Flops of any length, Gopakumar-Vafa invariants and 5d Higgs Branches

Andrés Collinuccia, Mario De Marcob, Andrea Sangiovannic and Roberto Valandroc

a Service de Physique Théorique et Mathématique, Université Libre de Bruxelles
and
International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles,
Belgium
b SISSA and INFN, Via Bonomea 265, I-34136 Trieste, Italy
c Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, I-34151
Trieste, Italy
and INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy

collinucci dot phys at gmail dot com mdemarco at sissa dot it

andrea dot sangiovanni at phd dot units dot it roberto dot valandro at ts dot
infn dot it

Abstract

The conifold is a basic example of a noncompact Calabi-Yau threefold that
admits a simple flop, and in M-theory, gives rise to a 5d hypermultiplet at
low energies, realized by an M2-brane wrapped on the vanishing sphere. We
develop a novel gauge-theoretic method to construct new classes of examples
that generalize the simple flop to so-called _length_ $\ell=1,\ldots,6$. The
method allows us to naturally read off the Gopakumar-Vafa invariants. Although
they share similar properties to the beloved conifold, these threefolds are
expected to admit M2-bound states of higher degree $\ell$. We demonstrate this
through our computations of the GV invariants. Furthermore we fully
characterize the associated Higgs branches by computing their dimensions and
flavor groups. With our techniques we extract more refined data such as the
charges of the hypers under the flavor group.

###### Contents

  1. 1 Introduction
  2. 2 ADE families and simultaneous resolution
  3. 3 Threefolds as ALE families, GV invariants and $\Phi$
    1. 3.1 The Higgs field from the simultaneous resolution
    2. 3.2 The threefold equation from $\Phi$
    3. 3.3 Gopakumar-Vafa invariants of $X$ from zero modes of $\Phi$
  4. 4 Simple flops with $l=1,...,6$ and their GV invariants
    1. 4.1 Simple flop with length $1$
    2. 4.2 Simple flop with length $2$
    3. 4.3 Simple flop with length $3$
    4. 4.4 Simple flop with length $4$
    5. 4.5 Simple flop with length $5$
    6. 4.6 Simple flop with length $6$
  5. 5 Conclusions
  6. A The threefold equation: The Katz-Morrison method
  7. B Casimirs for the equation of $E_{r}$ families
  8. C Milnor number and zero modes
  9. D Mathematica code for computing the zero modes

##  1 Introduction

Singular noncompact Calabi-Yau threefolds are interesting objects both from a
mathematical as well as a physics point of view. In physics, by defining
string theory or M-theory on them, they can provide the framework for modeling
effective supersymmetric gauge theories in four and five dimensions, with
gravity decoupled. This can happen via the _branes at singularities_ paradigm,
whereby a bunch of D3-branes probes such a singularity, or via the _geometric
engineering_ paradigm, whereby the field theory content emanates from
membranes wrapping the vanishing cycles of the singularity. We will focus on
the latter.

The prototypical example, the _conifold_ , is a threefold that can be
described as a hypersurface in $\mathbb{C}^{4}$. It admits a so-called _small
resolution_ , which means that the singularity is replaced by a
$\mathbb{P}^{1}$ (unlike other singularities where the exceptional locus is of
complex dimension two). This sphere can flop to negative volume, and a
homologically inequivalent sphere of positive area grows.

The conifold is but the first in a series of threefolds that admit _simple
flops_. Here, ‘simple’ refers to the fact that only one sphere grows after
resolution (as opposed to a chain of spheres). The conifold is known as a
simple flop of _length one_. Here, the _length_ of a flop refers to a
multiplicity that can be attributed to the exceptional $\mathbb{P}^{1}$. If
one pulls back the skyscraper sheaf corresponding to the singular point of the
conifold w.r.t. the blow-down map, one will get the structure sheaf of the
exceptional curve.

$\pi^{*}(\mathcal{O}_{p})=\mathcal{O}_{\mathbb{P}^{1}}\,.$ (1.1)

It has been known for decades that other _lengths_ are possible, up to $l=6$,
such that

$\pi^{*}(\mathcal{O}_{p})=\mathcal{O}_{\mathbb{P}^{1}}^{\oplus l}\,.$ (1.2)

Even though the above definition may appear rather obscure, the length has a
very concrete physical meaning: as we will see later, when we geometrically
engineer M-theory on simple threefold flops described as deformed ADE
singularities, we find that the length of the simple flop is nothing but the
maximal flavor charge of the 5d hypermultiplets arising from M2 branes wrapped
on the exceptional $\mathbb{P}^{1}$. We will get into the details of these
aspects in later sections.

In order to construct such higher length flops, the strategy in [1] has been
to consider families of deformed ADE surfaces. In other words, fibering local
K3’s over the deformation parameter space. In general, the total space of the
families may or may not be singular. Resolving the possible singularity
inflates part of the exceptional $\mathbb{P}^{1}$’s associated to the simple
roots of the ADE algebra associated with the family. Given such a family,
which will give rise to an $n$-fold, one can always cut out a complex three-
dimensional subvariety, which will be guaranteed to give a Gorenstein
threefold.

In this paper, following [2, 3], we develop a new method to construct simple
flops spanning all the possible values of the length of the exceptional
$\mathbb{P}^{1}$ (up to the maximal value $l=6$), based on a gauge-theoretic
approach. To this end, we consider M-theory on an ALE surface with an ADE
singularity. The 7d effective theory is a SYM theory with three adjoint
scalars and group $G=A,D,E$ with $\mathcal{N}=1$ supersymmetry. One can break
half of the supercharges by switching on a BPS vev for the scalar fields in
the following way:

  * •

Take the 7d spacetime to be $\mathbb{R}^{5}\times\mathbb{C}$. Call $w$ the
local coordinate on $\mathbb{C}$.

  * •

Split the tree real scalars into a complex adjoint scalar
$\Phi=\phi_{1}+i\phi_{2}$ and a real adjoint scalar $\phi_{3}$.

  * •

Take a holomorphic vev for $\Phi$ that depends on a coordinate $w$, transverse
to the local K3.

This preserves only a 5d Poincaré group. The 7d gauge group is broken to the
commutant of $\Phi$. The 7d $\mathcal{N}=1$ vector multiplet was made up of
the gauge field $A_{M}$ ($M=0,1,...,6$) and the three scalars $\phi_{i}$
($i=1,2,3$); after giving a vev to $\Phi$, only a 5d $\mathcal{N}=1$ vector
multiplet $(A_{\mu},\phi_{3})$ in the subalgebra $\mathcal{H}$ that commutes
with $\Phi$ survives (these vector multiplets still propagate in 7d, and are
then seen as background vector multiplets from the 5d point of view). As we
will see in detail in the main text, there are also localized $\Phi$-modes at
$w=0$ that organize in hypermultiplets charged under $\mathcal{H}$ and that
propagate in 5d.

The fields $\Phi$ and $\phi_{3}$ can be given two physical interpretations: In
the A or D series, the local K3 has a $\mathbb{C}^{*}$-fibration structure; we
can then reduce M-theory to IIA, and regard them as the three adjoint scalars
or a stack of D6-branes. For the E series, we can consider F-theory on an
elliptically fibered K3, giving rise to 8d SYM. Physically, $\Phi$ is the
adjoint Higgs on the stack of non-perturbative sevenbranes. By further
compactifying on a circle, we arrive at 7d SYM with exceptional gauge group.

The vev’s of $\Phi$ and $\phi$ encode the complex structure and Kähler moduli
of the local K3, respectively. When both are zero, the surface is singular.
Activating $\phi$ blows up spheres of non-zero Kähler volume. However,
activating $\Phi$ will switch on _versal deformations_ of the local K3, which
will unfold the singularity according to its Casimir invariants. From the
geometric point of view, we will obtain a fibration of deformed ALE surfaces
over the $w$-plane, giving rise to a threefold $X$ that admits a simple flop.
Reducing M-theory directly on $X$ produces the 5d spectrum of propagating
hypermultiplets and background vector multiplets outlined before.

Resolving the full fibration is known as a _simultaneous resolution_. Given
the Dynkin diagram of the gauge algebra in question, a simultaneous resolution
corresponds to a choice of simple roots. This selects, the subspace
$\mathcal{H}$ of the Cartan subalgebra $\mathfrak{t}$ of the original ADE
algebra where $\phi_{3}$ lives. The commutant of $\phi_{3}$ tells us what is
the form of a generic $\langle\Phi\rangle$ producing the sought-after family.
The M-theory threefold is then obtained by choosing the $w$-dependence of the
Casimir invariants of $\Phi$, which uniquely fix the defining equation of the
threefold.

At this point, one can compute the number of hypermultiplets of the 5d
effective theory, by counting the zero modes of the complex adjoint $\Phi$
that are localized around the singularity. Their number corresponds to the BPS
index of M2-states wrapping the vanishing $\mathbb{P}^{1}$’s, aka the
Gopakumar-Vafa invariants. In this paper, we are able to compute the charge of
each hypermultiplet with respect to the flavor group, which gives us not only
the multiplicity of states, but the actual refined information of the GV
invariants, which are organized by homology classes of the threefold. This
furthermore allows us to compute detailed information about the 5d Higgs
branch: Its dimension is equal to the sum of all GV invariants while the
action of the flavor group is determined by the charges of the hypers.

The paper is organized as follows. In Section 2 we review the families of
deformed ADE singularities. In Section 3 we explain how the choice of spheres
that are blown up in the simultaneous resolution determines the vev for
$\Phi$; by using this $\Phi$ we then show how to extract the hypersurface
equation of the threefold and its GV invariants. In Section 4 we apply our
method to simple flops, by constructing explicitly threefolds with any length
from $1$ to $6$. In Section 5 we draw our conclusions. We add four appendices:
in Appendix A we outline the Katz-Morrison method to derive the threefold
equation; in Appendix B we write some long formulae for the exceptional cases,
necessary to compute the corresponding threefold equations; in Appendix C we
illustrate a check of our computations by using the Milnor number of the
considered singularities; in Appendix D we expose the Mathematica code we
worked out to compute the zero modes once one chooses a specific $\Phi$.

##  2 ADE families and simultaneous resolution

In this section, we briefly summarize the philosophy and method of [1] for
building simultaneous resolutions, by fibering ALE surfaces over the Cartan
torus of an ADE group. Consider an ALE surface in $\mathbb{C}^{3}$ with an ADE
singularity, defined by the singular hypersurface equation $f(x,y,z)=0$. The
cases in the ADE classification can be presented as:

$\begin{split}&A_{r}:\hskip 28.45274ptx^{2}+y^{2}+z^{r+1}=0\\\ &D_{r}:\hskip
28.45274ptx^{2}+zy^{2}+z^{r-1}=0\\\ &E_{6}:\hskip
28.45274ptx^{2}+y^{3}+z^{4}=0\\\ &E_{7}:\hskip
28.45274ptx^{2}+y^{3}+yz^{3}=0\\\ &E_{8}:\hskip
28.45274ptx^{2}+y^{3}+z^{5}=0\\\ \end{split}$ (2.1)

One can deform these equations by adding a certain number of monomials:

$f+\sum_{i=1}^{k}\mu_{i}g_{i}=0,$ (2.2)

where the monomials $g_{i}$ belong to the ring:

$R=\frac{\mathbb{C}(x,y,z)}{\left(\frac{\partial f}{\partial x},\frac{\partial
f}{\partial y},\frac{\partial f}{\partial z}\right)}.$ (2.3)

The coefficients $\mu_{i}$ of the monomials $g_{i}$ are coordinates on the
space of deformations. One can construct a space by fibering the deformed ALE
space over the deformation space: at the origin (where all deformation
coefficients are zero), the ALE space develops the ADE singularity. The total
space of the fibration may or may not be singular, depending on the actual
fibration structure.

To illustrate the setup in more detail, we focus on an $A_{r-1}$ type ALE
family. This machinery is well-developed in [1]. The equation defining the
$A_{r-1}$ singularity is111Obtained from (2.1) by the change of variables
$x=\frac{v-u}{2},y=\frac{v+u}{2i}$.

$uv=z^{r}\>.$ (2.4)

The versal deformation is

$uv=z^{r}+\sum_{i=2}^{r}(-1)^{i-1}\sigma_{i}z^{n-i}\>.$ (2.5)

Promoting the $\sigma_{i}$’s to coordinates, this equation describes a
fibration with fiber given by the (deformed) $A_{r-1}$ surface and base the
space $\mathfrak{t}/\mathcal{W}$ of gauge invariant coordinates $\sigma_{i}$
($i=2,...,r$)222For $A_{r-1}$, the invariant $\sigma_{i}$ is the $i$-th
elementary symmetric polynomial in the eigenvalues of an element of the Lie
algebra. on the Lie algebra $\mathfrak{sl}(r)$, where $\mathfrak{t}$ is the
Cartan torus and $\mathcal{W}$ the Weyl group.

The space (2.5), a fibration over $\mathfrak{t}/\mathcal{W}$, is non-singular.
However, by making a _base change_ , one obtains a singular space whose
resolution blows up a subset of the roots of the central ALE fiber. Let us
consider the simplest case, i.e. $r=2$. The equation (2.5) becomes

$uv=z^{2}+\sigma_{2}\>,$ (2.6)

which is perfectly smooth as a total space, even though the central fiber over
$\sigma_{2}=0$ is a singular $A_{1}$-surface. If we make the base change
$\sigma_{2}=t^{2}$, i.e. we pull-back the ALE fibration w.r.t. the map:

$\displaystyle\mathfrak{t}$
$\displaystyle\longrightarrow\mathfrak{t}/\mathbb{Z}_{2}$ (2.7) $\displaystyle
t$ $\displaystyle\mapsto\sigma_{2}=t^{2}\,,$ (2.8)

where, in this case, $\mathfrak{t}\cong\mathbb{C}$, we obtain a threefold with
a conifold singularity:

$uv=z^{2}+t^{2}\>.$ (2.9)

The small resolution of the conifold blows up the simple root of $A_{1}$ in
the central fiber. This is called a _simultaneous resolution_. The family is
now fibered over the base $\mathfrak{t}$, whose $t$ is a coordinate.

For generic $r$ one can make an analogous base change by pulling back the
family w.r.t. to the map

$\mathfrak{t}\longrightarrow\mathfrak{t}/\mathcal{W}\qquad
t_{i}\mapsto\sigma_{k}(t_{i})$ (2.10)

by taking $\sigma_{k}$’s written as $\mathcal{W}$-invariant functions of
coordinates $t_{i}$ on $\mathfrak{t}$. This resolves all the simple roots of
$A_{r-1}$ in the central fiber. One can also choose a different base change
where

$\displaystyle\mathfrak{t}$ $\displaystyle\longrightarrow$
$\displaystyle\mathfrak{t}/\mathcal{W}^{\prime}$
$\displaystyle\longrightarrow$ $\displaystyle\mathfrak{t}/\mathcal{W}$ (2.11)
$\displaystyle t_{i}$ $\displaystyle\mapsto$ $\displaystyle\varrho_{i}(t_{j})$
$\displaystyle\mapsto$ $\displaystyle\sigma_{i}(t_{j})$ (2.12)

where $\mathcal{W}^{\prime}\subset\mathcal{W}$. In this case, the resolution
of the family blows up the roots that are left invariant by
$\mathcal{W}^{\prime}$, in the central fiber. The base of the fibration is now
parametrized by the $r-1$ $\mathcal{W}^{\prime}$ invariants, that we call
$\varrho_{i}$ ($i=1,...,r-1$). The $\varrho_{i}$’s can also be associated with
$\mathcal{W}^{\prime}$-invariant functions of coordinates $t_{i}$ on
$\mathfrak{t}$.

This line of reasoning can be generalized [1] to all the cases in the ADE
classification. Let us consider the family fibered over $\mathfrak{t}$. At the
origin of $\mathfrak{t}$, the surface develops the corresponding ADE
singularity. At a generic point of $\mathfrak{t}$, the singularity is
deformed: the surface admits $n$ non-vanishing $S^{2}$ intersecting in the
same pattern as the nodes of the Dynkin diagram and whose volume is measured
by the holomorphic (2,0)-form. In the deformed ALE surface, the holomorphic
(2,0)-form is along an element ${\bf t}$ of the Cartan torus, such that ${\rm
vol}_{\alpha_{i}}=\int_{\alpha_{i}}\Omega_{2,0}=\alpha_{i}[{\bf t}]$. We
choose a set of coordinates $t_{i}$ of $\mathfrak{t}$ such that the volumes of
the simple roots are given by

$\begin{split}&A_{r}:\quad{\rm vol}_{\alpha_{i}}=t_{i}-t_{i+1}\qquad
i=1,...,r\\\ &D_{r}:\quad{\rm
vol}_{\alpha_{i}}=\left\\{\begin{array}[]{l}t_{i}-t_{i+1}\qquad i=1,...,r-1\\\
t_{r-1}+t_{r}\qquad i=r\end{array}\right.\\\ &E_{r}:\quad{\rm
vol}_{\alpha_{i}}=\left\\{\begin{array}[]{l}t_{i}-t_{i+1}\qquad i=1,...,r-1\\\
-t_{1}-t_{2}-t_{3}\qquad i=r\end{array}\right.\end{split}$ (2.13)

It is useful to state the explicit form of the versal deformations of the
singularities (2.1) as fibrations of deformed ADE surfaces over the base
$\mathfrak{t}$ (with simultaneous resolution of all the simple roots):

$\begin{split}&A_{r}:\quad
x^{2}+y^{2}+\prod_{i=1}^{r+1}\left(z-t_{i}\right)=0\quad\quad\sum_{i=1}^{r+1}t_{i}=0\\\
&D_{r}:\quad
x^{2}+zy^{2}+\frac{\prod_{i=1}^{r}\left(z+t_{i}^{2}\right)-\prod_{i=1}^{r}t_{i}^{2}}{z}+2\prod_{i=1}^{r}t_{i}y=0\\\
&E_{6}:\quad
x^{2}+z^{4}+y^{3}+\epsilon_{2}yz^{2}+\epsilon_{5}yz+\epsilon_{6}z^{2}+\epsilon_{8}y+\epsilon_{9}z+\epsilon_{12}=0\\\
&E_{7}:\quad
x^{2}+y^{3}+yz^{3}+\tilde{\epsilon}_{2}y^{2}z+\tilde{\epsilon}_{6}y^{2}+\tilde{\epsilon}_{8}yz+\tilde{\epsilon}_{10}z^{2}+\tilde{\epsilon}_{12}y+\tilde{\epsilon}_{14}z+\tilde{\epsilon}_{18}=0\\\
&E_{8}:\quad
x^{2}+y^{3}+z^{5}+\hat{\epsilon}_{2}yz^{3}+\hat{\epsilon}_{8}yz^{2}+\hat{\epsilon}_{12}z^{3}+\hat{\epsilon}_{14}yz+\hat{\epsilon}_{18}z^{2}+\hat{\epsilon}_{20}y+\hat{\epsilon}_{24}z+\hat{\epsilon}_{30}=0,\end{split}$
(2.14)

where the $\epsilon_{i},\tilde{\epsilon}_{i},\hat{\epsilon}_{i}$ are known
functions of the parameters $t_{i}\in\mathfrak{t}$ (see [1] for the explicit
expression of $\epsilon_{i}$, $\tilde{\epsilon}_{i}$ and an algorithm to
compute $\hat{\epsilon}_{i}$).

The forms (2.14) admit the blow up of a collection of $\mathbb{P}^{1}$s in the
central fiber dual to all the simple roots of the corresponding ADE algebra.
Exactly as for the $A_{r}$-families, one can make a simultaneous resolution of
a subset of the simple roots, by choosing a subgroup
$\mathcal{W}^{\prime}\subset\mathcal{W}$ that leaves these roots invariant.
The corresponding family is fibered over the space parametrized by the
coordinates $\varrho_{i}$ of $\mathfrak{t}/\mathcal{W}^{\prime}$.

##  3 Threefolds as ALE families, GV invariants and $\Phi$

The families of ALE surfaces described in the previous section are a natural
starting point for constructing CY three-folds: the coordinates
$\varrho_{i}\in\mathfrak{t}/\mathcal{W}^{\prime}$ can be chosen so as to
depend (linearly)333Other dependences are indeed possible. However, we will
stick to linear in order to avoid creating further singularities in the
resulting threefolds. on a single complex parameter $w$. Such a threefold has
then the structure of an ALE fibration over the complex plane $\mathbb{C}_{w}$
and it will be defined by an equation such as

$F\left(x,y,z,\varrho_{1}(w),...,\varrho_{r}(w)\right)=0\>,$ (3.1)

where $F\left(x,y,z,\varrho_{1},...,\varrho_{r}\right)=0$ is the defining
equation of the $(r+2)$-dimensional family ($r$ is the rank of the ADE
algebra). We will restrict on threefolds $X$ that have isolated singularities,
whose exceptional locus does not contain compact divisors.444One would need a
severe degeneration of the ALE surface at the origin, e.g. the ALE surface
should split into several components, while in our cases the ALE surface just
develops a singularity.

In the following we are going to show how to build a threefold family of
deformed ADE surfaces by starting from the requirement that the ALE fibration
over $\mathbb{C}_{w}$ presents a specific partial simultaneous resolution,
i.e. a choice of resolved simple roots, say $\alpha_{1},...\alpha_{\ell}$. As
we will see, this method can actually produce in principle the full ADE family
over $\mathfrak{t}/\mathcal{W}^{\prime}$.

###  3.1 The Higgs field from the simultaneous resolution

The ADE families over $\mathfrak{t}/\mathcal{W}^{\prime}$ and the
corresponding threefolds have been derived and studied using several methods,
such as [1, 4, 5, 6, 7, 8]. We are going to use a different (more physical)
approach to construct these spaces that is based on the choice of an element
$\Phi$ of the ADE algebra, depending on the coordinate $w$. Our starting point
is M-theory reduced on an ALE surface with an ADE singularity of type
$\mathfrak{g}$. M-theory on an ADE singularity gives rise to a
$\mathcal{N}=1$, $d=7$ gauge theory with gauge group $G_{ADE}$ and three
adjoint scalars $\phi_{i}$ ($i=1,2,3$). With a choice of a complex structure,
a vev for $\Phi\equiv\phi_{1}+i\phi_{2}$ corresponds geometrically to a
deformation of the singularity, while a vev for $\phi_{3}$ corresponds to a
resolution, with $[\langle\Phi\rangle,\langle\phi_{3}\rangle]=0$. In
particular the coefficients of the deforming monomials are associated with the
Casimir invariants of $\Phi$.

We fiber this background over the $\mathbb{C}_{w}$-plane by switching on a
non-zero $w$-dependent vev for the complex adjoint scalar $\Phi$.
Geometrically this corresponds to deforming the singularity differently at
different values of $w$, obtaining a threefold $X$ that is an ALE fibration
over $\mathbb{C}_{w}$.

We now want to select a vev for $\Phi$ that allows the (simultaneous)
resolution at the origin only of a choice of simple roots
$\alpha_{1},...\alpha_{\ell}$ of $\mathfrak{g}$. Rephrased in the 7d field
theory language, $\langle\Phi\rangle$ must be compatible with switching on a
vev for $\phi_{3}$ along the subalgebra

$\mathcal{H}=\langle\alpha_{1}^{*},...\alpha_{\ell}^{*}\rangle\>.$ (3.2)

Since $[\langle\Phi\rangle,\langle\phi_{3}\rangle]=0$, then $\Phi$ must live
in the commutant of $\mathcal{H}$, that we call $\mathcal{L}$. Such a
subalgebra, that is the commutant of diagonal elements, is called a Levi
subalgebra of $\mathfrak{g}$.555At the level of the effective 7d theory, the
vev $\langle\Phi\rangle$ breaks the gauge algebra to its commutant
$\mathcal{H}$. It is important that $\mathcal{H}$ is a subspace of the Cartan
subalgebra of $\mathfrak{g}$. In fact, if the preserved algebra contained a
simple factor $\mathfrak{g}^{\prime}$, the vev $\langle\Phi\rangle$ would not
deform the ADE singularity completely, but the fiber over generic $w$ would
have a singularity of type $\mathfrak{g}^{\prime}$, i.e. $X$ would have a non-
isolated singularities.

Summing up, the choice of the blown up simple roots select an abelian
subalgebra $\mathcal{H}\subset\mathfrak{g}$. This defines a Levi subalgebra
$\mathcal{L}\subset\mathfrak{g}$, that is of the form

$\mathcal{L}=\bigoplus_{h}\mathcal{L}_{h}\oplus\mathcal{H}$ (3.3)

with $\mathcal{L}_{h}$ simple Lie algebras. The Higgs field666For ease of
notation, from now on we call $\Phi$ the vev of the field corresponding to a
given family of ADE deformed singularities. $\Phi$ corresponding to the ALE
fibration should be a generic element of $\mathcal{L}$.

The Casimir invariants of the Higgs field $\Phi$ tell us how the ALE fiber is
deformed. Since at the origin of $\mathbb{C}_{w}$ the fiber presents the full
ADE singularity, we expect that all the Casimir invariants of $\Phi$ vanish at
$w=0$. This means that at $w=0$ the Higgs field should be a non-zero nilpotent
element of $\mathcal{L}$. In particular, when restricted on each summand
$\mathcal{L}_{h}$ of the Levi subalgebra, $\Phi$ must be in the corresponding
principal nilpotent orbit, if we do not want terminal singularities.777If this
does not happen, there will be neutral zero modes of $\Phi$, that, as we will
see, will correspond to hypermultiplets that cannot get mass by going to the
Coulomb branch; this corresponds to non-resolvable singularities.

We call the Higgs at the origin $X_{+}$. The generic Higgs field has then the
following form

$\Phi=X_{+}+w\,Y$ (3.4)

where $Y\in\mathcal{L}$ can in principle depend on $w$; in this paper we
mainly consider cases with constant $Y$.

The Higgs field $\Phi$ must deform the singularity outside the origin. At
least in a neighborhood of $w=0$, we demand that the ALE fiber does not
develop any singularity. This happens when $\Phi$ restricted on each summand
of $\mathcal{L}$ is a non-zero semisimple element of that summand. For
$\mathcal{L}_{h}=A_{m-1}$, the generic form of $\Phi$, up to gauge
transformations is in the form of a _reconstructible Higgs_ [9].

$\Phi|_{A_{m-1}}=\left(\begin{array}[]{ccccc}0&1&0&\cdots&0\\\ 0&0&1&0&0\\\
\vdots&0&\ddots&\ddots&0\\\ 0&0&0&0&1\\\
(-1)^{m-1}\hat{\sigma}_{m}&(-1)^{m-2}\hat{\sigma}_{m-1}&\cdots&-\hat{\sigma}_{2}&0\\\
\end{array}\right)$ (3.5)

with $\hat{\sigma}_{j}$ ($j=2,...,m$) the Casimirs of $\Phi|_{A_{m-1}}$ (we
called them the partial Casimir of $\mathfrak{g}$). There are analogous
canonical forms when the summand is a different Lie algebra. Collecting the
Casimirs $\hat{\sigma}_{j}$’s for each summand $\mathcal{L}_{h}$ and the
coefficient deformations along $\mathcal{H}$ one obtains the set of invariant
coordinates $\varrho_{i}$ that span the base of the family with simultaneous
partial resolution. The total Casimirs of $\Phi$ (that appear as coefficients
of the deforming monomials in the versal deformation of the ADE singularity)
can be written as functions of the $\varrho_{i}$’s. Notice that by formulae
like (3.5), we give $\Phi$ as a function of the partial Casimirs
$\varrho_{i}$. The choice of a dependence of $\varrho_{i}$ on $w$ produces a
threefold.

We finally notice that the ADE Lie Algebra $\mathfrak{g}$ can be decomposed
into representations of the Levi subalgebra:

$\mathfrak{g}=\mathcal{L}\oplus...=\bigoplus_{p}R_{p}^{\mathcal{L}}\>,$ (3.6)

where the irreps $R_{p}^{\mathcal{L}}$ include the terms in the decomposition
(3.3). This will turn out useful in the following.

####  $A_{3}$ example

Let us see how the choice of $\Phi$ works in a simple example: a family of
deformed $A_{3}$ singularity where only the central simple root $\alpha_{2}$
is simultaneously resolved at the origin (see Figure 1).

Figure 1: Partial resolution of $A_{3}$

We then have $\mathcal{H}=\langle\alpha_{2}^{*}\rangle$ and
$\mathcal{W}^{\prime}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The commutant of
$\mathcal{H}$ is

$\mathcal{L}=A_{1}^{(1)}\oplus A_{1}^{(3)}\oplus\langle\alpha_{2}^{*}\rangle$
(3.7)

where $A_{1}^{(i)}$ ($i=1,3$) is generated by the triplet
$e_{\pm\alpha_{i}},h_{i}$ normalized s.t. the commutation relations are
$h_{i}=[e_{\alpha_{i}},e_{-\alpha_{i}}]$ and $[h_{i},e_{\pm\alpha_{i}}]=\pm
2e_{\pm\alpha_{i}}$. Following the prescription (3.5) for each $A_{1}$ summand
we have

$\Phi|_{A_{1}^{(i)}}=\begin{pmatrix}0&1\\\
\varrho_{i}&0\end{pmatrix}=e_{\alpha_{i}}+\varrho_{i}e_{-\alpha_{i}}\qquad
i=1,3$ (3.8)

where $\varrho_{i}$ ($i=1,3$) is the only Casimir of the $s\ell_{2}$ algebra
$A_{1}^{(i)}$. $\Phi$ can also have a component along the Cartan
$\alpha_{2}^{*}$, with generic coefficient $\varrho_{2}$.
$\varrho_{1},\varrho_{2},\varrho_{3}$ are the partial Casimirs relative to the
chosen partial simultaneous resolution, i.e. they are invariant under the Weyl
subgroup $\mathcal{W}^{\prime}$. In matrix form, the chosen Higgs field vev
is:

$\Phi=\begin{pmatrix}\varrho_{2}&1&&\\\ \varrho_{1}&\varrho_{2}&&\\\
&&-\varrho_{2}&1\\\ &&\varrho_{3}&-\varrho_{2}\\\ \end{pmatrix}$ (3.9)

One can then take $\varrho_{i}=\varrho_{i}(w)$ in order to obtain a fibration
over $\mathbb{C}_{w}$.

###  3.2 The threefold equation from $\Phi$

In order to obtain the explicit equation of the threefold we first derive the
expression of the family of ADE surfaces fibered over
$\mathfrak{t}/\mathcal{W}^{\prime}$ with coordinates $\varrho_{i}$. We then
let these coordinates depend on $w\in\mathbb{C}$.

Katz and Morrison [1] developed a method for obtaining the hypersurface
equation for an ADE family with a given simultaneous resolution, which we
review in Appendix A. In the following, instead, we propose a novel method to
read off the explicit equation of the simple flops in the ADE classification
using the Higgs field $\Phi$.888Actually, our method works for any CY
threefold derived from an ADE family. Although the resulting expressions match
those of [1], this approach tackles the problem from a different perspective,
unifying the physical description via the Higgs field $\Phi$ and the algebraic
expressions of the deformed ADE singularities.

We first consider the adjoint quotient map, that associates an element
$\mathrm{g}$ of the ADE algebra $\mathfrak{g}$ of rank $r$ with $r$
independent polynomials $\chi_{i}$, whose value gives a point in
$\mathfrak{t}/\mathcal{W}$:

$\chi:\quad\mathfrak{g}\rightarrow\mathfrak{t}/\mathcal{W}:\mathrm{g}\mapsto(\chi_{1}(\mathrm{g}),\cdots,\chi_{r}(\mathrm{g})).$
(3.10)

The polynomials $\chi_{i}(\mathrm{g})$ are the Casimirs of $\mathfrak{g}$ and
are defined in the following way. By choosing a representation of the Lie
algebra $\mathfrak{g}$, $\mathrm{g}$ can be put in a matrix form. Its Casimirs
are then given by specific invariant polynomials of this matrix: For the $A$
and $D$ series, the canonical Casimirs of $\mathrm{g}$ are (with $\mathrm{g}$
a matrix in the fundamental/vector representation):

$\begin{array}[]{c|l}\bm{A_{r}}&\bm{D_{r}}\\\
\hline\cr\chi^{A}_{i}(\mathrm{g})=\tfrac{1}{i+1}\text{Tr}(\mathrm{g}^{i+1}),\hskip
5.69046pti=1,\ldots,r&\chi^{D}_{i}(\mathrm{g})=\text{Tr}(\mathrm{g}^{2i}),\hskip
5.69046pti=1,\ldots,r-1\\\
&\chi^{D}_{r}(\mathrm{g})=\text{Pfaff}(\mathrm{g})\\\ \end{array}$ (3.11)

For the exceptional algebras $E_{r}$, one takes $\mathrm{g}$ in the following
representations: $\bm{27}$ for $E_{6}$, $\bm{133}$ for $E_{7}$ and $\bm{248}$
for $E_{8}$. One then defines the Casimirs of $\mathrm{g}$ as[10, 11]:

$\begin{array}[]{c|cl}\bm{E_{6}}&\chi^{E_{6}}_{i}(\mathrm{g})=\text{Tr}(\mathrm{g}^{k_{i}})&\text{for
}k_{i}=2,5,6,8,9,12\\\
\bm{E_{7}}&\chi^{E_{7}}_{i}(\mathrm{g})=\text{Tr}(\mathrm{g}^{k_{i}})&\text{for
}k_{i}=2,6,8,10,12,14,18\\\
\bm{E_{8}}&\chi^{E_{8}}_{i}(\mathrm{g})=\text{Tr}(\mathrm{g}^{k_{i}})&\text{for
}k_{i}=2,8,12,14,18,20,24,30\\\ \end{array},$ (3.12)

and $i=1,...,r$.

We now need to provide the definition of Slodowy slices. Consider a nilpotent
element $\mathrm{x}\in\mathfrak{g}$ belonging to some nilpotent orbit
$\mathcal{O}$: the Jacobson-Morozov theorem ensures that there exists a
standard triple $\\{\mathrm{x},\mathrm{y},\mathrm{h}\\}$ of elements in
$\mathfrak{g}$ satisfying the $\mathfrak{su}(2)$ algebra relations.999The
triple related to $\mathrm{x}$ is unique up to conjugation. Now, we define the
_Slodowy slice_ through the point $\mathrm{x}$ as those Lie algebra elements
satisfying:

$\mathcal{S}_{\mathrm{x}}=\\{\mathrm{z}\in\mathfrak{g}\hskip 5.69046pt|\hskip
5.69046pt[\mathrm{z}-\mathrm{x},\mathrm{y}]=0\\}.$ (3.13)

For our purposes, we are interested in the so-called subregular nilpotent
orbit $\mathcal{O}_{\text{subreg}}$, and in the Slodowy slice passing through
it.101010In general, the subregular nilpotent orbit of $\mathfrak{g}$ is
defined as the only orbit of dimension
$\text{dim}(\mathcal{O}_{\text{subreg}})=\text{dim}(\mathfrak{g})-\text{rank}(\mathfrak{g})-2$.

Let us immediately clarify the definitions given so far with a simple example.
Consider the $A_{3}$ Lie algebra, and a nilpotent element $\mathrm{x}$ lying
in its subregular nilpotent orbit. The element $\mathrm{x}$ and its Slodowy
slice are

$\mathrm{x}=\begin{pmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix}\>,\qquad\mathcal{S}_{\text{subreg}}=\left\\{\left(\begin{array}[]{cccc}{a}&{1}&{0}&{0}\\\
{b}&{a}&{1}&{0}\\\ {c}&{b}&{a}&{d}\\\
{e}&{0}&{0}&{-3a}\end{array}\right)\hskip 5.69046pt\Bigg{|}\hskip
5.69046pta,b,c,d,e\in\mathbb{C}\right\\}.$ (3.14)

The crucial fact for us is the following: Given an algebra $\mathfrak{g}$ in
the ADE classification, and an element
$\mathrm{x}\in\mathcal{O}_{\text{subreg}}$, the intersection of the Slodowy
slice through $\mathrm{x}$ with the fiber of the adjoint quotient map (3.10)
is isomorphic to the versal deformation of the corresponding ADE singularity,
namely:

$\mathcal{S}_{\text{subreg}}\cap\chi^{-1}(\bm{u})\cong\text{versal deformation
of }\mathbb{C}^{2}/\Gamma_{\rm ADE},$ (3.15)

where $\bm{u}$ is a point in $\mathfrak{t}/\mathcal{W}\simeq\mathbb{C}^{r}$
with coordinates $u_{i}$. In particular, the intersection of
$\mathcal{S}_{\text{subreg}}$ and the nilpotent cone111111I.e. the set of all
nilpotent elements. $\mathcal{N}=\chi^{-1}(0)$ is isomorphic to the ADE
singularity.

The isomorphism (3.15) is telling us that the coefficients of the monomials in
the versal deformation can be written in terms of the coordinates $u_{i}$
related to the polynomials $\chi_{i}$. In order to see this explicitly, let us
come back to the $A_{3}$ example. In the $A$ cases, the Casimir invariants
$\chi_{i}$ are given in (3.11). Considering the elements in the Slodowy slice
(3.14) through the subregular nilpotent orbit, we find:

$\displaystyle\chi_{1}(\mathcal{S}_{\text{subreg}})$ $\displaystyle=$
$\displaystyle 6a^{2}+2b,$
$\displaystyle\chi_{2}(\mathcal{S}_{\text{subreg}})$ $\displaystyle=$
$\displaystyle-8a^{3}+4ab+c,$ (3.16)
$\displaystyle\chi_{3}(\mathcal{S}_{\text{subreg}})$ $\displaystyle=$
$\displaystyle 21a^{4}+6a^{2}b+3ac+2b^{2}+de\>.$

We now compute $\mathcal{S}_{\text{subreg}}\cap\chi^{-1}(\bm{u})$:

$\left\\{\begin{array}[]{l}6a^{2}+2b=u_{1}\\\ -8a^{3}+4ab+c=u_{2}\\\
21a^{4}+6a^{2}b+3ac+2b^{2}+de=u_{3}\\\
\end{array}\right.\Rightarrow\,-cd=(3a)^{4}-u_{1}(3a)^{2}+u_{2}(3a)-u_{3}+\frac{u_{1}^{2}}{2}\>,$
(3.17)

that is the usual presentation of the deformed $A_{3}$ singularity: if we set
$u=c,v=-d,z=3a$, we obtain

$uv=z^{4}-u_{1}z^{2}+u_{2}z-u_{3}+\frac{u_{1}^{2}}{2}$ (3.18)

that matches with (2.5) (up to an invertible redefinition of the coordinates
in $\mathfrak{t}/\mathcal{W}$).121212The match works for
$\sigma_{2}=u_{1},\sigma_{3}=u_{2},\sigma_{4}=u_{3}-u_{1}^{2}/2$.

We now want to describe the ALE families of deformed ADE singularities over
$\mathbb{C}_{w}$. We have constructed them by the choice of a Higgs field
$\Phi(w)$, that through its Casimirs is telling us how the deformation is
performed on top of each point of $\mathbb{C}_{w}$. In other words, take
$w\in\mathbb{C}_{w}$; to see which is the deformed ALE surface over $w$, we
pick $\Phi(w)$ and compute its Casimirs $\chi_{i}(\Phi)$. Their values select
a specific point $\bm{u}\in\mathfrak{t}/\mathcal{W}$ and then a specific
deformed ALE surface (with precise volumes of the non-holomorphic spheres
(2.13)). The equation of the threefold is obtained imposing the relations

$\chi_{i}(\mathcal{S}_{\text{subreg}})=\chi_{i}(\Phi(w))\,,\qquad
i=1,...,r\>,$ (3.19)

with $\chi_{i}(\mathrm{g})$ defined in (3.11) and (3.12), and substituting the
resulting $\chi_{i}(\mathcal{S}_{\text{subreg}})$ into the expressions
defining
$\mathcal{S}_{\text{subreg}}\cap\chi^{-1}(\chi_{i}(\mathcal{S}_{\text{subreg}}))$.

In general, if we keep $\Phi(w)$ unspecified, (3.19) gives us the versal
deformations of the ADE singularity, with deformation parameters depending on
the Casimirs of $\Phi$. We have worked out these relations for all the ADE
algebras. For the $A_{r}$ and $D_{r}$ singularities, one obtains (up to
coordinate redefinition) the compact and known form

$A_{r}:\qquad x^{2}+y^{2}+\text{det}(z\mathbb{1}-\Phi)=0$ (3.20)

and

$D_{r}:\qquad
x^{2}+zy^{2}-\frac{\sqrt{\text{det}(z\mathbb{1}+\Phi^{2})}-\text{Pfaff}^{2}(\Phi)}{z}+2y\hskip
2.84544pt\text{Pfaff}(\Phi)=0.$ (3.21)

For the $E_{r}$ singularities the expression of the deformation parameters
$\epsilon_{i},\tilde{\epsilon}_{i},\hat{\epsilon}_{i}$ (appearing in (2.14))
in terms of the Casimirs (3.12) are given in Appendix B.

Hence, given a Higgs field $\Phi(w)$, one obtains the threefold equation by
simply computing its Casimirs and then inserting them into the relations
(3.20) and (3.21) for the $A$ and $D$ cases, or using the formulae in Appendix
B for the $E$ cases.

####  $A_{3}$ example

In order to make the procedure explicit, let us apply it to the example of
$A_{3}$. Take the Higgs field (3.9) and compute its Casimirs:

$\displaystyle\chi_{1}(\Phi)$ $\displaystyle=$
$\displaystyle\varrho_{1}+\varrho_{3}+2\varrho_{2}^{2}$
$\displaystyle\chi_{2}(\Phi)$ $\displaystyle=$ $\displaystyle
2\varrho_{2}(\varrho_{1}-\varrho_{3})$ (3.22) $\displaystyle\chi_{3}(\Phi)$
$\displaystyle=$
$\displaystyle\frac{1}{2}(\varrho_{1}+\varrho_{3}+2\varrho_{2}^{2})^{2}-(\varrho_{2}^{2}-\varrho_{1})(\varrho_{2}^{2}-\varrho_{3})$

We substitute $u_{i}$ with $\chi_{i}(\Phi)$ in (3.18), obtaining

$uv=\left((z+\varrho_{2})^{2}-\varrho_{1}\right)\left((z-\varrho_{2})^{2}-\varrho_{3}\right)\>.$
(3.23)

One can check that the same equation can be obtained using directly (3.20).
Later we will be interested into the threefold given by
$\varrho_{1}=-\varrho_{3}=w$ and $\varrho_{2}=0$, that leads to
$uv=z^{4}-w^{2}$.

###  3.3 Gopakumar-Vafa invariants of $X$ from zero modes of $\Phi$

In the non-compact Calabi-Yau threefolds studied in this paper, we have
isolated singularities whose exceptional locus is given by a bunch of
$\mathbb{P}^{1}$’s. These (genus-zero) holomorphic curves are rigid and the
BPS M2-branes wrapped on them generate massless hypermultiplets in the 5d
theory coming from reduction of M-theory in $X$. The genus-zero and degree
$\mathbf{d}=(d_{1},...,d_{\ell})$ Gopakumar-Vafa invariants
$n^{g=0}_{d_{1},...,d_{\ell}}$ count such states. In particular, we say that

$n^{g=0}_{\mathbf{d}}=\\#\mbox{ 5d hypers with charges $(d_{1},...,d_{\ell})$
under the flavor group generated by }\mathcal{H}\>.$ (3.24)

These numbers are reproduced by counting the zero modes of the Higgs field
$\Phi$, that are localized at $w=0$.

Given a vev for $\Phi$, the zero modes are the deformation
$\varphi\in\mathfrak{g}$ of the Higgs field up to the (linearized) gauge
transformations

$\delta_{g}\varphi=[\Phi,g]\qquad\mbox{with}\qquad g\in\mathfrak{g}\>.$ (3.25)

We take $\Phi=X_{+}+w\,Y$ as explained at page 3.4, where
$X_{+}\in\mathcal{L}$ is in the principal nilpotent orbit of each
$\mathcal{L}_{h}$. We need to work out which components of the deformation
$\varphi$ can be set to zero by a gauge transformation (3.25). One then tries
to solve the equation

$\varphi+\delta_{g}\varphi=0\,,\qquad\mbox{with}\qquad\delta_{g}\varphi=[X_{+}+w\,Y,g]$
(3.26)

with unknown $g\in\mathfrak{g}$. As we will see, at special points in
$\mathbb{C}_{w}$, there can be components of $\varphi$ that cannot be gauge-
fixed to zero: these directions in the Lie algebra $\mathfrak{g}$ support zero
modes.

Since the irreducible representations $R^{\mathcal{L}}$ of $\mathcal{L}$ are
invariant under the action of $\Phi$, we implement the decomposition (3.6) and
we solve the equation (3.26) in each representation $R^{\mathcal{L}}$ at a
time, where now $g,\varphi\in R^{\mathcal{L}}\subset\mathfrak{g}$. We can
write more explicitly the representation $R^{\mathcal{L}}$ of
$\mathcal{L}=\mathcal{H}\oplus\mathcal{L}_{1}\oplus\mathcal{L}_{2}\oplus...$
as

$R^{\mathcal{L}}=\left(\,\,\,R^{\mathcal{L}_{1}}\,\,,\,\,R^{\mathcal{L}_{2}}\,\,,\,\,...\,\,\,\right)_{q_{1},...,q_{\ell}}$
(3.27)

where $R^{\mathcal{L}_{h}}$ is an irreducible representation of the simple
summand $\mathcal{L}_{h}$ and $(q_{1},...,q_{\ell})$ are the charges under the
$U(1)^{\ell}$ group generated by $\mathcal{H}$. If there are $n$ 5d modes in
the representation $R^{\mathcal{L}}$, there will be other $n$ 5d modes in the
conjugate representation $\bar{R}^{\mathcal{L}}$; together these generate $n$
massless hypermultiplets in the 5d theory localized at the singularity. By
using the correspondence (3.24), we can then say that the GV invariant with
degrees $(q_{1},...,q_{\ell})$ is

$n^{g=0}_{q_{1},...,q_{\ell}}=\tfrac{1}{2}\cdot\\#\,\mbox{localized zero-modes
from }\,R^{\mathcal{L}}\oplus\bar{R}^{\mathcal{L}}$ (3.28)

with
$R^{\mathcal{L}}=\left(R^{\mathcal{L}_{1}},R^{\mathcal{L}_{2}},...\right)_{q_{1},...,q_{\ell}}$.

Let us describe our algorithm to compute the number and the charges of zero
modes. For each representation $R^{\mathcal{L}}$ of $\mathcal{L}$ with
dimension $d_{R}$, we choose a basis $e_{1},...,e_{d_{R}}$ of
$R^{\mathcal{L}}$. In this basis, the equation (3.26) becomes

$\left(A+w\,B\right)\bm{\rho}=-\bm{\phi}$ (3.29)

where $\bm{\rho}$ and $\bm{\phi}$ are the $d_{R}$-column vectors of
coefficients of $g$ and $\varphi$ in the given basis and $A,B$ are the
constant $d_{R}\times d_{R}$ matrices representing the linear operators
$X_{+}$ and $Y$ respectively.

If $A+w\,B$ is invertible, then there exists a vector $\bm{\rho}$ (i.e. a
$g\in R^{\mathcal{L}}$) that completely gauge fixes $\varphi\in
R^{\mathcal{L}}$ to zero at generic $w$. At the values of $w$ where the rank
of $A+w\,B$ decreases, there will be vectors $\bm{\phi}$ that cannot be set to
zero, leaving a zero mode localized at that points.

With the chosen $X_{+}$, we immediately see that such a special point is (by
construction) the origin $w=0$. Here the matrix $A+w\,B$ reduces to the
nilpotent matrix $A$, that has non-trivial kernel.131313In particular, the
kernel is spanned by the vectors $|j,j\rangle$, when writing $R^{\mathcal{L}}$
in terms of $sl_{2}$ representations, where $sl_{2}$ is generated by the
Jacobson-Morozov standard triple associated with $X_{+}$. In the following we
only use the fact that $A$ has rank $r<d_{R}$; hence, our conclusions are
valid also when $A$ is not necessarily nilpotent. What we are going to say of
course applies also for a nilpotent $A$.

We choose the basis $e_{1},...,e_{d_{R}}$ of $R^{\mathcal{L}}$ such that $A$
is in the Jordan form. If the rank of $A$ is $r$, we then have $d_{R}-r$ rows
of zeros and $d_{R}-r$ columns of zeros. We can rearrange rows and columns
such that $A$ takes the block diagonal form

$A=\left(\begin{array}[]{l|l}A_{u}&{\bm{0}}_{r\times(d_{R}-r)}\\\
\hline\cr{\bm{0}}_{(d_{R}-r)\times r}&{\bm{0}}_{(d_{R}-r)\times(d_{R}-r)}\\\
\end{array}\right)\>,\mbox{ with $A_{u}$ invertible.}$ (3.30)

Doing the same operations on $B$, we obtain

$B=\left(\begin{array}[]{l|l}B_{u}&B_{r}\\\ \hline\cr B_{l}&B_{d}\\\
\end{array}\right)\>.$ (3.31)

The equations (3.29) now read

$\left\\{\begin{array}[]{rcl}(A_{u}+wB_{u})\bm{\rho}_{u}+wB_{r}\bm{\rho}_{d}&=&-\bm{\phi}_{u}\\\
wB_{l}\bm{\rho}_{u}+wB_{d}\bm{\rho}_{d}&=&-\bm{\phi}_{d}\\\
\end{array}\right.$ (3.32)

Since $(A_{u}+wB_{u})$ is invertible (at least in the vicinity of $w=0$), from
the first set of equations we see that we can always gauge fix the
$\bm{\phi}_{u}$ components to zero,141414These correspond to all states except
$|j,-j\rangle$. by setting

$\bm{\rho}_{u}=-(A_{u}+w\,B_{u})^{-1}\left(\bm{\phi}_{u}+w\,B_{r}\bm{\rho}_{d}\right)\>.$
(3.33)

Substituting in the second set of equations we obtain

$\,w\,\left[B_{d}-w\,B_{l}(A_{u}+w\,B_{u})^{-1}B_{r}\right]\bm{\rho}_{d}=-\bm{\phi}_{d}+w\,B_{l}(A_{u}+w\,B_{u})^{-1}\bm{\phi}_{u}\>.$
(3.34)

We see that the components $\bm{\phi}_{d}$ cannot be fixed identically to
zero: at $w=0$ there can be a remnant, i.e. a localized mode. Said
differently, the best we can do is to cancel from $\bm{\phi}_{d}$ its
dependence on $w$, leaving a constant entry (instead of a generic polynomial
in $w$). This is possible for all components of $\bm{\phi}_{d}$ only when the
matrix $B_{d}$ has maximal rank, i.e. rank equal to $d_{R}-r$. In this case,
the number of zero modes is

$\\#=d_{R}-r\>,$

because each component of $\bm{\phi}_{d}$ has now a constant entry, i.e. one
degree of freedom.

If $B_{d}$ is not invertible, we can iterate what we have done so far, in the
following way. Let us define for simplicity
$\bm{\rho}^{\prime}\equiv\bm{\rho}_{d}$, $\bm{\phi}^{\prime}_{\rm
tot}\equiv\bm{\phi}_{d}-w\,B_{l}(A_{u}+w\,B_{u})^{-1}\bm{\phi}_{u}$,
$A^{\prime}\equiv B_{d}$ and $B^{\prime}\equiv-
B_{l}(A_{u}+w\,B_{u})^{-1}B_{r}$. We can decompose $\bm{\phi}^{\prime}_{\rm
tot}=\bm{\phi}^{\prime}_{0}+w\bm{\phi}^{\prime}$, where
$\bm{\phi}^{\prime}_{0}$ is $\bm{\phi}^{\prime}_{\rm tot}$ evaluated at $w=0$.
We can then rewrite the equation (3.34) as

$\left(A^{\prime}+w\,B^{\prime}\right)\bm{\rho}^{\prime}=-\bm{\phi}^{\prime}\>.$
(3.35)

This has the same form as (3.29), so we can again change the basis such that
$A^{\prime}\equiv B_{d}$ is in the Jordan form and write the equations in this
basis. We will obtain a set of equations in the form (3.32) where we have to
substitute $(A,B)_{u,l,r,d}\rightarrow(A^{\prime},B^{\prime})_{u,l,r,d}$ and
$(\bm{\rho},\bm{\phi})\rightarrow(\bm{\rho}^{\prime},\bm{\phi}^{\prime})$.

The matrix $A^{\prime}$ will now have rank $r^{\prime}<d_{R}-r$. There will
then be $r^{\prime}$ components of $\bm{\phi}^{\prime}$ that can be gauge
fixed to zero; we correspondingly have $r^{\prime}$ zero modes along the
corresponding components of $\bm{\phi}_{d}$. If the matrix $B^{\prime}_{d}$
has maximal rank (i.e. $d_{R}-r-r^{\prime}$), then the other
$d_{R}-r-r^{\prime}$ components of $\bm{\phi}_{d}$ will be of the form $a+bw$
and hence each hosts _two_ zero modes. In this case the number of zero modes
is

$\\#=r^{\prime}+2(d_{R}-r-r^{\prime})\>.$ (3.36)

On the other hand, if $B^{\prime}_{d}$ has rank
$r^{\prime\prime}<d_{R}-r-r^{\prime}$, then we have to iterate once more the
algorithm above and, provided $B^{\prime\prime}_{d}$ has maximal rank (i.e.
$d_{R}-r-r^{\prime}-r^{\prime\prime}$) we obtain

$\\#=r^{\prime}+2r^{\prime\prime}+3(d_{R}-r-r^{\prime}-r^{\prime\prime})\>.$

We now have the factor "$3$" because the $d_{R}-r-r^{\prime}-r^{\prime\prime}$
directions of $\bm{\phi}_{d}$ are of the form $a+bw+cw^{2}$, i.e. they host
_three_ zero modes each.

In conclusion, let us assume that the algorithm stops at the $N$-th step and
let us call $r^{(k)}$ the rank of the matrix $A$ at the step $k$, then the
number of zero modes is

$\\#_{\rm zero\,\
modes}=\sum_{k=0}^{N}k\,r^{(k)}\qquad\mbox{with}\qquad\sum_{k=0}^{N}r^{(k)}=d_{R}$
(3.37)

where $r^{(0)}=r$.

If there are other values of $w$, say $w=w_{0}$, where the rank of $A+wB$ is
not maximal, one can shift $w\mapsto w+w_{0}$ ending out with the same
situation as above, where the new $A$ is now $A+w_{0}B$. Applying the
algorithm that we have just outlined, one computes the zero modes localized at
$w=w_{0}$. In this case the matrix at $w=w_{0}$ is not necessarily nilpotent.

Notice that this algorithm could never end. This is the case for example when
the $A+wB$ matrix is identically zero at one step. The corresponding
directions of $\varphi$ cannot be gauge fixed at any order in $w$, leaving a
zero mode that lives in 7d.

In conclusion, in this section we have shown that one can reduce the problem
of finding the zero modes to a simple exercise in linear algebra. These
computations are algorithmic and can be done by a calculator in a reasonable
amount of time. In Appendix D we describe the implementation of the algorithm
in Mathematica, that we used for our computations.

##  4 Simple flops with $l=1,...,6$ and their GV invariants

One-parameter deformations of ADE surfaces admitting a small simultaneous
resolution blowing up a single $\mathbb{P}^{1}$ are known as simple threefold
flops. From a mathematical point of view, they can be classified according to
a variety of invariants.

A rich classification of simple flops can be obtained employing the so-called
length invariant, first introduced in [12], that we defined in Section 1. It
was proven (see [1]) that the length of a simple flop can only assume discrete
values ranging from 1 to 6, and that examples of any length indeed exist.

From a Lie-algebraic point of view, the length of a simple flop corresponds to
the dual Coxeter label of the node of the Dynkin diagram that is being
resolved by the small simultaneous resolution. Given an ADE algebra
$\mathfrak{g}$ of rank $r$, a set of simple roots $\alpha_{i}$, with
$i=1,\ldots r$, and the highest root $\theta$, the dual Coxeter label of a
node is the multiplicity of the corresponding simple root in the decomposition
of the highest root. In other words, given a node corresponding to a simple
root $\alpha_{i_{0}}$ and the decomposition of the highest root

$\theta=c_{1}\alpha_{1}+\ldots+c_{i_{0}}\alpha_{i_{0}}+\ldots c_{r}\alpha_{r}$
(4.1)

then $c_{i_{0}}$ is the dual Coxeter label of the node. As we will use this
fact extensively in the explicit constructions of simple threefold flops of
length up to 6, it is useful to report the Dynkin diagrams of all the ADE
cases, along with the dual Coxeter labels of their nodes in Figure 2,

Figure 2: ADE Dynkin diagrams and dual Coxeter labels of the nodes

where we have highlighted in black the nodes that are being resolved in the
simple threefold flops that we will analyze in the following sections.
The classification of simple threefold flops based on the length can be
further refined introducing the Gopakumar-Vafa (GV) invariants [13, 14, 15].
These invariants can be used to distinguish between simple flops of the same
length.151515Even though we will not use it in this paper, it is worth
mentioning an even subtler invariant that can be associated to a simple flop,
namely its contraction algebra. It has been proven [16] that there exist
simple flops with the same normal bundle, same length, same Gopakumar-Vafa
invariants and different contraction algebra. Physically, the contraction
algebra can be understood, for example, as describing the quiver relations of
the theory on a D3 brane in type IIB probing the singularity, and explicit
constructions of contraction algebras at all lengths can be found in [5].

In the following, we apply the method discussed in Section 3 to construct
threefolds with a simple flop. The threefold will be obtained from a family of
deformed ADE singularities in which only the black node in Figure 2 is
simultaneously resolved. Let us call it $\alpha_{c}$. The subalgebra
$\mathcal{H}$ is then generated by $\alpha_{c}^{*}$, i.e.

$\mathcal{H}=\langle\alpha_{c}^{*}\rangle\>,$ (4.2)

and the Higgs field will correspondingly be chosen in the commutant
$\mathcal{L}$ of $\mathcal{H}$, i.e. the Levi subalgebra corresponding to the
chosen partial simultaneous resolution. From Figure 2, we see that the simple
summands $\mathcal{L}_{h}$ of $\mathcal{L}$ are of $A$-type. The Higgs $\Phi$
restricted on these spaces is then of the form (3.5), and collecting the
$\hat{\sigma}$’s from each summand $\mathcal{L}_{h}$ gives the partial
Casimirs $\varrho_{i}$’s that parametrize the base
$\mathfrak{t}/\mathcal{W}^{\prime}$. The threefold is obtained by setting
$\varrho_{i}=\varrho_{i}(w)$.

We will construct threefold with different values of _length_ from $1$ to $6$.
For each manifold we give the Higgs field $\Phi$ that produces the desired
simple flop threefold $X$. This allows us to build the 5d theory realized from
reducing M-theory on $X$. In particular the flavor group will always be the
$U(1)$ group generated by $\alpha_{c}^{*}$. The number of hypermultiplets and
their charges under the $U(1)$ flavor group, namely the GV invariants of $X$
and their degrees, will be derived by counting the zero modes of $\Phi$.

###  4.1 Simple flop with length $1$

#### The simplest Example: the Conifold

The Conifold threefold is given by

$u\,v=z^{2}-w^{2}\>.$ (4.3)

This is actually a family of deformed $A_{1}$ surfaces over $\mathbb{C}_{w}$,
with the simultaneous resolution of the exceptional $\mathbb{P}^{1}$ at $w=0$.

It can be constructed following the previous sections in the following way:
$A_{1}$ has only one simple root $\alpha$. We require it to be blown up by the
simultaneous resolution (the only other choice is to blow up no sphere, that
would produce a non-singular threefold). The Levi subalgebra is now simply

$\mathcal{L}=\langle\alpha^{*}\rangle=\mathcal{H}.$ (4.4)

The partial Casimir is the coefficient $\varrho$ along the Cartan
$\alpha^{*}$. Choosing $\varrho=w$, the Higgs field is

$\Phi=\begin{pmatrix}w&0\\\ 0&-w\\\ \end{pmatrix}$ (4.5)

and the threefold equation is easily checked to be (4.3).

This is the simplest example of simple flop, where the flavor group (i.e. the
preserved 7d gauge group) is $U(1)$.

Zero modes. Notoriously, M-theory on the conifold gives a free 5d
hypermultiplet (localized at $w=0$). This can be checked by computing the zero
modes of $\Phi$. This computation has already been shown in [3], by explicitly
using the linearized equations of motion in holomorphic gauge,161616For the
conifold, i.e. a family of deformed $A_{1}$, that computation was enough.
However for more complicated algebras our method simplifies the calculations
and make them more systematic. as explained in [9]. In order to illustrate the
method outlined in Section 3.3, we apply it to the conifold case to reproduce
the result of [3].

The decomposition (3.6) of the $A_{1}$ algebra in representations of the Levi
subalgebra $\mathcal{L}=\langle\alpha^{*}\rangle$ is

$A_{1}={\bf 1}_{0}+{\bf 1}_{+}+{\bf 1}_{-}.$ (4.6)

Let us consider each representation individually. Remember that the matrices
$A$ and $B$ in Section 3.3 are the restriction of $X_{+}$ and $Y$ on the
considered representation, where $\Phi=X_{+}+w\,Y$.

${\bf 1}_{0}$: $\Phi$ restricted to this representation is zero. Hence, the
two ‘matrices’ $A$ and $B$ vanish, nothing is gauge fixed and then there is
one 7d mode.

${\bf 1}_{+}$: $\Phi|_{{\bf 1}_{+}}=2w$, so $A=0$ and it has rank zero, but
$B=2$ has rank one; then $d_{R}-r=1-0=1$ and this gives one localized mode at
$w=0$.

${\bf 1}_{-}$: $\Phi|_{{\bf 1}_{-}}=-2w$, so $A=0$ and $B=-2$ that again gives
a localized mode at $w=0$.

The two localized zero modes made up one hypermultiplet, as expected. Its
charge under the flavor $U(1)$ can be easily read from the representation
where the modes sit. The zero mode analysis correctly reproduces the GV
invariant of the conifold, that is $n_{1}^{g=0}=1$.

#### Threefolds with a simple flop of length $1$: generic case

We now generalize the conifold case, by starting from the Lie algebra
$A_{k-1}$. The simple roots are now $\alpha_{1},...,\alpha_{k-1}$. We require
that the only root that is simultaneously resolved in the threefold is
$\alpha_{p}$ for a given choice of $p\in\\{1,...,k-1\\}$ (without loss of
generality, we can take $p\geq\frac{k}{2}$). Consequently, we have
$\mathcal{H}=\langle\alpha_{p}^{*}\rangle$. Its commutant is

$\mathcal{L}=A_{p-1}\oplus A_{k-p-1}\oplus\langle\alpha_{p}^{*}\rangle\>.$
(4.7)

The Higgs field at $w=0$ is (in the principal nilpotent orbit when restricted
on the simple summands of $\mathcal{L}$)

$X_{+}=e_{\alpha_{1}}+\dots+e_{\alpha_{p-1}}+e_{\alpha_{p+1}}+\dots+e_{\alpha_{k-1}}\>.$
(4.8)

We choose the $w$-dependence of the partial Casimirs such that the Higgs
restricted on each block is171717A different choice would only complicate the
equation of the three-fold, without changing its salient features.

$\Phi|_{A_{p-1}}=\left(\begin{array}[]{ccccc}0&1&0&\cdots&0\\\ 0&0&1&0&0\\\
\vdots&0&\ddots&\ddots&0\\\ 0&0&0&0&1\\\ w&0&\cdots&0&0\\\
\end{array}\right)\qquad\mbox{and}\qquad\Phi|_{A_{k-p-1}}=\left(\begin{array}[]{ccccc}0&1&0&\cdots&0\\\
0&0&1&0&0\\\ \vdots&0&\ddots&\ddots&0\\\ 0&0&0&0&1\\\ -w&0&\cdots&0&0\\\
\end{array}\right)\>.$ (4.9)

This means that

$Y=e_{-\alpha_{1}-\alpha_{2}-...-\alpha_{p-1}}-e_{-\alpha_{p+1}-\alpha_{p+2}-...-\alpha_{2k-1}}\>.$
(4.10)

The equation of the threefolds is read form (3.20), by using the chosen
$\Phi=X_{+}+w\,Y$:

$u\,v=(z^{p}-w)(z^{k-p}+w)\>.$ (4.11)

When $k=2n$ is even and $p=n$, we have the _Reid Pagoda_ of degree $n$ (whose
Dynkin diagram for the simultaneous resolution is depicted in Figure 2).

Zero modes. Let us perform the zero mode computation in the case $p=n=2$, i.e.
for the Reid Pagoda with degree $2$. The Higgs field is actually given by the
$A_{3}$ example studied before (see (3.9)), where one chooses the following
dependence of $\varrho_{i}$ on $w$:

$\varrho_{1}=w,\qquad\varrho_{2}=0,\qquad\varrho_{3}=-w\>.$

The $A_{3}$ algebra decomposes in the following way in representations of the
Levi subalgebra $\mathcal{L}=s\ell_{2}^{(1)}\oplus
s\ell_{2}^{(3)}\oplus\langle\alpha_{2}^{*}\rangle$:

$A_{3}=({\bf 3},{\bf 1})_{0}\oplus({\bf 1},{\bf 3})_{0}\oplus({\bf 1},{\bf
1})_{0}\oplus({\bf 2},{\bf 2})_{+}\oplus({\bf 2},{\bf 2})_{-}\>.$ (4.12)

Let us consider each Levi representation $R^{\mathcal{L}}$ individually.

$({\bf 3},{\bf 1})_{0}$: the operator $X_{+}$ is represented in the basis
$\\{-e_{\alpha_{1}},\frac{1}{2}h_{1},\frac{1}{2}e_{-\alpha_{1}}\\}$ by the
matrix

$A_{({\bf 3},{\bf 1})_{0}}=\left(\begin{array}[]{c|cc}0&1&0\\\ 0&0&1\\\
\hline\cr 0&0&0\\\ \end{array}\right)\>,$ (4.13)

that has rank $r=2$. In the same basis $Y$ is represented by

$B_{({\bf 3},{\bf 1})_{0}}=\left(\begin{array}[]{c|cc}0&0&0\\\ 2&0&0\\\
\hline\cr 0&2&0\\\ \end{array}\right)\>.$ (4.14)

We plug them into the expression (3.29) and apply the algorithm: arranging the
rows and columns to arrive to the expression (3.30) is equivalent to taking
$\bm{\rho}_{u}=(\rho_{2},\rho_{3})$, $\bm{\rho}_{d}=\rho_{1}$,
$\bm{\phi}_{u}=(\phi_{1},\phi_{2})$ and $\bm{\phi}_{d}=\phi_{3}$. We can then
read

$B_{u}=\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix},\qquad
B_{r}=\begin{pmatrix}0\\\ 2\\\ \end{pmatrix},\qquad
B_{l}=\begin{pmatrix}2&0\\\ \end{pmatrix},\qquad B_{d}=0\>.$

In particular $B_{d}-wB_{l}(A_{u}+w\,B_{u})^{-1}B_{r}$ vanishes identically.
This means that at the second step $A^{\prime}+wB^{\prime}=0$ and the
corresponding zero mode left by the rank 2 matrix A is not localized at any
$w$. We have found a 7d zero mode.

$({\bf 1},{\bf 3})_{0}$: we obtain the same result as above, i.e. one 7d zero
mode.

$({\bf 1},{\bf 1})_{0}$: $X_{+}$ and $Y$ vanish on this one-dimensional
representation, leaving a 7d zero mode.

$({\bf 2},{\bf 2})_{+}$: the operator $X_{+}$ is represented in the basis
$\\{e_{\alpha_{1}+\alpha_{2}+\alpha_{3}},e_{\alpha_{1}+\alpha_{2}}+e_{\alpha_{2}+\alpha_{3}},e_{\alpha_{2}},e_{\alpha_{1}+\alpha_{2}}-e_{\alpha_{2}+\alpha_{3}}\\}$
by the matrix

$A_{({\bf 2},{\bf 2})_{+}}=\left(\begin{array}[]{c|cc|c}0&1&0&0\\\ 0&0&1&0\\\
\hline\cr 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ (4.15)

that has rank $r=2$. In the same basis $Y$ is represented by

$B_{({\bf 2},{\bf 2})_{+}}=\left(\begin{array}[]{c|cc|c}0&0&0&0\\\ 0&0&0&0\\\
\hline\cr 0&0&0&2\\\ 1&0&0&0\\\ \end{array}\right)\>.$ (4.16)

We plug them into the expression (3.29) and apply the algorithm: arranging the
rows and columns to arrive to the expression (3.30) is equivalent to taking
$\bm{\rho}_{u}=(\rho_{2},\rho_{3})$, $\bm{\rho}_{d}=(\rho_{1},\rho_{4})$,
$\bm{\phi}_{u}=(\phi_{1},\phi_{2})$ and $\bm{\phi}_{d}=(\phi_{3},\phi_{4})$.
We can then read

$B_{u}=\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix},\qquad
B_{r}=\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix},\qquad
B_{l}=\begin{pmatrix}0&0\\\ 0&0\\\ \end{pmatrix},\qquad
B_{d}=\begin{pmatrix}0&2\\\ 1&0\\\ \end{pmatrix}\>.$

In particular $B_{d}$ has maximal rank, equal to $d_{R}-r=4-2=2$, where $r$ is
the rank of $A$. This means that $\phi_{d}$ hosts two constant zero modes
localized at $w=0$. These have charge $+1$ with respect to the $U(1)$
generated by $\alpha_{2}^{*}$. Notice that det$(A+wB)=-2w^{2}$. Hence there
are no other points in the base $\mathbb{C}_{w}$ that host localized zero
modes.

$({\bf 2},{\bf 2})_{-}$: analogously to before, we have two zero modes
localized at $w=0$ with charge $-1$ under the preserved $U(1)$ group.

Hence, the number of localized modes at $w=0$ is $4$, that gives rise to two
hypermultiplets with charge $1$ with respect to the $U(1)$ flavor group. This
matches the results of [2, 3], that were obtained with a different procedure.

For generic $k$ and $p$, the zero mode counting proceeds analogously as for
the Pagoda with $n=2$. The $A_{k-1}$ algebra decomposes in the following way
in representations of the Levi subalgebra

$A_{k-1}=({\bm{p^{2}-1}},{\bf 1})_{0}\oplus({\bf
1},\bm{(k-p)^{2}-1})_{0}\oplus({\bf 1},{\bf
1})_{0}\oplus(\bm{p},\bm{\overline{k-p}})_{+}\oplus(\bm{\bar{p}},\bm{k-p})_{-}\>.$
(4.17)

The first three representations host 7d modes, but no localized one. Let us
concentrate on the charged representation $(\bm{p},\bm{\overline{k-p}})_{+}$
of dimension $p(k-p)$. With the choice $p\geq\frac{k}{2}$, we have $p\geq
k-p$. The matrix representing $X_{+}$ in this representation has kernel with
dimension equal to $k-p$, then in our algorithm $r=(p-1)(k-p)$. With a bit of
work, one can check that $B_{d}$ has rank $k-p=d_{R}-r$, that gives then $k-p$
modes localized at $w=0$ with charge $+1$ with respect to the flavor $U(1)$.
The other charge representation hosts again $k-p$ modes localized at $w=0$ and
with charge $-1$. In total we then have $k-p$ charged hypermultiplets, i.e.
the GV invariant is

$n^{g=0}_{1}=k-p\>.$ (4.18)

When $k=2n$ and $p=n$, we obtain $n$ hypers, that is the result we have found
for the Reid’s Pagodas in [2, 3], i.e. $n^{g=0}_{1}=n$.

###  4.2 Simple flop with length $2$

In this section we consider a family of flops of length 2 arising from a
$D_{4}$ singularity deformed over the $\mathbb{C}_{w}$ plane. The threefold is
singular at the origin (where the fiber exhibits a $D_{4}$ singularity) and
can only be partially resolved inflating a $\mathbb{P}^{1}$ corresponding to
the central root of the $D_{4}$ Dynkin diagram. As we can see from Figure 2
the central node has dual Coxeter label equal to 2, and thus its resolution
yields a flop of length 2. In Figure 3 we show our conventions for the
labeling of the simple roots.

Figure 3: $D_{4}$ Dynkin diagram

Since we wish to blow up only the central node, we have
$\mathcal{H}=\langle\alpha_{2}^{*}\rangle$. The Levi subalgebra $\mathcal{L}$
commuting with $\mathcal{H}$ is:

$\mathcal{L}=A_{1}^{(1)}\oplus A_{1}^{(3)}\oplus
A_{1}^{(4)}\oplus\langle\alpha_{2}^{*}\rangle,$ (4.19)

where the $A_{1}$ algebras correspond to the white “tails” in picture 3,
generated by the roots $\alpha_{1}$, $\alpha_{3}$ and $\alpha_{4}$
respectively.

Following the prescription (3.5) for each $A_{1}$ summand we have

$\Phi|_{A_{1}^{(i)}}=\begin{pmatrix}0&1\\\
\varrho_{i}&0\end{pmatrix}=e_{\alpha_{i}}+\varrho_{i}e_{-\alpha_{i}}\qquad
i=1,3,4\>,$ (4.20)

where $\varrho_{i}$ ($i=1,3,4$) is the Casimir of the $s\ell_{2}$ algebra
$A_{1}^{(i)}$. Moreover $\Phi$ can have a component along $\alpha_{2}^{*}$
with coefficient $\varrho_{2}$. Although not necessary for the employment of
our machinery, we report for the sake of visual clarity the explicit matrix
form of the adjoint Higgs field corresponding to the choice (4.19), employing
the standard basis of [17]:

$\Phi=\left(\begin{array}[]{cccc|cccc}\varrho_{2}&1&0&0&0&0&0&0\\\
\varrho_{1}&\varrho_{2}&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&1\\\
0&0&\varrho_{3}&0&0&0&-1&0\\\ \hline\cr
0&0&0&0&-\varrho_{2}&-\varrho_{1}&0&0\\\ 0&0&0&0&-1&-\varrho_{2}&0&0\\\
0&0&0&-\varrho_{4}&0&0&0&-\varrho_{3}\\\ 0&0&\varrho_{4}&0&0&0&-1&0\\\
\end{array}\right).$ (4.21)

The threefold is found by imposing

$\varrho_{i}(w)=w\,c_{i}(w)\qquad\mbox{for }i=1,2,3,4\>,$ (4.22)

where we take the $c_{i}(w)$’s such that $c_{i}(0)\neq 0$. Later we will
simply choose the $c_{i}(w)$’s to be constant in $w$.

The Higgs at the origin is then

$X_{+}=e_{\alpha_{1}}+e_{\alpha_{3}}+e_{\alpha_{4}},$ (4.23)

while $Y$ is

$Y=c_{1}e_{-\alpha_{1}}+c_{3}e_{-\alpha_{3}}+c_{4}e_{-\alpha_{4}}+c_{2}\langle\alpha_{2}^{*}\rangle\>.$
(4.24)

The threefold equation is simply obtained by taking the choice (4.22) and the
expression of $\Phi$ (4.21) and plugging them into the formula
(3.21):181818Notice that the threefold expression is not invariant under the
exchange of $c_{1},c_{3}$ and $c_{4}$, which are the Casimirs of the three
$A_{1}$ tails: this can be overcome by a change of variables. In any case, the
mode localization proceeds in a way that is invariant under the exchange of
$c_{1},c_{3},c_{4}$.

$\begin{split}&x^{2}+zy^{2}-z^{3}+w^{2}z\left[c_{1}^{2}+c_{3}^{2}+c_{4}^{2}+4c_{1}c_{3}+4c_{1}c_{4}-2c_{3}c_{4}-2c_{2}^{2}w(c_{1}-2c_{3}-2c_{4})+c_{2}^{4}w^{2}\right]+\\\
&-2w^{3}\left[c_{1}\left(c_{3}^{2}+c_{4}^{2}+c_{1}c_{3}+c_{1}c_{4}-2c_{3}c_{4}\right)+c_{2}^{2}w\left(c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4}\right)+c_{2}^{4}w^{2}(c_{3}+c_{4})\right]+\\\
&-2wz^{2}\left(c_{1}+c_{3}+c_{4}+c_{2}^{2}w\right)+2w^{2}y(c_{3}-c_{4})\left(c_{1}-c_{2}^{2}w\right)=0.\end{split}$

Let’s consider what happens when one of the $c_{i}$’s vanishes. If $c_{2}=0$,
the preserved gauge group after Higgsing is $SU(2)$ instead of $U(1)$. This
says that the ALE fiber has an $A_{1}$ singularity for all values of $w$, i.e.
the threefold has a non-isolated singularity. If $c_{i}=0$ with $i=1,3,4$,
then the preserved group is still $U(1)$. However, the threefold equation has
an $A_{1}$ singularity for generic $w\in\mathbb{C}_{w}$: in fact, the
threefold equation is the same one would obtain by taking
$\Phi|_{A_{1}^{(i)}}$ identically zero (the equation is insensitive to the
“$1$” in (4.20)). Such a nilpotent vev for the Higgs field is called a T-brane
[9].

Since we want to consider isolated singularities (with a simple flop),
avoiding T-brane configurations, we will take $c_{i}\neq 0$ $\forall i$.

Zero modes. We now analyze the 5d zero modes arising from M-theory reduced on
the flop of length 2 defined by (4.22). We keep the $c_{i}$’s as generic
constants.

As in the case of the flops of length 1, the first step consists in
determining the decomposition of the algebra $\mathfrak{g}=D_{4}$ into irreps
of the Levi subalgebra (4.19), obtaining:

$D_{4}=(\bm{3},\bm{1},\bm{1})_{0}\oplus(\bm{1},\bm{3},\bm{1})_{0}\oplus(\bm{1},\bm{1},\bm{3})_{0}\oplus(\bm{2},\bm{2},\bm{2})_{1}\oplus(\bm{2},\bm{2},\bm{2})_{-1}\oplus(\bm{1},\bm{1},\bm{1})_{2}\oplus(\bm{1},\bm{1},\bm{1})_{-2},$
(4.25)

where the numbers in parenthesis refer to representations of the three $A_{1}$
factors, and the subscript is the charge w.r.t. the Cartan
$\langle\alpha_{2}^{*}\rangle$. Let us examine the zero-mode content of the
Levi representations in (4.25) one by one:

$(\bm{3},\bm{1},\bm{1})_{0}$: for this representation the story flows
identically to the representation $(\bm{3},\bm{1})_{0}$ in the $A_{3}$
example, see (4.13). The operator $X_{+}$ can be represented in the basis
$\\{-e_{\alpha_{1}},\frac{1}{2}h_{1},\frac{1}{2}e_{-\alpha_{1}}\\}$:

$A_{(\bm{3},\bm{1},\bm{1})_{0}}=\left(\begin{array}[]{c|cc}0&1&0\\\ 0&0&1\\\
\hline\cr 0&0&0\\\ \end{array}\right).$ (4.26)

Proceeding as in (4.13) it is easy to show that this representation does not
host any localized 5d zero mode. The same holds for the representations
$(\bm{1},\bm{3},\bm{1})_{0}$ and $(\bm{1},\bm{1},\bm{3})_{0}$.

$(\bm{1},\bm{1},\bm{1})_{2}$: $X_{+}$ is represented by a 1-dimensional matrix
that, in the basis $e_{\alpha_{1}+2\alpha_{2}+\alpha_{3}+\alpha_{4}}$, reads

$A_{(\bm{1},\bm{1},\bm{1})_{2}}=\left(0\right).$ (4.27)

We also have:

$B_{(\bm{1},\bm{1},\bm{1})_{2}}=2c_{2}.$ (4.28)

As a result we find that $B$ has maximal rank, i.e. 1, and so we obtain _one_
localized 5d zero-mode with $U(1)$ charge 2. Analogously, the representation
$(\bm{1},\bm{1},\bm{1})_{-2}$ yields one 5d zero-mode of $U(1)$ charge $-2$.

$(\bm{2},\bm{2},\bm{2})_{1}$: $X_{+}$, once put in Jordan form in an
appropriate basis191919The basis explicitly reads:
$\\{-e_{\alpha_{1}+\alpha_{2}}-e_{\alpha_{2}+\alpha_{3}},e_{\alpha_{1}+\alpha_{2}+\alpha_{4}}-e_{\alpha_{2}+\alpha_{3}+\alpha_{4}},\frac{2e_{\alpha_{1}+\alpha_{2}}}{3}+\frac{e_{\alpha_{2}+\alpha_{3}}}{3}+\frac{e_{\alpha_{2}+\alpha_{4}}}{3},-\frac{1}{3}e_{\alpha_{1}+\alpha_{2}+\alpha_{3}}-\frac{1}{3}e_{\alpha_{1}+\alpha_{2}+\alpha_{4}}+\frac{2}{3}e_{\alpha_{2}+\alpha_{3}+\alpha_{4}},-6e_{\alpha_{2}},-2e_{\alpha_{1}+\alpha_{2}}+2e_{\alpha_{2}+\alpha_{3}}+2e_{\alpha_{2}+\alpha_{4}},e_{\alpha_{1}+\alpha_{2}+\alpha_{3}}+e_{\alpha_{1}+\alpha_{2}+\alpha_{4}}+e_{\alpha_{2}+\alpha_{3}+\alpha_{4}},e_{\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}}\\}$.,
is represented as the 8-dimensional matrix

$\scalebox{0.8}{\mbox{$\displaystyle
A_{(\bm{2},\bm{2},\bm{2})_{1}}=\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\right)$}},$ (4.29)

which has rank $r=5$. Using the same basis for $Y$ we get:

$\displaystyle
B_{(\bm{2},\bm{2},\bm{2})_{1}}=\left(\begin{array}[]{cccccccc}c_{2}&0&0&0&6c_{4}-6c_{3}&0&0&0\\\
c_{1}-c_{3}+c_{4}&c_{2}&\frac{2(c_{3}-c_{4})}{3}&0&0&2c_{4}-2c_{3}&0&0\\\
0&0&c_{2}&0&-6(c_{1}+c_{3}-2c_{4})&0&0&0\\\
2(c_{1}-c_{3})&0&\frac{-c_{1}+5c_{3}-c_{4}}{3}&c_{2}&0&-2(c_{1}+c_{3}-2c_{4})&0&0\\\
0&0&0&0&\rho&0&0&0\\\ 0&0&0&0&c_{1}+c_{3}+c_{4}&c_{2}&0&0\\\
\frac{c_{3}-c_{1}}{3}&0&\frac{2c_{1}-c_{3}-c_{4}}{9}&0&0&\frac{4(c_{1}+c_{3}+c_{4})}{3}&c_{2}&0\\\
0&c_{3}-c_{1}&0&\frac{2c_{1}-c_{3}-c_{4}}{3}&0&0&c_{1}+c_{3}+c_{4}&c_{2}\\\
\end{array}\right)\>.$

(4.30)

Let us pause for a moment and use the results just found to prove that there
are other isolated singularities in the threefold. In fact, these correspond
to values of $w$ where 5d localized modes appear. This happens in the
representation under study when the rank of
$A_{(\bm{2},\bm{2},\bm{2})_{1}}+wB_{(\bm{2},\bm{2},\bm{2})_{1}}$ drops. Its
determinant explicitly reads:

$\displaystyle\begin{split}&\text{det}(A_{(\bm{2},\bm{2},\bm{2})_{1}}+wB_{(\bm{2},\bm{2},\bm{2})_{1}})=w^{4}\left[(c_{1}^{2}+c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4})^{2}+\right.\\\
&\left.-4c_{2}^{2}w(c_{1}^{3}+c_{3}^{3}+c_{4}^{3}-c_{1}^{2}c_{3}-c_{1}^{2}c_{4}-c_{3}^{2}c_{1}-c_{3}^{2}c_{4}-c_{4}^{2}c_{1}-c_{4}^{2}c_{3}+10c_{1}c_{3}c_{4})+\right.\\\
&\left.+2c_{2}^{4}w^{2}(3c_{1}^{2}+3c_{3}^{2}+3c_{4}^{2}+2c_{1}c_{3}+2c_{1}c_{4}+2c_{3}c_{4})-4c_{2}^{6}w^{3}(c_{1}+c_{3}+c_{4})+c_{2}^{8}w^{4}\right].\end{split}$

(4.31)

It turns out that for generic $c_{i}$s the rank of
$A_{(\bm{2},\bm{2},\bm{2})_{1}}+wB_{(\bm{2},\bm{2},\bm{2})_{1}}$ drops on top
of $w=0$, as well as on further four distinct points with non-zero $w$. It can
be checked that these additional points correspond to conifold singularities
far from the origin. In addition, if the condition

$c_{1}^{2}+c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4}=0\>$ (4.32)

is satisfied, one of the additional singularities collides onto the origin: in
this case, the rank of
$A_{(\bm{2},\bm{2},\bm{2})_{1}}+wB_{(\bm{2},\bm{2},\bm{2})_{1}}$ drops on
$w=0$ as well as on three additional points outside the origin. This signals
the appearance of further localized modes at $w=0$, coming from the conifold
singularity that has collided onto the origin. We will explicitly check this
claim momentarily, deriving again condition (4.32).

Rearranging rows and columns to get to the form (3.30) we obtain:

$\displaystyle\begin{split}&B_{u}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&c_{2}&0&0\\\
0&0&\frac{4(c_{1}+c_{3}+c_{4})}{3}&c_{2}&0\\\ \end{array}\right)\\\
&B_{r}=\left(\begin{array}[]{ccc}c_{2}&0&6c_{4}-6c_{3}\\\
0&c_{2}&-6(c_{1}+c_{3}-2c_{4})\\\ 0&0&c_{2}\\\ 0&0&c_{1}+c_{3}+c_{4}\\\
\frac{c_{3}-c_{1}}{3}&\frac{2c_{1}-c_{3}-c_{4}}{9}&0\\\ \end{array}\right)\\\
&B_{l}=\left(\begin{array}[]{ccccc}c_{2}&0&2c_{4}-2c_{3}&0&0\\\
0&c_{2}&-2(c_{1}+c_{3}-2c_{4})&0&0\\\
c_{3}-c_{1}&\frac{2c_{1}-c_{3}-c_{4}}{3}&0&c_{1}+c_{3}+c_{4}&c_{2}\\\
\end{array}\right)\\\
&B_{d}=\left(\begin{array}[]{ccc}c_{1}-c_{3}+c_{4}&\frac{2(c_{3}-c_{4})}{3}&0\\\
2(c_{1}-c_{3})&\frac{-c_{1}+5c_{3}-c_{4}}{3}&0\\\ 0&0&0\\\
\end{array}\right)\\\ \end{split}$

(4.33)

Notice that the rank of $B_{d}$, which is surely non-maximal, depends on the
precise choice of the partial Casimirs. It drops to one when its determinant
is equal to zero. This happens when

$c_{1}^{2}+c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4}=0\>.$
(4.34)

Let us first examine the case in which the $c_{i}$’s are generic constants,
i.e. $B_{d}$ has rank 2. Afterwards we see the case when $B_{d}$ has rank 1.
Notice that $B_{d}$ cannot have rank zero, otherwise $c_{1}=c_{3}=c_{4}=0$,
that we excluded.

  * •

Let’s take generic $c_{i}$’s such that
$c_{1}^{2}+c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4}\neq 0$.
Renaming $A^{\prime}\equiv B_{d}$ and $B^{\prime}\equiv-
B_{l}(A_{u}+wB_{u})^{-1}B_{r}$ we can use equation (3.35) to rerun the
algorithm. $A^{\prime}$ is already in a form with a $2\times 2$ invertible
block and all other elements equal to zero, i.e. $r^{\prime}=2$. We can then
immediately read $B_{d}^{\prime}$ by computing the $(33)$ element of
$B^{\prime}$. It is

$\scalebox{0.8}{\mbox{$\displaystyle
B_{d}^{\prime}=3\left(c_{1}^{2}+c_{3}^{2}+c_{4}^{2}-2c_{1}c_{3}-2c_{1}c_{4}-2c_{3}c_{4}\right)+\frac{10}{3}wc_{2}^{2}(c_{1}+c_{3}+c_{4})-c_{2}^{4}w^{2}$}},$
(4.35)

that has rank 1. As a result, according to (3.36), we find that the total
number of zero modes is:

$\\#=r^{\prime}+2(d_{R}-r-r^{\prime})=2+2(8-5-2)=4,$ (4.36)

where we recall that $d_{R}$ is the dimension of the representation, $r$ is
the rank of (4.29) and $r^{\prime}$ is the rank of $A^{\prime}$. The zero-
modes have charge $+1$ with respect to the $U(1)$ generator.
Analogously, we find 4 localized zero-modes with charge $-1$ in the
$(\bm{2},\bm{2},\bm{2})_{-1}$ representation.

  * •

When the $c_{i}$’s fulfill (4.34), the rank of $B_{d}$ drops to 1. This
produces a change in the zero-mode counting. We can parametrize a solution of
(4.34) in terms of two parameters $q_{1},q_{4}$ as:

$c_{1}=q_{1}^{2},\qquad c_{3}=(q_{1}+\varepsilon q_{4})^{2},\qquad
c_{4}=q_{4}^{2}$ (4.37)

where $\varepsilon$ can take the values $\pm 1$. Now we have

$A^{\prime}=\left(\begin{array}[]{ccc}2q_{1}q_{4}&\frac{2}{3}q_{1}(q_{1}+2\varepsilon
q_{4})&0\\\ -2q_{4}(q_{4}+2\varepsilon
q_{1})&\frac{2}{3}(2q_{1}^{2}+5\varepsilon q_{1}q_{4}+2q_{4}^{2})&0\\\
0&0&0\\\ \end{array}\right),$ (4.38)

When $q_{1}^{2}+\varepsilon q_{1}q_{4}+q_{4}^{2}\neq 0$, the $2\times 2$
matrix is diagonalizable with the non-zero eigenvalue equal to
$\frac{4}{3}\left(q_{1}^{2}+\varepsilon q_{1}q_{4}+q_{4}^{2}\right)$. The
corresponding $B_{d}^{\prime}$ is

$B_{d}^{\prime}=\left(\begin{array}[]{cc}c_{2}^{2}&-\frac{12c_{2}q_{1}q_{4}(q_{1}+\varepsilon
q_{4})}{q_{1}^{2}+\varepsilon q_{1}q_{4}+q_{4}^{2}}\\\
4c_{2}q_{1}q_{4}(q_{1}+\varepsilon q_{4})&0\\\ \end{array}\right).$ (4.39)

This matrix has rank less than two only when one of the $c_{i}$’s vanishes
(and consequently the other two are equal to each other), that we excluded.

When $q_{1}^{2}+\varepsilon q_{1}q_{4}+q_{4}^{2}=0$ (i.e. all the eigenvalues
vanish) the $2\times 2$ matrix has still rank 1 and the corresponding
$B_{d}^{\prime}$ is also forced to have rank 2 (for non-vanishing $c_{i}$’s).

We can finally count the localized zero-modes using formula (3.36), finding:

$\\#=r^{\prime}+2(d_{R}-r-r^{\prime})=1+2(8-5-1)=5.$ (4.40)

Notice that, with respect to the case (4.36) in which the Casimirs were
totally generic, we have found an enhancement in the number of modes on a
specific locus in the space of the partial Casimirs. This is the same locus
where one conifold singularity that was at $w\neq 0$ collides onto the origin.

The representation $(\bm{2},\bm{2},\bm{2})_{-1}$ gives us further 5 zero-modes
of charge $-1$.

Let us summarize our findings for the modes localized at $w=0$ for the simple
flop of length 2 and partial Casimirs given by $\varrho_{i}(w)=w\,c_{i}$, with
$c_{i}$ constants.

  * •

For generic values of $c_{i}$’s, we get:

    * -

8 modes with charge $\pm 1$,

    * -

2 modes with charge $\pm 2$.

In terms of the GV invariants, this means

$n^{g=0}_{1}=4\qquad\mbox{ and }\qquad n^{g=0}_{2}=1\>.$ (4.41)

  * •

For $c_{i}$’s satisfying the constraint (4.34), we get:

    * -

10 modes with charge $\pm 1$,

    * -

2 modes with charge $\pm 2$.

In terms of the GV invariants, this means

$n^{g=0}_{1}=5\qquad\mbox{ and }\qquad n^{g=0}_{2}=1\>.$ (4.42)

For the other (non-zero) values of $w$ where there are localized modes, we
have conifold singularities and the flop is therefore not of length two: in
fact, at these values of $w$ the $D_{4}$ is still deformed to a smaller
singularity of $A$-type.

Non-constant $c_{i}$’s. For simplicity, we have analyzed cases when the
partial Casimirs $\varrho_{i}$ are just a constant $c_{i}$ multiplied by $w$.
Of course, one can also let $c_{i}$ depend on $w$ and rerun the algorithm.

One can in particular find the dependence of the $c_{i}(w)$’s such that the
threefold $X$ has only one isolated singularity at the origin. An easy
solution is when

$c_{1}=4a+b^{2}\,w\>,\qquad c_{2}=b\>,\qquad c_{3}=c_{4}=a\>.$ (4.43)

One can check that for this choice the determinant (4.31) is equal to
$-256a^{3}b^{2}w^{5}$, i.e. it vanishes only at $w=0$. The corresponding
threefold has $n^{g=0}_{1}=5$ and $n^{g=0}_{2}=1$. For $a=-1/4$ and $b=1/2$
one actually recovers the Brown-Wemyss threefold [16] in the form that
appeared in [3] (that has the expected GV invariants).

###  4.3 Simple flop with length $3$

In this section we engineer a threefold $X$ with a simple flop of length
three. Analogously to the previous sections, we are going to define a suitable
Higgs field, valued in the $E_{6}$ Lie algebra, that generates a family of
deformed $E_{6}$ surfaces with an $E_{6}$ singularity at $w=0$. The resolution
of the isolated singularity in the threefold $X$ will blow-up only the
trivalent node of the $E_{6}$ Dynkin diagram (see Figure 2). To achieve this
result, we pick the following Levi subalgebra

$\mathcal{L}=A^{(1,2)}_{2}\oplus A^{(4,5)}_{2}\oplus
A_{1}^{(6)}\oplus\langle\alpha_{3}^{*}\rangle,$ (4.44)

where the factors $A^{(i,j)}_{2}$ are associated, as subalgebras, to the roots
$\alpha_{i},\alpha_{j}$ of the $E_{6}$ Dynkin diagram (we follow the labels in
Figure 4) and $A_{1}^{(6)}$ is the algebra associated to the root
$\alpha_{6}$.

Figure 4: $E_{6}$ Dynkin diagram, with the root blown up in the length three
flop colored in black.

Again, we pick $X_{+}\equiv\Phi\rvert_{w=0}$ to be an element of the principal
nilpotent orbit of each simple factor of $\mathcal{L}$. The partial Casimirs
relative to $\mathcal{L}$ are the total Casimirs of each simple factor of
(4.44), plus the coefficient along the Cartan element
$\langle\alpha_{3}^{*}\rangle$. I.e. the generic $\Phi$ will be such
that202020To match the conventions of Section 2, one takes
$\\{\varrho_{i}|\,\,i=1,...,6\\}=\\{\varrho_{1}^{(3)},\varrho_{2}^{(6)}\\}\cup\\{\varrho_{3}^{(i,j)},\varrho_{2}^{(i,j)}|\,\,(i,j)=(1,2),(4,5)\\}$.

$\Phi\rvert_{A_{2}^{(i,j)}}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
\varrho^{(i,j)}_{3}&\varrho^{(i,j)}_{2}&0\\\
\end{array}\right)=e_{\alpha_{i}}+e_{\alpha_{j}}+\varrho^{(i,j)}_{2}e_{-\alpha_{j}}+\varrho^{(i,j)}_{3}[e_{-\alpha_{j}},e_{-\alpha_{i}}],\qquad
i<j,$ (4.45) $\Phi\rvert_{A_{1}^{(6)}}=\left(\begin{array}[]{ccc}0&1\\\
\varrho^{(6)}_{2}&0\\\
\end{array}\right)=e_{\alpha_{6}}+\varrho^{(6)}_{2}e_{-\alpha_{6}}\qquad\mbox{and}\qquad\Phi\rvert_{\langle\alpha_{3}^{*}\rangle}=\varrho_{1}^{(3)}\langle\alpha_{3}^{*}\rangle\>.$

We now explicitly construct a threefold, by making the choice

$\begin{array}[]{lcl}\varrho_{1}^{(3)}&=&w\,c_{3}\\\
\varrho_{2}^{(6)}&=&w\,c_{6}\\\ \varrho_{2}^{(1,2)}&=&0\\\
\varrho_{2}^{(4,5)}&=&0\\\ \varrho_{3}^{(1,2)}&=&w\,c_{12}\\\
\varrho_{3}^{(4,5)}&=&w\,c_{45}\\\ \end{array}$ (4.46)

with $c_{3},c_{6},c_{12},c_{45}$ constant numbers.

By plugging this choice into the Higgs field vev $\Phi$, and following the
procedure described in Section 3.1, one obtains the threefold as an
hypersurface of $(x,y,z,w)\in\mathbb{C}^{4}$.
As an example, if we pick $c_{3}=0,c_{6}=-3,c_{12}=1,c_{45}=-1$, one gets the
following threefold, which is singular at the origin (as well as at other
three points with non-zero $w$):

$x^{2}+y^{3}+z^{4}+\frac{27w^{6}}{32}+18w^{5}+\left(12w^{3}-\frac{27w^{4}}{16}\right)y+2\left(w^{2}-\frac{9w^{3}}{8}\right)z^{2}+3wyz^{2}=0\>.$
(4.47)

Via a change of coordinates, this exactly coincides with the length 3
threefold explicitly presented by [5].

Zero modes. We now proceed (with the same procedure of the previous sections)
to the mode counting. The branching of the adjoint representation 78 of
$E_{6}$ w.r.t $\mathcal{L}$ in (4.44) is given by212121It can be better
understood starting from the one of the maximal subalgebra
$A_{2}^{(1,2)}\oplus A_{2}^{(4,5)}\oplus A_{2}^{\prime}$ (with
$A_{2}^{\prime}$ containing $e_{\alpha_{6}}$):
$\textbf{78}=\textbf{(8,1,1)}\oplus\textbf{(1,8,1)}\oplus{\textbf{(1,1,8)}}\oplus{(\textbf{3},\overline{\textbf{3}},\textbf{3})}\oplus{(\overline{\textbf{3}},\textbf{3},\overline{\textbf{3}})}.$
One then selects the subalgebra $A_{1}^{(6)}\subset A_{2}^{\prime}$, and
correspondingly branches each term of the sum.

78 $\displaystyle=$
$\displaystyle\textbf{(8,1,1)}_{0}\oplus\textbf{(1,8,1)}_{0}\oplus{\textbf{(1,1,3)}_{0}\oplus(\textbf{1},\textbf{1},\textbf{2})_{3}\oplus(\textbf{1},\textbf{1},\textbf{2})_{-3}\oplus(\textbf{1},\textbf{1},\textbf{1})_{0}}\oplus$
(4.48)
$\displaystyle\oplus{(\textbf{3},\overline{\textbf{3}},\textbf{2})_{1}\oplus(\textbf{3},\overline{\textbf{3}},\textbf{1})_{-2}}\oplus{(\overline{\textbf{3}},\textbf{3},\textbf{2})_{-1}\oplus(\overline{\textbf{3}},\textbf{3},\textbf{1})_{2}}\>,$

where the subscripts denote the charges under $\langle\alpha_{3}^{*}\rangle$.

For the $E$-cases the explicit computations done for length one and two become
convoluted. We present here only the results. We have worked out a Mathematica
routine, presented in Appendix D, that implements the algorithm described in
Section 3.3 and that can be used to check the results. Running this code for a
generic choice of the parameters $c_{6},c_{3},c_{12},c_{45}$, we obtained, for
each irreducible representation appearing in (4.48), the 5d modes shown in
Table 1. In the table, we also write how many elements of the given
representation support a mode localized in $\mathbb{C}[w]/(w^{k})$, for each
$k$; we find that $k\leq 2$.

$R^{\mathcal{L}}$ |  $\mathbb{C}[w]/(w)$ |  $\mathbb{C}[w]/(w^{2})$ |  $\\#_{\rm zero\,modes}$
---|---|---|---
$(\textbf{8},\textbf{1},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{8},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{1},\textbf{3})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{1},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{3},\overline{\textbf{3}},\textbf{2})_{1}$ |  4 |  1 |  6
$(\overline{\textbf{3}},\textbf{3},\textbf{2})_{-1}$ |  4 |  1 |  6
$(\overline{\textbf{3}},\textbf{3},\textbf{1})_{2}$ |  3 |  0 |  3
$(\textbf{3},\overline{\textbf{3}},\textbf{1})_{-2}$ |  3 |  0 |  3
$(\textbf{1},\textbf{1},\textbf{2})_{3}$ |  1 |  0 |  1
$(\textbf{1},\textbf{1},\textbf{2})_{-3}$ |  1 |  0 |  1
Table 1: 5d modes for $E_{6}$ length three simple flop.

We get a total of 20 5d modes:

  * •

one hyper with charge three, inside
$(\textbf{1},\textbf{1},\textbf{2})_{3}\oplus(\textbf{1},\textbf{1},\textbf{2})_{-3}$;

  * •

three hypers with charge two inside
$(\overline{\textbf{3}},\textbf{3},\textbf{1})_{2}\oplus(\textbf{3},\overline{\textbf{3}},\textbf{1})_{-2}$;

  * •

six hypers with charge one inside
$(\textbf{3},\overline{\textbf{3}},\textbf{2})_{1}\oplus(\overline{\textbf{3}},\textbf{3},\textbf{2})_{-1}$.

In terms of the GV invariants, one then reads

$n^{g=0}_{1}=6\,,\qquad n^{g=0}_{2}=3\qquad\mbox{ and }\qquad
n^{g=0}_{3}=1\>,$ (4.49)

which perfectly coincides with the results of [5].
We can finally check whether there are special choices of the parameters
$c_{12}$, $c_{45}$, $c_{6},c_{3}$ for which the number of 5d modes localized
at $w=0$ enhances. A necessary condition for the enhancement of the number of
modes is that the rank of the matrix $B_{d}$ drops for a special choice of the
partial Casimirs. By explicit computation, we find that the rank drops when
$c_{12}=c_{45}$ or $c_{6}=0$. However, these choices would create a non-
isolated singularity.

###  4.4 Simple flop with length $4$

In the following section we are going to engineer, by means of a Higgs field
$\Phi$ valued in the $E_{7}$ Lie algebra, a flop of length four. By looking at
the dual Coxeter labels of the $E_{7}$ Dynkin diagram in Figure 2, we see that
the simultaneous resolution should involve the trivalent node. Analogously to
the previous examples, this means that we have to pick the Higgs field in the
Levi subalgebra

$\mathcal{L}\equiv A_{3}^{(4,5,6)}\oplus A_{2}^{(1,2)}\oplus
A_{1}^{(7)}\oplus\langle\alpha_{3}^{*}\rangle,$ (4.50)

where the superscripts refer to the roots of the $E_{7}$ Dynkin diagram
numbered as in the Figure 5, and $\alpha_{3}$ is the trivalent root of
$E_{7}$.

Figure 5: $E_{7}$ Dynkin diagram, with the root blown up in the length four
flop colored in black.

Analogously to the $E_{6}$ case, we choose the Higgs field as follows:

$\Phi\rvert_{\langle\alpha_{3}^{*}\rangle}=c_{3}w\langle\alpha_{3}^{*}\rangle$
(4.51)

and

$\displaystyle\Phi\rvert_{A_{1}^{(7)}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&1\\\ c_{7}w&0\\\
\end{array}\right)=e_{\alpha_{7}}+c_{7}w\,e_{-\alpha_{7}},$ (4.54)
$\displaystyle\Phi\rvert_{A_{2}^{(1,2)}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ c_{12}w&0&0\\\
\end{array}\right)=e_{\alpha_{1}}+e_{\alpha_{2}}+c_{12}w\,[e_{-\alpha_{1}},e_{-\alpha_{2}}],$
(4.58) $\displaystyle\Phi\rvert_{A_{3}^{(4,5,6)}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
c_{456}w&0&0&0\\\
\end{array}\right)=e_{\alpha_{4}}+e_{\alpha_{5}}+e_{\alpha_{6}}+c_{456}w\Big{[}[e_{-\alpha_{4}},e_{-\alpha_{5}}],e_{-\alpha_{6}}\Big{]}.$
(4.63)

The corresponding threefold is a hypersurface in $\mathbb{C}^{4}$, that is a
family of deformed $E_{7}$ singularities over $\mathbb{C}_{w}$. To make the
equation of the threefold more readable, we set the parameters to specific
values, picking $c_{3}=0,c_{7}=3,c_{12}=\frac{1}{2},c_{456}=-\frac{1}{2}$,
obtaining

$\begin{split}&x^{2}-y^{3}+yz^{3}+3wy^{2}z+y^{2}\frac{81w^{2}}{16}-yz\frac{w^{2}}{12}+z^{2}\frac{5w^{3}}{8}-y\frac{w^{3}}{108}+z\frac{w^{4}}{3}+\frac{w^{5}}{144}=0.\end{split}$
(4.64)

where we neglected terms of high degree, irrelevant for the singularity at
$w=0$.

Zero modes. We now proceed with the modes counting. We will again perform the
gauge-fixing separately in each irreducible representation of the branching of
the adjoint representation 133 of $E_{7}$ under the subalgebra
$\mathcal{L}$:222222The first entry of each summand is a representation of
$A_{3}^{(4,5,6)}$, the second one is a representation of $A_{2}^{(1,2)}$, and
the third on a representation of $A_{1}^{(7)}$. The subscript is the charge
under $\langle\alpha_{3}^{*}\rangle$.

133 $\displaystyle=$
$\displaystyle(\textbf{15},\textbf{1},\textbf{1})_{0}\oplus(\textbf{1},\textbf{8},\textbf{1})_{0}\oplus(\textbf{1},\textbf{1},\textbf{3})_{0}\oplus(\textbf{1},\textbf{1},\textbf{1})_{0}\oplus$
(4.65)
$\displaystyle(\overline{\textbf{4}},\textbf{3},\textbf{2})_{-1}\oplus(\textbf{4},\overline{\textbf{3}},\textbf{2})_{1}\oplus(\textbf{6},\overline{\textbf{3}},\textbf{1})_{-2}\oplus(\textbf{6},\textbf{3},\textbf{1})_{2}\oplus$
$\displaystyle(\textbf{4},\textbf{1},\textbf{2})_{-3}\oplus(\overline{\textbf{4}},\textbf{1},\textbf{2})_{3}\oplus(\textbf{1},\textbf{3},\textbf{1})_{-4}\oplus(\textbf{1},\overline{\textbf{3}},\textbf{1})_{4}.$

Running the Mathematica routine described in Appendix D, we find the results
displayed in table 2. As in the $E_{6}$ case, there are no five-dimensional
modes localized in $\mathbb{C}[w]/(w^{k})$, with $k>2$.

$R^{\mathcal{L}}$ |  $\mathbb{C}[w]/(w)$ |  $\mathbb{C}[w]/(w^{2})$ |  $\\#_{\rm zero\,modes}$
---|---|---|---
$(\textbf{15},\textbf{1},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{8},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{1},\textbf{3})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{1},\textbf{1})_{0}$ |  0 |  0 |  0
$(\overline{\textbf{4}},\textbf{3},\textbf{2})_{-1}$ |  6 |  0 |  6
$(\textbf{4},\overline{\textbf{3}},\textbf{2})_{1}$ |  6 |  0 |  6
$(\textbf{6},\overline{\textbf{3}},\textbf{1})_{-2}$ |  3 |  1 |  5
$(\textbf{6},\textbf{3},\textbf{1})_{2}$ |  3 |  1 |  5
$(\textbf{4},\textbf{1},\textbf{2})_{-3}$ |  2 |  0 |  2
$(\overline{\textbf{4}},\textbf{1},\textbf{2})_{3}$ |  2 |  0 |  2
$(\textbf{1},\textbf{3},\textbf{1})_{-4}$ |  1 |  0 |  1
$(\textbf{1},\overline{\textbf{3}},\textbf{1})_{4}$ |  1 |  0 |  1
Table 2: five-dimensional modes for $E_{7}$ length four simple flop.

In total, we find 28 modes localized at $w=0$:

  * •

one hyper with charge four, inside
$(\textbf{1},\textbf{3},\textbf{1})_{-4}\oplus(\textbf{1},\overline{\textbf{3}},\textbf{1})_{4}$;

  * •

two hypers with charge three inside
$(\textbf{4},\textbf{1},\textbf{2})_{-3}\oplus(\overline{\textbf{4}},\textbf{1},\textbf{2})_{3}$;

  * •

five hypers with charge two inside
$(\textbf{6},\overline{\textbf{3}},\textbf{1})_{-2}\oplus(\textbf{6},\textbf{3},\textbf{1})_{2}$.

  * •

six hypers with charge one inside
$(\overline{\textbf{4}},\textbf{3},\textbf{2})_{-1}\oplus(\textbf{4},\overline{\textbf{3}},\textbf{2})_{1}$.

In terms of the GV invariants, one then reads

$n^{g=0}_{1}=6\,,\qquad n^{g=0}_{2}=5\,,\qquad n^{g=0}_{3}=2\qquad\mbox{ and
}\qquad n^{g=0}_{4}=1\>.$ (4.66)

Finally, we find (as in the $E_{6}$ case) that no particular choice of the
constants $c_{i}$ can enhance the number of zero modes at $w=0$ (without
generating non-isolated singularities).

###  4.5 Simple flop with length $5$

A flop with length 5 is obtained from an $E_{8}$ family over $\mathbb{C}_{w}$.
The node that should be simultaneously resolved at $w=0$ is depicted in Figure
6.

Figure 6: $E_{8}$ Dynkin diagram, with the root blown up in the length five
flop colored in black.

We then have $\mathcal{H}=\langle\alpha_{4}^{*}\rangle$ and

$\mathcal{L}=A_{3}^{(5,6,7)}\oplus
A_{4}^{(1,2,3,8)}\oplus\langle\alpha_{4}^{*}\rangle.$ (4.67)

We make the simple choice

$\displaystyle\Phi\rvert_{\langle\alpha_{4}^{*}\rangle}$ $\displaystyle=$
$\displaystyle c_{4}w\,\langle\alpha_{4}^{*}\rangle$
$\displaystyle\Phi\rvert_{A_{3}^{(5,6,7)}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
c_{567}w&0&0&0\end{array}\right)=e_{\alpha_{5}}+e_{\alpha_{6}}+e_{\alpha_{7}}+c_{567}w\left[[e_{-\alpha_{5}},e_{-\alpha_{6}}],e_{-\alpha_{7}}\right]$
(4.72) $\displaystyle\Phi\rvert_{A_{4}^{(1,2,3,8)}}$ $\displaystyle=$

$\displaystyle\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\\\ c_{1238}w&0&0&0&0\\\
\end{array}\right)=e_{\alpha_{1}}+e_{\alpha_{2}}+e_{\alpha_{3}}+e_{\alpha_{8}}-c_{1238}w\left[\Big{[}[e_{-\alpha_{1}},e_{-\alpha_{2}}],e_{-\alpha_{3}}\right],e_{-\alpha_{8}}\Big{]}$

(4.78)

with constant $c$’s. We obtain our threefold as an hypersurface in
$\mathbb{C}^{4}$. To make the equation more readable, we pick explicit values
for the parameters, setting $c_{4}=0,c_{567}=1,c_{1238}=-1$:

$x^{2}+y^{3}+z^{5}+w^{7}+\frac{w^{6}}{864}-\frac{23w^{5}z}{36}-\frac{w^{4}y}{48}-\frac{187w^{4}z^{2}}{36}-\frac{13}{3}w^{3}yz-\frac{2w^{3}z^{3}}{27}-\frac{1}{3}w^{2}yz^{2}=0\>.$
(4.79)

Zero modes. We can explicitly perform the branching of the adjoint
representation 248 of $E_{8}$ under the chosen $\mathcal{L}$:232323The first
number denotes the dimension of the representation of $A_{3}^{(5,6,7)}$, the
second under $A_{4}^{(1,2,3,8)}$ and the subscript is the charge under the
Cartan $\alpha_{4}^{*}$.

248 $\displaystyle=$
$\displaystyle(\textbf{1},\textbf{24})_{0}\oplus(\textbf{15},\textbf{1})_{0}\oplus(\textbf{1},\textbf{1})_{0}\oplus$
(4.80)
$\displaystyle(\textbf{4},\overline{\textbf{10}})_{1}\oplus(\overline{\textbf{4}},\textbf{10})_{-1}\oplus(\textbf{6},\textbf{5})_{2}\oplus(\textbf{6},\overline{\textbf{5}})_{-2}\oplus$
$\displaystyle(\overline{\textbf{4}},\overline{\textbf{5}})_{3}\oplus(\textbf{4},\textbf{5})_{-3}\oplus(\textbf{1},\textbf{10})_{4}\oplus(\textbf{1},\overline{\textbf{10}})_{-4}\oplus$
$\displaystyle(\textbf{4},\textbf{1})_{5}\oplus(\overline{\textbf{4}},\textbf{1})_{-5}.$

The result of the zero mode counting is displayed in Table 3. There are no
modes localized in $\mathbb{C}[w]/(w^{k}),$ with $k>2$. We find $48$ modes
localized at $w=0$:

$R^{\mathcal{L}}$ |  $\mathbb{C}[w]/(w)$ |  $\mathbb{C}[w]/(w^{2})$ |  $\\#_{\rm zero\,modes}$
---|---|---|---
$(\textbf{1},\textbf{24})_{0}$ |  0 |  0 |  0
$(\textbf{15},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{1},\textbf{1})_{0}$ |  0 |  0 |  0
$(\textbf{4},\overline{\textbf{10}})_{1}$ |  6 |  1 |  8
$(\overline{\textbf{4}},\textbf{10})_{-1}$ |  6 |  1 |  8
$(\textbf{6},\textbf{5})_{2}$ |  6 |  0 |  6
$(\textbf{6},\overline{\textbf{5}})_{-2}$ |  6 |  0 |  6
$(\overline{\textbf{4}},\overline{\textbf{5}})_{3}$ |  4 |  0 |  4
$(\textbf{4},\textbf{5})_{-3}$ |  4 |  0 |  4
$(\textbf{1},\textbf{10})_{4}$ |  2 |  0 |  2
$(\textbf{1},\overline{\textbf{10}})_{-4}$ |  2 |  0 |  2
$(\textbf{4},\textbf{1})_{5}$ |  1 |  0 |  1
$(\overline{\textbf{4}},\textbf{1})_{-5}$ |  1 |  0 |  1
Table 3: five-dimensional modes for $E_{8}$ length five simple flop.

  * •

one hyper with charge five, inside
$(\textbf{4},\textbf{1})_{5}\oplus(\overline{\textbf{4}},\textbf{1})_{-5}$;

  * •

two hyper with charge four, inside
$(\textbf{1},\textbf{10})_{4}\oplus(\textbf{1},\overline{\textbf{10}})_{-4}$;

  * •

four hypers with charge three inside
$(\overline{\textbf{4}},\overline{\textbf{5}})_{3}\oplus(\textbf{4},\textbf{5})_{-3}$;

  * •

six hypers with charge two inside
$(\textbf{6},\textbf{5})_{2}\oplus(\textbf{6},\overline{\textbf{5}})_{-2}$;

  * •

eight hypers with charge one inside
$(\textbf{4},\overline{\textbf{10}})_{1}\oplus(\overline{\textbf{4}},\textbf{10})_{-1}$.

In terms of the GV invariants, one then reads

$n^{g=0}_{1}=8\,,\qquad n^{g=0}_{2}=6\,,\qquad n^{g=0}_{3}=4\,,\qquad
n^{g=0}_{4}=2\qquad\mbox{ and }\qquad n^{g=0}_{5}=1\>.$ (4.81)

Again, we notice that we can not enhance the number of zero-modes at $w=0$
without generating a non-isolated singularity.

###  4.6 Simple flop with length $6$

In this section we conclude our analysis of simple flops by dealing with the
highest length case, i.e. a flop of length 6 arising from a $E_{8}$
singularity deformed over the plane $\mathbb{C}_{w}$. We choose the Higgs
$\Phi\in E_{8}$ in such a way to resolve only the central node of the $E_{8}$
Dynkin diagram as depicted in Figure 7.

Figure 7: $E_{8}$ Dynkin diagram

According to the principles outlined in previous sections, the Higgs field
resolving the central node must lie in the Levi subalgebra defined by:

$\mathcal{L}=A_{4}^{(4,5,6,7)}\oplus A_{2}^{(1,2)}\oplus
A_{1}^{(8)}\oplus\langle\alpha_{3}^{*}\rangle,$ (4.82)

where as usual the upper indices label the simple roots. Again we choose
$\Phi$ of the following form:

$\begin{split}&\Phi|_{{A_{4}}^{(4,5,6,7)}}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\ c_{4567}w&0&0&0&0\\\
\end{array}\right)=e_{\alpha_{4}}+e_{\alpha_{5}}+e_{\alpha_{6}}+e_{\alpha_{7}}-c_{4567}w\left[\left[\left[e_{-\alpha_{1}},e_{-\alpha_{2}}\right],e_{-\alpha_{3}}\right],e_{-\alpha_{4}}\right],\\\
&\Phi|_{{A_{2}}^{(1,2)}}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
c_{12}w&0&0\\\
\end{array}\right)=e_{\alpha_{1}}+e_{\alpha_{2}}+c_{12}w\left[e_{-\alpha_{1}},e_{-\alpha_{2}}\right],\\\
&\Phi|_{{A_{1}}^{(8)}}=\left(\begin{array}[]{cc}0&1\\\ c_{8}w&0\\\
\end{array}\right)=e_{\alpha_{8}}+c_{8}w\hskip 2.84544pte_{-\alpha_{8}},\\\
&\Phi\rvert_{\langle\alpha_{3}^{*}\rangle}=c_{3}w\,\langle\alpha_{3}^{*}\rangle.\end{split}$
(4.83)

To make the equation more readily understandable, we set the parameters to a
specific value $c_{3}=0,c_{8}=1,c_{12}=-1,c_{4567}=1$. In this way we obtain
the threefold

$\displaystyle
x^{2}+y^{3}+z^{5}+\frac{89w^{7}}{144}+\frac{w^{6}}{864}+\frac{55w^{5}y}{12}-\frac{w^{4}y}{48}-\frac{17w^{4}z^{2}}{24}-\frac{11w^{3}z^{3}}{12}-\frac{7}{2}w^{2}yz^{2}-wyz^{3}=0\>,$

(4.84)

where we neglected terms of high degree, irrelevant for the singularity at
$w=0$.

Zero modes. We perform the mode counting explicitly, independently for each
irreducible representation arising from the adjoint $\bm{248}$ of $E_{8}$,
branched under the Levi subalgebra (4.82). The decomposition reads:

$\begin{split}\bm{248}\hskip 5.69046pt=\hskip
5.69046pt&(\bm{24},\bm{1},\bm{1})_{0}\oplus(\bm{1},\bm{8},\bm{1})_{0}\oplus(\bm{1},\bm{1},\bm{3})_{0}\oplus(\bm{1},\bm{1},\bm{1})_{0}\hskip
2.84544pt\oplus\\\
&(\bm{5},\bm{\overline{3}},\bm{2})_{1}\oplus(\bm{\overline{5}},\bm{3},\bm{2})_{-1}\oplus(\bm{10},\bm{3},\bm{1})_{2}\oplus(\bm{\overline{10}},\bm{\overline{3}},\bm{1})_{-2}\hskip
2.84544pt\oplus\\\
&(\bm{\overline{10}},\bm{1},\bm{2})_{3}\oplus(\bm{10},\bm{1},\bm{2})_{-3}\oplus(\bm{\overline{5}},\bm{\overline{3}},\bm{1})_{4}\oplus(\bm{5},\bm{3},\bm{1})_{-4}\hskip
2.84544pt\oplus\\\
&(\bm{1},\bm{3},\bm{2})_{5}\oplus(\bm{1},\bm{\overline{3}},\bm{2})_{-5}\oplus(\bm{5},\bm{1},\bm{1})_{6}\oplus(\bm{\overline{5}},\bm{1},\bm{1})_{-6}\end{split}$
(4.85)

Applying the Mathematica routine presented in Appendix D, we find the zero
modes in Table 4.

$R^{\mathcal{L}}$ |  $\mathbb{C}[w]/(w)$ |  $\mathbb{C}[w]/(w^{2})$ |  $\\#_{\rm zero\,modes}$
---|---|---|---
$(\bm{24},\bm{1},\bm{1})_{0}$ |  0 |  0 |  0
$(\bm{1},\bm{8},\bm{1})_{0}$ |  0 |  0 |  0
$(\bm{1},\bm{1},\bm{3})_{0}$ |  0 |  0 |  0
$(\bm{1},\bm{1},\bm{1})_{0}$ |  0 |  0 |  0
$(\bm{5},\bm{\overline{3}},\bm{2})_{1}$ |  6 |  0 |  6
$(\bm{\overline{5}},\bm{3},\bm{2})_{-1}$ |  6 |  0 |  6
$(\bm{10},\bm{3},\bm{1})_{2}$ |  6 |  0 |  6
$(\bm{\overline{10}},\bm{\overline{3}},\bm{1})_{-2}$ |  6 |  0 |  6
$(\bm{\overline{10}},\bm{1},\bm{2})_{3}$ |  4 |  0 |  4
$(\bm{10},\bm{1},\bm{2})_{-3}$ |  4 |  0 |  4
$(\bm{\overline{5}},\bm{\overline{3}},\bm{1})_{4}$ |  3 |  0 |  3
$(\bm{5},\bm{3},\bm{1})_{-4}$ |  3 |  0 |  3
$(\bm{1},\bm{3},\bm{2})_{5}$ |  2 |  0 |  2
$(\bm{1},\bm{\overline{3}},\bm{2})_{-5}$ |  2 |  0 |  2
$(\bm{5},\bm{1},\bm{1})_{6}$ |  1 |  0 |  1
$(\bm{\overline{5}},\bm{1},\bm{1})_{-6}$ |  1 |  0 |  1

.

Table 4: Five-dimensional modes for $E_{8}$ length six simple flop.

We find a total of 44 localized modes:

  * •

one hyper with charge six, inside
$(\textbf{5},\textbf{1},\textbf{1})_{6}\oplus(\overline{\textbf{5}},\textbf{1},\textbf{1})_{-6}$;

  * •

two hypers with charge five, inside
$(\textbf{1},\textbf{3},\textbf{2})_{5}\oplus(\textbf{1},\overline{\textbf{3}},\textbf{2})_{-5}$;

  * •

three hypers with charge four, inside
$(\overline{\textbf{5}},\overline{\textbf{3}},\textbf{1})_{4}\oplus(\textbf{5},\textbf{3},\textbf{1})_{-4}$;

  * •

four hypers with charge three inside
$(\overline{\textbf{10}},\textbf{1},\textbf{2})_{3}\oplus(\textbf{10},\textbf{1},\textbf{2})_{-3}$;

  * •

six hypers with charge two inside
$(\textbf{10},\textbf{3},\textbf{1})_{2}\oplus(\overline{\textbf{10}},\overline{\textbf{3}},\textbf{1})_{-2}$;

  * •

six hypers with charge one inside
$(\textbf{5},\overline{\textbf{3}},\textbf{2})_{1}\oplus(\overline{\textbf{5}},\textbf{3},\textbf{2})_{-1}$.

In terms of the GV invariants, one then reads

$n^{g=0}_{1}=6,\qquad n^{g=0}_{2}=6,\qquad n^{g=0}_{3}=4,\qquad
n^{g=0}_{4}=3,\qquad n^{g=0}_{5}=2\quad\mbox{ and }\quad n^{g=0}_{6}=1\>.$

Finally, analyzing the rank of the matrix $B_{d}$, we find that no enhancement
in the number of localized modes at $w=0$ is feasible without generating a
non-isolated singularity.

##  5 Conclusions

In the present work, we constructed and studied the GV invariants of examples
of threefold simple flops of any length. Building on our previous results
based on the M-theory/Type IIA duality [3, 2], we introduced techniques to
deal with intrinsically non-perturbative cases, where the Type IIA limit is
not available, involving one-parameter families of deformed
$E_{6},E_{7},E_{8}$ singularities. By means of a group-theoretic computational
algorithm (included in an ancillary Mathematica file), the determination of
the GV invariants for flops of any length has been reduced to a linear algebra
problem. Our method is explicit, and also enables us to detect special cases
of threefold simple flops exhibiting enhanced GV invariants.
Furthermore, from a physical perspective, our work allows to study the
dynamics of M-theory on threefold simple flops, which are notably non-toric,
characterizing the Higgs Branches of the resulting 5d SCFTs as complex
algebraic varieties. This is possible thanks to the correspondence between the
GV invariants of threefold simple flops and BPS states arising from M2-branes
wrapped on the curve inflated by the resolution of the simple flops. Indeed,
M2-brane states descend to hypermultiplets in 5d, whose charges under the
flavor group are determined by the degrees of the GV invariants, which we
computed explicitly in flops of all lengths. In particular, we are able to
fully characterize the action of the flavor symmetries on the Higgs Branch.
These results pave the way for a range of engaging perspectives. For example,
it would be interesting to apply our methods to the so-called $(A_{j},E_{k})$
(with $k=6,7,8$) singularities, which require an intrinsically non-
perturbative description. Furthermore, achieving a more thorough understanding
of the construction of general Higgs backgrounds would prove hugely profitable
for the study of more general one-parameter fibrations of exceptional
singularities.

## Acknowledgments

A.C. is a Research Associate of the Fonds de la Recherche Scientifique
F.N.R.S. (Belgium). The work of A.C. is partially supported by IISN - Belgium
(convention 4.4503.15), and supported by the Fonds de la Recherche
Scientifique - F.N.R.S. under Grant CDR J.0181.18. A.S. and R.V. acknowledge
support by INFN Iniziativa Specifica ST&FI. M.D.M. acknowledges support by
INFN Iniziativa Specifica GAST.

##  Appendix A The threefold equation: The Katz-Morrison method

Let us explain the method for obtaining a simultaneous resolution developed by
Katz and Morrison [1].

One can start from the family over $\mathfrak{t}$ (2.14), that is the
expression of the versal deformation of the ADE singularity in which the
deformation parameters are functions of the $t_{i}$’s that encode the volumes
of the 2-cycles. To write down the equation of the fibration over
$\mathfrak{t}/\mathcal{W}^{\prime}$, one first needs to write the
$\mathcal{W}^{\prime}$ invariant coordinates $\varrho_{i}$’s in terms of the
$t_{i}$’s; inverting these relations and plugging the result in (2.14) one
obtains the hypersurface equation of the partial simultaneous resolution [1].

Let us immediately clarify this statement with the example of the deformed
$A_{3}$ singularity we have introduced before. According to (2.14) the
defining equation reads:

$x^{2}+y^{2}+(z-t_{1})(z-t_{2})(z-t_{3})(z-t_{4})=0\quad\quad
t_{1}+t_{2}+t_{3}+t_{4}=0.$ (A.1)

Now suppose again that we wish to produce a simple flop in which only the
central node is resolved. The original Weyl group $\mathcal{W}=S_{4}$, acts as
the permutation group on the four $t_{i}$’s. Here, the desired simultaneous
resolution defines for us the subgroup
$\mathcal{W}^{\prime}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, which acts by
exchange of $t_{1}\leftrightarrow t_{2}$ and $t_{3}\leftrightarrow t_{4}$,
respectively. It is easy to see that the $\mathcal{W}^{\prime}$-invariant
coordinates are:

$\tilde{s}_{1}=t_{1}\cdot
t_{2}\,,\qquad\tilde{s}_{2}=t_{1}+t_{2}=-t_{3}-t_{4}\,,\qquad\tilde{s}_{3}=-t_{1}t_{3}-t_{2}t_{3}-t_{3}^{2}=t_{3}\cdot
t_{4}$ (A.2)

and comparing them to the expression (A.1) in terms of the $t_{i}$ it is a
matter of mechanical computation242424In practice, once one has the expression
of the $\tilde{s}_{i}$ as functions of the $t_{i}$, one can write down the
most general deformed $A_{3}$ singularity invariant under
$\mathcal{W}^{\prime}$ in terms of the $\varrho_{i}$ with unfixed
coefficients. Comparing its terms degree by degree in the $t_{i}$ with
expression (A.1) the correct coefficients can be easily extracted. to extract
the fibration over $\mathfrak{t}/\mathcal{W}^{\prime}$:

$x^{2}+y^{2}+(z^{2}-\tilde{s}_{2}z+\tilde{s}_{1})(z^{2}+\tilde{s}_{2}z+\tilde{s}_{3})=0.$
(A.3)

Finally, choosing a dependence $\tilde{s}_{i}(w)$ one gets a simple threefold
flop. A well-known example is _Reid’s pagoda_ , obtained by choosing:

$\tilde{s}_{2}=0\,,\quad\tilde{s}_{1}=-\tilde{s}_{3}=w$ (A.4)

which gives the hypersurface $x^{2}+y^{2}+z^{4}-w^{2}$.

In principle, this procedure can be carried out for any deformed ADE
singularity and any resolution pattern, although for the exceptional cases it
can get particularly time-consuming.

##  Appendix B Casimirs for the equation of $E_{r}$ families

Let us consider the deformed $E_{n}$ singularities in equation (2.14). In each
one, we have $n$ deformation parameters:

$\begin{array}[]{c|cl}\bm{E_{6}}&\epsilon_{i}&\text{for }i=2,5,6,8,9,12\\\
\bm{E_{7}}&\tilde{\epsilon}_{i}&\text{for }i=2,6,8,10,12,14,18\\\
\bm{E_{8}}&\hat{\epsilon}_{i}&\text{for }i=2,8,12,14,18,20,24,30\\\
\end{array}$ (B.1)

Following Section 3.2, one computes their expression in terms of the Casimir
$\chi_{i}^{E_{r}}(\Phi)$, with $\chi_{i}^{E_{r}}$ defined in (3.12).

Let us define (see (3.12))

$c_{k_{i}}\equiv\chi_{i}^{E_{6}}(\Phi),\qquad\tilde{c}_{k_{i}}\equiv\chi_{i}^{E_{7}}(\Phi),\qquad\hat{c}_{k_{i}}\equiv\chi_{i}^{E_{8}}(\Phi)\>.$
(B.2)

The result for the $E_{6}$ case is:

$\displaystyle\begin{split}&\epsilon_{2}=-\frac{c_{2}}{24}\\\
&\epsilon_{5}=\frac{c_{5}}{60}\\\
&\epsilon_{6}=\frac{c_{2}^{3}}{13824}-\frac{c_{6}}{144}\\\
&\epsilon_{8}=-\frac{c_{2}^{4}}{110592}+\frac{13c_{2}c_{6}}{8640}-\frac{c_{8}}{240}\\\
&\epsilon_{9}=\frac{c_{9}}{756}-\frac{c_{2}^{2}c_{5}}{11520}\\\
&\epsilon_{12}=-\frac{c_{12}}{3240}+\frac{109c_{2}^{6}}{4299816960}-\frac{847c_{2}^{3}c_{6}}{134369280}+\frac{109c_{2}^{2}c_{8}}{3732480}+\frac{13c_{2}c_{5}^{2}}{466560}+\frac{61c_{6}^{2}}{933120}.\\\
\end{split}$

(B.3)

For the $E_{7}$ case:

$\displaystyle\begin{split}\tilde{\epsilon}_{2}&=\frac{\tilde{c}_{2}}{18}\\\
\tilde{\epsilon}_{6}&=\frac{\tilde{c}_{2}^{3}}{139968}-\frac{\tilde{c}_{6}}{72}\\\
\tilde{\epsilon}_{8}&=-\frac{7\tilde{c}_{2}^{4}}{25194240}+\frac{11\tilde{c}_{2}\tilde{c}_{6}}{16200}-\frac{\tilde{c}_{8}}{300}\\\
\tilde{\epsilon}_{10}&=-\frac{2\tilde{c}_{10}}{315}+\frac{\tilde{c}_{2}^{5}}{151165440}-\frac{17\tilde{c}_{2}^{2}\tilde{c}_{6}}{583200}+\frac{\tilde{c}_{2}\tilde{c}_{8}}{1400}\\\
\tilde{\epsilon}_{12}&=-\frac{16\tilde{c}_{10}\tilde{c}_{2}}{1148175}+\frac{\tilde{c}_{12}}{12150}-\frac{149\tilde{c}_{2}^{6}}{10579162152960}+\frac{167\tilde{c}_{2}^{3}\tilde{c}_{6}}{3401222400}+\frac{737\tilde{c}_{2}^{2}\tilde{c}_{8}}{881798400}-\frac{31\tilde{c}_{6}^{2}}{437400}\\\
\tilde{\epsilon}_{14}&=\frac{8303\tilde{c}_{10}\tilde{c}_{2}^{2}}{14935460400}-\frac{2201\tilde{c}_{12}\tilde{c}_{2}}{217314900}+\frac{4\tilde{c}_{14}}{62601}+\frac{11083\tilde{c}_{2}^{7}}{24082404724998144}-\frac{11609\tilde{c}_{2}^{4}\tilde{c}_{6}}{5530387622400}\\\
&-\frac{1289\tilde{c}_{2}^{3}\tilde{c}_{8}}{1433804198400}+\frac{353\tilde{c}_{2}\tilde{c}_{6}^{2}}{142242480}-\frac{31\tilde{c}_{6}\tilde{c}_{8}}{1463400}\\\
\tilde{\epsilon}_{18}&=\frac{12182634587\tilde{c}_{10}\tilde{c}_{2}^{4}}{77806514663884339200}-\frac{564449\tilde{c}_{10}\tilde{c}_{2}\tilde{c}_{6}}{3418744644000}+\frac{1844\tilde{c}_{10}\tilde{c}_{8}}{3956880375}-\frac{27233975\tilde{c}_{12}\tilde{c}_{2}^{3}}{11321053720935552}\\\
&+\frac{301\tilde{c}_{12}\tilde{c}_{6}}{452214900}+\frac{307855\tilde{c}_{14}\tilde{c}_{2}^{2}}{13588370378352}-\frac{2\tilde{c}_{18}}{1507383}-\frac{886993691\tilde{c}_{2}^{9}}{313644160640867419847393280}\\\
&+\frac{4713945967\tilde{c}_{2}^{6}\tilde{c}_{6}}{72026602145995788288000}-\frac{14715122551\tilde{c}_{2}^{5}\tilde{c}_{8}}{2334195439916530176000}-\frac{579011753\tilde{c}_{2}^{3}\tilde{c}_{6}^{2}}{23156700792822720000}\\\
&+\frac{2313866297\tilde{c}_{2}^{2}\tilde{c}_{6}\tilde{c}_{8}}{222355151645760000}-\frac{77393\tilde{c}_{2}\tilde{c}_{8}^{2}}{3376537920000}-\frac{15011\tilde{c}_{6}^{3}}{97678418400}.\\\
\end{split}$

(B.4)

For the $E_{8}$ case:

$\displaystyle\begin{split}\hat{\epsilon}_{2}&=\frac{\hat{c}_{2}}{120}\\\
\hat{\epsilon}_{8}&=\frac{13\hat{c}_{2}^{4}}{24883200000}-\frac{\hat{c}_{8}}{5760}\\\
\hat{\epsilon}_{12}&=\frac{\hat{c}_{12}}{181440}+\frac{101\hat{c}_{2}^{6}}{3224862720000000}-\frac{\hat{c}_{2}^{2}\hat{c}_{8}}{64512000}\\\
\hat{\epsilon}_{14}&=-\frac{71\hat{c}_{12}\hat{c}_{2}}{798336000}+\frac{\hat{c}_{14}}{1108800}-\frac{2531\hat{c}_{2}^{7}}{9029615616000000000}+\frac{103\hat{c}_{2}^{3}\hat{c}_{8}}{696729600000}\\\
\hat{\epsilon}_{18}&=-\frac{4451\hat{c}_{12}\hat{c}_{2}^{3}}{689762304000000}+\frac{1523\hat{c}_{14}\hat{c}_{2}^{2}}{12454041600000}-\frac{\hat{c}_{18}}{47174400}-\frac{26399\hat{c}_{2}^{9}}{2080423437926400000000000}\\\
&+\frac{4747\hat{c}_{2}^{5}\hat{c}_{8}}{722369249280000000}+\frac{331\hat{c}_{2}\hat{c}_{8}^{2}}{1672151040000}\\\
\hat{\epsilon}_{20}&=\frac{191071\hat{c}_{12}\hat{c}_{2}^{4}}{2121019084800000000}+\frac{127\hat{c}_{12}\hat{c}_{8}}{174569472000}-\frac{1165063\hat{c}_{14}\hat{c}_{2}^{3}}{612738846720000000}+\frac{236627\hat{c}_{18}\hat{c}_{2}}{434023349760000}\\\
&+\frac{10249681\hat{c}_{2}^{10}}{61414099887587328000000000000}-\frac{2994007\hat{c}_{2}^{6}\hat{c}_{8}}{35540567064576000000000}-\frac{323371\hat{c}_{2}^{2}\hat{c}_{8}^{2}}{82269831168000000}-\frac{\hat{c}_{20}}{220809600}\\\
\hat{\epsilon}_{24}&=-\frac{193\hat{c}_{12}^{2}}{17793312768000}+\frac{228270563\hat{c}_{12}\hat{c}_{2}^{6}}{29320967828275200000000000}+\frac{234189517\hat{c}_{12}\hat{c}_{2}^{2}\hat{c}_{8}}{945465467240448000000}\\\
&-\frac{9171869023\hat{c}_{14}\hat{c}_{2}^{5}}{52675933174824960000000000}-\frac{23281\hat{c}_{14}\hat{c}_{2}\hat{c}_{8}}{9150846566400000}+\frac{561557071\hat{c}_{18}\hat{c}_{2}^{3}}{8291582073815040000000}\\\
&+\frac{8268193432181\hat{c}_{2}^{12}}{580761207304971815485440000000000000000}-\frac{20976434911\hat{c}_{2}^{8}\hat{c}_{8}}{3055351469407469568000000000000}\\\
&-\frac{16935675593\hat{c}_{2}^{4}\hat{c}_{8}^{2}}{33005339947302912000000000}-\frac{666323\hat{c}_{2}^{2}\hat{c}_{20}}{721337268326400000}+\frac{\hat{c}_{24}}{10061694720}-\frac{593\hat{c}_{8}^{3}}{887354818560000}\\\
\hat{\epsilon}_{30}&=-\frac{636328729\hat{c}_{12}^{2}\hat{c}_{2}^{3}}{367646783551116410880000000}-\frac{189107437\hat{c}_{12}\hat{c}_{14}\hat{c}_{2}^{2}}{277976001893990400000000}+\frac{2521\hat{c}_{12}\hat{c}_{18}}{31907254579200000}\\\
&+\frac{122785779721089347\hat{c}_{12}\hat{c}_{2}^{9}}{5354576379380206927872000000000000000000}+\frac{374760114643099\hat{c}_{12}\hat{c}_{2}^{5}\hat{c}_{8}}{685159914799807856640000000000000}\\\
&-\frac{199931513\hat{c}_{12}\hat{c}_{2}\hat{c}_{8}^{2}}{94458563710156800000000}+\frac{28501673\hat{c}_{14}^{2}\hat{c}_{2}}{3860777804083200000000}-\frac{1634513578407571229\hat{c}_{14}\hat{c}_{2}^{8}}{3206548401263100769075200000000000000000}\\\
&-\frac{3442332938170993\hat{c}_{14}\hat{c}_{2}^{4}\hat{c}_{8}}{593805259493166809088000000000000}+\frac{1223\hat{c}_{14}\hat{c}_{8}^{2}}{112201334784000000}+\frac{15587535288859801\hat{c}_{18}\hat{c}_{2}^{6}}{76346390506264304025600000000000000}\\\
&-\frac{1051350791\hat{c}_{18}\hat{c}_{2}^{2}\hat{c}_{8}}{1243310844834938880000000}+\frac{38736013334814563129113\hat{c}_{2}^{15}}{919171413254131073937239231692800000000000000000000000}\\\
&-\frac{966205043352894287\hat{c}_{2}^{11}\hat{c}_{8}}{46497194159854305977303040000000000000000000}-\frac{53516928494297557\hat{c}_{2}^{7}\hat{c}_{8}^{2}}{42002885419922588958720000000000000000}\\\
&-\frac{2159242595767\hat{c}_{2}^{5}\hat{c}_{20}}{737984035215212544000000000000}+\frac{21328481\hat{c}_{2}^{3}\hat{c}_{24}}{58332071437516800000000}+\frac{225239997090599\hat{c}_{2}^{3}\hat{c}_{8}^{3}}{119591548765057371340800000000000}\\\
&+\frac{72667\hat{c}_{2}\hat{c}_{20}\hat{c}_{8}}{4518107320320000000}-\frac{\hat{c}_{30}}{1978376400000}.\\\
\end{split}$

(B.5)

##  Appendix C Milnor number and zero modes

From the point of view of the 5d theory coming from M-theory reduced on the
singular space $X$, the number of massless hypermultiplets gives the dimension
$d_{HB}$ of the Higgs Branch (HB). This is due to the fact that for the
threefolds we consider in this paper, there is no gauge symmetry (as the
exceptional set has no codimension-1 loci).

The dynamics of M-theory on $X$ admits an alternative description in terms of
the complex structure deformations of the threefold. When the singularity is
deformed, a number $\mu$ of compact three-cycles blow up. Some of them
intersect each other pairwise (these are called paired), while the other do
not intersect any other compact three-cycle (these are called unpaired). $\mu$
is called the Milnor number of the singularity $F(x,y,z,w)=0$ and it is the
dimension, as complex vector space, of the Jacobian ring

$\mathcal{J}\equiv\frac{\mathbb{C}[x,y,w,z]}{(F,\partial_{x}F,\partial_{y}F,\partial_{w}F,\partial_{z}F)}.$
(C.1)

The dimension of the Higgs branch is given by counting appropriately the
number of independent, dynamical complex structure deformations that smooth
the singularity. For non-compact Calabi-Yau varieties, the number $d_{HB}$ of
independent, dynamical complex deformations coincides with [18, 19, 20]

$d_{HB}=\frac{\mu-\ell}{2}+\ell,$ (C.2)

where $\mu$ is the Milnor number and $\ell$ the number of “unpaired” three-
cycles.

On general grounds, $\ell$ coincides with the number of two-cycles inflated in
the resolution, that in our case coincides with the dimension of $\mathcal{H}$
in (3.2). For the simple flop cases studied in this paper $\ell=1$ and hence
$d^{\rm simple\,flop}_{HB}=\frac{\mu-1}{2}+1$.

We have used this formula to check our results: we first compute the Milnor
number for the examples in Section 4 to compute the Higgs branch dimension
$d^{\rm simple\,flop}_{HB}$; we then checked that this number coincided with
the total number of hypermultiplets we computed.

##  Appendix D Mathematica code for computing the zero modes

In this section we will describe the ancillary Mathematica code, which can be
found at the arXiv page of this paper. We loaded, together with the paper, a
zipped folder. The folder contains nine text files containing the positive and
negative roots of the exceptional algebras252525For the $E_{6}$ algebra, the
root vectors are in the 27 representation. For the $E_{7},$ $E_{8}$ algebras
the root vectors are in the adjoint representation. and a basis of the Cartan
subalgebras. We used, to produce these matrices, the results of [21, 22, 23].
These text files should be saved in one of the folders of the variable $Paths
of Mathematica. In the folder, one can find also the notebook file
"CodeHiggsBranchData.nb" which is divided into two sections. The first section
“Main Code” contains the routines that we used to produce the Higgs branch
data having, as input, the Higgs field describing the type IIA dual of
M-theory on the considered threefold. The second sections “Examples” contains
the application of the functions defined in the section “Main Code” to the
simple flops we analyzed in this paper. We now describe the main routines of
the ancillary Mathematica code.

##### HbData function

The main routine contained in the code is

$\footnotesize{\text{{HbData[ADE, rank, listhiggs, coeffhiggs, cartanhiggs,
coeffcartan, listlevi]}}}.$

The arguments of the function are

  * •

ADE: is a Symbol to be picked among "A, DD, E6, E7, E8" and specifies the type
of ADE algebra associated to the threefold.

  * •

rank: is a positive Integer that specifies the rank of the ADE algebra
associated to the threefold.

  * •

listhiggs: is a List of Lists. Each sublist represents a root such that the
Higgs field has a non-zero coefficient along the corresponding root-vector in
$\mathfrak{g}$. The root vectors considered here do not lie in the Cartan
subalgebra: the elements in the Cartan subalgebra will be separately input
with the variables cartanhiggs and coeffcartan. The roots are described by
their integer coefficients decomposition with respect to the simple roots. For
example, keeping the notations in section 4.5, the root vector
$v=\left[\left[e_{-\alpha_{5}},e_{-\alpha_{6}}\right]e_{-\alpha_{7}}\right]$
appearing in (4.5) is expressed as

$\left\\{0,0,0,0,-1,-1,-1,0\right\\}.$

To order the roots, we followed the labelling of Figures 4, 5, 6 and 7 for the
exceptional algebras. We labelled roots from the left to the right in the
$A_{r}$ Dynkin diagram. For the $D_{r}$ case, the first $r-3$ integers are
associated to the $A_{r-3}$ subalgebra associated to the longest tail of the
$D_{r}$ Dynkin diagram (again, labelling roots from left to right), the third
last integer number appearing in the sublists of listhiggs is associated to
the trivalent root, the last two integers are associated to the two $A_{1}$
short tails of the $D_{r}$ diagram.

For example, if we again consider the Higgs field of section 4.5, we have to
insert, as input

listhiggs $\displaystyle\scriptstyle=$ $\displaystyle\Big{\\{}\text{
}\scriptstyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\left\\{0,0,0,0,1,0,0,0\right\\},\left\\{0,0,0,0,0,1,0,0\right\\},\left\\{0,0,0,0,0,0,1,0\right\\}}},$
$\displaystyle\quad\scriptstyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\left\\{0,0,0,0,-1,-1,-1,0\right\\}}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\left\\{1,0,0,0,0,0,0,0\right\\},\left\\{0,1,0,0,0,0,0,0\right\\}}},$
$\displaystyle\quad\scriptstyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\left\\{0,0,1,0,0,0,0,0\right\\},\left\\{0,0,0,0,0,0,0,1\right\\},\left\\{-1,-1,-1,0,0,0,0,-1\right\\}}}\Big{\\}},$

where the red elements correspond to $\Phi\rvert_{A_{3}^{(5,6,7)}}$, and the
blue elements to $\Phi\rvert_{A_{4}^{(1,2,3,8)}}$ in (4.5).

The root system, in our convention, can be printed on screen calling the
function PrintRootSystem[ADE,rank] (the first argument being again the ADE
type of $\mathfrak{g}$, and the second argument its rank).

  * •

coeffhiggs: is a List containing the coefficients corresponding to the
elements of listhiggs. If we again consider the Higgs field of (4.5) we have

$\text{{coeffhiggs}}=\left\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{1,1,1,wc_{567}}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{1,1,1,1,wc_{1238}}}\right\\}.$

  * •

cartanhiggs: is a List of positive Integers $n_{i}$, with $n_{i}$ = 1, …,
rank, describing the elements of the Cartan subalgebra $\mathcal{C}$ of
$\mathfrak{g}$ along which the Higgs field has a non-zero coefficient. The
generators of the Cartan subalgebra has been chosen as the dual elements
$\alpha_{j}^{*}$ of the simple roots. For example, let’s consider again the
$E_{8}$ algebra. We know by construction that the Higgs field has to preserve
the Cartan element dual to $\alpha_{4}$ in Figure 6. This Cartan element is
defined as the one that commutes with all the simple root vectors different
from $e_{\alpha_{4}}$, and

$\left[\alpha_{4}^{*},e_{\alpha_{4}}\right]=1.$

In order to pick a Higgs field with a non-zero component along $\alpha_{4}$ we
then input

$\text{{cartanhiggs}}=\left\\{4\right\\}.$

For non-simple flops, where we have more than one $\mathbb{P}^{1}$ resolved,
the input

$\text{{cartanhiggs}}=\left\\{n_{1},...,n_{f}\right\\},$

corresponds to turning on a Higgs v.e.v. along the roots
$\alpha_{n_{1}}^{*},...,\alpha_{n_{f}}^{*}$.

  * •

coeffcartan: is a List of Symbols describing the coefficients corresponding to
the elements of cartanhiggs. In the previous example, if we input

$\text{{cartanhiggs}}=\left\\{4\right\\},\quad\text{{coeffcartan}}=\left\\{wc_{4}\right\\},$

we picked the Higgs to have a coefficient $wc_{4}$ along $\alpha_{4}^{*}$.

  * •

listlevi: is a List of Integers describing the simple roots that labels the
Levi subalgebra. In the previous example, we picked the root vectors all
residing in the Levi subalgebra labeled by all the roots of $E_{8}$ but the
resolved one. The fact that the Levi subalgebra associated to the Higgs field
(4.67) is $L=A_{3}^{(5,6,7)}\oplus
A_{4}^{(1,2,3,8)}\oplus\langle\alpha_{4}^{*}\rangle$ is explained in details
in section 4.5 but can be also read off from our choice of the variable
listhiggs. Indeed, the simple roots labeling the Levi subalgebra (in this
case, $\left\\{1,2,3,5,6,7,8\right\\}$) are the minimum amount of simple roots
we need to generate (with integer coefficients) all the root we input in
listhiggs. All the Levi subalgebras contains [17] the Cartan subalgebra, hence
the data cartanhiggs and coeffcartan do not modify the Levi subalgebra datum.

The function has a void output and prints all the data we need to determine
the action of the spontaneously broken flavor symmetry on the Higgs Branch
(or, in other words, the Gopakumar-Vafa invariants of the small resolution of
the singularity, labeled by their degrees).

As an example we report here a part of the output of the length-five simple
flop case we analyzed in section 4.5. The whole output is contained in the
Examples section of the ancillary Mathematica file. In this case, the output
contains many blocks (one for each irreducible representation of the branching
of $\mathfrak{g}$ with respect to the Levi subalgebra (4.67)) of the following
type:

The first three lines report the number of five-dimensional modes localized in
the considered irreducible representation. In this case, we read the list
$\left\\{6,1,0\right\\}$, this means that we have six modes localized in
$\mathbb{C}[w]/(w)$, one mode localized in $\mathbb{C}[w]/(w^{2})$ and zero in
$\mathbb{C}[w]/(w^{k})$ with $k>2$. The overall number of complex-valued modes
is, hence, $6*1+1*2=8$ (indeed, a mode inside $\mathbb{C}[w]/(w^{k})$ counts
for $k$ complex-valued five-dimensional degrees of freedom). The last three
lines tell us:

  * •

The complex dimension of the considered irreducible representation. In the
example, we have $40$.

  * •

The Dynkin indices of the highest weight state of the representation. In the
example, we read

$\left\\{0,0,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{1}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{1}},0,0,0\right\\}.$

Since the variable listlevi is
$\left\\{1,2,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{3}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{5}},6,7,8\right\\}$,
the output is telling us that the lowest weight has weight $1$ with respect to
the third root of Figure 6, $1$ with respect to the fourth root and zero with
respect to all the other roots labelling the Levi subalgebra.

  * •

The charges of these modes with respect to the flavor group generators. The
generators of the flavor group are the Catan elements $\alpha_{i}^{*}$ that
are dual to the roots that gets resolved. In this case, we see that all the
five-dimensional modes in the considered irreducible representation have
weight one with respect to the $\mathfrak{u}(1)$ flavor symmetry associated to
the resolved node of Figure 6. This implies, in terms of the Higgs Branch
geometry, that there is a spontaneously broken $\mathfrak{u}(1)$ symmetry,
that acts with charge one on the modes localized in the considered irreducible
representation.

##### ExtractRootDec function

The second main routine is

$\text{{ExtractRootDec[ADE, rank, higgs]}}.$

The routine can be used to analyze, in the language of this paper, the Higgs
fields we presented in [3, 24]. The first two arguments of the function are
again, as in the HbData function, the ADE type of $\mathfrak{g}$ and its rank.
The third argument is a matrix, representing the Higgs field. The Higgs field
has to be input

  * •

in the fundamental representations for the $A_{r}$, $D_{r}$ cases (following
the notations in [17]);

  * •

in the 27 representation for the $E_{6}$ case;

  * •

in the adjoint representation for the $E_{7}$, $E_{8}$ cases.

The output is a List containing the arguments of the function HbData (ordered
as in the function call process). In other words, the first argument of the
output List will be ADE, the second will be rank, the third listhiggs and so
on. The function does not output the datum listlevi, which has to be added
manually by the user when HbData is called. We remark again that the simple
roots labelling the Levi subalgebra are the minimum amount of simple roots we
need to generate (with integer coefficients) all the roots in listhiggs
(namely, appearing in the third element of the output of ExtractRootDec[ADE,
rank, higgs]).

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