Symmetries of Calabi-Yau Prepotentials with Isomorphic Flops
Andre<EMAIL_ADDRESS>Fabian
<EMAIL_ADDRESS>

a Rudolf Peierls Centre for Theoretical Physics, University of Oxford
Parks Road, Oxford OX1 3PU, UK
b Department of Physics & Department of Mathematics, Northeastern University
360 Huntington Avenue, Boston, MA 02115, United States
c The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions
Boston, MA, United States

###### Abstract

Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have
an extended Kähler cone made up from an infinite number of individual Kähler
cones. These cones are related by reflection symmetries across flop walls. We
study the implications of this cone structure for mirror symmetry, by
considering the instanton part of the prepotential in Calabi-Yau threefolds.
We show that such isomorphic flops across facets of the Kähler cone boundary
give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori
cone, non-flopping curve classes that are identified under these groups have
the same Gopakumar-Vafa invariants. This leads to instanton prepotentials
invariant under Coxeter groups, which we make manifest by introducing
appropriate invariant functions. For some cases, these functions can be
expressed in terms of theta functions whose appearance can be linked to an
elliptic fibration structure of the Calabi-Yau manifold.

###### Contents

  1. 1 Introduction
  2. 2 A simple warm-up example
    1. 2.1 Single-flop cases with Picard rank two
    2. 2.2 Example for a single flop
  3. 3 Infinitely many flops for Picard rank two
    1. 3.1 Group structure
    2. 3.2 Cones
    3. 3.3 The prepotential
    4. 3.4 Example with infinitely many flops
  4. 4 Manifolds with $\boldsymbol{h^{1,1}(X)>2}$ and two symmetry generators
    1. 4.1 Group structure
    2. 4.2 The prepotential
    3. 4.3 Examples
  5. 5 Coxeter groups, triangle groups, and reflection symmetries
  6. 6 Conclusions
  7. A Appendix: Special symmetry groups and elliptic fibrations
    1. A.1 Type 1
    2. A.2 Type 2

##  1 Introduction

Topology change is an interesting and characteristic feature of string theory
which has been studied for some time [1, 2, 3], mostly in the context of
string compactifications on Calabi-Yau (CY) threefolds. There are two main
types of topology changing transitions for CY threefolds: a milder form known
as flop transitions and a more severe form, the conifold transitions. In the
paper, we will be interested in the former.

Flop transitions of CY threefolds $X$ are known to leave the Hodge numbers
$h:=h^{1,1}(X)$ and $h^{2,1}(X)$ unchanged but they can change more refined
topological invariants, such as the intersection form and the second Chern
class of the tangent bundle. Recently, it has been noted [4, 5] that topology-
preserving flop transition $X\rightarrow X^{\prime}$ between two isomorphic CY
threefolds $X$ and $X^{\prime}$ are by no means rare. Such isomorphic flops,
as we will call them, and their implications are the main topic of this paper.

Suppose an isomorphic flop arises at a boundary facet of the Kähler cone
$\mathcal{K}$ of $X$. Then, there is an involution, which, relative to a
suitable basis $(D_{i})$ of divisor classes, can be described by an $h\times
h$ matrix $\tilde{\mathcal{M}}$ (satisfying
$\tilde{\mathcal{M}}^{2}=\mathbbm{1}_{h\times h}$). If two divisors
$D=k^{i}D_{i}$ and $D^{\prime}={k^{\prime}}^{i}D_{i}$ are related by this
involution, that is, $k^{\prime}=\tilde{\mathcal{M}}k$, then it turns out that
their associated linear systems have the same dimension [6], so $h^{0}({\cal
O}_{X}(D))=h^{0}({\cal O}_{X}(D^{\prime}))$. Isomorphic flops can arise across
more than one facet of the Kähler cone boundary. In this case, we have
multiple involutions and corresponding matrices
$\tilde{\mathcal{M}}_{1},\ldots,\tilde{\mathcal{M}}_{k}$ with
$\tilde{\mathcal{M}}_{1}^{2}=\cdots=\tilde{\mathcal{M}}_{k}^{2}=\mathbbm{1}_{h\times
h}$, generating groups $\tilde{G}$. Such groups generated by reflections were
introduced by Coxeter [7] and further studied by Tits and Vinberg [8].
Especially the latter studies reflections along the walls of polyhedral cones,
and shows that these correspond to Coxeter groups. This structure gives rise
to infinite sequences of isomorphic flops generated by repeatedly reflecting
the Kähler cone along a flop wall and its reflection images under
$\tilde{\mathcal{M}}_{i}$. The union $\mathcal{K}_{\rm
ext}=\cup_{\alpha}\mathcal{K}_{\alpha}$ of their Kähler cones is referred to
as the extended Kähler cone and is mirror-dual to the complex structure moduli
space of the mirror of $X$ [2]. It turns out, the zeroth cohomology of line
bundles on $X$ is invariant under the entire group $\tilde{G}$ and this fact
can be immensely helpful for deriving formulae for cohomology [6]. In the
context of infinite flop chains acting on divisors, the authors of [9]
recently studied Euclidean D3 branes and noted that theta functions also
appear in the non-perturbative superpotential of Type IIB.

In the present paper we are interested in the implications of this symmetry
for Gopakumar-Vafa (GV) invariants [10, 11] and the instanton prepotential for
Kähler moduli. To this end, we introduce a (dual) basis $(C^{i})$ of curve
classes and represent arbitrary classes $C$ by $h$-dimensional integer vectors
$d$ such that $C=d_{i}C^{i}$. Their GV invariants are denoted by $n_{d}$. On
these curve classes, the involutions act via the matrices
$\mathcal{M}_{a}=\tilde{\mathcal{M}}_{a}^{T}$ and the entire group via the
dual $G$ of $\tilde{G}$, generated by the matrices $\mathcal{M}_{a}$. Our main
observation is that classes $d$ which do not flop at any of the facets of the
(possibly infinite sequence of) CYs $X_{\alpha}$, the GV invariants are
unchanged under the action of $G$, so $n_{d}=n_{gd}$ for all $g\in G$. This
implies that a part of the instanton prepotential for the Kähler moduli
(specifically, the part associated to non-flopping curve classes) is
$\tilde{G}$-invariant and can be expressed in terms of $\tilde{G}$-invariant
functions

$\displaystyle\psi_{d}^{G}(T)=\sum_{g\in G}e^{2\pi
i(gd)\cdot{T}}=\sum_{\tilde{g}\in\tilde{G}}e^{2\pi
id\cdot(\tilde{g}{T})}\qquad\Rightarrow\qquad\psi_{d}^{G}(\tilde{g}{T})=\psi_{d}^{G}({T})\quad\forall\;\tilde{g}\in\tilde{G}\,,$
(1.1)

where ${T}=\chi+it$ are the complexifications of the Kähler parameters
$t^{i}$. As far as we are aware, these functions, invariant under certain
representations of Coxeter groups, have not been introduced and studied in
this context. We will show that, for certain special cases, depending on the
underlying CY manifold $X$, they can be expressed in terms of Jacobi theta
functions whose appearance can be traced to an elliptic fibration structure of
$X$. For explicit examples, we will work with complete intersection CYs in
products of projective spaces (CICYs) [12].

The plan of the paper is as follows. In Section 2, we start with a simple
warm-up example, a CY manifold with $h^{1,1}(X)=2$ and only a single flop
boundary, leading to a finite symmetry
$\tilde{G}=\langle\tilde{M}_{1}\rangle\cong\mathbb{Z}_{2}$. The insight from
this example is used in Section 3 to study CY manifolds with $h^{1,1}(X)=2$
and two flop boundaries, with the associated groups
$\tilde{G}=\langle\tilde{M}_{1},\tilde{M}_{2}\rangle$ isomorphic to universal
Coxeter groups with two generators. In Section 4 we generalize the discussion
to manifolds with $h^{1,1}(X)>2$ but still with two isomorphic flop boundaries
and group $\tilde{G}=\langle\tilde{M}_{1},\tilde{M}_{2}\rangle$. As we will
show, this case exhibits new features, compared to the $h^{1,1}(X)=2$ case,
and, in particular, we find that the $\tilde{G}$ invariant functions
$\psi_{d}^{G}$ can sometimes be expressed in terms of Jacobi theta functions.
The general case with arbitrary $h^{1,1}(X)$ and arbitrary number of flop
boundaries, and its relation to Coxeter groups, is discussed in Section 5. We
present our conclusions in Section 6. Appendix A contains details about the
relation of CYs with infinitely many flops, the occurrence of Jacobi theta
functions in the prepotential, and the presence of elliptic fibrations.

##  2 A simple warm-up example

We are mostly interested in cases where the group $G$ is of infinite order.
However, there are two complications, related to the structure of $G$, in
analyzing such cases. First, it is difficult to express elements of Coxeter
groups with more than two generators in a sufficiently systematic way in terms
of the generators, which we rely on to facilitate computations such as working
out the sums in (1.1). Secondly, even for Coxeter groups with only two
generators, where writing group elements in terms of generators is relatively
simple, the actual matrices in $G$ become complicated. For this reason, we
start our discussion with the simple single-flop case where
$G\cong\mathbb{Z}_{2}$, thereby cutting out the above-mentioned complications,
and leaving us to focus on other issues, such as the relevant cone structures
and properties of GV invariants. Cases with infinite order groups $G$ will be
discussed in the subsequent sections.

###  2.1 Single-flop cases with Picard rank two

Consider a CY manifold $X$ with $h:=h^{1,1}(X)=2$, a basis $(D_{1},D_{2})$ of
divisor classes generating the Kähler cone, corresponding Kähler parameters
$t=(t^{1},t^{2})$ and Kähler forms $J_{X}=t^{i}D_{i}$, where $t^{i}\geq 0$.
So, relative to our chosen divisor basis, the Kähler cone is then the positive
quadrant, $\mathcal{K}=\\{t_{1}e_{1}+t_{2}e_{2}\,|\,t^{i}\geq 0\\}$ (where
$e_{i}$ denote the standard unit vectors). We assume that $\\{t^{2}=0\\}$ is a
boundary of the effective cone111The discussion would not change if it was
just the boundary of the extended Kähler cone. while $\\{t^{1}=0\\}$ is a
boundary with an isomorphic flop. Such a structure can be easily detected from
the triple intersection numbers $\lambda_{ijk}=D_{i}\cdot D_{j}\cdot D_{k}$ by
introducing the quantities

$\displaystyle m_{1}=\frac{2\lambda_{122}}{\lambda_{222}}\,,\qquad
m_{2}=\frac{2\lambda_{112}}{\lambda_{111}}\,.$ (2.1)

An isomorphic flop at $\\{t^{1}=0\\}$ occurs (for generic complex structure
choice) if $m_{1}$ is finite and integral, and $\\{t^{2}=0\\}$ ends the
effective cone if $m_{2}$ does not satisfy these requirements. In this case,
passing through $t^{1}=0$ leads to an isomorphic CY $X^{\prime}$ and an
involution, $\tilde{G}=\langle\tilde{M}_{1}\rangle\cong\mathbb{Z}_{2}$
generated by

$\displaystyle\tilde{M}_{1}=\begin{pmatrix}-1&0\\\ m_{1}&1\\\
\end{pmatrix}\,.$ (2.2)

We reserve the symbol $M$ for the $2\times 2$ reflection matrix and later,
when $h^{1,1}(X)>2$, use the symbol $\mathcal{M}$ for higher-dimensional
matrices.

The Kähler cone
$\mathcal{K}^{\prime}=\mathbb{R}^{+}(\tilde{v}_{1},\tilde{v}_{2})$ of
$X^{\prime}$ is generated by

$\displaystyle\tilde{v}_{1}=\tilde{M}_{1}e_{1}=\left(\begin{array}[]{r}-1\\\
m_{1}\end{array}\right)\,,\qquad\tilde{v}_{2}=\tilde{M}_{1}e_{2}=e_{2}\,,$
(2.5)

and the two Kähler cones are exchanged by the involution,
$\mathcal{K}^{\prime}=\tilde{M}_{1}\mathcal{K}$. In particular, since the
boundary along $e_{1}$ is mapped into the one along $v_{1}$ the latter just as
the former must be a boundary of the effective cone, so that $\mathcal{K}_{\rm
eff}={\mathcal{K}}\cup{\mathcal{K}}^{\prime}=\\{t_{1}^{\text{eff}}\tilde{v}_{1}+t_{2}^{\text{eff}}e_{2}\,|\,t_{i}^{\rm
eff}\geq 0\\}$.

Homology classes $C=d_{1}C^{1}+d_{2}C^{2}$ are labeled by integer vectors
$d=(d_{1},d_{2})$, relative to a dual basis $(C^{i})$, so $C^{i}\cdot
D_{j}=\delta^{i}_{j}$, and their GV invariants are denoted by $n_{d}$. Curve
classes are acted on by the dual group $G=\langle
M_{1}\rangle\cong\mathbb{Z}_{2}$, with generator

$\displaystyle M_{1}=\tilde{M}_{1}^{T}=\left(\begin{array}[]{cc}-1&m_{1}\\\
0&1\end{array}\right)\,,$ (2.8)

and the dual (Mori) cones222The Mori cones are denoted by $\mathcal{M}$,
$\mathcal{M}^{\prime}$, etc., while reflections are denoted by
$\mathcal{M}_{1}$, $\mathcal{M}_{2}$, etc. Despite this slight clash of
notation, it should hopefully be clear which object is meant in any given
context.

$\displaystyle{\mathcal{M}}=\\{c_{1}e_{1}+c_{2}e_{2}\,|\,c_{i}\geq
0\\}\,,\qquad{\mathcal{M}}^{\prime}=\\{c^{\prime}_{1}v_{1}+c^{\prime}_{2}v_{2}\,|\,c^{\prime}_{i}\geq
0\\}\,,\qquad v_{1}=\left(\begin{array}[]{c}m_{1}\\\
1\end{array}\right)\,,\quad v_{2}=-e_{1}$ (2.11)

are exchanged by the action of $M_{1}$. We call their intersection
$\mathcal{M}_{\rm
restr}=\mathcal{M}\cap\mathcal{M}^{\prime}=\\{c_{1}^{\text{restr}}v_{1}+c_{2}^{\text{restr}}e_{2}\,|\,c_{i}^{\rm
restr}\geq 0\\}$ the restricted cone.

To discuss what happens to GV invariants under the action of $G$, we should
distinguish between flopping and non-flopping curve classes.

Non-flopping curve classes are those for which all curves retain a finite
volume under a flop transition, that is, all curve classes that are not in the
codimension 1 facet of the Mori cone dual to the flop wall of the Kähler cone.
These must have $G$-invariant GV invariants, $n_{gd}=n_{d}$ for all $g\in G$,
since the number of curves in such a class is not affected by the flop. Non-
flopping classes split into two groups, namely those in the restricted cone
$\mathcal{M}_{\rm restr}$ and those outside. Under the action of $G$, non-
flopping classes in $d\in\mathcal{M}\setminus\mathcal{M}_{\rm restr}$ are
mapped outside the Mori cone of $X$ and consequently their GV invariants must
vanish. In other words, all non-flopping classes with positive GV invariants
reside in the ($G$-invariant) restricted cone. Within the restricted cone, we
can identify a fundamental region of the $G$-action,
$\mathcal{M}_{f}\subset\mathcal{M}_{\rm restr}$. For the present case, this
can be chosen as
$\mathcal{K}_{f}=\\{c_{1}^{f}w+c_{2}^{f}e_{2}\,|\,c_{i}^{f}\geq 0\\}$. We
illustrate this cone structure for the example (to be discussed in more detail
in the following sub-section) in Figure 1.

Flopping curve classes are classes with $n_{d}>0$ that contain curves
shrinking to zero volume, $d_{i}t^{i}\to 0$, at the flop locus. In the present
case, the flop wall is $\\{t^{1}=0\\}$, so they must be of the form
$d=(d_{1},0)$. In fact, among those, the only classes which can have non-zero
GV invariants [4] are $(1,0)$ and $(2,0)$, so $n_{(d_{1},0)}=0$ for $d_{1}>2$.
We collect all flopping classes in a set $\mathcal{B}$, which in this example
is simply $\mathcal{B}\subset\\{(1,0),(2,0)\\}$. The GV invariants for these
flopping classes are not $G$-invariant: The image of a curve class $d$ with
$n_{d}\neq 0$ is $M_{1}(d_{1},0)^{T}=(-d_{1},0)^{T}$, which is outside the
cone of effective curves $\mathcal{M}$ and, hence, its GV should vanish. Of
course, from the point of view of the manifold $X^{\prime}$ with Kähler
parameter $t_{1}^{\prime}=-t_{1}$ the situation is reversed and its GV
invariants satisfy $n^{\prime}_{(d_{1},0)}=0$ for all $d_{1}>0$ and
$n^{\prime}_{(-d_{1},0)}>0$ for $d_{1}\in\\{1,2\\}$.

Now we have enough information to discuss the implications of the symmetry $G$
for the instanton prepotential [13]

$\displaystyle\mathcal{F}_{\text{inst}}=\sum_{d}n_{d}\;\text{Li}_{3}(e^{2\pi
id\cdot{T}})\,.$ (2.12)

Splitting the sum into flopping and non-flopping classes and introducing
$G$-orbits for the latter we have

$\displaystyle\mathcal{F}_{\text{inst}}=\left(\sum_{d\in\mathcal{B}}n_{d}+\sum_{d\in\mathcal{M}_{\rm
restr}}n_{d}\right)\;\text{Li}_{3}(e^{2\pi
id\cdot{T}})=\sum_{d\in\mathcal{B}}n_{d}\,\text{Li}_{3}(e^{2\pi
id\cdot{T}})+\sum_{d\in\mathcal{M}_{f}}n_{d}\Psi_{d}^{G}({T})$ (2.13)

where we have introduced the $G$-invariant functions

$\displaystyle\Psi_{d}^{G}({T})=\sum_{g\in G}\text{Li}_{3}(e^{2\pi
i(gd)\cdot{T}})=\text{Li}_{3}(e^{2\pi id\cdot{T}})+\text{Li}_{3}(e^{2\pi
i(M_{1}d)\cdot{T}})\,.$ (2.14)

While the first term in Eq. (2.13) for the flopping classes is not
invariant333This part of can be made $G$-invariant by extending
$\mathcal{F}_{\text{inst}}$ to the entire effective cone which amounts to
replacing the first sum in Eq. (2.13) by
$\sum_{d\in\mathcal{B}}n_{d}\left(\theta(t^{1})\text{Li}_{3}(e^{2\pi
id\cdot{T}})+\theta(-t^{1})\text{Li}_{3}(e^{2\pi i(M_{1}d)\cdot{T}}\right)$.
under the action of $\tilde{G}$ on ${T}$, the second term for the non-flopping
classes is, thanks to the functions $\Psi_{d}$ satisfying

$\displaystyle\Psi_{d}^{G}(\tilde{g}{T})=\Psi_{d}^{G}({T})\qquad\mbox{for
all}\qquad\tilde{g}\in\tilde{G}\,.$ (2.15)

Of course, the present situation with a finite group
$\tilde{G}\cong\mathbb{Z}_{2}$ is relatively simple and the resulting
constraints on the prepotential rather mild. Things will become significantly
more restrictive (and the functions $\Psi_{d}^{G}$ will become more
interesting) for the cases with symmetries $\tilde{G}$ of infinite order.
Before we move on to this we illustrate the single-flop case with an explicit
example.

###  2.2 Example for a single flop

Figure 1: Kähler and Mori cone for CICY 7887. Left: Kähler cones $\mathcal{K}$
(blue) and $\mathcal{K}^{\prime}$ (orange). The extended cone is
$\mathcal{K}_{\text{ext}}=\mathcal{K}\cup\mathcal{K}^{\prime}$. Center: The
(dual) Mori cones $\mathcal{M}$ of $\mathcal{K}$ (blue and green) and
$\mathcal{M}^{\prime}$ of $\mathcal{K}^{\prime}$ (orange and green). Note that
the cones overlap in the restricted cone
$\mathcal{M}_{\text{restr}}=\mathcal{M}\cap\mathcal{M}^{\prime}$ (green).
Right: Similar to the center figure, but we have included the GV invariants as
dots. Green dots indicate non-zero GV invariants, red dots indicate zero GV
invariants, and yellow dots indicate (non-zero) GV invariants in curve classes
that are flopped. A choice for the fundamental cone
$\mathcal{M}_{f}\subset\mathcal{M}_{\text{restr}}$ is the hatched region
between the two purple lines.

As an example for the single flop case, we discuss the $h^{1,1}(X)=2$ complete
intersection Calabi-Yau manifolds $X$ given as hypersurfaces of degree $(2,4)$
in the ambient space $\mathbbm{P}^{1}\times\mathbbm{P}^{3}$ (CICY 7887 in the
list of Ref. [12]). Choosing the divisors $(D_{1},D_{2})$ as the Poincare
duals of the standard Kähler forms of the ambient projective factors
restricted to $X$, the Kähler cone relative to this basis is indeed given by
the positive quadrant, $\mathcal{K}=\\{t_{1}e_{1}+t_{2}e_{2}\,|\,t^{i}\geq
0\\}$. The non-zero intersection numbers $\lambda_{ijk}=D_{i}\cdot D_{j}\cdot
D_{k}$ are $\lambda_{122}=4$ and $\lambda_{222}=2$. Inserting these numbers
into Eqs. (2.1) gives $m_{1}=4$ while $m_{2}\not\in\mathbbm{Z}$, which
indicates we have indeed a flop boundary at $\\{t^{1}=0\\}$ while
$\\{t^{2}=0\\}$ is a boundary of the effective cone. As a consequence, we have

$\displaystyle M_{1}=\begin{pmatrix}-1&4\\\ 0&1\end{pmatrix}\,,\qquad
G=\langle\left.M_{1}~{}~{}\right|~{}~{}M_{1}^{2}=\mathbbm{1}_{2\times
2}\rangle\,.$ (2.16)

The resulting Kähler cone structure

$\displaystyle\mathcal{K}=\\{t_{1}e_{1}+t_{2}e_{2}\,|\,t_{i}\geq
0\\}\,,\quad\mathcal{K}^{\prime}=\\{t_{1}^{\prime}\tilde{v}_{1}+t_{2}^{\prime}e_{2}\,|\,t_{i}^{\prime}\geq
0\\}\,,\quad\mathcal{K}_{\rm eff}=\\{t^{\rm eff}_{1}e_{1}+t^{\rm
eff}_{2}\tilde{v}_{1}\,|\,t_{i}^{\rm eff}\geq 0\\}\,,$ (2.17)

where $\tilde{v}_{1}=(-1,4)^{T}$, is shown in Figure 1 on the left. For the
dual cones this means

$\displaystyle\begin{array}[]{rclcrcl}\mathcal{M}&=&\\{c_{1}e_{1}+c_{2}e_{2}\,|\,c_{i}\geq
0\\}&&\mathcal{M}^{\prime}&=&\\{c_{1}^{\prime}v_{1}-c_{2}^{\prime}e_{1}\,|\,c_{i}^{\prime}\geq
0\\}\\\ \mathcal{M}_{\rm restr}&=&\\{c_{1}^{\rm restr}v_{1}+c_{2}^{\rm
restr}e_{2}\,|\,c_{i}^{\rm restr}\geq
0\\}&&\mathcal{M}_{f}&=&\\{c_{1}^{f}w+c_{1}^{f}e_{2}\,|\,c_{i}^{f}\geq
0\\}\end{array}\,,$ (2.20)

where $v_{1}=(4,1)^{T}$ and $w=(2,1)^{T}$. The dual cones and the restricted
cones are illustrated in Figure 1 in the middle. The structure of GV
invariants is indicated in Figure 1 on the right. Flopping classes correspond
to yellow dots, so for this example there is only a single flopping class,
$\mathcal{B}=\\{(1,0)\\}$, with GV invariant $n_{(1,0)}=64$. Red points
indicate classes with vanishing GV invariants while the GV invariants for the
green points, contained in the restricted cone, can be positive. The green,
hatched region is the fundamental region $\mathcal{K}_{f}$. Inserting this
into Eq. (2.13) with $\Psi_{d}^{G}$ defined in Eq. (2.14), we get the
instanton prepotential

$\displaystyle\mathcal{F}_{\rm inst}=64\,{\rm Li}_{3}\left(e^{2\pi
i{T}_{1}}\right)+\sum_{d\in\mathcal{K}_{f}}n_{d}\Psi_{d}^{G}({T})=64\,{\rm
Li}_{3}\left(e^{2\pi
i{T}_{1}}\right)+\left(\sum_{d\in\mathcal{K}_{f}}n_{d}\sum_{g\in\\{\mathbbm{1},M_{1}\\}}{\rm
Li}_{3}\left(e^{2\pi i(gd)\cdot T}\right)\right)\,,$ (2.21)

where the summation range $\mathcal{K}_{f}$ is as in Eq. (2.20).

Out of the $36$ manifolds with $h^{1,1}(X)=2$, defined as complete
intersections in product of projective spaces, there are 20, including the
above example, with a single flop, as can be seen from the Appendix A of [5].
There are also $36$ Kreuzer-Skarke CY manifolds with $h^{1,1}(X)=2$ and
Appendix B of [5] shows at least $16$ of those are single-flop cases. All of
these manifolds have a prepotential with a structure similar to Eq. (2.21).

##  3 Infinitely many flops for Picard rank two

We now turn to manifolds with $h^{1,1}(X)=2$ that admit infinitely many flops.
For these manifolds, there exists an infinite symmetry group $G$ and,
correspondingly, the orbits of homology classes with identical GV invariants
are infinite and the definition of the $G$-invariant functions $\Psi_{d}^{G}$
will involve an infinite sum.

The basic set-up and conventions are exactly as in Section 2.1 but, unlike for
the single-flop case, we now assume that both $m_{1}$ and $m_{2}$ in (2.1) are
integers. In this case, the Kähler cone $\mathcal{K}$ has two flop boundaries
at $\\{t^{1}=0\\}$ and $\\{t^{2}=0\\}$ with corresponding involutions
generated by

$\displaystyle\tilde{M}_{1}=\left(\begin{array}[]{cc}-1&0\\\
m_{1}&1\end{array}\right)\,,\qquad\tilde{M}_{2}=\left(\begin{array}[]{cc}1&m_{2}\\\
0&-1\end{array}\right)\,.$ (3.5)

These matrices do not commute and lead to a group
$\tilde{G}=\langle\tilde{M}_{1},\tilde{M}_{2}\rangle$ as well as its dual
$G=\langle M_{1},M_{2}\rangle$ generated by

$\displaystyle M_{1}=\tilde{M}_{1}^{T}=\left(\begin{array}[]{cc}-1&m_{1}\\\
0&1\end{array}\right)\,,\qquad
M_{2}=\tilde{M}_{2}^{T}=\left(\begin{array}[]{cc}1&0\\\
m_{2}&-1\end{array}\right)\,.$ (3.10)

The structure of $G$ depends on the values of $m_{1}$ and $m_{2}$ or, more
precisely, on the product $m_{1}m_{2}$. There are three cases which lead to
qualitatively different results for $G$, namely $m_{1}m_{2}<4$, $m_{1}m_{2}>4$
and the limiting case $m_{1}m_{2}=4$. It turns out only the second case is
realized for CY manifolds with $h^{1,1}(X)=2$. In the first case, the
effective Kähler cone would not be convex, while in the last case, it would be
rational polyhedral, that is, it would end with its final wall still
containing integral divisors, whose dual curve would, however, not be
floppable. Since these inconsistencies do not necessarily occur for
$h^{1,1}(X)>2$, it is nevertheless useful to cover all cases.

###  3.1 Group structure

To uncover the structure of $G$, it is convenient to introduce the matrices

$\displaystyle S=M_{1}\,,\qquad Q=M_{1}\,M_{2}\qquad\Rightarrow\qquad
Q^{-1}=SQS=M_{2}M_{1}\,.$ (3.11)

Then, every group element can be written in the form $Q^{k}S^{\alpha}$, where
$\alpha\in\\{0,1\\}$ and $k\in\mathbb{Z}$. Clearly, $S$ has order two and
generates a sub-group of $G$ isomorphic to $\mathbb{Z}_{2}$. On the other
hand, the order $N$ of $Q$ can be finite or infinite, depending on the value
of $m_{1}m_{2}$, and the group $\mathbb{Z}_{N}$ (with
$\mathbb{Z}_{\infty}=\mathbb{Z}$) it generates is normal in $G$. So in
conclusion, we have $G\cong\mathbb{Z}_{N}\rtimes\mathbb{Z}_{2}$. It remains to
determine the order $N$ and the precise form of the matrices $Q^{k}S^{\alpha}$
and to do this we need to distinguish the three cases mentioned above.

$(m_{1},m_{2})$ | order $N$ of $Q$ | $G\cong$
---|---|---
$(1,1)$ | 3 | $\mathbbm{Z}_{3}\rtimes\mathbbm{Z}_{2}$
$(1,2)$ | 4 | $\mathbbm{Z}_{4}\rtimes\mathbbm{Z}_{2}$
$(1,3)$ | 6 | $\mathbbm{Z}_{6}\rtimes\mathbbm{Z}_{2}$
Table 1: Choices for $(m_{1},m_{2})$ with $m_{1}m_{2}<4$ and the resulting
group structure.

#### The case $\boldsymbol{m_{1}m_{2}<4}$

In these cases the order $N$ of $Q$ is finite, so
$G\cong\mathbb{Z}_{N}\rtimes\mathbb{Z}_{2}$, and every group element can be
written uniquely as $Q^{k}S^{\alpha}$, where $k\in\\{0,\ldots,N-1\\}$ and
$\alpha\in\\{0,1\\}$. Specifically, we have the three sub-cases listed in
Table 1. As mentioned, this case is not realized for CY manifolds with
$h^{1,1}(X)=2$, but can occur for $h^{1,1}(X)>2$. We will see an example in
Section 4.3.2.

#### The case $\boldsymbol{m_{1}m_{2}>4}$

In these cases, the matrix $Q$ has infinite order so that
$G\cong\mathbbm{Z}\rtimes\mathbbm{Z}_{2}$ and every element of $G$ can be
uniquely written in the form $Q^{k}S^{\alpha}$, where $k\in\mathbbm{Z}$ and
$\alpha\in\\{0,1\\}$. The powers of the matrix $Q$ can be computed explicitly
and are given by

$\displaystyle Q^{k}=\left(\begin{array}[]{cc}a(k)&b(k)\\\
c(k)&d(k)\end{array}\right)$ (3.14)

where

$\displaystyle\begin{array}[]{rclcrcl}a(k)&=&\sum_{i=0}^{k}(-1)^{k+i}\left(\begin{array}[]{c}k+i\\\
2i\end{array}\right)(m_{1}m_{2})^{i}\,,&&d(k)&=&-a(k-1)\,,\\\\[14.22636pt]
b(k)&=&\sum_{i=1}^{k}(-1)^{k+i+1}\left(\begin{array}[]{c}k+i-1\\\
k-i\end{array}\right)m_{1}^{i}m_{2}^{i-1}\,,&&c(k)&=&-\frac{m_{2}}{m_{1}}b(k)\,,\end{array}$
(3.21)

and $d(0)=1$ by definition. Note that the entries of $Q^{k}$ are power series
in $m_{i}$ whose exponents grow with $k$.

#### The case $\boldsymbol{m_{1}m_{2}=4}$

The boundary case $m_{1}m_{2}=4$ is, in a sense, the most interesting one. The
order of $Q$ is still infinite, so $G\cong\mathbbm{Z}\rtimes\mathbbm{Z}_{2}$
and every group element can be uniquely written as $Q^{k}S^{\alpha}$, where
$k\in\mathbb{Z}$ and $\alpha\in\\{0,1\\}$. However, the matrices $Q^{k}$
behave very differently compared to the case $m_{1}m_{2}>4$. Their entries,
rather than being power series with exponents growing with $k$, are linear
polynomials in $k$. Concretely, for the three possible choices of
$(m_{1},m_{2})$ we find

$\displaystyle\begin{array}[]{cclcl}(m_{1},m_{2})=(2,2)\;:&&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-2k\\\
2k&1-2k\end{array}\right)\\\\[14.22636pt]
(m_{1},m_{2})=(1,4)\,:&&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-k\\\
4k&1-2k\end{array}\right)\\\\[14.22636pt]
(m_{1},m_{2})=(4,1)\,:&&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-4k\\\
k&1-2k\end{array}\right)\,.\end{array}$ (3.31)

As mentioned, this case is only realized for $h^{1,1}(X)>2$. As we will see
later, the simple form of $Q^{k}$ for $m_{1}m_{2}=4$ and its linear growth
with $k$ allows re-writing the $G$-invariant functions $\Psi_{d}^{G}$ in terms
of Jacobi theta functions.

###  3.2 Cones

For the remainder of this section, we will focus on the case $m_{1}m_{2}>4$,
the only one actually realized for $h^{1,1}(X)=2$. The group $G$ consists of
elements $Q^{k}S^{\alpha}$, where $k\in\mathbb{Z}$ and $\alpha\in\\{0,1\\}$,
with the matrices $S$ and $Q$ from Eq. (3.11) and it is isomorphic to
$\mathbbm{Z}\rtimes\mathbbm{Z}_{2}$. In addition to the “central” Kähler cone
$\mathcal{K}$ of $X$ we have an infinite number of isomorphic CYs with
adjacent Kähler cones $\mathcal{K}_{\tilde{g}}=\tilde{g}\mathcal{K}$ to either
side of $\mathcal{K}$. The union of these forms the effective cone

$\displaystyle{\mathcal{K}}_{\rm
eff}=\bigcup_{\tilde{g}\in\tilde{G}}\mathcal{K}_{\tilde{g}}=\\{t_{1}^{\text{eff}}\tilde{v}_{1}+t_{2}^{\text{eff}}\tilde{v}_{2}\,|\,t_{i}^{\rm
eff}>0\\}\,,\qquad\tilde{v}_{1}=\left(\begin{array}[]{c}\mu_{2}\\\
-1\end{array}\right)\,,\quad\tilde{v}_{2}=\left(\begin{array}[]{c}-1\\\
\mu_{1}\end{array}\right)\,,$ (3.36)

where

$\displaystyle\mu_{i}=\frac{m_{i}}{2}\left(1+\sqrt{1-\frac{4}{m_{1}m_{2}}}\right)\,.$
(3.37)

Note that for the case $m_{1}m_{2}>4$ the closure of this cone is irrational.
For a specific example, the extended cone is shown in Fig. 2 on the left. The
Mori cones dual of the Kähler cones $\mathcal{K}_{\tilde{g}}$ are denoted by
$\mathcal{M}_{g}=g\mathcal{M}$ and their intersection, which is the dual of
the effective cone, forms the restricted cone

$\displaystyle\mathcal{M}_{\rm restr}=\bigcap_{g\in
G}\mathcal{M}_{g}=\check{\mathcal{K}}_{\rm
eff}=\\{c_{1}^{\text{restr}}v_{1}+c^{\text{restr}}_{2}v_{2}\,|\,c_{i}\geq
0\\}\,,\qquad v_{1}=\left(\begin{array}[]{c}1\\\
\mu_{2}\end{array}\right)\,,\quad v_{2}=\left(\begin{array}[]{c}\mu_{1}\\\
1\end{array}\right)\,.$ (3.42)

An example of a restricted cone is shown in Figure 2, in the middle.

In our discussion of the GV invariants we will distinguish again between
flopping and non-flopping classes. The flopping classes $\mathcal{B}$ of the
original manifold $X$ are contained in
$\mathcal{B}\subset\\{(1,0),(2,0),(0,1),(0,2)\\}$, straightforwardly
generalizing what we have seen in the single-flop case. As before, for a
flopping class $d\in\mathcal{B}$ of $X$, its images $gd$ under the group $G$
are also flopping classes, but rather for one of the isomorphic manifolds
$X^{\prime}$ instead of $X$. Hence, for every flopping class $d\in\mathcal{B}$
of $X$ we have an entire infinite $G$-orbit of flopping classes (while, for
the single flop case where $G\cong\mathbb{Z}_{2}$ the orbits only had length
two). We have already seen that GV invariants are not constant along these
orbits. Rather, for a flopping class $d\in\mathbb{B}$ of $X$ we have

$\displaystyle n_{gd}=\left\\{\begin{array}[]{cl}n_{d}&\mbox{for
}gd\in\mathcal{M}\\\ 0&\mbox{for }gd\notin\mathcal{M}\end{array}\right.\,.$
(3.45)

As before, non-flopping classes $d$ outside the restricted cone always have
vanishing GV invariants, $n_{d}=0$. The restricted cone is $G$-invariant and
it splits into $G$-orbits on which the GV invariants are constant, so
$n_{gd}=n_{d}$ for $d\in\mathcal{M}_{\rm restr}$ and all $g\in G$. Within the
restricted cone we can choose a fundamental region for the $G$-action as

$\displaystyle\mathcal{M}_{f}=\\{c_{1}^{f}w_{1}+c_{2}^{f}w_{2}\,|\,c_{i}^{f}\geq
0\\}\,,\qquad w_{1}=\left(\begin{array}[]{c}2\\\
\mu_{2}\end{array}\right)\,,\quad v_{2}=\left(\begin{array}[]{c}\mu_{1}\\\
2\end{array}\right)\,.$ (3.50)

For our example, the structure of the various cones and the GV invariants is
indicated in Figure 2, on the right.

###  3.3 The prepotential

Now we have everything in place to write down an expression for the instanton
prepotential. As before, we split the sum into flopping and non-flopping
classes, using (3.45) for the former and $G$-invariance of the GV invariants
for the latter. This leads to

$\displaystyle\mathcal{F}_{\rm
inst}=\sum_{d\in\mathcal{B}}n_{d}\sum_{\begin{subarray}{c}g\in G\\\
gd\in\mathcal{M}\end{subarray}}{\rm Li}_{3}\left(e^{2\pi
i(gd)\cdot{T}}\right)+\sum_{d\in\mathcal{K}_{f}}n_{d}\Psi_{d}^{G}({T})$ (3.51)

with the $G$-invariant functions

$\displaystyle\Psi_{d}^{G}({T})=\sum_{g\in G}{\rm Li}_{3}\left(e^{2\pi
i(dg)\cdot{T}}\right)=\sum_{k\in\mathbb{Z}}\left({\rm Li}_{3}\left(e^{2\pi
i(Q^{k}d)\cdot{T}}\right)+{\rm Li}_{3}\left(e^{2\pi
i(Q^{k}Sd)\cdot{T}}\right)\right)\,.$ (3.52)

The functions (3.52) contain the matrix powers $Q^{k}$ from (3.14) in their
exponents and, hence, these exponents increase with powers of $k$. Functions
that behave in this way are not modular – the exponents for modular functions
depend quadratically on $k$. To the best of our knowledge, the functions
$\Psi_{d}^{G}$ in Eq. (3.52) have not been studied before and they do not have
an established name. It would be interesting to analyze their properties in
more detail.

###  3.4 Example with infinitely many flops

Figure 2: Kähler and Mori cone for CICY 7863. Left: Kähler cone $\mathcal{K}$
(blue) of $X$ and the extended Kähler cone $\mathcal{K}_{\text{ext}}$ (blue
and orange). Center: The (dual) Mori cone $\mathcal{M}$ of $\mathcal{K}$ (blue
and green) and the union of all Mori cones $\mathcal{M}^{\prime}$ of
$\mathcal{K}^{\prime}$ (orange, blue, green).The restricted cone is
$\mathcal{M}_{\text{restr}}=\mathcal{M}\cap\mathcal{M}^{\prime}$ (green).
Right: We included GV invariants. Green dots indicate non-zero GV invariants,
red dots indicate zero GV invariants, and yellow dots indicate (non-zero) GV
invariants in curve classes that are flopped. A choice for the fundamental
cone $\mathcal{M}_{f}\subset\mathcal{M}_{\text{restr}}$ is the hatched region
between the two purple lines.

We would like to illustrate the above structure with a specific example, the
complete intersection CY manifold described by the configuration matrix

$\displaystyle X\in\left[\begin{array}[]{c|ccc}\mathbb{P}^{3}&2&1&1\\\
\mathbb{P}^{3}&2&1&1\end{array}\right]^{2,66}_{-128}\,.$ (3.55)

This is CICY 7863 in the standard list of Ref. [12] and it has Hodge numbers
$(h^{1,1},h^{2,1})=(2,68)$. The above configuration matrix indicates that the
manifold $X$ is defined as the common zero locus of three polynomials in
$\mathbb{P}^{3}\times\mathbb{P}^{3}$ with bi-degrees $(2,2)$, $(1,1)$ and
$(1,1)$. The Kähler cone $\mathcal{K}$ (relative to the standard divisor basis
$(D_{1},D_{2})$ associated to pullbacks of the hyperplane classes of the
ambient space $\mathbb{P}^{3}$ factors) is the positive quadrant and the non-
zero intersection numbers are given by $\lambda_{111}=\lambda_{222}=2$ and
$\lambda_{112}=\lambda_{122}=6$. Equation (2.1) implies that $m_{1}=m_{2}=6$
and hence both boundaries, $\\{t^{1}=0\\}$ and $\\{t^{2}=0\\}$, exhibit
isomorphic flops, resulting in an infinite flop sequence. The various cones
are shown in Figure 2.

The flopping classes of $X$ are given by
$\mathcal{B}=\\{(1,0),(2,0),(0,1),(0,2)\\}$ (indicated by yellow dots in
Figure 2 on the right) with $n_{(1,0)}=n_{(0,1)}=80$ and
$n_{(2,0)}=n_{(0,2)}=4$. Hence, the first sum in (3.51) runs over four
(partial) $G$-orbits. For the elements of $G$ we have

$\displaystyle
Q^{k}=\left(\begin{array}[]{cc}\frac{\delta_{+}\alpha_{+}^{k}-\delta_{-}\alpha_{-}^{k}}{4\sqrt{2}}&\frac{\alpha_{-}^{k}-\alpha_{+}^{k}}{4\sqrt{2}}\\\
\frac{\alpha_{+}^{k}-\alpha_{-}^{k}}{4\sqrt{2}}&\frac{\delta_{+}\alpha_{-}^{k}-\delta_{-}\alpha_{+}^{k}}{4\sqrt{2}}\\\
\end{array}\right)\,,\quad
Q^{k}S=\left(\begin{array}[]{cc}\frac{\delta_{-}\alpha_{-}^{k}-\delta_{+}\alpha_{+}^{k}}{4\sqrt{2}}&\frac{\alpha_{+}^{k+1}-\alpha_{-}^{k+1}}{4\sqrt{2}}\\\
\frac{\alpha_{-}^{k}-\alpha_{+}^{k}}{4\sqrt{2}}&\frac{\delta_{+}\alpha_{+}^{k}-\delta_{-}\alpha_{-}^{k}}{4\sqrt{2}}\\\
\end{array}\right)\,,$ (3.60)

where we have defined

$\displaystyle\alpha_{\pm}=17\pm 12\sqrt{2}\,,\qquad\beta_{\pm}=4\pm
3\sqrt{2}\,,\qquad\gamma_{\pm}=24\pm 17\sqrt{2}\,,\quad\delta_{\pm}=3\pm
2\sqrt{2}\,.$ (3.61)

Inserting this into (3.52), we get the the $G$-invariant functions
$\Psi_{d}^{G}$

$\displaystyle\Psi_{d}^{G}({T})=$ $\displaystyle\sum_{k\in\mathbb{Z}}{\rm
Li}_{3}\left(e^{\frac{1}{4}i\pi\left(\alpha_{-}^{k}\left(d_{2}\left(\beta_{+}{T}_{2}+\sqrt{2}{T}_{1}\right)-d_{1}\left(\sqrt{2}{T}_{2}-\beta_{-}{T}_{1}\right)\right)+\alpha_{+}^{k}\left(d_{1}\left(\beta_{+}{T}_{1}+\sqrt{2}{T}_{2}\right)-d_{2}\left(\sqrt{2}{T}_{1}-\beta_{-}{T}_{2}\right)\right)\right)}\right)+$
$\displaystyle\sum_{k\in\mathbb{Z}}{\rm
Li}_{3}\left(e^{\frac{1}{4}i\pi\left(\alpha_{-}^{k}\left(d_{2}\left(\beta_{-}{T}_{2}+\gamma_{-}{T}_{1}\right)+d_{1}\left(\sqrt{2}{T}_{2}-\beta_{-}{T}_{1}\right)\right)+\alpha_{+}^{k}\left(d_{2}\left(\beta_{+}{T}_{2}+\gamma_{+}{T}_{1}\right)-d_{1}\left(\beta_{+}{T}_{1}+\sqrt{2}{T}_{2}\right)\right)\right)}\right)$
(3.62)

As can be seen from Appendix A of Ref. [5], among the $h^{1,1}(X)=2$ CICYs,
there are six manifolds with infinite flop sequences (including the above
example) which realize the values

$\displaystyle(m_{1},m_{2})\in\\{(4,4),(3,8),(5,8),(6,6),(7,7)\\}\,.$ (3.63)

They can be analyzed along the same lines as the example above.

##  4 Manifolds with $\boldsymbol{h^{1,1}(X)>2}$ and two symmetry generators

We would now like to generalize our discussion to manifolds with Picard number
$h=h^{1,1}(X)>2$. As usual, we introduce a basis $(D_{1},\ldots,D_{h})$ of
divisor classes generating the Kähler cone (assuming, for simplicity, that the
Kähler cone is simplicial) and we write Kähler forms as $J_{X}=t^{i}D_{i}$,
with Kähler parameters $t^{i}\geq 0$. The presence of isomorphic flop
boundaries can then be detected from the intersection numbers
$\lambda_{ijk}=D_{i}\cdot D_{j}\cdot D_{k}$ as follows. An isomorphic flop
across the Kähler cone boundary $\\{t^{i}=0\\}$ occurs iff there is an integer
vector $u_{i}$ with components $u_{i}^{a}$, $a=1,2,\ldots,h$, that satisfies

$\displaystyle u_{i}^{i}=2\,,\qquad u_{i}^{a}\leq
0\text{~{}~{}~{}for~{}~{}~{}}a\neq i\,,\qquad
d_{abc}u_{i}^{a}=0\text{~{}~{}~{}for all~{}~{}~{}}\,b,c\neq i\,.$ (4.1)

If such a vector exists then we have an involution acting “across” the
boundary $\\{t^{i}=0\\}$ which is generated by

$\displaystyle\tilde{\mathcal{M}}_{i}=\mathbbm{1}_{h}-(\vec{0},\ldots,\vec{0},u_{i},\vec{0},\ldots,\vec{0})\,,\qquad\Rightarrow\qquad\mathcal{M}_{i}=\mathcal{\tilde{M}}^{T}_{i}=\mathbbm{1}_{h}-\begin{pmatrix}\vec{0}^{\,T}\\\
\vdots\\\ \vec{0}^{\,T}\\\ u_{i}\\\ \vec{0}^{\,T}\\\ \vdots\end{pmatrix}$
(4.2)

where the vector $u_{i}$ in the second $h\times h$ matrix appears in the
$i^{\rm th}$ row. The symmetry group $\tilde{G}$ is then generated by all such
matrices, $\tilde{G}=\langle\tilde{\mathcal{M}}_{i}\rangle$, while the
matrices $\mathcal{M}_{i}$ generate the dual
$G=\langle\mathcal{M}_{i}\rangle$. It can be shown that matrices of the type
(4.2) are indeed reflections, that is, they satisfy
$\tilde{\mathcal{M}}_{i}^{2}=\mathbbm{1}_{h\times h}$ and they have precisely
one eigenvector with eigenvalue $-1$ and $(h-1)$ eigenvectors with eigenvalues
$+1$, such that $\det{\tilde{\mathcal{M}}_{i}}=-1$. The integers $m_{1}$ and
$m_{2}$ introduced previously are given by $m_{1}=-u_{1}^{2}$ and
$m_{2}=-u_{2}^{1}$.

For a manifold with Picard number $h$ (and a simplicial Kähler cone) we can
have at most $h$ isomorphic flop boundaries and corresponding involutions.
They are all isomorphic to Coxeter groups as we will discuss in section 5. In
this section, we will focus on the case of just two involutions. This can
arise either because the manifold in question only has two isomorphic flop
boundaries or, if there are three or more, because we focus on the Coxeter
subgroup generated by just two involutions. It might seem that this situation
is not significantly different from the two generator case for $h^{1,1}(X)=2$
we have discussed in the previous section. However, it turns out interesting
new features do arise since all of the cases for $m_{1}m_{2}$ listed in
Section 3.1 can actually be realized for appropriate choices of manifolds. In
particular, there are examples for the interesting limiting case
$m_{1}m_{2}=4$ where the elements of $G$ have a particularly simple form, cf.
Eq. (3.31). As we will see, for such cases the $G$-invariant functions
$\Psi_{d}^{G}$ can be expressed in terms of Jacobi theta functions.

###  4.1 Group structure

We can assume that the two flop boundaries are $\\{t^{1}=0\\}$ and
$\\{t^{2}=0\\}$, so that the corresponding generators can be written as

$\displaystyle\begin{array}[]{rclcrcl}\mathcal{M}_{1}&=&\left(\begin{array}[]{c:c}M_{1}&U_{1}\\\
\hdashline
0&\mathbbm{1}_{(h-2)\times(h-2)}\end{array}\right)&&\mathcal{M}_{2}&=&\left(\begin{array}[]{c:c}M_{2}&U_{2}\\\
\hdashline 0&\mathbbm{1}_{(h-2)\times(h-2)}\end{array}\right)\\\\[14.22636pt]
U_{1}&=&\left(\begin{array}[]{l}u_{1}\\\
\vec{0}^{\,T}\end{array}\right)&&U_{2}&=&\left(\begin{array}[]{l}\vec{0}^{\,T}\\\
u_{2}\end{array}\right)\end{array}\,.$ (4.13)

These $(h\times h)$ matrices $\mathcal{M}_{i}$ are block matrices consisting
of a $2\times 2$ block $M_{i}$ from Eq. (3.10), a $2\times(h-2)$ block
$U_{i}$, an $(h-2)\times 2$ block with the zero-matrix, and a
$(h-2)\times(h-2)$ identity matrix. The $1\times(h-2)$-dimensional row vectors
$u_{1}$ and $u_{2}$ are defined in terms of the GLSM charges as explained in
Ref. [4] and reviewed in Appendix A. If we define new generators

$\displaystyle\mathcal{S}=\mathcal{M}_{1}\,,\qquad\mathcal{Q}=\mathcal{M}_{1}\mathcal{M}_{2}=\left(\begin{array}[]{c:c}Q&U\\\
\hdashline 0&\mathbbm{1}_{(h-2)\times(h-2)}\end{array}\right)\,,\qquad
U=M_{1}U_{2}+U_{1}=\left(\begin{array}[]{c}u_{1}+m_{1}u_{2}\\\
u_{2}\end{array}\right)$ (4.18)

the elements of $G=\langle\mathcal{M}_{1},\;\mathcal{M}_{2}\rangle$ can be
uniquely written as $\mathcal{Q}^{k}\mathcal{S}^{\alpha}$, where
$k\in\mathbbm{Z}$ and $\alpha\in\\{0,1\\}$. More explicitly, these matrices
are

$\displaystyle\mathcal{Q}^{k}=\left(\begin{array}[]{c:c}Q^{k}&R_{k}U\\\
\hdashline
0&\mathbbm{1}_{(h-2)\times(h-2)}\end{array}\right)\,,\;\;\mathcal{Q}^{k}\mathcal{S}=\left(\begin{array}[]{c:c}Q^{k}S&Q^{k}U_{1}+R_{k}U\\\
\hdashline 0&\mathbbm{1}_{(h-2)\times(h-2)}\end{array}\right)\,,$ (4.23)

where

$\displaystyle
R_{k}=\left\\{\begin{array}[]{rl}\sum_{j=0}^{k-1}Q^{j}&\mbox{for }k\geq 0\\\
-\sum_{j=k}^{-1}Q^{j}&\mbox{for }k<0\end{array}\right.\,,\qquad
Q=M_{1}M_{2}\,,\qquad S=M_{1}\,,$ (4.26)

and the powers $Q^{k}$ of $Q$ have already been computed in Section 3.1. For
the $k<0$ case, note that $M_{i}u_{i}=-u_{i}$ and
$M_{i}^{2}=\mathbbm{1}_{2\times 2}$. The interesting observation is that the
qualitative behavior of these matrices with $k$ is controlled by the behavior
of the $2\times 2$ matrices $M_{1}$, $M_{2}$ and, hence, by the product
$m_{1}m_{2}$. This means we can essentially use the classification from
Section 3.1.

#### The case $\boldsymbol{m_{1}m_{2}=0}$

One can show that $m_{1}=1$ iff $m_{2}=0$. Then $Q$ as well as $\mathcal{Q}$
are of order two, so that $G\cong\mathbbm{Z}_{2}\rtimes\mathbbm{Z}_{2}$. This
can only happen for direct product manifolds.

#### The case $\boldsymbol{0<m_{1}m_{2}<4}$

We have the choices $(m_{1},m_{2})\in\\{(1,1),(1,2),(1,3),(2,1),(3,1)\\}$ and
it can be checked by explicit computation of the matrices $R_{k}$, that the
order of $\mathcal{Q}$ remains the same as the order of $Q$ for all cases.
This means we obtain the same groups as in Table 1. As we explain in Appendix
A, the cases $(1,2)$ and $(1,3)$ are not realized, at least not by complete
intersections in projective spaces.

#### The case $\boldsymbol{m_{1}m_{2}>4}$

We know that the order of $Q$ is infinite for this case, which implies the
same is true for $\mathcal{Q}$. This means that the group structure is
$G\cong\mathbb{Z}\rtimes\mathbb{Z}_{2}$. Further, it is clear that (at least
some of) the entries of $\mathcal{Q}$ increase as powers of $k$ so that the
associated $G$-invariant functions $\Psi_{d}^{G}$ are complicated, non-modular
functions analogous to the example in (3.4).

#### The case $\boldsymbol{m_{1}m_{2}=4}$

In this case, the order of $Q$ is infinite so that
$G\cong\mathbb{Z}\rtimes\mathbb{Z}_{2}$, but we know from (3.31) that the
matrix elements of $Q^{k}$ only grow linearly with $k$. Computing the matrices
$R_{k}$,

$\displaystyle\begin{array}[]{clcrcrcl}(m_{1},m_{2})=(2,2)\;:&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-2k\\\
2k&1-2k\end{array}\right)\,,&&R_{k}&=&\left(\begin{array}[]{cc}k^{2}&k(1-k)\\\
k(k-1)&k(2-k)\\\ \end{array}\right)\,,\\\\[14.22636pt]
(m_{1},m_{2})=(1,4)\,:&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-k\\\
4k&1-2k\end{array}\right)\,,&&R_{k}&=&\left(\begin{array}[]{cc}k^{2}&\frac{k}{2}(1-k)\\\
2k(k-1)&k(2-k)\\\ \end{array}\right)\,,\\\\[14.22636pt]
(m_{1},m_{2})=(4,1)\,:&Q^{k}&=&\left(\begin{array}[]{cc}2k+1&-4k\\\
k&1-2k\end{array}\right)\,,&&R_{k}&=&\left(\begin{array}[]{cc}k^{2}&2k(1-k)\\\
\frac{k}{2}(k-1)&k(2-k)\\\ \end{array}\right)\,,\end{array}$ (4.42)

we see that the entries of $\mathcal{Q}^{k}$ are at most quadratic polynomials
in $k$. This feature means that the functions $\Psi_{d}^{G}$ can be expressed
in terms of theta functions, as we will show below.

###  4.2 The prepotential

The general structure of the prepotential is given by the same equation (3.51)
as for Picard number two, with the obvious generalizations of the various
quantities involved. The cases $0\leq m_{1}m_{2}<4$ lead to a finite symmetry
group $G$ (at least when focusing on a rank 2 subgroup), so that the functions
$\Psi_{d}^{G}$ are given by a finite sum of exponential terms similar to
(2.13). Likewise, we will not discuss the case $m_{1}m_{2}>4$ explicitly – we
have already encountered this case for $h^{1,1}(X)=2$ and it it leads to
complicated functions $\Psi_{d}^{G}$ similar to (3.4). Instead, we focus on
the structure of the $G$-invariant functions $\Psi_{d}^{G}$ for
$m_{i}m_{j}=4$. In this case, the group
$G\cong\mathbb{Z}_{2}\rtimes\mathbb{Z}$ is infinite and the matrices in $G$
depend (at most) quadratically on $k$, as Eq. (4.42) shows.

In general, expanding the tri-logarithm and using Eq. (1.1), the $G$-invariant
functions can be written as

$\displaystyle\Psi({T})=\sum_{l=1}^{\infty}\frac{1}{l^{3}}\psi_{ld}^{G}({T})\,,\qquad\psi_{ld}^{G}({T})=\sum_{g\in
G}e^{2\pi il(gd)\cdot{T}}=\sum_{k\in\mathbbm{Z}}\left(e^{2\pi
il(\mathcal{Q}^{k}d)\cdot{T}}+e^{2\pi
il(\mathcal{Q}^{k}\mathcal{S}d)\cdot{T}}\right)\,.$ (4.43)

Of course, the functions $\psi_{d}^{G}$ are also $\tilde{G}$-invariant, that
is, they satisfy $\psi_{d}^{G}(\tilde{g}{T})=\psi_{d}^{G}({T})$ for all
$\tilde{g}\in\tilde{G}$. To re-write these expressions, we split all vectors
into directions parallel and perpendicular to the two flop directions $i=1,2$,
so we write

$\displaystyle{T}=\left(\begin{array}[]{l}{T}_{\parallel}\\\
{T}_{\perp}\end{array}\right)\text{~{}~{}~{}with~{}~{}~{}}{T}_{\parallel}=\left(\begin{array}[]{l}{T}_{1}\\\
{T}_{2}\end{array}\right)\,,\quad\quad
d=\left(\begin{array}[]{l}d_{\parallel}\\\
d_{\perp}\end{array}\right)\text{~{}~{}~{}with~{}~{}~{}}d_{\parallel}=\left(\begin{array}[]{l}d_{1}\\\
d_{2}\end{array}\right)\,.$ (4.52)

From (4.23), the exponents in (4.43) can then be written as

$\displaystyle\begin{split}(\mathcal{Q}^{k}d)\cdot{T}&=(Q^{k}d_{\parallel}+R_{k}Ud_{\perp})\cdot{T}_{\parallel}+d_{\perp}\cdot{T}_{\perp}\\\
(\mathcal{Q}^{k}\mathcal{S}d)\cdot{T}&=(Q^{k}Sd_{\parallel}+(R_{k}U+Q^{k}U_{1})d_{\perp})\cdot{T}_{\parallel}+d_{\perp}\cdot{T}_{\perp}\,\end{split}$
(4.53)

Defining $\delta_{i}:=u_{i}\cdot d_{\perp}$ and using the matrices from
(4.42), we find for the case $(m_{1},m_{2})=(2,2)$

$\displaystyle(\mathcal{Q}^{k}d)\cdot{T}=\frac{1}{2}k^{2}\tau+kz+d\cdot{T}\,,\qquad(\mathcal{Q}^{k}\mathcal{S}d)\cdot{T}=\frac{1}{2}k^{2}\tau+k\tilde{z}+y+d\cdot{T}\,,$
(4.54)

where

$\displaystyle\begin{array}[]{rl}\tau&=2(\delta_{1}+\delta_{2})({T}_{1}+{T}_{2})\\\
z&=2(d_{1}-d_{2})({T}_{1}+{T}_{2})+\delta_{2}{T}_{1}-\delta_{1}{T}_{2}\\\
\tilde{z}&=z-2(\delta_{1}+\delta_{2}){T}_{1}\\\
y&=(\delta_{1}-2d_{1}+2d_{2}){T}_{1}\,.\end{array}$ (4.59)

Using these results in Eq. (4.43), we find that the $G$-invariant functions
$\psi_{d}^{G}$ can be re-written as

$\displaystyle\psi_{d}^{G}({T})=e^{2\pi id\cdot{T}}\vartheta(z;\tau)+e^{2\pi
i(d\cdot{T}+y)}\vartheta(\tilde{z};\tau)\,.$ (4.60)

where

$\displaystyle\vartheta(z;\tau):=\sum_{k\in\mathbb{Z}}e^{\pi ik^{2}\tau+2\pi
ikz}\,.$ (4.61)

is the Jacobi theta function. A similar calculation for $(m_{1},m_{2})=(1,4)$,
using the appropriate matrices from Eq. (3.31), gives a result with the same
structure as in (4.54) and (4.60), but with

$\displaystyle\begin{array}[]{rl}\tau&=(2\delta_{1}+\delta_{2})({T}_{1}+2{T}_{2})\,,\\\
z&=(2d_{1}-d_{2})({T}_{1}+2{T}_{2})+\frac{1}{2}\delta_{2}{T}_{1}-2\delta_{1}{T}_{2}\,,\\\
\tilde{z}&=z-(2\delta_{1}+\delta_{2}){T}_{1}\,,\\\
y&=(-2d_{1}+d_{2}+\delta_{1}){T}_{1}\,.\end{array}$ (4.66)

Of course, the case $(m_{1},m_{2})=(4,1)$ leads to a similar result as the
present case $(m_{1},m_{2})=(1,4)$, but with the exchange $1\leftrightarrow 2$
of indices on all quantities. Hence, for groups $G$ with two generators and
$m_{1}m_{2}=4$, we have shown that the $G$-invariant functions which make up
the prepotential can be expressed in terms of Jacobi theta functions. We give
an overview of how many CICYs (up to $h^{1,1}=7$) realize the various cases
discussed above in Table 2.

$h^{1,1}(X)$ | # CICYs |  $\geq$ 2 involutions | $m_{1}m_{2}<4$ | $m_{1}m_{2}=4$ | $m_{1}m_{2}>4$
---|---|---|---|---|---
3 | 155 | 81 | 38 | 23 | 20
4 | 425 | 186 | 90 | 96 | 0
5 | 837 | 179 | 124 | 55 | 0
6 | 1140 | 74 | 74 | 0 | 0
7 | 1112 | 38 | 38 | 0 | 0
Table 2: The number of CICYs, CICYs with at least two involutions and numbers
which realize the three cases for $m_{1}m_{2}$ for Picard numbers
$h^{1,1}(X)\leq 7$.

The table shows larger values of $m_{1}m_{2}$ become rarer as $h^{1,1}(X)$
increases, to the extent that $h^{1,1}(X)=7$ CICYs with at least two flop
boundaries all satisfy $m_{1}m_{2}<4$ and, hence, lead to finite symmetry
groups only. We do not know if this “decline” of the group order with
increasing $h^{1,1}(X)$ is a feature of the CICY construction or a feature of
CY manifolds more generally. In this context, it would be interesting to
analyze other classes of CY manifolds, such as those obtained from the
Kreuzer-Skarke classification.

###  4.3 Examples

####  4.3.1 Example with three Kähler moduli and two isomorphic flops

We would like to discuss some example manifolds with two symmetry generators
and modularity. The smallest Picard rank which allows for such examples is
three and a relevant example at $h^{1,1}(X)=3$ is the degree $(2,2,3)$
hypersurface in $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{2}$,
which is the CICY with number 7880. Its Kähler cone is the positive octant,
$\mathcal{K}=\\{(t^{1},t^{2},t^{3})\,|\,t^{i}\geq 0\\}$ and its intersection
form

$\displaystyle\kappa=\lambda_{ijk}t^{i}t^{j}t^{k}=18t_{1}t_{2}t_{3}+6t_{1}{t_{3}}^{2}+6t_{2}{t_{3}}^{2}$
(4.67)

indicates the presence of two isomorphic flop boundaries at $\\{t^{1}=0\\}$
and $\\{t^{2}=0\\}$ (while $\\{t^{3}=0\\}$ is the end of the effective cone)
with associated involutions generated by

$\displaystyle M_{1}=\left(\begin{array}[]{ccc}-1&2&3\\\ 0&1&0\\\ 0&0&1\\\
\end{array}\right)\,,\qquad M_{2}=\left(\begin{array}[]{ccc}1&0&0\\\ 2&-1&3\\\
0&0&1\\\ \end{array}\right)\,.$ (4.74)

These shows that $(m_{1},m_{2})=(2,2)$, so this is indeed a case which leads
to modularity. Comparison with the general form (4.13) of the matrices further
shows that $U_{1}=(3,0)^{T}$, $U_{2}=(0,3)^{T}$, which means
$\delta_{1}=\delta_{2}=3d_{3}$. Hence, the $G$-invariant functions
$\psi_{d}^{G}$ are of the form (4.60) with

$\displaystyle\begin{array}[]{rl}\tau&=12d_{3}({T}_{1}+{T}_{2})\\\
z&=2(d_{1}-d_{2})({T}_{1}+{T}_{2})+3d_{3}({T}_{1}-{T}_{2})\\\
\tilde{z}&=z-12d_{3}{T}_{1}\\\
y&=(-2d_{1}+2d_{2}+3d_{3}){T}_{1}\,.\end{array}$ (4.79)

We present the Coxeter diagram for this example in Figure 3 on the left.

####  4.3.2 Example with five Kähler moduli and five isomorphic flops

As another example we discuss the complete intersection CY defined by two
equations of multi-degree $(1,1,1,1,1)$ inside $(\mathbbm{P}^{1})^{5}$, which
is the CICY with number 7447. This is the mirror dual [14] of the Hulek-
Verrill manifold [15], which has recently received attention in the context of
the study of rank two attractor points [16] and its relation to Feynman loop
integrals [17]. Using the techniques from Appendix A, we see that it has
isomorphic flops along all five Kähler cone boundaries, and all corresponding
$M_{i}$ have $m_{i}=1$. Hence, the subgroup $G$ of the full symmetry group
that is generated by any two of the reflections
$G=\langle\mathcal{M}_{i},\,\mathcal{M}_{j}\rangle$ with $i\neq j$ is finite
and the prepotential is not modular. In fact, from Table 1 we see that
$Q^{3}=(M_{i}M_{j})^{3}=\mathbbm{1}$. In particular, we can easily write down
all the words of $G$ and the sum over the group elements in (1.1) explicitly.

However, if we consider the subgroup generated by three of the
$\mathcal{M}_{i}$, say
$G=\langle\mathcal{M}_{1},\,\mathcal{M}_{2},\,\mathcal{M}_{3}\rangle$, we can
write down infinitely many words, e.g. of the type
$(\mathcal{M}_{1}\mathcal{M}_{2}\mathcal{M}_{3})^{k}$. These contain entries
that are quadratic, linear, and constant in $k$ and could hence also lead to
modular behavior. A detailed study of this is beyond the scope of the current
paper, which focuses on the subgroups generated by two involutions, but it
would be very interesting to understand the implications of the involutions
and the resulting extended Kähler cone structure for the mirror. The Coxeter
diagram corresponding to this case is given in Figure 3 on the right.

##  5 Coxeter groups, triangle groups, and reflection symmetries

A Coxeter group is defined as a group with presentation

$\displaystyle
W=\langle\mathcal{M}_{1},\;\mathcal{M}_{2},\;\ldots,\mathcal{M}_{r}~{}~{}|~{}~{}(\mathcal{M}_{i}\mathcal{M}_{j})^{c_{ij}}=\mathbbm{1}\rangle\,.$
(5.1)

Here, $\mathcal{M}_{i}$ are generators, $c_{ij}=c_{ji}$, $c_{ii}=1$, and
$c_{ij}\geq 2$ for $i\neq j$. The condition $c_{ii}=1$ implies that
$(\mathcal{M}_{i}\mathcal{M}_{i})^{1}=\mathbbm{1}$, so all generators
$\mathcal{M}_{i}$ are involutions. If there is no relation of the form
$(\mathcal{M}_{i}\mathcal{M}_{j})^{c_{ij}}$, one sets $c_{ij}=\infty$. The
symmetric matrix $C$ with entries $c_{ij}$ is called the Coxeter matrix.
Conversely, any symmetric matrix with ones on the diagonal and entries
$2,3,\ldots,\infty$ otherwise defines a Coxeter group. The pair $(W,\Gamma)$
where $W$ is the Coxeter group and
$\Gamma=\\{\mathcal{M}_{1},\mathcal{M}_{2},\ldots,\mathcal{M}_{r}\\}$ are the
generators is called a Coxeter system.

A Coxeter group can be represented by its Coxeter diagram which is constructed
from the Coxeter matrix as follows. First, each generator is represented by a
node. Each two nodes $i$ and $j$ are connected with an edge labeled by
$c_{ij}$ if $c_{ij}\geq 3$; in the case where $c_{ij}=3$ the edge label is
omitted. The Coxeter matrix $C_{ij}$ is closely related to the Schläfli matrix
$S$ with entries $s_{ij}=-2\cos(\pi/c_{ij})$. If the eigenvalues of $S$ are
all positive, the Coxeter group is finite. If all eigenvalues are non-negative
(and at least one zero), the Coxeter group is affine. Otherwise, the Coxeter
group is of indefinite type. Indefinite Coxeter groups with $1$ negative and
$r-1$ positive eigenvalue are sometimes called hyperbolic Coxeter groups.

Coxeter [7] studied these groups originally in the context where the
involutions $M_{i}$ are reflections along hyperplanes. A canonical
representation of a Coxeter group is then in terms of reflection matrices.
This representation is constructed by associating to $\mathcal{M}_{i}$ a basis
$\mathcal{M}_{i}\mapsto e_{i}$ of $\mathbbm{R}^{n}$ and by defining a bi-
linear form $\mathcal{I}$ and reflections $\sigma_{i}$ in $\mathbbm{R}^{n}$
via the Schläfli matrix

$\displaystyle
S_{ij}:=\mathcal{I}(e_{i},e_{j})=-2\cos\frac{\pi}{c_{ij}}=s_{ij}\,,\qquad\sigma_{i}(v)=v-s_{ij}e_{i}\,.$
(5.2)

If $c_{ij}=\infty$, we set $s_{ij}=-2$. Then, the representation
$W\to\text{GL}(n,\mathbbm{R})$ defined by $\mathcal{M}_{i}\mapsto\sigma_{i}$
is faithful. Moreover, it can be checked that the reflections $\sigma_{i}$ are
isometries of the bi-linear form $\mathcal{I}$, so
$\mathcal{I}(\sigma_{i}(v),\sigma_{j}(w))=\mathcal{S}(v,w)$ for all
$v,w\in\mathbbm{R}^{n}$.

The question of when a group generated by involutions that are not necessarily
reflections along hyperplanes) is a Coxeter group is more difficult, but
criteria (like the exchange condition) that describe the behavior of reduced
words, are known. Vinberg generalized the concept of reflecting along
hyperplanes to involutions across polyhedral cones [8]. This is precisely the
setup we need for our discussion.

Figure 3: Coxeter graphs of the groups $G$ for the examples discussed in this
paper. Left: Coxeter diagram of CICY 7863 from the example in Section 3.4 and
CICY 7880 from the example in Section 4.3.1. Right: Coxeter diagram of CICY
7447, the mirror of the Hulek-Verrill manifold, from the example in Section
4.3.2.

Vinberg starts from involutions $\mathcal{M}_{i}$ of the form (4.1) and shows
that the Cartan matrix 444The matrices constructed this way are the Cartan
matrices of semisimple Lie algebras, where the reflections are those along the
simple roots that generate the Weyl group. $A$, whose columns are given by the
vectors $u_{i}$ in (4.1), satisfies $u_{ij}=0$ if $u_{ji}=0$. This has been
used in Section 4.1 to argue that the case $m_{1}m_{2}=0$ implies
$m_{1}=m_{2}=0$. Vinberg also shows that either $u_{ij}\geq 4$ (in which case
the entry $c_{ij}$ in the Coxeter matrix is $c_{ij}=\infty$) or
$u_{ij}u_{ji}=s_{ij}s_{ji}$. This precisely reproduces Table 1. From this
discussion, we see that the groups $G$ (or $\tilde{G}$) acting on the Mori
cone (or Kähler cone) are Coxeter groups with generators $\mathcal{M}_{i}$.
Their rank $r$ corresponds to the number of facets of the Kähler cone across
which isomorphic flops occur.

The boundary case $m_{i}m_{j}=4$ has a geometric realization as reflections
along hyperplanes as studied by Coxeter, while $m_{i}m_{j}>4$ corresponds to
more general reflections as considered by Vinberg. In either case, we have
$c_{ij}=\infty$ for $i\neq j$ and the Schläfli matrix has only entries of 2
(positive on the diagonal, negative elsewhere). The eigenvalues of this matrix
are $\lambda_{1}=\lambda_{2}=\ldots=\lambda_{r-1}=4$ and $\lambda_{r}=r-2$.
Hence, the case $r=2$, corresponds to an affine Coxeter algebra called
$\tilde{I}_{1}$ (see left part of Figure 3), while the cases with $r>2$ are
hyperbolic Coxeter groups. Thus, in these cases, the Coxeter groups are given
by fully connected graphs where each edge carries the label $\infty$. These
are precisely the universal Coxeter groups, given by the free product of $r$
$\mathbbm{Z}_{2}$ factors. For the examples with two and five generators
studied earlier, the Coxeter graphs are shown in Figure 3. In this paper, we
have focused on (sub-)groups $\tilde{G}$ with two generators, so all the
infinite reflection groups in this paper are isomorphic to the affine case
$\tilde{I}_{1}$. For such cases, terminology for CY moduli spaces can be
translated into terminology for Coxeter groups. Specifically, under the group
isomorphism $\tilde{G}\rightarrow\tilde{I}_{1}$, the Kähler cone is identified
with the fundamental chamber, the Kähler cone walls that admit flops to
isomorphic CYs are identified with the walls of the fundamental chamber, and
the extended Kähler cone is identified with the Tits cone.

Next, we want to briefly study the group elements of a Coxeter group. In
general, two or more words can represent the same group element (irrespective
of their lengths). This is even true for reduced words, which means that
consecutive inverses are deleted. In the context of Coxeter groups, each
generator is self-inverse, which means that in reduced words, the same letter
does not appear more than once consecutively. For universal Coxeter groups,
each reduced word corresponds to a unique group element. Hence, each word $w$
corresponding to a $g\in W$ can be written as
$\mathcal{M}_{i_{1}}\mathcal{M}_{i_{2}}\cdots\mathcal{M}_{i_{n}}$ with
$i_{k}\neq i_{k+1}$. This is reflected (no pun intended) in our choice of
generators $\mathcal{S}=\mathcal{M}_{1}$ and
$\mathcal{Q}=\mathcal{M}_{1}\mathcal{M}_{2}$ for the group $G$.

Finally, let us briefly comment on the relation to triangle groups. Triangle
groups are groups which are realized by reflections along the sides of a
triangle. They are specified in terms of three integers $(l,m,n)$ which
correspond to angles $\pi/l$, $\pi/m$, $\pi/n$ of the triangle. If
$1/l+1/m+1/n=1$, the triangles tessellate the Euclidean plane, if
$1/l+1/m+1/n>1$ they tessellate the unit sphere, and if $1/l+1/m+1/n<1$ they
tessellate the hyperbolic plane. As mentioned above, the non-finite cases with
$m_{a}m_{b}\geq 4$ and $r\geq 3$ correspond to hyperbolic Coxeter groups.
These are related to the hyperbolic triangle groups (for $r=3$) and their
generalizations: The integers $(k,l,m)$ correspond to the entries $c_{1,2}$,
$c_{1,3}$ and $c_{2,3}$ of the Coxeter matrix. This means that the general
case, which leads to universal Coxeter groups, corresponds to the triangle
group $(\infty,\infty,\infty)$.

Figure 4: The triangle group $(\infty,\infty,\infty)$ on the Poincaré disk,
which corresponds to the universal Coxeter group of rank 3. In the context of
the infinite flop examples for Picard rank 3 CYs discussed here, each arc
corresponds to a flop wall. Arcs with the same color are reflection images of
one another.

As can be seen from the examples discussed in Section 4, or just by noting how
$(\mathcal{Q}^{T})^{k}$ behaves, the $\tilde{G}$-images of the original CICY
Kähler cone become thinner and thinner slivers (in the cohomology basis of the
CICY Kähler cone, which is the positive octant), accumulating towards the
boundaries of the extended Kähler cone. They can be resolved better in
hyperbolic space, for example by plotting them on the Poincaré disk, whose
boundary represents infinity and where straight lines are arcs (or diameters
of the disk). We plot the triangle group $(\infty,\infty,\infty)$ in Figure 4.
Under the isomorphism between Coxeter groups and reflections along flop walls
of the Kähler cone, each arc corresponds to one Kähler cone wall. Walls that
are identified under the group $\tilde{G}$ are plotted in the same color.

##  6 Conclusions

In this paper, we have studied CY three-folds with isomorphic flops, the
resulting reflection symmetries and their implications for the Gopakumar-Vafa
(GV) invariants and the Kähler moduli prepotential. We have seen that such
isomorphic flops lead to symmetry groups $\tilde{G}$ on the Kähler moduli
space, and dual symmetry groups $G$ acting on curves, which are isomorphic to
Coxeter groups. These groups can be of finite or infinite order, depending on
the manifold, and the rank of the Coxeter group is given by the number of
Kähler cone facets across which flops occur. The key observation is that GV
invariants $n_{d}$ for non-flopping curve classes $d$ are $G$-invariant, that
is, $n_{gd}=n_{d}$ for all $g\in G$. This means the (non-flopping part of the)
instanton prepotential can be written in terms of suitable $\tilde{G}$
invariant functions, as defined in Eq. (1.1). To our knowledge, these
functions, which are invariant under certain representations of Coxeter
groups, have not been studied before.

We have analyzed in some detail the case of $h^{1,1}(X)=2$ and isomorphic
flops across both facets of the Kähler cone which leads to a symmetry group
$G$ controlled by two integers $m_{1}$ and $m_{2}$. These integers can be
computed from the triple intersection numbers via Eq. (2.1). For Picard rank
two these numbers always satisfy $m_{1}m_{2}>4$ and this leads to a symmetry
group $\tilde{G}\cong\mathbb{Z}_{2}\rtimes\mathbb{Z}$, isomorphic to a
universal Coxeter group of rank two. In this case, the associated $\tilde{G}$
invariant functions are non-modular.

It turns out, for Picard rank $h^{1,1}(X)>2$ and two isomorphic flop
boundaries, all three cases, $m_{1}m_{2}<4$, $m_{1}m_{2}=4$ and $m_{1}m_{2}>4$
can be realized by appropriate CY manifolds. Cases with $m_{1}m_{2}<4$ lead to
a finite Coxeter group $\tilde{G}$, while for $m_{1}m_{2}\geq 4$ the group
$\tilde{G}$ is a universal Coxeter group of rank two. If $m_{1}m_{2}>4$ the
$\tilde{G}$ invariant functions are non-modular, as they were for Picard rank
two. The interesting new case is the limiting one, $m_{1}m_{2}=4$, where the
$\tilde{G}$ invariant functions can be expressed in terms of Jacobi theta
functions. We also argued that the appearance of theta functions is related to
the presence of elliptic fibrations in the underlying CYs of the type studied
in Ref. [4].

For manifolds with more than two isomorphic flop boundaries the structure is
more complicated. While the group $G$ is isomorphic to a Coxeter group, it is
not clear how to write down the group elements systematically. It is,
therefore, difficult to work out the form of the $\tilde{G}$ invariant
functions more explicitly. However, we have argued, for the example of the
Hulek-Verrill manifold, that modularity can arise for (sub-)groups $\tilde{G}$
with more than two generators. Moreover, the example of the Hulek-Verrill
manifold illustrates that the full symmetry group can have infinitely many
elements even though the rank two subgroup has $m_{1}=m_{2}=1$ and is thus
finite. The detailed analysis of manifolds with such larger symmetry groups
and their prepotentials is an interesting direction for future work.

### Acknowledgments

We thank Callum Brodie, Albrecht Klemm, Wolfgang Lerche, Guglielmo Lockhart,
and Thorsten Schimmanek for useful discussions. The work of FR is supported by
the NSF grant PHY-2210333, by the NSF under Cooperative Agreement PHY-2019786
(The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions), and by startup funding from Northeastern University.

##  Appendix A Appendix: Special symmetry groups and elliptic fibrations

In this appendix we would like to study the CICYs with $m_{1}m_{2}\leq 4$
somewhat more systematically and we will argue that these cases lead to
elliptic fibrations. To do so, we distinguish the two types of CICYs that lead
to infinitely many isomorphic flops, as identified in Ref. [4], separately. As
we will show, type 1 CICYs lead to $(m_{1},m_{2})\in\\{(1,1),(2,2)\\}$ only,
while type 2 CICYs allow for $(m_{1},m_{2})\in\\{(1,4),(4,1),(2,2)\\}$. The
cases $(m_{1},m_{2})\in\\{(1,2),(1,3)\\}$, which also appear in Table 1, are
not realized by CICYs. One example of this is the mirror dual of the Hulek-
Verrill manifold as discussed in Section (4.3.2). It is intriguing that theta
functions also appear for this manifold, albeit only once three or more
reflections are considered.

###  A.1 Type 1

For type 1 CICYs, the configuration matrix is of the form

$\displaystyle\left[\begin{array}[]{c|cccccc}\mathbbm{P}^{n}&1&1&\ldots&1&0&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}&\vec{q}&\ldots&\vec{q}&\vec{q}_{n+1}&\ldots\end{array}\right]$
(A.3)

The reflection matrix $\mathcal{M}_{1}$ is given by (4.2) with

$\displaystyle u=(2,-n\vec{q}^{\;T})\,.$ (A.4)

Now we want this row to contain an entry $m_{1}\leq 4$, such that, when
combined with the second flop that will generate the infinite flop chain, we
can have $m_{1}m_{2}\leq 4$. Since $n$ is the dimension of the ambient space
$\mathbbm{P}^{n}$ factor, we know that $n\geq 1$. Since all $\vec{q}$ have to
be the same in type 1 flops, they actually cannot be combined with a type 2
flop in a sensible way: In a type 2 flop, the entries that occur more than
once are 0’s and 1’s. If all $\vec{q}$ contain a zero row, this means that the
configuration matrix is block-diagonal and hence the complete intersection is
a direct product. More precisely, we would get two linear equations in
$\mathbbm{P}^{1}$ times the CICY defined by
$[\vec{\mathbbm{P}}~{}|~{}\vec{q}_{n+1}~{}~{}\ldots]$, which has no solution
for generic CS, or just reproduces the original CICY over special points in CS
space. Either one is not interesting. If the $\vec{q}$ contain a row with 1’s,
we already get an entry $m_{2}$ that is at least $m_{2}=2+2\cdot 2=6$
according to (A.15). So we can focus on the case where the second flop is of
the same type as the first flop.

In such a setup, because of (A.4), we can focus on $1\leq n\leq 3$, and
possible configuration matrices are of the form

$\displaystyle\left[\begin{array}[]{c|cccc}\mathbbm{P}^{1}&1&1&0&\ldots\\\
\mathbbm{P}^{1}&1&1&0&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}&\vec{q}&\vec{q}_{3}&\ldots\end{array}\right]\,,\quad\left[\begin{array}[]{c|ccccc}\mathbbm{P}^{2}&1&1&1&0&\ldots\\\
\mathbbm{P}^{2}&1&1&1&0&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}&\vec{q}&\vec{q}&\vec{q}_{4}&\ldots\end{array}\right]\,,\hskip
56.9055pt$ (A.11)

giving rise to $(m_{1},m_{2})=(1,1)$ and $(m_{1},m_{2})=(2,2)$, respectively.

We can argue next that CICYs of the type (A.11) are always elliptically
fibered. To do so, we use a criterion due to Oguiso and Kollar. This comes
down to showing that there exists an effective divisor $D_{*}$ such that
$D_{*}.C\geq 0$, $D_{*}.D_{*}\neq 0$, and $D_{*}.D_{*}.D_{*}=0$ for any
algebraic curve $C\subset X$. So in essence we get a basis of divisors by
pullbacks from the ambient space, compute their triple intersection numbers,
and look for a positive linear combination
$D_{*}=\sum_{i=1}^{h^{11}}a_{i}D_{i}$ such that $D_{*}^{2}\neq 0$ and
$D_{*}^{3}=0$ ($D.C\geq 0$ is automatic for effective divisors and curves).
For CICYs of the first form, the divisors $D_{1}$ and $D_{2}$ corresponding to
the two ambient space $\mathbbm{P}^{1}$ factors are good candidates. They
automatically have $D^{3}=0$. However, they also have $D^{2}=0$. This can be
easily cured by taking the linear combination $D_{*}=D_{1}+D_{2}$. Now
$D_{*}^{2}=(D_{1}+D_{2})^{2}=2D_{1}.D_{2}$ fixes a point in
$\mathbbm{P}^{1}\times\mathbbm{P}^{1}$ on the CY, and will intersect some
other effective divisor with non-zero intersection number, i.e., we have an
elliptic fibration. For CICY’s of the last type, either of the divisors
$D_{1}$ and $D_{2}$ satisfies $D_{i}^{2}\neq 0$ and $D_{i}^{3}=0$ on
dimensional grounds. The other cases cannot be treated in generality.

###  A.2 Type 2

For type 2 CICYs, the configuration matrix is of the form

$\displaystyle\left[\begin{array}[]{c|cccccc}\mathbbm{P}^{n}&2&1&\ldots&1&0&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}_{1}&\vec{q}_{2}&\ldots&\vec{q}_{n}&\vec{q}_{n+1}&\ldots\end{array}\right]$
(A.14)

The reflection matrix $\mathcal{M}_{1}$ along the $\mathbbm{P}^{n}$ direction
will be given by (4.2) with

$\displaystyle u=(2,\;-\vec{q}_{1}^{\,T}-2\sum_{i=2}^{n}\vec{q}_{i}^{\,T})\,.$
(A.15)

In fact, since we want infinitely many flops, we need a second direction along
which we can flop (say the second ambient space factor. That means the
configuration matrix is of the form

$\displaystyle\left[\begin{array}[]{c|ccccccc}\mathbbm{P}^{n}&2&1&\ldots&1&0&\ldots\\\
\mathbbm{P}^{\tilde{n}}&q_{1}&q_{2}&\ldots&q_{n}&q_{n+1}&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}_{1}^{\;{}^{\prime}}&\vec{q}_{2}^{\;{}^{\prime}}&\ldots&\vec{q}_{n}^{\;{}^{\prime}}&\vec{q}_{n+1}^{\;{}^{\prime}}&\ldots\end{array}\right]$
(A.19)

The part with $\vec{\mathbbm{P}}$ is irrelevant for our subsequent discussion,
so we will focus on the first two rows. We want to find entries
$q_{1},q_{2},\ldots$ such that the resulting rows have entries $m_{1}$ and
$m_{2}$ with $m_{1}m_{2}\leq 4$. In principle we can choose
$q_{1},q_{2},\ldots$ such that we get a type 1 or a type 2 flop. For a type 1
flop, we need the $q_{i}$ to be either 1 or 0. If they are 1, then the rest of
the matrix has to be the same for all positions where a 1 occurs. This means
in particular that $q_{1}=0$, since there is only a single entry $2$ in the
first row. Similarly, since the $m_{1}$ and the $m_{2}$ should not “miss” each
other, we cannot use the entries $q_{n+1}$, $q_{n+2}$, etc. and can set these
to zero as well. In other words, we can choose a subset of $q_{2}$ to $q_{n}$
to be 1, and the rest is zero. However, as we can see from (A.15), the entries
$q_{2}$, $q_{3}$, $\ldots$, $q_{n}$ come with a factor of 2, so
$m_{1}m_{2}\leq 4$ would require only one of these being non-zero, which would
mean an ambient space $\mathbbm{P}^{0}$. So we cannot generate an infinite
flop chain with $m_{1}m_{2}\leq 4$ by using types 1 and types 2. This leaves
us with choosing the $q_{i}$ of the second row to be of type 2, i.e., a single
two, and a bunch of ones and zeros. By choosing the $q_{i}$ appropriately, we
can engineer examples with $m_{1}m_{2}=4$ in three different ways:

The first way is to set $q_{1}=2$, $q_{i}=0$ for $i=2,\ldots,n$, and choosing
the remaining $q_{i}$ for $i\geq n+1$ to be either 1 or 0 (to obtain a type 2
flop). This gives $(m_{1},m_{2})=(2,2)$ and a configuration matrix of the form

$\displaystyle\left[\begin{array}[]{c|ccccccc}\mathbbm{P}^{n}&2&1&\ldots&1&0&\ldots\\\
\mathbbm{P}^{\tilde{n}}&2&0&\ldots&0&q_{n+1}&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}_{1}^{\;{}^{\prime}}&\vec{q}_{2}^{\;{}^{\prime}}&\ldots&\vec{q}_{n}^{\;{}^{\prime}}&\vec{q}_{n+1}^{\;{}^{\prime}}&\ldots\end{array}\right]$
(A.23)

The second way is to set $q_{1}=1$, $q_{i}=0$ for $i=2,\ldots,n$, and choosing
one of the remaining $q_{i}$ for $i\geq n+1$ to be 2 and all others to be
either 1 or 0 (to obtain a type 2 flop). This gives $(m_{1},m_{2})=(1,4)$ and
a configuration matrix of the form

$\displaystyle\left[\begin{array}[]{c|cccccccc}\mathbbm{P}^{n}&2&1&\ldots&1&0&0&\ldots\\\
\mathbbm{P}^{\tilde{n}}&1&0&\ldots&0&2&q_{n+2}&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}_{1}^{\;{}^{\prime}}&\vec{q}_{2}^{\;{}^{\prime}}&\ldots&\vec{q}_{n}^{\;{}^{\prime}}&\vec{q}_{n+1}^{\;{}^{\prime}}&\vec{q}_{n+2}^{\;{}^{\prime}}&\ldots\end{array}\right]$
(A.27)

The third way is to to set $q_{1}=0$, precisely one of the $q_{i}$,
$i=2,\ldots,n$ to one, and choosing one of the remaining $q_{i}$ for $i\geq
n+1$ to be 2 and all others to be either 1 or 0 (to obtain a type 2 flop).
This also gives $(m_{1},m_{2})=(2,2)$ and a configuration matrix of the form

$\displaystyle\left[\begin{array}[]{c|ccccccccc}\mathbbm{P}^{n}&2&1&1&\ldots&1&0&0&\ldots\\\
\mathbbm{P}^{\tilde{n}}&0&1&0&\ldots&0&2&q_{n+2}&\ldots\\\
\vec{\mathbbm{P}}&\vec{q}_{1}^{\;{}^{\prime}}&\vec{q}_{2}^{\;{}^{\prime}}&\vec{q}_{3}^{\;{}^{\prime}}&\ldots&\vec{q}_{n}^{\;{}^{\prime}}&\vec{q}_{n+1}^{\;{}^{\prime}}&\vec{q}_{n+2}^{\;{}^{\prime}}&\ldots\end{array}\right]$
(A.31)

As examples for CICYs that realize these three possibilities, we can take the
tetra-quadric, CICY 5299 (these even have flops along all ambient space
factors), and CICY 6971, respectively.

$\displaystyle X_{\text{TQ}}\sim\left[\begin{array}[]{c|c}\mathbbm{P}^{1}&2\\\
\mathbbm{P}^{1}&2\\\ \mathbbm{P}^{1}&2\\\
\mathbbm{P}^{1}&2\end{array}\right]\,,\qquad
X_{5299}\sim\left[\begin{array}[]{c|ccc}\mathbbm{P}^{2}&2&1&0\\\
\mathbbm{P}^{2}&1&0&2\\\ \mathbbm{P}^{2}&0&2&1\end{array}\right]\,,\qquad
X_{\text{6971}}\sim\left[\begin{array}[]{c|ccc}\mathbbm{P}^{2}&2&1&0\\\
\mathbbm{P}^{2}&0&1&2\\\ \mathbbm{P}^{2}&1&1&1\end{array}\right]\,.$ (A.42)

Note that for all three possibilities, we get $(n+\tilde{n}-1)$ equations that
involve the ambient spaces $\mathbbm{P}^{n}\times\mathbbm{P}^{\tilde{n}}$.
This is because there are $n$ (resp. $\tilde{n}$) non-zero entries in the
first (resp. second) row, and precisely in one column, both the first and
second row have a non-zero entry. Hence we get $(n+\tilde{n}-1)$ equations
inside $\mathbbm{P}^{n}\times\mathbbm{P}^{\tilde{n}}$, i.e., a CY one-fold.
This is what the authors of Ref. [18] call an “obvious elliptic fibration”.

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# Strong rate of convergence of the Euler scheme for SDEs with irregular drift
driven by Lévy noise

Oleg Butkovsky Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany
<EMAIL_ADDRESS>Konstantinos Dareiotis University of Leeds,
Woodhouse, LS2 9JT Leeds, United Kingdom<EMAIL_ADDRESS>and Máté
Gerencsér TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract

We study the strong rate of convergence of the Euler–Maruyama scheme for a
multidimensional stochastic differential equation (SDE)

$dX_{t}=b(X_{t})dt+dL_{t},$

with irregular $\beta$-Hölder drift, $\beta>0$, driven by a Lévy process with
exponent $\alpha\in(0,2]$. For $\alpha\in[2/3,2]$ we obtain strong $L_{p}$ and
almost sure convergence rates in the whole range $\beta>1-\alpha/2$, where the
SDE is known to be strongly well–posed. This significantly improves the
current state of the art both in terms of convergence rate and the range of
$\alpha$. In particular, the obtained convergence rate does not deteriorate
for large $p$ and is always at least $n^{-1/2}$; this allowed us to show for
the first time that the the Euler–Maruyama scheme for such SDEs converges
almost surely and obtain explicit convergence rate. Furthermore, our results
are new even in the case of smooth drifts. Our technique is based on a new
extension of the stochastic sewing arguments.

##  1 Introduction

We consider the stochastic differential equation

$dX_{t}=b(X_{t})\,dt+dL_{t},\quad t\geq 0,\quad X_{0}=x_{0},$ (1.1)

driven by a $d$-dimensional Lévy process $L$. Here the coefficient $b$ is a
measurable function $\mathbb{R}^{d}\to\mathbb{R}^{d}$, and the initial
condition $x_{0}\in\mathbb{R}^{d}$. Throughout the article the dimension
$d\in\mathbb{N}=\\{1,2,\ldots\\}$ is arbitrary.

The ‘strength’ of a Lévy process can often be characterised by a single
parameter $\alpha\in(0,2]$ called the _stable index_ (for various examples see
LABEL:Sec:ex), with $\alpha=2$ corresponding to the usual Brownian motion.
This parameter can be used to describe the regularisation provided by $L$.
Indeed, assuming a natural nondegeneracy condition (which essentially requires
that for $\alpha\neq 2$ the span of possible jumps is $\mathbb{R}^{d}$), (1.1)
has a unique strong solution whenever $b$ belongs to the Hölder space
$\mathcal{C}^{\beta}(\mathbb{R}^{d})$, where $\beta$ satisfies

$\beta>1-\frac{\alpha}{2}.{\hbox{\rm(1.2)}\kern
5.69046pt\kern-5.69046pt}q:exponents-WP{=}e1.1{1.2}$