
(2.43)

with all $\varphi$ dependence captured in the numerator
$R_{p}\left(X_{\varphi};T\right)$, which is a degree $b_{3}=2h^{2,1}+2$
polynomial in $T$.

Let us stress the following point: for generic moduli $\varphi$ we obtain some
typically irreducible polynomial $R_{p}\left(X_{\varphi};T\right)$, but for
particular choices of moduli $R_{p}\left(X_{\varphi};T\right)$ can factorise
for each $p$. When this happens, the coefficients of quadratic factors are
conjecturally related to modular forms. We shall now explain this in greater
detail.

####  2.4.2 Modular forms and modularity of elliptic curves

For elliptic curves $E_{\varphi}$ (Calabi-Yau onefolds) over $\mathbbl{Q}$,
the zeta function numerator $Q_{p}$ is a quadratic function of $T$ of the form

$Q_{p}\left(X_{\varphi};T\right)=1-c_{p}(\varphi)pT+p^{3}T^{2}~{}.$ (2.44)

We remark that the Weil conjectures for elliptic curves were originally proved
by Hasse [39]. The modularity theorem for elliptic curves over $\mathbbl{Q}$,
proved in work by Breuil, Conrad, Diamond, Taylor, and Wiles [5, 6, 7]
guarantees that the $c_{p}$ so defined are the coefficients of $q^{p}$ in the
Fourier expansion of a weight-two modular form (in fact, a newform) for a
congruence subgroup $\Gamma_{0}(N)$, where $N$ is the conductor of the
elliptic curve $E_{\varphi}$. This number $N$ is divisible by each prime $p$
that has bad reduction, which means that the elliptic curve
$E_{\varphi}/\mathbbl{F}_{p}$ is singular.

The modular group $\Gamma\cong\text{PSL}(2,\mathbbl{Z})$ of $2\times 2$
matrices with integer components and determinant $\pm 1$ acts on the upper
half plane $\mathbbl{H}$ (complex numbers with positive imaginary part) by the
Möbius action:

$\gamma(\tau)=\frac{a\tau+b}{c\tau+d}~{},\qquad\text{where the matrix $\gamma$
has components}\qquad\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}~{}.$ (2.45)

We shall always take an elements $\gamma$ of $\Gamma$ to have matrix
components $a,\,b,\,c,\,d$ as above. A modular form of weight $k\geq 0$ is a
holomorphic function $f$ on the upper half plane with the following
transformation properties under the modular group:

$f\left(\gamma(\tau)\right)=\left(c\tau+d\right)^{k}f(\tau)~{}.$ (2.46)

There are functions that satisfy the above property not for $\Gamma$, but for
certain subgroups. Specifically, _congruence subgroups_ , which we now define.
The principal congruence subgroup of level $N$ is the group

$\Gamma(N)=\left\\{\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\Gamma:\ a,d=1\text{ and }b,c=0\text{ mod
}N\right\\}.$ (2.47)

A congruence subgroup is any subgroup of $\Gamma$ that contains a $\Gamma(N)$
as a subgroup. The examples that we shall make frequent return to, appearing
in the statement of the modularity theorem for elliptic curves that we gave at
the start of this subsection, are the groups

$\Gamma_{0}(N)=\left\\{\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in\Gamma:\ c=0\text{ mod }N\right\\}.$ (2.48)

A modular form of level $N\in\mathbbl{N}$ and weight $k\geq 0$ is a
holomorphic function obeying (2.46) for all $\gamma\in\Gamma_{0}(N)$. These
functions form a finite-dimensional vector space over $\mathbbl{C}$, and
newforms are particular basis elements of this space. These newforms are
tabulated in the L-Functions and Modular Forms Database, LMFDB, [40]. Newforms
are in particular cusp forms, which have 0 constant term in their Fourier
expansions. We shall make frequent use of the LMFDB labels when discussing
examples in later sections.

##### L functions

The Mellin transform of a modular form $f$ yields the associated $L$-function:

$L(s)=\frac{(2\pi)^{s}}{\Gamma(s)}\int_{0}^{\infty}\text{d}y\,f(\text{i}\,y)\,y^{s-1}~{}.$
(2.49)

Naturally, $\Gamma(s)$ is the Gamma function and not a group. In practice, one
usually has the first few hundred (or more) terms in a Fourier expansion of
$f$, which can be inserted into the above integral so as to compute values
$L(s)$. The accuracy improves with more terms in $f$’s Fourier expansion, but
a manipulation of the integrand using transformation properties of $f$ allows
for a quicker evaluation to a given accuracy. To do this, first break up the
integral like so:

$L(s)=\frac{(2\pi)^{s}}{\Gamma(s)}\left(\int_{\frac{1}{\sqrt{N}}}^{\infty}\,f(\text{i}\,y)\,y^{s-1}\,\text{d}y\;+\;\int_{0}^{\frac{1}{\sqrt{N}}}\,f(\text{i}\,y)\,y^{s-1}\,\text{d}y\right)~{}.$
(2.50)

Because the lower range of integration on the left-hand integral is nonzero,
the approximations to this term obtained by truncating $f$’s Fourier expansion
are much better than for the right-hand integral. To better approximate the
right-hand integral we will make use of the Fricke involution, a property of
cusp forms for $\Gamma_{0}(N)$. Such functions, with weight $k$, admit the
following transformation (Fricke involution):

$f\left(-\frac{1}{N\tau}\right)=\epsilon N^{k/2}\tau^{k}f(\tau)~{}.$ (2.51)

The quantity $\epsilon$ is the Fricke sign of the form $f$, which is equal to
plus or minus one. We now effect a change of variables
$\tau\mapsto-\frac{1}{N\tau}$ in the right-hand integral of (2.50), and then
apply the formula (2.51), to obtain

$L(s)=\frac{(2\pi)^{s}}{\Gamma(s)}\int_{\frac{1}{\sqrt{N}}}^{\infty}\text{d}y\,f(\text{i}\,y)\left(y^{s-1}+(-1)^{k/2}\epsilon
N^{k/2-s}y^{k-1-s}\right)~{}.$ (2.52)

This expression is well-approximated upon replacing $f$ by a truncation of its
Fourier expansion.

####  2.4.3 Modularity of Calabi-Yau threefolds

Whereas the modularity of elliptic curves over $\mathbbl{Q}$ is a proven fact
for all such curves, the same is not true of threefolds. The topic of Calabi-
Yau modularity was reviewed in [8].

A Calabi-Yau is rigid if it has a pointlike space of complex structures, i.e.
$h^{2,1}=0$. Then $b_{3}=2$ and the zeta function numerators $R_{p}(X;T)$ are,
by the Weil conjectures, automatically quadratic. It was proven by Gouvea and
Yui [41] that such threefolds are indeed modular, of weight-four. For these
Rigid threefolds

$R_{p}(X;T)=1-\beta_{p}T+p^{3}T^{2}$ (2.53)

and the integers $\beta_{p}$ are the coefficients of the $p^{th}$ terms in the
Fourier expansion of a weight-four modular form.

This is in contrast to manifolds $X_{\varphi}$ with $h^{2,1}\neq 0$. The most
general approach here, which we do not take, is to recognise in the zeta
function of $X_{\varphi}$ the coefficients of an automorphic form for a
symplectic group’s Langlands dual, in the style of the Langlands program [42,
43, 44]. The problem that we turn to is, for which moduli $\varphi$ is
$X_{\varphi}$ modular? This means that the zeta function numerator
$R_{p}(X_{\varphi};T)$ possesses for all good primes $p$ a quadratic factor
from which we can read off a modular form.

There are a number of threefolds proven to be weight-four modular. Certain
members of a family associated to the $A_{4}$ lattice were shown to be so by
Hulek and Verrill [45], Schoen proved modularity of singular quintic
threefolds in [46], and in [13] a number of conjecturally modular manifolds
were displayed, and among them it was proven that the mirror of
$\begin{matrix}\mathbbl{P}^{7}\end{matrix}\\!\left[\begin{matrix}2&2&2&2\end{matrix}\right]$
was modular at the conifold. In addition to these proven cases, a number of
manifolds are conjectured to be modular with evidence given by an extensive
computation of the zeta function. Such cases are included in the works [14,
47].

There are two problems in compactifications of type IIB superstring theory on
$X_{\varphi}$, solved by restrictions on the moduli $\varphi$ to values
$\varphi_{*}$, so that conjecturally the manifolds $X_{\varphi_{*}}$ are
modular. The first of these is the identification of _attractor points of
rank-two_ , a program instigated by Moore in [10, 9]. The relevant manifolds
here are conjecturally weight-four modular. The second such set of moduli are
those that give _supersymmetric flux vacua_. The manifolds here should be
weight-two, as conjectured by Kachru, Nally, and Yang [12, 11]. We will in
later sections exposit both of these, and display new results in both
directions. Here, we discuss some general features of modular Calabi-Yau
threefolds that bears relevance to both physical setups.

##### The Frobenius map

We review this background material following [14, 48, 49, 50]. Fermat’s little
theorem gives, for integer $c$,

$c^{p}=c\qquad\text{ Mod }p~{}.$ (2.54)

Further, when working modulo $p$ we have a simplified form of the binomial
theorem for $c$ and $d$ in $\mathbbl{F}_{p^{k}}$:

$(c+d)^{p}=c^{p}+d^{p}\qquad\text{ Mod }p~{}.$ (2.55)

When considering varieties over finite fields, we refer to the vanishing set
of some polynomials. Note that in light of the above two identities, we get a
result for polynomials $Q$ that have coefficients in $\mathbbl{F}_{p}$ and
indeterminates in any $\mathbbl{F}_{p^{k}}$ with $k\in\mathbbl{N}$. This is

$Q(x)=0\implies Q(x)^{p}=0\implies Q(x^{p})=0~{}.$ (2.56)

So, if we act on the coordinates $x$ of the ambient space containing
$X_{\varphi}$ by the Frobenius map

$\text{Frob}_{p}:x\mapsto x^{p}~{},$ (2.57)

we leave invariant the vanishing locus $X_{\varphi}$. In light of this, the
Frobenius map is an automorphism of any variety
$X_{\varphi}/\mathbbl{F}_{p^{k}}$ with coefficients in $\mathbbl{F}_{p}$.

The fixed points of this automorphism obey $x^{p}=x$, which means that $x$ is
defined in $\mathbbl{F}_{p}\subset\mathbbl{F}_{p^{k}}$. Fixed points of
$\left(\text{Frob}_{p}\right)^{2}$ have $x^{p^{2}}=x$, and so are defined in
$\mathbbl{F}_{p^{2}}\subset\mathbbl{F}_{p^{k}}$, and so on for each higher
field.

So one has that the zeroes of the defining polynomials of $X_{\varphi}$, which
we aim to count, are fixed points of a continuous automorphism of
$X_{\varphi}$. By providence, the number of fixed points of such an
automorphism appears in Lefschetz’s trace formula:

$\sum_{i=0}^{6}(-1)^{i}\text{Tr}\left((\text{Frob}_{p})^{k}|H^{i}(X_{\varphi})\right)=N_{p^{k}}(\varphi)~{}.$
(2.58)

Here, one uses the $p$-adic cohomology $H^{i}(X_{\varphi})$ (we do not discuss
this point further, instead referring to [50]). The 6 in the above formula is
the real dimension of the threefolds we study, for a complex $n$-fold one
would replace 6 by $2n$.

Using this formula, working from the exponential form (2.38) one can show that
the polynomials $P_{k}$ appearing in the rational form of the zeta function
(2.39) are in fact determinants:

$P_{k}(X_{\varphi};T)=\det\left(1-T\,\text{Frob}_{p}^{\phantom{-}-1}|H^{k}(X_{\varphi}\right)~{}.$
(2.59)

It should be noted that our notation for $P_{k}$ suppresses their dependence
on the prime $p$.

Counting the $\mathbbl{F}_{p^{k}}$-points of the varieties $X_{\varphi}$, as
done for example in [51], is a laborious process. In [50] a method was
developed, using the above determinant formula, to more efficiently compute
zeta function numerators.

##  Chapter 3 Reflections in the mirror

> And I saw how the stars of Heaven come out, and counted the Gates out of
> which they come, and wrote down all their outlets, for each one,
> individually, according to their number. And their names, according to their
> constellations, their positions, their times, and their months, as the Angel
> Uriel, who was with me, showed me.
>
> Enoch 33.3

Given a complex manifold $M$, a quantity of foremost interest in classical
algebraic geometry is the number of holomorphic embeddings of a Riemann
surface with given genus and degree in $M$. This statement requires some
delicacy, for such curves can lie in continuous families and so a more refined
notion of ‘counting’ is required. A major success of string theory is the
means it provides of obtaining these curve counts when $M$ is Calabi-Yau. This
is done by studying the partition function of two distinct topological string
theories, the A and B models. The A-model bears a direct relation to these
curve counts, the partition function is their generating function. A direct
computation of the A-model partition function is infeasible, but the B-model
partition function can be computed from the period functions of $M$ (there are
however complications with increasing genus). The two models are related by a
string duality, _mirror symmetry_ , which means that once either partition
function is computed, both are known. And so the ‘easy’ B-model computation
yields the sought curve counts.

To elaborate, as covered in the textbook [52] the A and B models are
topological twists of the two-dimensional $\mathcal{N}=(2,2)$ nonlinear sigma
model with target space $M$. The R-symmetry group of a 2d $\mathcal{N}=(2,2)$
theory is $\text{U}(1)_{A}\times\text{U}(1)_{V}$, the product of axial and
vector $\text{U}(1)$ symmetries. The vector $\text{U}(1)_{V}$ symmetry is
never anomalous, while the axial $\text{U}(1)_{A}$ is anomalous unless $M$ is
Calabi-Yau. The $A$ model is obtained by twisting the theory with respect to
the $U(1)_{V}$ R-symmetry, while the B-model comes from twisting with respect
to $U(1)_{A}$. The bosonic field content of the nonlinear sigma model includes
the map $\phi$ that gives the coordinates of an embedding of the worldsheet
$\Sigma$ into the target space $M$,

$\phi:\Sigma\mapsto M~{}.$ (3.1)

The path integral of a supersymmetric quantum field theory can be computed
from the field configurations where the supersymmetry variations of the
fermions vanish. This is the principle of supersymmetric localisation.

The characterisation of such field configurations is different in the A and B
twists, and is covered in the textbook [52]. The upshot is that in the A model
the fermionic variation vanishes for field configurations where the map $\phi$
is holomorphic, i.e.

$\partial_{\bar{z}}\phi=0$ (3.2)

where $(z,\bar{z})$ are complex coordinates on the Riemann surface $\Sigma$.
For such a field configuration, $\phi$ embeds the string _holomorphically_
into the target space. Since the closed string worldsheet is a genus-$g$
Riemann surface, the partition function will serve as a generating function
for counts of holomorphic embeddings, of various degrees, of genus-$g$ Riemann
surfaces in the target space.

On the other hand, in the B-model the path integral localises onto _constant_
maps. Very roughly speaking, the space of constant maps to $X$ is $X$, and so
the B-model partition function is computed by integrals over the target space,
using Hodge theory. We will now give a computationally-oriented account of
these identities.

###  3.1 The genus 0 prepotential

We will denote $m=h^{2,1}(X)=h^{1,1}(Y)$. In terms of the Frobenius periods
$\varpi$ as detailed in (2.2), the B-model prepotential at genus 0 is

$\mathcal{F}^{(0)}(\mathbf{\varphi})=\frac{1}{2}\left(-\varpi_{0}\varpi_{3}+\sum_{i=1}^{m}\varpi_{1,i}\varpi_{2,i}\right)~{},$
(3.3)

with each of the periods a function of the complex structure variables
$\mathbf{\varphi}$.

The A-model prepotential at genus 0 admits the expansion

$F^{(0)}=\frac{1}{6}Y_{ijk}t^{i}t^{j}t^{k}+\frac{1}{(2\pi\text{i})^{3}}\sum_{\mathbf{b}\geq
0}n^{(0)}_{\mathbf{b}}\text{Li}_{3}\left(\text{e}^{2\pi\text{i}\,\mathbf{b}\cdot\mathbf{t}}\right)~{},$
(3.4)

with the $Y_{ijk}$ being the triple-intersection numbers on the manifold $Y$ .
The quantity $\mathbf{b}\cdot\mathbf{t}$ should be understood as the integral
of the Kähler form over a two-cycle which, in the basis dual to canonical
generators of the Kähler cone, has nonnegative components $\mathbf{b}$. The
summation over $\mathbf{b}\in\mathbbl{Z}^{h^{1,1}(Y)}_{\geq 0}$ gives a sum
over all possible degree vectors that a rational curve embedded in $Y$ can
have (neglecting cases with torsion classes), and $\mathbf{b}\cdot\mathbf{t}$
is the area of such a curve.

Since $\mathcal{F}^{(0)}$ is given in terms of solutions to differential
equations (which we can find as explicit power series), the above pair of
formulae provide the means of determining the genus 0 instanton numbers
(integers also nomenclated genus 0 Gopakumar-Vafa invariants)
$n^{(0)}_{\beta}$ of $Y$. It only remains to find a suitable transformation of
coordinates

$\bm{\varphi}\to\mathbf{t}$ (3.5)

and make a compatible choice of gauge.

The complex structure moduli space is itself Kähler, with Kähler potential
$K=-\log\left(\text{i}\overline{\Pi}^{T}\Sigma\Pi\right)$ and metric
$G_{ij}=\partial_{\varphi^{i}}\partial_{\varphi^{j}}K$. The necessary choice
of coordinates is the flat one, in which the Christoffel symbols for the
metric $G$ vanish. This is

$t^{i}=\frac{1}{2\pi\text{i}}\frac{\varpi_{1,i}(\bm{\varphi)}}{\varpi_{0}(\bm{\varphi})}~{}.$
(3.6)

If $\varphi_{i}$ is moved in a circle about the MUM point $0$, we get a
monodromy transformation $t^{i}\mapsto t^{i}+1$. This is a symmetry of the
theory, and leaves the free energy $F^{(0)}$ invariant when the symplectic
transformations of the $Y_{ijk}$ are taken into account (this amounts to a
change of basis for $H_{2}(Y,\mathbbl{Z})$). A convenient repackaging of the
$\mathbf{t}$ in a new coordinate $\mathbf{q}$ makes this manifest:

$q^{i}=\exp\left(2\pi\text{i}\,t^{i}\right)\equiv\exp\left(\frac{\varpi_{1,i}(\bm{\varphi)}}{\varpi_{0}(\bm{\varphi})}\right)~{}.$
(3.7)

The appropriate change of scale is to multiply (3.3) by
$\frac{1}{(2\pi\text{i})^{3}\varpi_{0}^{2}}$, and so the numbers
$n^{(0)}_{\mathbf{b}}$ are fixed by

$\displaystyle(\mathbf{2}\pi\text{i})^{3}\,F^{(0)}(\mathbf{q})$
$\displaystyle=\frac{1}{2\varpi_{0}^{2}}\left(-\varpi_{0}\varpi_{3}+\sum_{i=1}^{m}\varpi_{1,i}\varpi_{2,i}\right)\bigg{|}_{\mathbf{\varphi}=\mathbf{\varphi}(\mathbf{q})}$
(3.8)
$\displaystyle=\frac{1}{6}Y_{ijk}\log(q^{i})\log(q^{j})\log(q^{k})+\sum_{\mathbf{b}\geq
0}n^{(0)}_{\mathbf{b}}\,\text{Li}_{3}\left(\mathbf{q}^{\mathbf{b}}\right)~{}.$

By $\mathbf{q}^{\mathbf{b}}$ we mean the product
$\prod_{i}\left(q^{i}\right)^{b_{i}}$. This form of $F^{(0)}(\mathbf{q})$
holds for manifolds whose homology groups lack torsion. Manifolds with torsion
homology classes were considered in [53, 54, 55] . The problem of efficiently
performing this computation was addressed first in the package INSTANTON [56],
and more recently in the library CYtools [57, 58].

#### Performing the computation efficiently

Let us make three points on the efficient computation of the genus 0
invariants.

One is that instead of replacing $\mathbf{\varphi}$ with the $q$-series
$\varphi(\mathbf{q})$ in the first line of (3.8), the appearances of $q^{i}$
in the second line can be replaced with $\mathbf{\varphi}$-series. This means
that, if only genus 0 numbers are sought, the inverse mirror map
$\mathbf{\varphi}(\mathbf{q})$ does not need to be computed.

Secondly, if $\mathbf{q}$ is replaced with $\mathbf{q}(\mathbf{\varphi})$ in
the bottom line of (3.8), the $\log(q^{i})$ terms can be expanded and one
finds (as necessitated) that the terms proportional to $\log(\varphi^{i})$ in
the top line (coming from the logarithmic parts of the Frobenius periods) can
be subtracted from both sides, so we need only work with the power series that
appear in each Frobenius period. Said another way, we can set up the
computation so that the logarithmic terms in each $\varpi$ are removed from
the start.

Finally, we draw attention to the use of symmetric polynomials in suitable
examples. Computing the instanton numbers up to degree vectors $\mathbf{b}$
with total $n$ involves finding the $n^{th}$ order Taylor series of the
prepotential, which can be found from the $n^{th}$ order series parts of each
Frobenius period. For one-parameter manifolds the number of terms in this
series is $n$. If the number of parameters is greater than 1 ($h^{1,1}(Y)\geq
2$), then these truncated Taylor series become very large much faster as $n$
is increased, with a number of new terms at order $n$ equal to the number of
partitions of $n$ with length $h^{1,1}(Y)$. The process of multiplying the
large intermediate polynomials is computationally expensive, and prevents one
from straightforwardly computing instanton numbers to very high degrees for
multiparameter manifolds.

This latter problem, of an exponential growth in the number of terms in each
power series, can to some extent be mitigated in examples with a symmetry in
their parameters. Suppose three (or more) of the parameters enter on symmetric
footing, for example the case where $Y$ is a CICY with three identical rows.
Then $\varpi_{0}$ and $F^{(0)}$ are symmetric functions of the three
parameters.

A symmetric polynomial in $n$ variables $a_{i}$ can be uniquely expressed in
terms of the elementary symmetric polynomials

$A_{i}=\sum_{\begin{subarray}{c}L\text{ a length-$i$}\\\ \text{subset of
\\{1,...,n\\}}\end{subarray}}\;\prod_{k\in L}a_{k}~{},$ (3.9)

with a convention $A_{0}=1$.

The utility here is that the number of distinct new monomials in the
polynomials $A_{i}$ at each degree is smaller than the number of new monomials
in the $a_{i}$, because $A_{i}$ has degree $i$. The asymptotic growth is still
exponential, but with a smaller base.

Now, a number of the functions involved in the genus 0 computation will not be
symmetric in these variables. However, if the coordinates
$\varphi^{j},\,\varphi^{k}$ enter the problem symmetrically then
$\varpi_{1,k},\,\varpi_{2,k}$ can be obtained from
$\varpi_{1,j},\,\varpi_{2,j}$ by effecting a swap
$\varphi^{j}\leftrightarrow\varphi^{k}$. So, the number of computations to
perform has decreased.

Moreover, identities between the $A_{i}$ and $a_{i}$ allow for further
simplification of the less symmetric functions like $\varpi_{1,i}$. We have
that, for each $k$,

$\sum_{i=1}^{m}(-a_{k})^{i}A_{m-i}=0~{}.$ (3.10)

This means that, for the power series of a function like $\varpi_{1,k}$ where
$a_{k}$ breaks the symmetry, we can uniquely express a truncation to a finite
degree as a polynomial in the $A_{i}$ and $a_{k}$ with degree less than $m$ in
$a_{k}$. This decreases the number of terms in the expression.

Using these ideas, and the ARC supercomputing resources [59], it was possible
in [2] to compute genus 0 instanton numbers up to degree 29 for the five-
parameter Mirror Hulek-Verrill model

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}1&1\\\
1&1\\\ 1&1\\\ 1&1\\\ 1&1\end{matrix}\right]~{},$ (3.11)

which possesses an $S_{5}$ symmetry corresponding to permuting rows of this
matrix.

To illustrate our claims anew here, we present in table 3.1 genus 0 numbers
for the family111We will, as and when useful in this section but not
throughout the thesis, write CICY matrices with a superscript displaying the
Hodge numbers $(h^{1,1},h^{2,1})$ and a subscript displaying the Euler
characteristic.

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&0&0&0&0&1\\\
0&1&0&0&0&1\\\ 0&0&1&0&0&1\\\ 0&0&0&1&0&1\\\ 0&0&0&0&1&1\\\
1&1&1&1&1&0\end{matrix}\right]^{(6,36)}_{-60}~{},$ (3.12)

which similarly possesses an $S_{5}$ symmetry originating in the freedom to
permute the first five rows. These numbers were computed on a 6-CPU desktop
machine with 16GB of RAM, and so we do not reach so high a degree as was done
for the five parameter model (3.11) in [2].

More can be done, and we also present in table 3.2 the genus 0 numbers up to
degree 13 for the family

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&0&0&0&0&1&0&0&0&0\\\
0&1&0&0&0&0&1&0&0&0\\\ 0&0&1&0&0&0&0&1&0&0\\\ 0&0&0&1&0&0&0&0&1&0\\\
0&0&0&0&1&0&0&0&0&1\\\ 1&1&1&1&1&0&0&0&0&0\\\
0&0&0&0&0&1&1&1&1&1\end{matrix}\right]^{(7,27)}_{-40}~{}.$ (3.13)

Another nice example is found in the family of tetraquadrics

$\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\ 2\\\
2\end{matrix}\right]^{(4,68)}_{-128}~{}.$ (3.14)

We present the genus 0 numbers up to degree 20 for this model in table 3.3.

We conclude this run of examples with a model previously studied by Hosono and
Takagi, and for which we will have more to say in our later discussions on
higher genus invariants and modularity. This is the 2-parameter family

$\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]^{(2,52)}_{-100}~{}.$ (3.15)

Genus 0 invariants up to total degree 18 were already given in [60]. We
provide them to degree 37 in table 3.4.

One motivation for studying these families is their presence in [61], along
with the mirror Hulek-Verrill manifold. These geometries are related by
splittings, and all possess certain symmetries that allow for quotient
constructions with small Hodge numbers.

The content of the following tables is available in electronic form [1].

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ 0,0,0,0,0,1 10 0,1,2,2,2,5 540
1,1,1,1,2,3 27492 1,1,2,2,4,5 964648 0,0,0,0,1,0 12 0,1,2,2,3,2 22 1,1,1,1,2,4
8892 1,1,2,2,5,2 36 0,0,0,0,1,1 12 0,1,2,2,3,3 2796 1,1,1,1,2,5 66 1,1,2,2,5,3
-400 0,0,0,1,1,0 1 0,1,2,2,3,4 8860 1,1,1,1,3,1 -24 1,1,2,2,5,4 2560
0,0,0,1,1,1 22 0,1,2,2,3,5 2796 1,1,1,1,3,2 180 1,1,2,2,6,2 -4 0,0,0,1,1,2 1
0,1,2,2,3,6 22 1,1,1,1,3,3 9444 1,1,2,2,6,3 30 0,0,1,1,1,1 56 0,1,2,2,4,3 56
1,1,1,1,3,4 9444 1,1,2,3,3,2 2 0,0,1,1,1,2 56 0,1,2,2,4,4 1344 1,1,1,1,3,5 180
1,1,2,3,3,3 139918 0,0,1,1,2,1 1 0,1,2,2,4,5 1344 1,1,1,1,3,6 -24 1,1,2,3,3,4
2449206 0,0,1,1,2,2 22 0,1,2,2,4,6 56 1,1,1,1,4,1 3 1,1,2,3,3,5 5705460
0,0,1,1,2,3 1 0,1,2,2,5,4 1 1,1,1,1,4,2 -22 1,1,2,3,4,3 2676 0,0,1,2,2,2 12
0,1,2,2,5,5 22 1,1,1,1,4,3 59 1,1,2,3,4,4 766084 0,0,1,2,2,3 12 0,1,2,3,3,3
1344 1,1,1,1,4,4 1024 1,1,2,3,5,3 -376 0,0,2,2,2,2 10 0,1,2,3,3,4 13968
1,1,1,1,4,5 59 1,1,2,4,4,3 20 0,0,2,2,2,3 60 0,1,2,3,3,5 13968 1,1,1,1,4,6 -22
1,1,3,3,3,3 112196 0,0,2,2,2,4 10 0,1,2,3,3,6 1344 1,1,1,1,4,7 3 1,1,3,3,3,4
4878540 0,0,2,2,3,3 12 0,1,2,3,4,3 22 1,1,1,2,2,2 10792 1,1,3,3,4,3 2775
0,0,2,2,3,4 12 0,1,2,3,4,4 2796 1,1,1,2,2,3 95648 1,2,2,2,2,2 27600
0,0,2,3,3,3 1 0,1,2,3,4,5 8860 1,1,1,2,2,4 95648 1,2,2,2,2,3 1585020
0,0,2,3,3,4 22 0,1,2,3,5,4 12 1,1,1,2,2,5 10792 1,2,2,2,2,4 8537376
0,0,2,3,3,5 1 0,1,2,4,4,4 540 1,1,1,2,3,2 276 1,2,2,2,2,5 8537376 0,0,3,3,3,4
56 0,1,3,3,3,3 798 1,1,1,2,3,3 43326 1,2,2,2,2,6 1585020 0,0,3,3,3,5 56
0,1,3,3,3,4 25662 1,1,1,2,3,4 125676 1,2,2,2,3,2 1344 0,0,3,3,4,4 1
0,1,3,3,3,5 69528 1,1,1,2,3,5 43326 1,2,2,2,3,3 1097052 0,0,3,3,4,5 22
0,1,3,3,4,3 12 1,1,1,2,3,6 276 1,2,2,2,3,4 14667648 0,1,1,1,1,1 174
0,1,3,3,4,4 6708 1,1,1,2,4,2 -48 1,2,2,2,3,5 32044488 0,1,1,1,1,2 756
0,2,2,2,2,2 756 1,1,1,2,4,3 432 1,2,2,2,4,2 -552 0,1,1,1,1,3 174 0,2,2,2,2,3
25662 1,1,1,2,4,4 22320 1,2,2,2,4,3 26364 0,1,1,1,2,1 12 0,2,2,2,2,4 69516
1,1,1,2,4,5 22320 1,2,2,2,4,4 4907940 0,1,1,1,2,2 540 0,2,2,2,2,5 25662
1,1,1,2,4,6 432 1,2,2,2,5,2 198 0,1,1,1,2,3 540 0,2,2,2,2,6 756 1,1,1,2,5,2 3
1,2,2,2,5,3 -3960 0,1,1,1,2,4 12 0,2,2,2,3,2 56 1,1,1,2,5,3 -22 1,2,2,2,6,2
-48 0,1,1,1,3,1 -2 0,2,2,2,3,3 13968 1,1,1,2,5,4 59 1,2,2,3,3,2 24 0,1,1,1,3,2
10 0,2,2,2,3,4 103000 1,1,1,2,5,5 1024 1,2,2,3,3,3 878696 0,1,1,1,3,3 96
0,2,2,2,3,5 103000 1,1,1,3,3,3 22680 1,2,2,3,3,4 28577520 0,1,1,1,3,4 10
0,2,2,2,3,6 13968 1,1,1,3,3,4 194664 1,2,2,3,4,2 -2 0,1,1,1,3,5 -2 0,2,2,2,4,3
756 1,1,1,3,3,5 194664 1,2,2,3,4,3 24120 0,1,1,2,2,2 474 0,2,2,2,4,4 25662
1,1,1,3,3,6 22680 1,2,3,3,3,3 823008 0,1,1,2,2,3 1852 0,2,2,2,4,5 69516
1,1,1,3,4,3 276 2,2,2,2,2,2 52740 0,1,1,2,2,4 474 0,2,2,2,5,4 540 1,1,1,3,4,4
43326 2,2,2,2,2,3 7545520 0,1,1,2,3,2 12 0,2,2,3,3,2 1 1,1,1,3,4,5 125676
2,2,2,2,2,4 80109420 0,1,1,2,3,3 540 0,2,2,3,3,3 8860 1,1,1,3,5,3 -24
2,2,2,2,2,5 166265920 0,1,1,2,3,4 540 0,2,2,3,3,4 176251 1,1,1,3,5,4 180
2,2,2,2,3,2 2928 0,1,1,2,3,5 12 0,2,2,3,3,5 425712 1,1,1,4,4,4 10792
2,2,2,2,3,3 6157524 0,1,1,2,4,3 1 0,2,2,3,4,3 540 1,1,2,2,2,2 15902
2,2,2,2,3,4 154063572 0,1,1,2,4,4 22 0,2,2,3,4,4 54504 1,1,2,2,2,3 369646
2,2,2,2,4,2 -1338 0,1,1,2,4,5 1 0,2,2,4,4,3 10 1,1,2,2,2,4 920760 2,2,2,2,4,3
178908 0,1,1,3,3,3 174 0,2,3,3,3,3 6708 1,1,2,2,2,5 369646 2,2,2,2,5,2 660
0,1,1,3,3,4 756 0,2,3,3,3,4 342924 1,1,2,2,2,6 15902 2,2,2,3,3,2 142
0,1,1,3,3,5 174 0,2,3,3,4,3 474 1,1,2,2,3,2 596 2,2,2,3,3,3 5734779
0,1,1,3,4,4 56 0,3,3,3,3,3 6204 1,1,2,2,3,3 210748 2,2,2,3,4,2 -24 0,1,1,3,4,5
56 1,1,1,1,1,1 700 1,1,2,2,3,4 1399712 0,1,1,4,4,4 1 1,1,1,1,1,2 8900
1,1,2,2,3,5 1399712 0,1,1,4,4,5 22 1,1,1,1,1,3 8900 1,1,2,2,3,6 210748
0,1,2,2,2,2 540 1,1,1,1,1,4 700 1,1,2,2,4,2 -176 0,1,2,2,2,3 6708 1,1,1,1,2,1
66 1,1,2,2,4,3 3558 0,1,2,2,2,4 6708 1,1,1,1,2,2 8892 1,1,2,2,4,4 360222 Table
3.1: The genus 0 instanton numbers of total degree $\leqslant 15$ for the
family (3.12). The numbers not in this list are either zero, or given by those
in the table after a permutation of the first five indices. The sixth index
cannot be exchanged with the others.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ 0,0,0,0,0,0,1 10 0,1,1,1,2,2,2 264 1,1,1,1,1,2,3 4120
0,0,0,0,1,0,0 6 0,1,1,1,2,2,3 264 1,1,1,1,1,2,4 330 0,0,0,0,1,0,1 6
0,1,1,1,2,2,4 6 1,1,1,1,1,3,3 4120 0,0,0,0,1,1,1 6 0,1,1,1,2,3,3 264
1,1,1,1,1,3,4 330 0,0,0,1,1,0,1 1 0,1,1,1,2,3,4 6 1,1,1,1,1,4,4 20
0,0,0,1,1,1,1 20 0,1,1,1,3,1,3 -2 1,1,1,1,2,1,2 15 0,0,0,1,1,1,2 1
0,1,1,1,3,2,3 10 1,1,1,1,2,1,3 36 0,0,1,1,1,1,1 28 0,1,1,1,3,3,3 80
1,1,1,1,2,1,4 15 0,0,1,1,1,1,2 28 0,1,1,1,3,3,4 10 1,1,1,1,2,2,2 1692
0,0,1,1,1,2,2 28 0,1,1,2,2,2,2 72 1,1,1,1,2,2,3 5478 0,0,1,1,2,1,2 1
0,1,1,2,2,2,3 330 1,1,1,1,2,2,4 1692 0,0,1,1,2,2,2 20 0,1,1,2,2,2,4 72
1,1,1,1,2,2,5 15 0,0,1,1,2,2,3 1 0,1,1,2,2,3,3 1192 1,1,1,1,2,3,3 16464
0,0,1,2,2,2,2 6 0,1,1,2,2,3,4 330 1,1,1,1,2,3,4 5478 0,0,1,2,2,2,3 6
0,1,1,2,3,2,3 6 1,1,1,1,3,1,3 -12 0,0,1,2,2,3,3 6 0,1,1,2,3,2,4 6
1,1,1,1,3,1,4 -12 0,0,2,2,2,2,3 10 0,1,1,2,3,3,3 264 1,1,1,1,3,2,3 90
0,0,2,2,2,3,3 40 0,1,2,2,2,2,2 6 1,1,1,1,3,2,4 90 0,0,2,2,2,3,4 10
0,1,2,2,2,2,3 264 1,1,1,1,3,3,3 4644 0,0,2,2,3,3,3 6 0,1,2,2,2,2,4 264
1,1,1,1,4,1,4 3 0,1,1,1,1,1,1 24 0,1,2,2,2,3,3 3090 1,1,1,2,2,2,2 480
0,1,1,1,1,1,2 126 0,1,2,2,3,2,3 1 1,1,1,2,2,2,3 4916 0,1,1,1,1,1,3 24
0,2,2,2,2,2,3 126 1,1,1,2,2,2,4 4916 0,1,1,1,1,2,2 504 1,1,1,1,1,1,1 20
1,1,1,2,2,3,3 42908 0,1,1,1,1,2,3 126 1,1,1,1,1,1,2 330 1,1,1,2,3,2,3 54
0,1,1,1,1,3,3 24 1,1,1,1,1,1,3 330 1,1,2,2,2,2,2 80 0,1,1,1,2,1,2 6
1,1,1,1,1,1,4 20 1,1,2,2,2,2,3 3293 0,1,1,1,2,1,3 6 1,1,1,1,1,2,2 4120
1,2,2,2,2,2,2 6 Table 3.2: The genus 0 instanton numbers of total degree
$\leqslant 13$ for the family (3.13). The numbers not in this list are either
zero, or given by those in the table after permutations of the first five and
last two indices. The sixth and seventh indices cannot be exchanged with the
first five.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ 1,0,0,0 48 5,4,0,0 48 6,5,4,3 70283944963932592401152
1,1,0,0 160 5,4,1,0 3265984 6,5,4,4 5129348718881933358532960 1,1,1,0 2432
5,4,1,1 14129065088 6,5,5,0 1335543626995200 1,1,1,1 86016 5,4,2,0 1017784976
6,5,5,1 15470217373819587584 2,1,0,0 48 5,4,2,1 3874428076928 6,5,5,2
11284911987863253245440 2,1,1,0 5056 5,4,2,2 1074177329475184 6,5,5,3
2276112606495732006202112 2,1,1,1 518784 5,4,3,0 59821118208 6,5,5,4
195588673772290431107673344 2,2,0,0 128 5,4,3,1 260287936814720 6,6,0,0 128
2,2,1,0 34640 5,4,3,2 82385600163793920 6,6,1,0 57521568 2,2,1,1 7037184
5,4,3,3 7471070109015318656 6,6,1,1 845112783616 2,2,2,0 507904 5,4,4,0
1268723299376 6,6,2,0 59821080576 2,2,2,1 171165840 5,4,4,1 6894039650704512
6,6,2,1 645375311627376 2,2,2,2 6547900416 5,4,4,2 2643331064630855200 6,6,2,2
435566317842866176 3,1,1,0 2432 5,4,4,3 291051848701788699648 6,6,3,0
9911375239392 3,1,1,1 899072 5,4,4,4 13862614815470765167056 6,6,3,1
108118212077621248 3,2,0,0 48 5,5,0,0 160 6,6,3,2 74937314945510009952 3,2,1,0
61824 5,5,1,0 11746432 6,6,3,3 13961655375523769282816 3,2,1,1 28639616
5,5,1,1 73320878080 6,6,4,0 549828820220928 3,2,2,0 2089008 5,5,2,0 5243776704
6,6,4,1 6736033668350881952 3,2,2,1 1297013760 5,5,2,1 27965750549248 6,6,4,2
5160552306598346379264 3,2,2,2 80605022416 5,5,2,2 10380731579151872 6,6,4,3
1080443741007758375555472 3,3,0,0 160 5,5,3,0 432161466624 6,6,4,4
95469681182710628821921792 3,3,1,0 308352 5,5,3,1 2545321829650432 6,6,5,0
13962620228584512 3,3,1,1 241754112 5,5,3,2 1042877238127097984 6,6,5,1
201987366061445211648 3,3,2,0 17677056 5,5,3,3 120304302520611954688 6,6,5,2
178213464042928851755872 3,3,2,1 17200647552 5,5,4,0 12709250594240 6,6,5,3
42979893686208656793686016 3,3,2,2 1557963029504 5,5,4,1 90222224188616064
6,6,6,0 191827050005069824 3,3,3,0 252810752 5,5,4,2 43400958203031979648
6,6,6,1 3375664486259351631456 3,3,3,1 352652451840 5,5,4,3
5917889244182186813696 6,6,6,2 3521325683147770386726912 3,3,3,2
43431181213824 5,5,4,4 347399539693965379619648 7,2,1,1 518784 3,3,3,3
1646607181615104 5,5,5,0 176223512332928 7,2,2,0 34640 4,1,1,0 160 5,5,5,1
1575672398623395840 7,2,2,1 1297013760 4,1,1,1 518784 5,5,5,2
922075476759511908864 7,2,2,2 1108729141920 4,2,1,0 34640 5,5,5,3
151692725887218685190144 7,3,1,0 2432 4,2,1,1 44662400 5,5,5,4
10726096697730587601447552 7,3,1,1 241754112 4,2,2,0 3265280 5,5,5,5
399456970261757309912334336 7,3,2,0 17677056 4,2,2,1 4042603552 6,1,1,1 2432
7,3,2,1 249714336000 4,2,2,2 426305384448 6,2,1,0 48 7,3,2,2 165296580806656
4,3,0,0 48 6,2,1,1 7037184 7,3,3,0 3808483584 4,3,1,0 508160 6,2,2,0 507904
7,3,3,1 39529406119936 4,3,1,1 789382400 6,2,2,1 4042603552 7,3,3,2
24136954745543808 4,3,2,0 57521568 6,2,2,2 1514352078848 7,3,3,3
3672469607846903808 4,3,2,1 94371041920 6,3,1,0 61824 7,4,1,0 508160 4,3,2,2
13002686858864 6,3,1,1 789382400 7,4,1,1 14129065088 4,3,3,0 1424625024
6,3,2,0 57521568 7,4,2,0 1017784976 4,3,3,1 3031512413312 6,3,2,1 342823329792
7,4,2,1 12163074768640 4,3,3,2 527889850255360 6,3,2,2 125771915722896 7,4,2,2
8102781267426048 4,3,3,3 27526058324060160 6,3,3,0 5243776704 7,4,3,0
188051992832 4,4,0,0 128 6,3,3,1 30008993843584 7,4,3,1 1984799528343552
4,4,1,0 2089008 6,3,3,2 11575415612576768 7,4,3,2 1294128535920058368 4,4,1,1
4985888864 6,3,3,3 1203234568961692800 7,4,3,3 219681085367035373952 4,4,2,0
360824832 6,4,1,0 2089008 7,4,4,0 9911375239392 4,4,2,1 871452197440 6,4,1,1
19804670208 7,4,4,1 112338620588205056 4,4,2,2 166647022068736 6,4,2,0
1424616960 7,4,4,2 79594696896349282688 4,4,3,0 13391051328 6,4,2,1
9173232641616 7,4,4,3 15110868372965862621696 4,4,3,1 39844405855232 6,4,2,2
3846287763468288 7,4,4,4 1189045428745766422260768 4,4,3,2 9199812356321968
6,4,3,0 141793830384 7,5,1,0 11746432 4,4,3,3 627205272329861504 6,4,3,1
938779827042688 7,5,1,1 255547666432 4,4,4,0 188051928064 6,4,3,2
419700792333851408 7,5,2,0 18178255616 4,4,4,1 741058165634496 6,4,3,3
51546803704385918976 7,5,2,1 233839970568832 4,4,4,2 217057048330727424
6,4,4,0 4650991239168 7,5,2,2 176514847485662208 4,4,4,3 18647281489625577504
6,4,4,1 35938232545114592 7,5,3,0 3600925105536 4,4,4,4 699794636853614635008
6,4,4,2 18502170216819978240 7,5,3,1 43715576259518464 5,1,1,1 86016 6,4,4,3
2653048822718080612368 7,5,3,2 32805238780261668736 5,2,1,0 5056 6,4,4,4
161629518930438998396928 7,5,3,3 6480417642541472432128 5,2,1,1 28639616
6,5,0,0 48 7,5,4,0 221844802863872 5,2,2,0 2089008 6,5,1,0 17677056 7,5,4,1
2932079676057142912 5,2,2,1 5834714624 6,5,1,1 188011416064 7,5,4,2
2387966888748874810112 5,2,2,2 1108729141920 6,5,2,0 13391051328 7,5,4,3
523912487781610422477568 5,3,1,0 308352 6,5,2,1 107087122937856 7,5,4,4
47981652609161486365088768 5,3,1,1 1154723840 6,5,2,2 55530466229981712
7,5,5,0 6036801603750144 5,3,2,0 84056832 6,5,3,0 1651763971584 7,5,5,1
92621472022310780928 5,3,2,1 249714336000 6,5,3,1 13704006681994624 7,5,5,2
85897234681261670146944 5,3,2,2 54663940002816 6,5,3,2 7493044028521115648
7,5,5,3 21540883712581909578244096 5,3,3,0 3808483584 6,5,3,3
1120675249090550046720 7,6,0,0 48 5,3,3,1 12944311033856 6,5,4,0
69188204803344 7,6,1,0 84056832 5,3,3,2 3295227059205504 6,5,4,1
661446106005076224 7,6,1,1 2010040031104 5,3,3,3 240139807447941120 6,5,4,2
409473371380697240000 7,6,2,0 141793830384 Table 3.3: The genus 0 instanton
numbers of total degree $\leqslant 20$ for the family (3.14). The numbers not
in this list are either zero, or given by those in the table after permuting
indices.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ 7,6,2,1 2208595900084224 8,6,3,1 1462074329960965376
9,7,1,1 23624438407168 7,6,2,2 2021884379122016048 8,6,3,2
1771264367116721723008 9,7,2,0 1651763971584 7,6,3,0 33789217322496 8,6,3,3
542307422558027394152960 9,7,2,1 70930323016569984 7,6,3,1 503495581721955456
8,6,4,0 7456149925576704 9,7,2,2 150750588373893541888 7,6,3,2
455218569908331757568 8,6,4,1 162132923614385349360 9,7,3,0 1071736381056384
7,6,3,3 107719074363142059583360 8,6,4,2 203054957441346394726912 9,7,3,1
37743466871928029184 7,6,4,0 2566483940906640 8,6,5,0 343945562331791376
9,7,4,0 191827050036601856 7,6,4,1 41379145350412949504 8,6,5,1
8263402711793164324608 9,8,0,0 48 7,6,4,2 40123559676936131346720 8,6,6,0
8277365987776740864 9,8,1,0 1424625024 7,6,4,3 10410193126310745548984320
8,7,0,0 48 9,8,1,1 142582803775232 7,6,5,0 87046894784712960 8,7,1,0 360828928
9,8,2,0 9911375239392 7,6,5,1 1607540052866407577728 8,7,1,1 18129723322752
9,8,2,1 497106580728095936 7,6,5,2 1749693070862210564431872 8,7,2,0
1268723299376 9,8,3,0 7456149932740608 7,6,6,0 1576912001012723760 8,7,2,1
36298079877646592 9,9,0,0 160 7,6,6,1 34462954830661856456704 8,7,2,2
55987725237356145792 9,9,1,0 3808483584 7,7,0,0 160 8,7,3,0 549828822442752
9,9,1,1 487860526727168 7,7,1,0 252810752 8,7,3,1 14005384303898305024 9,9,2,0
33789217322496 7,7,1,1 8087692763136 8,7,3,2 20171041734451334944000 10,2,2,1
34640 7,7,2,0 567589415680 8,7,4,0 71290241748070896 10,2,2,2 6547900416
7,7,2,1 11584959011321216 8,7,4,1 1864208769268632504064 10,3,1,1 2432 7,7,2,2
13491384176760961024 8,7,5,0 3988612138839842816 10,3,2,0 48 7,7,3,0
176223512332928 8,8,0,0 128 10,3,2,1 1297013760 7,7,3,1 3369477175185817600
8,8,1,0 1017784976 10,3,2,2 13002686858864 7,7,3,2 3792683314958877584384
8,8,1,1 66748924807328 10,3,3,0 17677056 7,7,3,3 1099277650161884440166400
8,8,2,0 4650991239168 10,3,3,1 3031512413312 7,7,4,0 17175761255142144 8,8,2,1
170418886235419696 10,3,3,2 11575415612576768 7,7,4,1 348295401563531685632
8,8,2,2 328173467451404489600 10,3,3,3 7079306295276204288 7,7,4,2
412395106599944043177984 8,8,3,0 2566483940906640 10,4,1,1 44662400 7,7,5,0
740932627227834624 8,8,3,1 82211057906371385344 10,4,2,0 3265280 7,7,5,1
16875996286670980751360 8,8,4,0 417176124089319424 10,4,2,1 871452197440
7,7,6,0 16999688846773790208 9,2,2,1 7037184 10,4,2,2 3846287763468288 8,2,1,1
5056 9,2,2,2 80605022416 10,4,3,0 13391051328 8,2,2,0 128 9,3,1,1 899072
10,4,3,1 938779827042688 8,2,2,1 171165840 9,3,2,0 61824 10,4,3,2
2508525839463392816 8,2,2,2 426305384448 9,3,2,1 17200647552 10,4,3,3
1332539766738727936000 8,3,1,1 28639616 9,3,2,2 54663940002816 10,4,4,0
4650991239168 8,3,2,0 2089008 9,3,3,0 252810752 10,4,4,1 219342311907269104
8,3,2,1 94371041920 9,3,3,1 12944311033856 10,4,4,2 487713798985291233280
8,3,2,2 125771915722896 9,3,3,2 24136954745543808 10,5,1,0 5056 8,3,3,0
1424625024 9,3,3,3 8791828001566988288 10,5,1,1 14129065088 8,3,3,1
30008993843584 9,4,1,0 160 10,5,2,0 1017784976 8,3,3,2 30743146145765888
9,4,1,1 789382400 10,5,2,1 107087122937856 8,3,3,3 7079306295276204288 9,4,2,0
57521568 10,5,2,2 347926332689829040 8,4,1,0 34640 9,4,2,1 3874428076928
10,5,3,0 1651763971584 8,4,1,1 4985888864 9,4,2,2 8102781267426048 10,5,3,1
86320414207198848 8,4,2,0 360824832 9,4,3,0 59821118208 10,5,3,2
204999947699695388672 8,4,2,1 9173232641616 9,4,3,1 1984799528343552 10,5,4,0
439063136203008 8,4,2,2 10353839703716352 9,4,3,2 3120841582270747392 10,5,4,1
18557339208009894656 8,4,3,0 141793830384 9,4,3,3 1094190143621388343680
10,5,5,0 38952597058723520 8,4,3,1 2538964516666880 9,4,4,0 9911375239392
10,6,1,0 2089008 8,4,3,2 2508525839463392816 9,4,4,1 273504607073902144
10,6,1,1 845112783616 8,4,3,3 602927912105481193728 9,4,4,2
400065629964453039984 10,6,2,0 59821080576 8,4,4,0 12709250181888 9,4,4,3
141106319957374906048512 10,6,2,1 4533354437642480 8,4,4,1 219342311907269104
9,5,1,0 308352 10,6,2,2 13486794785676976128 8,4,4,2 219737543096535650304
9,5,1,1 73320878080 10,6,3,0 69188204803344 8,4,4,3 56255611905947864806512
9,5,2,0 5243776704 10,6,3,1 3368380766765580288 8,4,4,4
5785403639953201842686976 9,5,2,1 233839970568832 10,6,4,0 17175761244601344
8,5,1,0 3265984 9,5,2,2 435172594319284224 10,7,1,0 84056832 8,5,1,1
188011416064 9,5,3,0 3600925105536 10,7,1,1 18129723322752 8,5,2,0 13391051328
9,5,3,1 108024514779267072 10,7,2,0 1268723299376 8,5,2,1 302207140853120
9,5,3,2 167816512674800089472 10,7,2,1 88462669038163968 8,5,2,2
347926332689829040 9,5,3,3 61564588462323994361856 10,7,3,0 1335543626995200
8,5,3,0 4650991497216 9,5,4,0 549828822442752 10,8,1,0 1017784976 8,5,3,1
86320414207198848 9,5,4,1 15172240325024688384 10,8,1,1 182970871579264
8,5,3,2 91592941667768938752 9,5,4,2 22894439058514947334656 10,8,2,0
12709250181888 8,5,3,3 24378159414883707592448 9,5,5,0 31786723861681536
10,9,0,0 48 8,5,4,0 439063136203008 9,5,5,1 913426712060194627584 10,9,1,0
5243776704 8,5,4,1 8247835975441925056 9,6,1,0 17677056 10,10,0,0 128 8,5,4,2
9033581369124138593840 9,6,1,1 2010040031104 11,2,2,2 171165840 8,5,4,3
2573204027887680476759296 9,6,2,0 141793830384 11,3,2,1 28639616 8,5,5,0
17175761255142144 9,6,2,1 5744002412760576 11,3,2,2 1557963029504 8,5,5,1
356496506323192211328 9,6,2,2 10966073503132221968 11,3,3,0 308352 8,5,5,2
428090722447387987896384 9,6,3,0 87594475420800 11,3,3,1 352652451840 8,6,1,0
57521568 9,6,3,1 2738079662657568640 11,3,3,2 3295227059205504 8,6,1,1
2668638725632 9,6,3,2 4572692498129982156800 11,3,3,3 3672469607846903808
8,6,2,0 188051928064 9,6,4,0 13962620228584512 11,4,1,1 518784 8,6,2,1
4533354437642480 9,6,4,1 420332107728557286912 11,4,2,0 34640 8,6,2,2
5860852677002366976 9,6,5,0 895919580397273344 11,4,2,1 94371041920 8,6,3,0
69188204803344 9,7,1,0 252810752 11,4,2,2 1074177329475184

table 3.3 continued.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ 11,4,3,0 1424625024 11,6,2,0
13391051328 12,4,2,1 4042603552 13,3,3,1 241754112 11,4,3,1 260287936814720
11,6,2,1 2208595900084224 12,4,2,2 166647022068736 13,4,2,1 44662400 11,4,3,2
1294128535920058368 11,6,3,0 33789217322496 12,4,3,0 57521568 13,4,3,0 508160
11,4,4,0 1268723299376 11,7,1,0 11746432 12,4,3,1 39844405855232 13,5,1,1
86016 11,4,4,1 112338620588205056 11,7,1,1 8087692763136 12,4,4,0 188051928064
13,5,2,0 5056 11,5,1,1 1154723840 11,7,2,0 567589415680 12,5,1,1 28639616
11,5,2,0 84056832 11,8,1,0 360828928 12,5,2,0 2089008 11,5,2,1 27965750549248
12,2,2,2 507904 12,5,2,1 3874428076928 11,5,2,2 176514847485662208 12,3,2,1
61824 12,5,3,0 59821118208 11,5,3,0 432161466624 12,3,2,2 80605022416 12,6,1,0
48 11,5,3,1 43715576259518464 12,3,3,0 160 12,6,1,1 19804670208 11,5,4,0
221844802863872 12,3,3,1 17200647552 12,6,2,0 1424616960 11,6,1,0 61824
12,3,3,2 527889850255360 12,7,1,0 508160 11,6,1,1 188011416064 12,4,1,1 160
13,3,2,2 1297013760

table 3.3 continued.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$
$n^{(0)}_{\bm{p}}$ 1,0 50 8,8 2218998811196105750 17,5 75885200 1,1 650 9,7
782114760930236000 12,11 1169273852253720661047855850 2,1 1475 10,6
28126794522576400 13,10 261422628471008778452895000 2,2 29350 11,5
50034381769600 14,9 11425471345666372778573625 3,1 650 12,4 545403950 15,8
71553701937328489430500 3,2 148525 9,8 30429684503634827875 16,7
34680269311023701250 4,1 50 10,7 4075722566708421875 17,6 334030085380350 3,3
3270050 11,6 51642034298930775 18,5 148525 4,2 250550 12,5 24352783493100
12,12 26334146932509192721297606250 4,3 24162125 13,4 24162125 13,11
12925735995730366743674988500 5,2 148525 9,9 688579463588598857500 14,10
1443695931340763707228964750 4,4 545403950 10,8 270605922599775866950 15,9
30455096636986392995454400 5,3 75885200 11,7 14322131205924119500 16,8
83451873384004446556500 6,2 29350 12,6 63157566038079800 17,7
14322131205924119500 5,4 5048036025 13,5 7175800860250 18,6 27995704239850 6,3
110273275 14,4 250550 13,12 418749190393922926454264339775 7,2 1475 10,9
9910287533252141060075 14,11 105192221331381719953056030200 5,5 114678709000
11,8 1632070561204989561850 15,10 5983442985255839039237107500 6,4 22945154050
12,7 34680269311023701250 16,9 60960783194314781836252175 7,3 75885200 13,6
51642034298930775 17,8 71553701937328489430500 6,5 1231494256550 14,5
1231494256550 18,7 4075722566708421875 7,4 55531376500 15,4 50 19,6
1231494256550 8,3 24162125 10,10 223872593965525056524000 13,13
9418685010993246523213147309050 6,6 27995704239850 11,9
96088214450066089180650 14,12 4871005171529900672353774747800 7,5
7175800860250 12,8 6879784715166845894000 15,11 644225116339930387252675093800
8,4 74278763500 13,7 58660895139129344250 16,10 18858521597882017598430587450
9,3 3270050 14,6 28126794522576400 17,9 92220396289894438953276250 7,6
334030085380350 15,5 114678709000 18,8 45029496161343522802000 8,5
24352783493100 11,10 3351508924925685769008400 19,7 782114760930236000 9,4
55531376500 12,9 652460665889917943662000 20,6 22945154050 10,3 148525 13,8
20691735554891324819375 14,13 153346515556322207530993642061375 7,7
7584889119913750 14,7 69837157468295256300 15,12
42551462569251226858792953023050 8,6 2329042266808650 15,6 10084936612321850
16,11 3018482957837620078514600231550 9,5 50034381769600 16,5 5048036025 17,10
45641749489863534673387285075 10,4 22945154050 11,11
75590773394298108275641400 18,9 105822944845470819411145650 11,3 650 12,10
34897260129076709170702250 19,8 20691735554891324819375 8,7 97887416945961075
13,9 3187856349624109686861750 20,7 97887416945961075 9,6 10084936612321850
14,8 45029496161343522802000 21,6 110273275 10,5 63477362571125 15,7
58660895139129344250 14,14 3445063666410127667138401567917450 11,4 5048036025
16,6 2329042266808650 15,13 1863829015771384379547064210671650 Table 3.4: The
genus 0 instanton numbers of total degree $\leqslant 37$ for the family
(3.15). The numbers not in this list are either zero, or given by those in the
table after permuting indices.

$\bm{p}$ $n^{(0)}_{\bm{p}}$ $\bm{p}$ $n^{(0)}_{\bm{p}}$ 16,12
284748690462403703849327898236400 22,11 194151779259881472757738870486275
17,11 10956684127545224855242712323800 23,10 45641749489863534673387285075
18,10 85415237718946312165876907100 24,9 652460665889917943662000 19,9
92220396289894438953276250 25,8 97887416945961075 20,8 6879784715166845894000
26,7 1475 21,7 7584889119913750 17,17
187534310501258236886420065561355215672850 22,6 29350 18,16
112636401191272489352275415887745870490500 15,14
57245804146829441141855075686421600 19,15
23921213512060867824603015906296302482550 16,13
17311291067627079301898110530922200 20,14
1687381561533043508236480530888841957000 17,12
1482101901841193380170774085390975 21,13
35280439004752280205968167087919443350 18,11 31105645711336079302777239697100
22,12 182005705950504851032041523111610550 19,10
124182064589288217124451139225 23,11 173303334724056406174320632632850 20,9
60960783194314781836252175 24,10 18858521597882017598430587450 21,8
1632070561204989561850 25,9 96088214450066089180650 22,7 334030085380350 26,8
2329042266808650 15,15 1284701496853760180092631899781959250 18,17
3267643622168623942896097738545068717322400 16,14
722903234302220208838481980961898100 19,16
1206865467127098649476297800122829258555125 17,13
125101067315642934987038992528141500 20,15
158613328551370248346752880868799119011250 18,12
6071452882998806212881647096382400 21,14
6853883162475990624130911021611665107450 19,11
69561055478912921075310779919000 22,13 85346765578856941178005038357338030125
20,10 140638929443068626672454410250 23,12
249140684221048318682166766371696025 21,9 30455096636986392995454400 24,11
123171917767680954127111199469800 22,8 270605922599775866950 25,10
5983442985255839039237107500 23,7 7175800860250 26,9 9910287533252141060075
16,15 21731435735419411130551512688193970350 27,8 24352783493100 17,14
7084069788870929694102349973792379125 18,18
73136802934944577315316758502397717744613800 18,13
713360358445986060308899284173421250 19,17
45145186660990785137737197513727989291648750 19,12
19755724290719957639765719857823000 20,16
10440390902521863107237190843251830811038400 20,11
123171917767680954127111199469800 21,15
858330748648887482867022141394382548080200 21,10
124182064589288217124451139225 22,14 22822795902093633748803493742442397239000
22,9 11425471345666372778573625 23,13 168970246230273909199356496016178585000
23,8 30429684503634827875 24,12 276580864121735152873483395276585800 24,7
55531376500 25,11 69561055478912921075310779919000 16,16
487219827311876979652523699533587321150 26,10 1443695931340763707228964750
17,15 283802089997587090105839672112883654150 27,9 688579463588598857500 18,14
54760100433813765734980222782145239000 28,8 74278763500 19,13
3245572721974319554949303862415944250 19,18
1290315264071203979827100216377332535448538000 20,12
51423738528601615934359979687767300 20,17
502246809291235370794554217924188280591690325 21,11
173303334724056406174320632632850 21,16
73738372223409643492419300864401743615326225 22,10
85415237718946312165876907100 22,15
3821053219977680771955490543838867117725025 23,9 3187856349624109686861750
23,14 62641363244422442148059754716581069033750 24,8 2218998811196105750 24,13
274635658050542709324085728636859818125 25,7 75885200 25,12
249140684221048318682166766371696025 17,16
8371985898150331724830984679777619591325 26,11
31105645711336079302777239697100 18,15
2915743577162092867565871545455088037975 27,10 261422628471008778452895000
19,14 338273641516001759928541421158949393625 28,9 30429684503634827875 20,13
11885728626129063303448250398108564900 29,8 24162125 21,12
107659493684116349217356173601610375

table 3.4 continued.

###  3.2 Genus 1 mirror symmetry: Counting elliptic curves

The B-model prepotential was determined in [16] and is given in the
topological limit ${\bar{t}\rightarrow-\text{i}\infty}$ by

$\mathcal{F}^{(1)}=\log\left[\varpi_{0}^{-\left(3+h^{2,1}(X)+\chi(X)/12\right)/2}\text{det}\left(\frac{\partial\bm{\varphi}}{\partial\mathbf{t}}\right)^{1/2}f(\bm{\varphi})\right]~{},$
(3.16)

with the function $f$ giving the correct behaviour at singularities of the
moduli space, to be fixed after imposing a boundary condition that imposes
consistency with the A-model expansion. The large complex structure expansion
of the A-model genus 1 prepotential is

$F^{(1)}=\frac{1}{2}Y_{00i}t^{i}+\sum_{\mathbf{b}\geq
0}\left(n^{(1)}_{\mathbf{b}}+\frac{n^{(0)}_{\mathbf{b}}}{12}\right)\,\text{Li}_{1}\left(\mathbf{q}^{\mathbf{b}}\right)~{}.$
(3.17)

The function $f$ should have a factor that gives zeroes at the roots of the
discriminant locus $\Delta$, the set of moduli $\bm{\varphi}$ for which the
manifold $X$ becomes singular. Additionally, $f$ should vanish at the large
complex structure222so named because this point is taken by the mirror map to
the large volume point $t=\text{i}\infty$. (LCS) point $\bm{\varphi}=\bm{0}$,
with the order of vanishing such that the leading behaviour of $F^{(1)}$ is
recovered. These considerations fix $f$ to be of the form

$f=\frac{1}{\Delta^{c}\prod_{i=1}^{h^{2,1}(X)}\left(\varphi^{i}\right)^{(1-Y_{00i})/2}}~{}.$
(3.18)

We remark that the determinant in (3.16) goes like
$\left(\varphi^{i}\right)^{1/2}$ as $\varphi^{i}\to 0$, hence the exponents
$\frac{1-Y_{00i}}{2}$ above. The exponent $c$ is specified by the kind of
singularities encoded by $\Delta$. For a conifold singularity, $c$ equals
$1/12$. More complicated behaviours are possible at orbifold singularities,
see for instance the computation of genus 1 numbers on quotient manifolds in
[14], informed by [62]. For our purposes, $\Delta$ is always a conifold locus
and we work with $c=1/12$. For clarity, we remark that while $f$ encapsulates
a number of singularities, there is only one $\mathcal{F}^{(1)}$, and this
behaves appropriately at each different singularity through $f$’s appearance
in (3.16).

The polynomial $\Delta$ is in the variables $\varphi^{i}$, and we explain this
choice of complex structure coordinates in Appendix §A. $\Delta$ encodes
singularities not of the CICYs that we display, but of their mirrors $X$.
Appendix §A discusses the construction of those mirror families $X$, and their
parametrisation in terms of $\varphi^{i}$.

We shall display some genus 1 numbers for two families, the Tetraquadric
(3.14) and the maximally split Quintic (3.15), whose mirrors respectively have
discriminants

$\displaystyle\begin{matrix}\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\\\
\mathbbl{P}^{1}\\\ \mathbbl{P}^{1}\end{matrix}\\!\left[\begin{matrix}2\\\ 2\\\
2\\\ 2\end{matrix}\right]^{(4,68)}_{-128}~{},$
$\displaystyle\qquad\Delta=\prod_{\epsilon_{i}\in\\{1,-1\\}}\left(1+2\epsilon_{1}\sqrt{\varphi^{1}}+2\epsilon_{2}\sqrt{\varphi^{2}}+2\epsilon_{3}\sqrt{\varphi^{3}}+2\epsilon_{4}\sqrt{\varphi^{4}}\right)~{};$
(3.19) $\displaystyle\begin{matrix}\mathbbl{P}^{4}\\\
\mathbbl{P}^{4}\end{matrix}\\!\left[\begin{matrix}1&1&1&1&1\\\
1&1&1&1&1\end{matrix}\right]^{(2,52)}_{-100}~{},$
$\displaystyle\qquad\Delta=\prod_{1\leq n_{1},n_{2}\leq
5}\left(1-\text{e}^{\frac{2\pi\text{i}}{5}n_{1}}\left(\varphi^{1}\right)^{1/5}-\text{e}^{\frac{2\pi\text{i}}{5}n_{2}}\left(\varphi^{2}\right)^{1/5}\right)~{}.$

The $q$-series that we have used for obtaining genus-1 curve counts
$n^{(1)}_{\mathbf{b}}$, formula (3.17), is that given by the Gopakumar-Vafa
prescription [18, 19]. In earlier work by Bershadsky, Cecotti, Ooguri and Vafa
[16], the prescription differed slightly (counting what they called “primitive
elliptic curves”) and so some care is needed when comparing curve counts
across the literature. In either case the B-model prepotential is the same,
and the two sets of data are equivalent.

The content of the following tables is available in electronic form [1].

$\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$
$n^{(1)}_{\bm{p}}$ 2,2,0,0 4 5,5,1,0 -631296 6,6,4,0 -267545068581728 2,2,1,0
-96 5,5,1,1 -193064960 6,6,4,1 98216197349510400 2,2,1,1 1984 5,5,2,0
-831856256 6,6,4,2 1487879821243945966432 2,2,2,0 -10272 5,5,2,1 156514373632
6,6,4,3 546260879195635250051616 2,2,2,1 269280 5,5,2,2 951193768367744
6,6,4,4 66049985769182760347092224 2,2,2,2 93388992 5,5,3,0 -114273286144
6,6,5,0 -8478485684300928 3,2,1,0 -320 5,5,3,1 23771145744384 6,6,5,1
3465741933227797120 3,2,1,1 10496 5,5,3,2 165492284557610240 6,6,5,2
63508834166133397153792 3,2,2,0 -69568 5,5,3,3 33247249326979840000 6,6,5,3
26872551390971959525026816 3,2,2,1 2765056 5,5,4,0 -4598298755008 6,6,6,0
-136960390214771328 3,2,2,2 1812768672 5,5,4,1 1075385950542336 6,6,6,1
64863757395907531392 3,3,1,0 -4864 5,5,4,2 9348192801960947328 6,6,6,2
1466437377855662330826240 3,3,1,1 112640 5,5,4,3 2224378492069768184832
7,2,2,0 -96 3,3,2,0 -1046656 5,5,4,4 177840753331297192191808 7,2,2,1 2765056
3,3,2,1 54265088 5,5,5,0 -78791963521792 7,2,2,2 35189119872 3,3,2,2
57695471872 5,5,5,1 21862949555982336 7,3,1,1 112640 3,3,3,0 -25362432 5,5,5,2
243574478608897658880 7,3,2,0 -1046656 3,3,3,1 1683640320 5,5,5,3
70047199737530626793472 7,3,2,1 1021728256 3,3,3,2 2698701579264 5,5,5,4
6756842515156306030742016 7,3,2,2 10223759052288 3,3,3,3 173800712052736
5,5,5,5 310274010609799385697632256 7,3,3,0 -578600960 4,2,1,0 -96 6,2,1,1
1984 7,3,3,1 276199923712 4,2,1,1 20096 6,2,2,0 -10272 7,3,3,2
2739373720985856 4,2,2,0 -124616 6,2,2,1 9983872 7,3,3,3 748137868720717824
4,2,2,1 9983872 6,2,2,2 49952569216 7,4,1,0 -10112 4,2,2,2 12011885472 6,3,1,0
-320 7,4,1,1 -14325760 4,3,1,0 -10112 6,3,1,1 241408 7,4,2,0 -127854976
4,3,1,1 241408 6,3,2,0 -4386240 7,4,2,1 65476152960 4,3,2,0 -4386240 6,3,2,1
1445166080 7,4,2,2 723323393538432 4,3,2,1 353758784 6,3,2,2 7576252775296
7,4,3,0 -45623185536 4,3,2,2 621519800480 6,3,3,0 -831856256 7,4,3,1
18148629097216 4,3,3,0 -188206400 6,3,3,1 205607705344 7,4,3,2
206208348590980608 4,3,3,1 17552569088 6,3,3,2 1238547915076352 7,4,3,3
62079704012640705280 4,3,3,2 43026117967872 6,3,3,3 226786087463462912 7,4,4,0
-3510327932032 4,3,3,3 3851778847706880 6,4,1,0 -69568 7,4,4,1
1335576781170688 4,4,0,0 4 6,4,1,1 -25295744 7,4,4,2 17562256414556246528
4,4,1,0 -69568 6,4,2,0 -188206960 7,4,4,3 5884952965650513045248 4,4,1,1
-1580992 6,4,2,1 48618551008 7,4,4,4 636300785891465292898752 4,4,2,0
-38445312 6,4,2,2 323138860266848 7,5,1,0 -631296 4,4,2,1 3987616000 6,4,3,0
-33378648000 7,5,1,1 -1115140096 4,4,2,2 10558546637760 6,4,3,1 8232411529408
7,5,2,0 -3380586368 4,4,3,0 -2397896960 6,4,3,2 61783631376376704 7,5,2,1
1393481688832 4,4,3,1 286009826304 6,4,3,3 13300730517821622784 7,5,2,2
20022577101423360 4,4,3,2 999142624314816 6,4,4,0 -1541331810048 7,5,3,0
-1166173618944 4,4,3,3 117575052436736704 6,4,4,1 404052229218240 7,5,3,1
469840676702208 4,4,4,0 -45623190048 6,4,4,2 3723977589477458176 7,5,3,2
6554880932704430336 4,4,4,1 6666610324992 6,4,4,3 935179511020340807456
7,5,3,3 2281737394795361083392 4,4,4,2 31633436508163392 6,4,4,4
77803599946908880718016 7,5,4,0 -100934478032512 4,4,4,3 4709891230826927232
6,5,1,0 -1046656 7,5,4,1 41094723073773568 4,4,4,4 239082475099257093312
6,5,1,1 -734128384 7,5,4,2 656136930137156080768 5,2,1,1 10496 6,5,2,0
-2397896960 7,5,4,3 253382111085115792710400 5,2,2,0 -69568 6,5,2,1
626843119104 7,5,4,4 31839291301539012129070080 5,2,2,1 15218816 6,5,2,2
5792732588373728 7,5,5,0 -3469762473325056 5,2,2,2 35189119872 6,5,3,0
-497787993344 7,5,5,1 1531922193657643008 5,3,1,0 -4864 6,5,3,1
139578486511360 7,5,5,2 29300021039056213398016 5,3,1,1 286720 6,5,3,2
1362193196667713536 7,5,5,3 12921015476002583834605568 5,3,2,0 -6904192
6,5,3,3 356453634018444596992 7,6,1,0 -6904192 5,3,2,1 1021728256 6,5,4,0
-28770544038624 7,6,1,1 -16517906944 5,3,2,2 3034052551424 6,5,4,1
8683903357985664 7,6,2,0 -33378648000 5,3,3,0 -578600960 6,5,4,2
101579932691307744640 7,6,2,1 13324886863872 5,3,3,1 83573229568 6,5,4,3
30534454347465167308416 7,6,2,2 271114676514738784 5,3,3,2 317438354195968
6,5,4,4 3044826118116035108486016 7,6,3,0 -13264263786752 5,3,3,3
40174633448570880 6,5,5,0 -692429114488064 7,6,3,1 5997956867302144 5,4,1,0
-124608 6,5,5,1 237977798080726528 7,6,3,2 106978781273942999040 5,4,1,1
-14325760 6,5,5,2 3452295445836449045632 7,6,3,3 44494307569970341498112
5,4,2,0 -127854976 6,5,5,3 1220175570920202576920832 7,6,4,0 -1392349606298016
5,4,2,1 19566085376 6,5,5,4 143356009888676753465771648 7,6,4,1
649011301399128064 5,4,2,2 80961860336992 6,6,0,0 4 7,6,4,2
12931687111454709126208 5,4,3,0 -12801118336 6,6,1,0 -4386240 7,6,4,3
5900151322919540445706752 5,4,3,1 2116485979904 6,6,1,1 -5444761792 7,6,5,0
-59273706883766400 5,4,3,2 10738139481866240 6,6,2,0 -12801129824 7,6,5,1
29832962488756485888 5,4,3,3 1691948070016716544 6,6,2,1 3899397189472 7,6,5,2
699633783726226960540672 5,4,4,0 -372907450720 6,6,2,2 52718694526710016
7,6,6,0 -1270394426831547360 5,4,4,1 71025289697408 6,6,3,0 -3510327932032
7,6,6,1 719306357328985736192 5,4,4,2 466267539878814528 6,6,3,1
1211618774875136 7,7,1,0 -25362432 5,4,4,3 89381762755780427776 6,6,3,2
15814322874224859712 7,7,1,1 -93294921728 5,4,4,4 5783834578826599864896
6,6,3,3 5160195188934834782144 7,7,2,0 -154245226112 Table 3.5: The genus 1
instanton numbers of total degree $\leqslant 20$ for the family (3.14). The
numbers not in this list are either zero, or given by those in the table after
permuting indices.

$\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$
$n^{(1)}_{\bm{p}}$ 7,7,2,1 67556863891200 8,7,1,1 -249012852224 9,7,2,2
26201449192835298304 7,7,2,2 2038282247914091008 8,7,2,0 -372907450720 9,7,3,0
-547093483682304 7,7,3,0 -78791963521792 8,7,2,1 201277836622720 9,7,3,1
513891279029764096 7,7,3,1 42886489670017024 8,7,2,2 9200312079392193664
9,7,4,0 -136960390218830336 7,7,3,2 1004665124573369920512 8,7,3,0
-267545068649344 9,8,1,0 -188206400 7,7,3,3 512053569223265100480512 8,7,3,1
185891068420360448 9,8,1,1 -2888853456640 7,7,4,0 -10569489713182080 8,7,3,2
5838261073702086594688 9,8,2,0 -3510327932032 7,7,4,1 5915250606113727488
8,7,4,0 -47957368939626656 9,8,2,1 2188585915040384 7,7,4,2
149886488946568628294656 8,7,4,1 33506015096031958400 9,8,3,0
-4345925673853440 7,7,5,0 -572125762297261056 8,7,5,0 -3380856350018782208
9,9,1,0 -578600960 7,7,5,1 340591525223538673664 8,8,0,0 4 9,9,1,1
-12088456933376 7,7,6,0 -15558927658117177088 8,8,1,0 -127854976 9,9,2,0
-13264263786752 8,2,2,0 4 8,8,1,1 -1181782901504 10,2,2,1 -96 8,2,2,1 269280
8,8,2,0 -1541331810048 10,2,2,2 93388992 8,2,2,2 12011885472 8,8,2,1
844741431788416 10,3,2,1 2765056 8,3,1,1 10496 8,8,2,2 59548327861798208892
10,3,2,2 621519800480 8,3,2,0 -69568 8,8,3,0 -1392349606298016 10,3,3,0
-1046656 8,3,2,1 353758784 8,8,3,1 1140095263765735424 10,3,3,1 17552569088
8,3,2,2 7576252775296 8,8,4,0 -311703356855240960 10,3,3,2 1238547915076352
8,3,3,0 -188206400 9,2,2,1 1984 10,3,3,3 1507518654558092544 8,3,3,1
205607705344 9,2,2,2 1812768672 10,4,1,1 20096 8,3,3,2 3556448533895424
9,3,2,0 -320 10,4,2,0 -124616 8,3,3,3 1507518654558092544 9,3,2,1 54265088
10,4,2,1 3987616000 8,4,1,0 -96 9,3,2,2 3034052551424 10,4,2,2 323138860266848
8,4,1,1 -1580992 9,3,3,0 -25362432 10,4,3,0 -2397896960 8,4,2,0 -38445312
9,3,3,1 83573229568 10,4,3,1 8232411529408 8,4,2,1 48618551008 9,3,3,2
2739373720985856 10,4,3,2 418143162721593312 8,4,2,2 942605235833872 9,3,3,3
1899437094349107200 10,4,3,3 419022629321006302208 8,4,3,0 -33378648000
9,4,1,1 241408 10,4,4,0 -1541331810048 8,4,3,1 23534563926272 9,4,2,0 -4386240
10,4,4,1 2691518925004800 8,4,3,2 418143162721593312 9,4,2,1 19566085376
10,4,4,2 119867722557338716800 8,4,3,3 181018160115934668416 9,4,2,2
723323393538432 10,5,1,1 -14325760 8,4,4,0 -4598298768072 9,4,3,0 -12801118336
10,5,2,0 -127854976 8,4,4,1 2691518925004800 9,4,3,1 18148629097216 10,5,2,1
626843119104 8,4,4,2 51542075860983487936 9,4,3,2 527901584611872640 10,5,2,2
41383783201391104 8,4,4,3 23528568383214020784160 9,4,3,3
340169678300685140992 10,5,3,0 -497787993344 8,4,4,4 3349905544323212030802336
9,4,4,0 -3510327932032 10,5,3,1 955800484656896 8,5,1,0 -124608 9,4,4,1
3390956750776448 10,5,3,2 45752405840566558208 8,5,1,1 -734128384 9,4,4,2
97201242671174343424 10,5,4,0 -210165664436480 8,5,2,0 -2397896960 9,4,4,3
61931896048581461894144 10,5,4,1 280378657159159808 8,5,2,1 1809944501504
9,5,1,0 -4864 10,5,5,0 -25243927618418816 8,5,2,2 41383783201391104 9,5,1,1
-193064960 10,6,1,0 -69568 8,5,3,0 -1541331760640 9,5,2,0 -831856256 10,6,1,1
-5444761792 8,5,3,1 955800484656896 9,5,2,1 1393481688832 10,6,2,0
-12801129824 8,5,3,2 19486899779598178176 9,5,2,2 52565098296996352 10,6,2,1
27081415941696 8,5,3,3 9230145138666014389504 9,5,3,0 -1166173618944 10,6,2,2
2036173541408295680 8,5,4,0 -210165664436480 9,5,3,1 1207709952303104 10,6,3,0
-28770544038624 8,5,4,1 120646194440285120 9,5,3,2 37014997608595211776
10,6,3,1 42837663308834816 8,5,4,2 2670176886160388673088 9,5,3,3
24481664831474093735936 10,6,4,0 -10569489712041568 8,5,4,3
1347917974198473929771648 9,5,4,0 -267545068649344 10,7,1,0 -6904192 8,5,5,0
-10569489713182080 9,5,4,1 227415315078834432 10,7,1,1 -249012852224 8,5,5,1
6203840748464569344 9,5,4,2 7109119409066563817856 10,7,2,0 -372907450720
8,5,5,2 158349506217985346295936 9,5,5,0 -20339595410138112 10,7,2,1
463223659993600 8,6,1,0 -4386240 9,5,5,1 16444600976022953984 10,7,3,0
-692429114488064 8,6,1,1 -23596592896 9,6,1,0 -1046656 10,8,1,0 -127854976
8,6,2,0 -45623190048 9,6,1,1 -16517906944 10,8,1,1 -3867646677632 8,6,2,1
27081415941696 9,6,2,0 -33378648000 10,8,2,0 -4598298768072 8,6,2,2
840745818190013632 9,6,2,1 34157599278208 10,9,1,0 -831856256 8,6,3,0
-28770544038624 9,6,2,2 1634935031609724864 10,10,0,0 4 8,6,3,1
18089515225187456 9,6,3,0 -37107163518144 11,2,2,2 269280 8,6,3,2
449402422813104411520 9,6,3,1 34589813440859136 11,3,2,1 10496 8,6,3,3
243302732065835955711744 9,6,3,2 1221567939348425063424 11,3,2,2 57695471872
8,6,4,0 -4345925673694464 9,6,4,0 -8478485684300928 11,3,3,0 -4864 8,6,4,1
2674825644762726656 9,6,4,1 7170055316812400128 11,3,3,1 1683640320 8,6,4,2
71095410940695469670768 9,6,5,0 -699282768851250048 11,3,3,2 317438354195968
8,6,5,0 -254128031458899136 9,7,1,0 -25362432 11,3,3,3 748137868720717824
8,6,5,1 162347641909980632448 9,7,1,1 -342488981504 11,4,2,0 -96 8,6,6,0
-7295475375082553200 9,7,2,0 -497787993344 8,7,1,0 -38443520 9,7,2,1
377316002640896

table 3.5 continued.

$\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$
$n^{(1)}_{\bm{p}}$ 11,4,2,1 353758784 11,6,2,0 -2397896960 12,4,4,0
-45623190048 11,4,2,2 80961860336992 11,6,2,1 13324886863872 12,5,1,1 10496
11,4,3,0 -188206400 11,6,3,0 -13264263786752 12,5,2,0 -69568 11,4,3,1
2116485979904 11,7,1,0 -631296 12,5,2,1 19566085376 11,4,3,2
206208348590980608 11,7,1,1 -93294921728 12,5,3,0 -12801118336 11,4,4,0
-372907450720 11,7,2,0 -154245226112 12,6,1,1 -25295744 11,4,4,1
1335576781170688 11,8,1,0 -38443520 12,6,2,0 -188206960 11,5,1,1 286720
12,2,2,2 -10272 12,7,1,0 -10112 11,5,2,0 -6904192 12,3,2,1 -320 13,3,2,2
2765056 11,5,2,1 156514373632 12,3,2,2 1812768672 13,3,3,1 112640 11,5,2,2
20022577101423360 12,3,3,1 54265088 13,4,2,1 20096 11,5,3,0 -114273286144
12,3,3,2 43026117967872 13,4,3,0 -10112 11,5,3,1 469840676702208 12,4,2,1
9983872 11,5,4,0 -100934478032512 12,4,2,2 10558546637760 11,6,1,0 -320
12,4,3,0 -4386240 11,6,1,1 -734128384 12,4,3,1 286009826304

table 3.5 continued.

$\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$ $n^{(1)}_{\bm{p}}$ 3,3 1475 10,5
3573290410020 4,3 29350 11,4 46911250 4,4 2669500 8,8 428216622053327300 5,3
148525 9,7 135767820281303350 5,4 46911250 10,6 3417167213249325 6,3 250550
11,5 2731112702750 5,5 2311178040 12,4 2669500 6,4 303610050 9,8
7360276988409757150 7,3 148525 10,7 817523002761866550 6,5 38756326500 11,6
6658383337394000 7,4 882636150 12,5 1207298050100 8,3 29350 13,4 29350 6,6
1477879258975 9,9 213796802016132371125 7,5 298784327925 10,8
77723709111160034550 8,4 1249719025 11,7 3186381984770132650 9,3 1475 12,6
8301844531611000 7,6 24724246516200 13,5 298784327925 8,5 1207298050100 10,9
3733143718641168532250 9,4 882636150 11,8 534623718661493240750 7,7
824125289385950 12,7 8273823575633968400 8,6 217335663077200 13,6
6658383337394000 9,5 2731112702750 14,5 38756326500 10,4 303610050 10,10
104214421442680518762070 8,7 13948250904141600 11,9 42137416928528774899500
9,6 1103600201154950 12,8 2489528873573792774625 Table 3.6: The genus 1
instanton numbers of total degree $\leqslant 37$ for the family (3.15). The
numbers not in this list are either zero, or given by those in the table after
permuting indices.

$\bm{p}$ $n^{(1)}_{\bm{p}}$ $\bm{p}$ $n^{(1)}_{\bm{p}}$ 13,7
14571606313456936800 18,10 80361250809900507464623047700 14,6 3417167213249325
19,9 60438328652210496631088500 15,5 2311178040 20,8 2489528873573792774625
11,10 1846950755735426212125500 21,7 824125289385950 12,9
322658009930052499145200 15,14 98295022688621627634296681347227000 13,8
8054854261423242104500 16,13 28288279315284624001518199057961000 14,7
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217335663077200 17,13 220457095613704667909688795083550075 17,5 148525 18,12
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77723709111160034550 16,7 8273823575633968400 23,7 298784327925 17,6
24724246516200 16,15 46546285854998069136664551148160029750 12,12
23797576472047430629503926275 17,14 14543614745639013135983680186883590850
13,11 11242357248165502750651190625 18,13
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32177823874685392221678086500438300 15,9 18802673258937852338243175 20,11
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118872243137920021974764986250 17,7 3186381984770132650 22,9
6680798279094720126093800 18,6 1477879258975 23,8 7360276988409757150 13,12
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101324272018566859752134278000 16,16 1163420080671892184401603599755019719900
15,10 4956305470261096852879492520 17,15
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80361250809900507464623047700 14,12 5634805973327046211368971793400 23,9
1732761485763012342489325 15,11 675189575619716190356450822375 24,8