The Geometry of GTPs and 5d SCFTs

Guillermo Arias-Tamargoa<EMAIL_ADDRESS>Sebastián
Francob,c,d<EMAIL_ADDRESS>and Diego Rodríguez-Gómeze,f
<EMAIL_ADDRESS>
aTheoretical Physics Group, The Blackett Laboratory, Imperial College London,

Prince Consort Road London, SW7 2AZ, UK

bPhysics Department, The City College of the CUNY,

160 Convent Avenue, New York, NY 10031, USA

cPhysics Program and dInitiative for the Theoretical Sciences,

The Graduate School and University Center, The City University of New York

365 Fifth Avenue, New York NY 10016, USA

eDepartment of Physics, Universidad de Oviedo,

C/ Federico Garcia Lorca 18, 33007 Oviedo, Spain

fInstituto Universitario de Ciencias y Tecnologias Espaciales de Asturias
(ICTEA),

C/ de la Independencia 13, 33004 Oviedo, Spain

Abstract

We make progress in understanding the geometry associated to the Generalized
Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories
(SCFTs), by exploiting the connection between Hanany-Witten transitions and
the mathematical notion of polytope mutations. From this correspondence, it
follows that the singular geometry associated to a GTP is identical to that
obtained by regarding it as a standard toric diagram, but with some of its
resolutions frozen in way that can be determined from the invariance of the
so-called period under mutations. We propose the invariance of the period as a
new criterion for distinguishing inequivalent brane webs, which allows us to
resolve a puzzle posed in the literature. A second mutation invariant is the
Hilbert Series of the geometry. We employ this invariant to perform
quantitative checks of our ideas by computing the Hilbert Series of the BPS
quivers associated to theories related by mutation. Lastly, we discuss the
physical interpretation of a mathematical result ensuring the existence of a
flat fibration over $\mathbb{P}^{1}$ interpolating between geometries
connected by mutation, which we identify with recently introduced deformations
of the corresponding BPS quivers.

###### Contents

  1. 1 Introduction
  2. 2 Geometric engineering and brane webs
    1. 2.1 M-theory on CY3
      1. 2.1.1 The $E_{1}$ example
    2. 2.2 Brane webs and GTPs
      1. 2.2.1 The $E_{1}$ example
    3. 2.3 Mirror construction
      1. 2.3.1 The $E_{1}$ example
  3. 3 Hanany-Witten transitions and polytope mutations
    1. 3.1 Hanany-Witten in the mirror
      1. 3.1.1 The $E_{1}$ example
    2. 3.2 Polytope mutations
    3. 3.3 Mutation invariants
      1. 3.3.1 The $E_{1}$ example
  4. 4 5d SCFTs, GTPs, and the period
    1. 4.1 The period and the classification of brane webs
    2. 4.2 Hanany-Witten and geometric deformations
    3. 4.3 Seiberg-Witten curves for GTPs
  5. 5 BPS Quivers and the Hilbert series
    1. 5.1 The $E_{1}$ example
  6. 6 Additional examples
    1. 6.1 The $E_{2}$ theory
    2. 6.2 The $E_{0}$ theory
    3. 6.3 The $\tilde{E}_{1}$ theory
  7. 7 Discussion
    1. 7.1 Open questions

##  1 Introduction

Constructing interacting UV complete Quantum Field Theories (QFTs) in $d>4$ is
notoriously hard. In the particular case of $d=5$ it is fair to say that the
existence of consistent non-supersymmetric interacting QFTs has not been
clearly established (see e.g. [1, 2, 3, 4, 5, 6]). The situation is much
better for the case of supersymmetric theories, since in that case, using the
power of String/M theory, it is possible to conclusively construct 5d
interacting and consistent SCFTs. It turns out that these theories are very
interesting. Since 5d SCFTs do not have marginal deformations, they are
intrinsically strongly coupled and often exhibit rather exotic properties,
including for instance global symmetries of exceptional type. Moreover, upon
compactification, they can provide new perspectives on strong coupling
phenomena in lower dimensions.

Within String/M theory there are various approaches to constructing 5d SCFTs,
perhaps most saliently through geometric engineering them in M-theory on a 3d
Calabi-Yau (CY) $\mathcal{M}$ [7, 8, 9] (see e.g. [10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] for recent work) and on the
worldvolume of 5-brane webs in Type IIB String Theory [28] (see also [29, 30,
31, 32, 33, 34]). In recent years, significant efforts have been devoted to
understanding 5d QFTs using these methods, as well analytic tools such as
supersymmetric localization.

Regarding the 5-brane web avatar, it has been realized that, in order to
explicitly show properties of 5d SCFTs such as the full Higgs branch, it is
necessary to think of the web as ending on 7-branes. This has two immediate
consequences. On one hand, when more than one 5-brane ends on a 7-brane, the
7-brane imposes non-trivial supersymmetric boundary conditions –which go by
the name of s-rule– which constrain the Coulomb branch of the 5d SCFTs. On the
other hand, since the position of the 7-brane along the prong of the web
defined by the 5-branes ending on it (the leg of the web) is not a parameter
in the 5d SCFT, one may imagine crossing the 7-brane to the other side of the
web. Since 7-branes come with a branch cut, as it sweeps part of the web it
changes the type of a subset of the remaining 7-branes and 5-branes, sometimes
leading to 5-brane creation through the celebrated Hanany-Witten (HW) effect
[35]. Thus, one may have two different looking webs describing the same 5d
SCFT. Indeed, due to this fact, the program of classifying 5d SCFTs through
classifying the possible brane webs has proven to be very hard.

When all of external legs of the web consist of a single 5-brane ending on the
corresponding 7-brane, the latter can actually be neglected. In that case, it
turns out that there is a relation with the geometric engineering approach
when $\mathcal{M}$ is toric. Reducing on a $T^{2}$ inside the toric fiber,
M-theory becomes Type IIB String Theory with 5-branes along the locus where
the $T^{2}$ pinches off, which is precisely the corresponding 5-brane web.
Thus, the brane web is the toric skeleton of the CY3, and can be regarded as
the diagram graph-dual to the toric diagram of $\mathcal{M}$. However, in the
generic case of an arbitrary number of 5-branes ending on each 7-brane, the
geometric engineering description is not known. The notion of Generalized
Toric Polygon (GTP) has been developed to describe these more general cases
[29, 36, 37]. GTPs look like standard toric diagrams but are decorated with
white and black dots, encoding how 5-branes may terminate on the same 7-brane
if the corresponding external segments in the GTP are separated by a white
dot. Moreover, a rule for supersymmetrically triangulate (actually
“polygonate”) the interior of a GTP according to the $s$-rule has been
proposed. While GTPs look very close to standard toric diagrams (and reduce to
these when all dots are black), their geometric interpretation has not been
fully established.111Recent progress, on which we will elaborate, includes
[38, 39, 40, 41]. The purpose of this paper is to make progress on these
questions, notably by establishing a connection to a branch of Mathematics
that has emerged over the last decade, largely in parallel with the interest
in the Physics community regarding 5d SCFTs, and deals with essentially the
same problem.

Specifically, as suggested in [39], in this paper we will argue that the
mathematical notion of mutation turns out to precisely correspond to HW
crossing of branes. Note that HW transitions typically lead to GTPs, even if
the starting point is a standard toric diagram. Yet, from the mathematical
point of view, the decoration of the GTP with white dots is irrelevant: it
simply describes the singular toric variety as if no decoration was present.
Moreover, two toric varieties whose toric polytopes are related by mutation
can be regarded as members of a flat fibration over $\mathbb{P}^{1}$, which
implies that, geometrically, the HW transition can be regarded as a
deformation of the starting CY3. This establishes that the (singular) toric
variety geometrically engineering the 5d SCFT in M-theory is simply that
associated to the GTP forgetting its white dot decoration. Exploiting the
established invariants under mutation developed in the mathematical literature
–which will also enable us to refine previous classifications of 5-brane webs–
we will demonstrate that the white dot decoration has the effect of freezing
possible resolutions of the geometry in a precise way.

Moreover, we will offer quantitative evidence of this proposal by studying the
BPS quiver of the 5d SCFT compactified on $S^{1}$. This quiver is identical to
the fractional brane quiver for Type IIB D3-branes probing the toric CY3
variety $\mathcal{M}$. Thus, following the proposal above, for an arbitrary 5d
theory encoded in an arbitrary GTP, we will read off the BPS quiver by simply
forgetting the decoration of the GTP and regarding it as a standard toric
diagram. This is an easy task, given the substantial brane tiling machinery
developed for this purpose [42, 43]. We will see that the partition function
counting gauge-invariant operators of the BPS quiver matches, upon using the
appropriate prescription, the Hilbert series of the variety. The latter has
been shown to be invariant under mutation, and the agreement of quiver
partition functions will be a non-trivial check of our ideas.

This paper is organized as follows. In Section 2 we give a lightning review of
the two approaches to construct 5d SCFTs which we consider, namely
geometrically engineering them in M-theory and on 5-brane webs in Type IIB
String Theory. In Section 3, we describe the identification of Hanany-Witten
transitions in 5-brane webs with the mathematical notion of polytope mutation.
This allows us to introduce two quantities, the so-called period and the
Hilbert series, which are invariant under this transformation. In Section 4,
we explore the physical implications of the invariance of the period. This
will allow us to recover, from first principles, the recipe for the Seiberg-
Witten (SW) curves in [44], as well as to resolve a puzzle posed in [45]
concerning the classification of 5-brane webs. In Section 5, we turn to the
invariance of the Hilbert series, which provides a quantitative consistency
check of the perspective advocated in this paper. We offer further examples in
Section 6. We conclude in Section 7 with a summary of our results and a
discussion of the open problems and future perspectives raised by our work.

##  2 Geometric engineering and brane webs

In this section we review three of the main approaches to construct 5d
$\mathcal{N}=1$ SCFTs and their interrelation. First we begin by considering
their geometric engineering via M-theory on a local Calabi-Yau threefold.
Second, we study their construction via webs of 5-branes in flat space; in
simple cases the correspondence with the geometric setup is well known. Last,
we will review the mirror construction of the original M-theory CY.

###  2.1 M-theory on CY3

Let us now focus on the first of these three setups, namely consider M-theory
on a three (complex) dimensional, non-compact, canonical, Gorenstein
singularity. Following the standard terminology in the literature, we will
refer to it as a local CY3, and denote it by $\mathcal{M}$. Since the geometry
is non-compact, gravity is decoupled, and this system is described by a 5d
SCFT living in the 5 directions transverse to the singularity [7, 8, 9].

In this situation, there is a correspondence between the field theory and the
geometrical data. Let us denote by $\widetilde{\mathcal{M}}$ a (partial)
crepant resolution of $\mathcal{M}$. This geometry has a set of non-compact
divisors that arise directly from those in $\mathcal{M}$, as well as a set of
compact exceptional divisors that arise from the resolution. Together, they
form the extended Kähler cone of $\widetilde{\mathcal{M}}$, which we denote
$\mathcal{K}(\widetilde{\mathcal{M}})$. In the low energy field theory
description, they correspond to the 5d extended Coulomb branch; more precisely
compact cycles correspond to proper Coulomb branch VEVs, while non-compact
cycles map to (real) mass deformations. In fact, real masses can be understood
as VEVs for scalars in background vector multiplets for global symmetries,
hence the name of extended Coulomb branch.

In the rest of the paper, we will be focusing on the case where the Calabi-Yau
is toric, which translates to the field theory having at least a
$\mathrm{U}(1)^{3}$ global symmetry. This means that generically complicated
geometric notions can be simplified in terms of combinatorics, since the
complexified torus $(\mathbb{C}^{*})^{3}$ acts on a dense subvariety of
$\mathcal{M}$. Indeed, among many other features, this allows to trade the
defining equations of $\mathcal{M}$ as an algebraic variety for its toric fan,
which is a collection of vectors in a $\mathbb{Z}^{3}$ lattice specifying the
weights of the torus action. In these terms, the Calabi-Yau condition is seen
as the fact that all these vectors end on the same 2d plane. This further
streamlines the combinatorics, as it is now sufficient to focus on the 2d
polygon defined by the endpoints of the vectors in the toric fan.

In these terms, the discussion regarding the extended Coulomb branch of the 5d
theory simplifies. The singular Calabi-Yau $\mathcal{M}$ corresponds to a
polygon such that only external lattice points (that is, along the perimeter)
are joined by edges. Its resolution $\widetilde{\mathcal{M}}$ corresponds to a
triangulation of the polytope for $\mathcal{M}$; the variety is completely
resolved if every internal point is joined by lines of the polytope. The
lattice points themselves correspond to divisors of the geometry.222One should
note that when modding out by principal divisors in order to find the divisor
class group, not all points correspond to independent divisor classes. From
the discussion above, it follows that internal points are mapped to proper
Coulomb branch deformations of the 5d theory (compact cycles), while external
points correspond to mass deformations (non-compact cycles). The number of
internal points of the polygon is the rank of the 5d SCFT.

BPS states of the 5d theory can be understood in M-theory. M2-branes wrapping
certain holomorphic 2-cycles are identified with W-bosons and instanton
particles, and M5-branes wrapping 4-cycles give rise to instanton strings.
Making this map more precise would require introducing a ruling for the
compact divisors, but we will not do that here as we will not make use of it.
Instead, it is easier to look at the BPS states from the dual Type IIA picture
as follows.

We can consider compactifying one of the field theory directions on a circle,
obtaining a so-called 4d KK theory, described by Type IIA String Theory on the
CY3. The different BPS states of the theory correspond to the bound states of
D0-D2-D4 branes supersymmetrically wrapped in $\widetilde{\mathcal{M}}$. Such
bound states are captured by the BPS quiver, which coincides with the
fractional brane quiver of the CY3 singularity [46]. This object has been
thoroughly studied in the past in the case of toric CY3’s. This allows us to
import the heavy machinery developed to describe Type IIB D3-branes probing
toric CY3’s in terms of brane tilings [43] (see [47, 48] for comprehensive
reviews) to construct the BPS quiver for (compactified) 5d SCFTs geometrically
engineered by the CY3.

One interesting consequence of this correspondence is that it is possible to
go both ways. We just discussed how to reach the BPS quiver starting from the
geometry. But taking the BPS quiver as a starting point, one can also obtain
the CY3 geometry $\mathcal{M}$ that engineers the 5d SCFT in M-theory. Indeed,
given a BPS quiver we may regard it as a 4d $\mathcal{N}=1$ quiver for
D3-branes probing a CY3, and computing the geometry of the moduli space of
vacua of the 4d theory, we recover the original Calabi-Yau.

####  2.1.1 The $E_{1}$ example

Let us illustrate these concepts with some detail in the simple example of the
$E_{1}$ theory, to which we will come back throughout this work. This theory
is engineered by M-theory on the complex cone over the Hirzebruch surface
$\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$, and will be denoted by
$C(\mathbb{F}_{0})$. Both $\mathbb{F}_{0}$ and $C(\mathbb{F}_{0})$ are toric
varieties.333In the Physics literature CY3 geometries which are a complex cone
over a 2d base $C(B)$ are often denoted by simply the 2d base $B$. The (dual
of the) toric fan of $C(\mathbb{F}_{0})$ can be projected to the hyperplane at
height 1 obtaining a 2d integral polygon known as the toric diagram, which we
depict in Figure 1.

Figure 1: Toric diagram for the $E_{1}$ theory before and after the
resolution.

This specifies for us the weights of the toric action, which is then
$(u,v,w,t)\to(t_{1}t_{3}u,t_{2}t_{3}v,\frac{t_{3}}{t_{1}}w,\frac{t_{3}}{t_{2}}t)$.
From this we recognize the equations corresponding to two copies of
$\mathbb{P}^{1}$ over the complex plane. This variety can also be regarded as
a $\mathbb{Z}_{2}$ orbifold of the conifold.

This singular variety has four toric divisors $D_{i},i=1,\dots,4$ given by
setting each of the 4 complex coordinates equal to zero. After the blowup, we
have an additional exceptional divisor $D_{e}$. These divisors are not all
linearly independent: after modding out by principal divisors, one finds the
relations

$\displaystyle[D_{1}]-[D_{3}]=0\,,$
$\displaystyle[D_{2}]-\left[D_{4}\right]=0\,,$ (2.1)
$\displaystyle[D_{e}]+[D_{1}]+[D_{2}]+[D_{3}]+[D_{4}]=0\,.$

Therefore we conclude that there are two linearly independent divisors, which
we take to be $[D_{e}]$ and, say, $[D_{1}]$. One can compute their volumes by
integrating the Kähler form of $C(\mathbb{F}_{0})$. The volume of the
(compact) exceptional divisor corresponds to the proper Coulomb branch
modulus, and the volume of the (non-compact) toric divisor to the extended
Coulomb branch modulus; together, they generate the extended Kähler cone of
$C(\mathbb{F}_{0})$.444More precisely the Kähler parameters live in
$H^{2}(C(\mathbb{F}_{0}),\mathbb{R})$. The divisors live in
$H_{4}(C(\mathbb{F}_{0}),\mathbb{Z})$ and their Poincaré duals in
$H^{2}(C(\mathbb{F}_{0}),\mathbb{Z})$, and taking into account the volume
leads to the desired coefficients for the cohomology group. Indeed, we find
that its dimension is 2, as expected for a theory with gauge and global
symmetry of rank 1.

The BPS quiver coincides with the fractional brane quiver for D3-branes
probing $C(\mathbb{F}_{0})$. This quiver and its superpotential can be easily
computed using the standard technology of dimer models, as shown in Figure 2.

Figure 2: Brane tiling and quiver diagram for $C(\mathbb{F}_{0})$.

The corresponding superpotential is

$W=-X_{12}X_{23}X_{34}X_{41}+X_{41}Y_{12}X_{23}Y_{34}+X_{12}Y_{23}X_{34}Y_{41}-Y_{12}Y_{23}Y_{34}Y_{41}\,.$
(2.2)

From the field content and the superpotential, one can compute the complete
F-terms satisfied by the chiral fields, and it is possible to see that the
moduli space of the quiver in 2 precisely reproduces the toric diagram in 1.

It is worth noting that the fractional brane quiver associated to a toric CY
cone is not unique, as there can be multiple Seiberg dual phases. In the
example of $C(\mathbb{F}_{0})$ at hand there is another phase, obtaining by
Seiberg-dualizing any node in the quiver in 2 (all nodes are equivalent).

###  2.2 Brane webs and GTPs

The duality between M-theory on $T^{2}$ and Type IIB on $S^{1}$ implies that a
5d theory can be engineered both via pure geometry in M-theory, as described
above, or through a brane web in flat space in Type IIB [49]. Translating
between the two setups is a well understood problem when the geometry is a
toric CY3. In this case, the edges of the toric diagram, which correspond to
2d faces of the toric fan, signal that two of the three circles in
$(\mathbb{C}^{*})^{3}$ are pinched. In the dual Type IIB setup, this pinching
is seen as a discontinuity of the $B_{2}$ and $C_{2}$ fields which, in turn,
signals the presence of a charged 5-brane.

More precisely, if a $(p,q)$ complex 2-cycle of the torus pinches, it will
correspond to a $(p,q)$-5-brane. The conclusion is that, given a toric diagram
for the M-theory CY3, the dual of this graph (sometimes referred to as the
toric skeleton) is directly the 5-brane web in Type IIB. As an illustration,
Figure 3 shows the toric diagram and brane web for $SU(2)$ SQCD with 1 flavor.

Figure 3: Toric diagram and brane web for $SU(2)$ SQCD with 1 flavor.

More generally, 5-brane webs in Type IIB String Theory have been thoroughly
studied by themselves starting with [28]. Choosing the axio-dilaton $\tau=i$,
a $(p,q)$ 5-brane wraps $(01234)$ and looks like a segment or line with slope
$q/p$ on the $(56)$ plane. 5-branes can join, provided $(p,q)$ charge is
conserved at every intersection, forming a web on the $(56)$ plane. Note that
if $(p,q)$ are not coprime, the corresponding segment describes
$n=\rm{gcd}(p,q)$ 5-branes with charges $(p/n,q/n)$. To fix some notation, we
will always consider the 5-brane charges $(p,q)$ to be coprime, and we will
refer to the number of parallel 5-branes $n$ as the _multiplicity_ of the leg.

These configurations can be extended by adding 7-branes spanning $(01234789)$
and located at a point on the plane of the web with no further breaking of
supersymmetry, such that every leg of the web can terminate on an appropriate
7-brane. To be explicit, a 7-brane is described by its $[p,q]$ charge. Once
again we assume ${\rm gcd}(p,q)=1$ so that we describe a single 7-brane, on
which several $(p,q)$ 5-branes can end; and we write the total charge vector
of the leg as $n(p,q)$. The addition of the 7-branes allows the visualization
of the Higgs branch of the 5d theory as “sliding” sub-webs in the $(789)$
directions. Moreover, the 7-branes impose boundary conditions on the web which
restrict the number of Coulomb branch deformations and are constrained by the
supersymmetry of the configuration through the so-called $s$-rule. To see
this, let us consider a 5-brane web with $L$ external legs, and where each
external leg ends on the corresponding $[p_{i},q_{i}]$ 7-brane and has
multiplicity $n_{i}$, with $i=1,\cdots,L$. Thus, we can denote each 7-brane by
a vector $\ell_{i}=[p_{i},q_{i}]$, and for definiteness, choose to label them
anti-clock wise. Note that in these conventions, charge conservation reduces
to $\sum n_{i}\ell_{i}=0$. The self-intersection of the web is defined as [50,
51]555It is important to stress that the formula in (2.3) is valid for
irreducible webs, that is, those which are not the superimposition of various
individual webs.

$\mathcal{I}=\left|\sum_{i\leq
j}n_{i}\,n_{j}\,\langle\ell_{i}|\ell_{j}\rangle\right|-\sum_{i}n_{i}^{2}\,,\qquad\langle\ell_{i}|\ell_{j}\rangle={\rm
det}(\mathbf{\ell}_{i},\mathbf{\ell}_{j})\,.$ (2.3)

The condition for the web to be supersymmetric is

$\displaystyle\mathcal{I}\geq-2\,.$ (2.4)

There is a relation between the self-intersection of the web and the dimension
of the Coulomb branch of the 5d theory, given by

$\displaystyle d_{C}=\frac{\mathcal{I}+2}{2}\,,$ (2.5)

so that the so that the SUSY condition translates to the Coulomb branch having
non-negative dimension $d_{C}\geq 0$.

Since 7-branes are point like in the $(56)$ plane, they come with a branch cut
for the axio-dilaton that they source, which is specified by their $[p,q]$
charge. The associated monodromy matrix is666We use conventions such that when
the branch cut sweeps counter-clockwise an $[r,s]$ 7-brane, the latter gets
transformed into an $M_{(p,q)}[r,s]^{T}$ 7-brane. If, instead, the cut sweeps
the 7-brane clockwise, it acts with $M_{(p,q)}^{-1}$.

$M_{(p,q)}=\left(\begin{array}[]{cc}1-pq&p^{2}\\\
-q^{2}&1+pq\end{array}\right)\,.$ (2.6)

We assume a “standard presentation” for the web, where all branch cuts are
assumed to run away from the web, not crossing any brane. For instance, we can
take them to run “radially” along the direction of the prong corresponding to
the leg.

Given a web ending on 7-branes, there are two possible types of motions of the
7-branes: a) we can move the 7-branes changing the asymptotic positions of the
external legs attached to them, or b) we can move the 7-branes along the
external legs without changing their asymptotic position. Let us discuss the
consequences of each of these alternatives.

  1. (a)

Moving the 7-branes changing the asymptotic position of the external legs
corresponds to a mass deformation of the 5d SCFT. Interestingly, these
deformations may open Higgs branch directions. Indeed, moving two 7-branes
such that two or more external legs of the brane web coincide leads to a
5-brane segment which is free to move in the direction perpendicular to the
plane of the web, corresponding to a Higgs branch VEV in the 5d theory, as
discussed above. Integrating out this massive mode we are left with a web
where two parallel 5-branes end in the same 7-brane. This results on some
moduli of the theory becoming frozen: in fact, together with the requirement
that every time that 5-branes meet the junction needs to be supersymmetric,
more moduli can get frozen beyond the position of the external legs. As an
example, consider the webs shown in Figure 4. Requiring that the rightmost
junction of the web on the RHS satisfies $\mathcal{I}\geq-2$ implies that
there is only one free Coulomb branch moduli.

Figure 4: Brane web where two 5-branes end on the same 7-brane after a Higgs
branch flow.

  2. (b)

Consider instead moving a 7-brane along the corresponding external leg. Such
motion does not correspond to an observable of the 5d theory. In particular,
we can consider moving the 7-brane across the web. In order to keep its
monodromy cut pointing in outwards, we have to rotate it into the opposite
direction, sweeping in the process “half of the web” and transforming the
7-branes that are crossed accordingly. To be more explicit, suppose we cross
the $i$-th brane with charge vector $\ell_{i}$. Then, for the transformed
7-branes, their charge vector transforms as $\ell_{j}\rightarrow
M_{\ell_{i}}\ell_{j}=\ell_{j}+\langle\ell_{i}|\ell_{j}\rangle\,\ell_{j}$, so
that the full transformation of the web is

$\ell_{j}\rightarrow\begin{cases}\ell_{j}\,,\qquad j\leq i\,,\\\
-\ell_{i}\,,\qquad j=i\,,\\\
\ell_{j}+\langle\ell_{i}|\ell_{i}\rangle\,\ell_{j}\,,\qquad j>i\,.\end{cases}$
(2.7)

As a result, through the Hanany-Witten effect, the number of 5-branes ending
on the 7-brane needs to be changed so as to satisfy charge conservation. More
explicitly, the multiplicities must transform as

$n_{j}\rightarrow\begin{cases}n_{j}\,,\qquad j\leq i\,,\\\
-n_{i}+\sum_{k>j}n_{k}\,\langle\ell_{i}|\ell_{k}\rangle\,,\qquad j=i\,,\\\
n_{j}\,,\qquad j>i\,.\end{cases}$ (2.8)

Figure 5 shows an example of this process, which starts from the web for the
$T_{3}$ theory and moves one of the [1,0] 7-branes from the left to the right
of the web.

Figure 5: Brane web where two 5-branes end on the same 7-brane after a HW
transition.

We see that we have found two different ways to arrive at the same brane web.
Note that the two webs in Figure 5 are physically equivalent, while the ones
in Figure 4 are not (a Higgs branch flow has occurred).

It is possible to associate a cousin of the toric diagram for the case when
multiple 5-branes end on the same 7-brane [29]. This object has been dubbed
Generalized Toric Polygon (GTP) in the literature and is obtained by replacing
a black dot with a white dot whenever it splits a segment corresponding to two
5-branes that are stuck together as in Figure 6, corresponding to the web in
Figure 5. Note that in terms of the web, the effect of the s-rule due to the
ending on 7-branes is to constrain the possible Coulomb branch deformations.
Rules to account for that in terms of a “poligonation” of the GTP have been
proposed in [38, 29].

Figure 6: GTP for the brane web in Figures 4 and 5.

Returning to the relation to geometric engineering, the duality between
M-theory on a CY3 and a 5-brane web in Type IIB along the directions of the
pinching of the torus only holds for webs without the 7-branes. In the case of
webs with all external legs of multiplicity equal to 1, the external 7-branes
can be neglected as they do not impose any (supersymmetry) constraint on the
possible deformations. Thus, we can alternatively describe the corresponding
SCFTs through a 5-brane web in Type IIB or as M-theory on the CY3 whose toric
skeleton coincides with the web. However, for generic multiplicities the
question is: what does the GTP correspond to in the M-theory geometric
engineering language? Progress in this direction was made in [38]: in the
particular case where white dots appear only on one side of the GTP, it is
possible to understand them by exploiting the reduction from M-theory to Type
IIA. In the toric case, one side of the toric diagram corresponds to a (real)
codimension-4 A-type du Val singularity, and after reduction on $S^{1}$ it
appears as the D6 brane of Type IIA. In this setup, it is possible to give a
nilpotent VEV to the stack of branes which are then forced to remain stuck
together (a construction known as a T-brane [52]). After lifting back to
M-theory, the conjecture is that the defining equations for the geometry look
the same as for the toric diagram, except that fewer deformations are
available.

In this work, we will argue that this is also the case for more generic GTPs,
i.e. containing white dots in several sides as well as the interior. We will
do so by exploiting mirror symmetry. We will see that while the geometric
counterpart of the HW transition (analogous to Figure 5) is complicated to
write down for the original M-theory CY3 $\mathcal{M}$, it is straightforward
to implement it for its mirror. This, in turn, allows us to identify the
moduli that become frozen in general (analogous to Figure 4).

####  2.2.1 The $E_{1}$ example

Let us illustrate this discussion using the $E_{1}$ example. The corresponding
web is shown in Figure 7.

Figure 7: Web for the $E_{1}$ theory.

In this case, all external legs have multiplicity 1, which means that we are
in the realm of standard toric diagrams. Indeed, the dual graph to Figure 7 is
the resolved toric diagram of Figure 1. The proper Coulomb branch modulus
corresponds to the vertical distance between the two parallel (1,0) 5-branes
in the opened-up face; this is the mass of the fundamental string stretching
between them. The extended Coulomb branch modulus corresponds to a deformation
that moves the external legs horizontally, changing the mass of the D-string
stretched between the two (0,1) 5-branes. Both proper and extended Coulomb
moduli are Kähler moduli. They are distinguished because in one case we are
not modifying the boundary conditions of the web (the face can open up while
keeping the 7-branes fixed), while in the second one we are modifying them
(moving some of the 7-branes transverse to the corresponding leg). Note also
that while there are four external legs, two of their positions are fixed by
charge conservation at every 5-brane junction, plus a third one from an
overall translation, leaving us in total with two parameters for the extended
Coulomb branch, which matches the geometric engineering result.

We may imagine now moving one of the 7-branes along the corresponding leg
until it crosses the web. Without loss of generality (all 7-branes are
equivalent), we choose the $[-1,1]$ one to cross. Moving the branch cut clock-
wise, the $[1,1]$ 7-brane turns into a $[-1,3]$ 7-brane as it is swept by it.
The resulting web is shown in Figure 8, together with the dual diagram, which
is the toric diagram for $C(\mathbb{F}_{2})$.

Figure 8: Equivalent web for the $E_{1}$ theory after the Hanany-Witten
transition.

###  2.3 Mirror construction

Let us first focus on the realm of standard toric diagrams. Given a toric CY3
$\mathcal{M}$ there is a well-known construction for its Hori-Vafa mirror
partner $\mathcal{W}$ [53]. The toric geometry of the original $\mathcal{M}$
is encoded in its toric diagram, which can be regarded as a convex rational
polytope $\Delta$ in the lattice $\mathbb{Z}^{2}$. Associated to $\Delta$ one
can construct a Laurent polynomial by assigning to each point in
$\Delta\cap\mathbb{Z}^{2}$ with coordinates $(m,n)$ the monomial $x^{m}y^{n}$
(where $x,y\in\mathbb{C}^{*}$) with a generic coefficient $c_{(m,n)}$. The
resulting polynomial is

$\displaystyle
P(x,y)=\sum_{(m,n)\in\Delta\cap\mathbb{Z}^{2}}c_{(m,n)}x^{m}y^{n}\,.$ (2.9)

The mirror geometry is then constructed as a double fibration over a complex
plane $\mathbb{C}$ parametrized by $w$ as

$\mathcal{W}=\begin{cases}P(x,y)=w\,,\\\ uv=w\,.\end{cases}$ (2.10)

There are three remarks. First, strictly speaking, this construction only
works for $\Delta$’s with at least one internal point. These are called Fano
polytopes, and in the remainder of this paper we will stick to that case,
which implies that the 5d theory has rank greater or equal than 1. Second,
note also that the Laurent polynomial $P(x,y)$ depends on the choice of origin
of the lattice. While the choice of the origin is irrelevant for many
applications and therefore hardly ever considered, it will become important in
our discussion below, where it plays a crucial role in defining mutations of
said polynomials. Finally, while it looks that we have as many free
coefficients $c_{(m,n)}$ as points in the polytope, in fact we can always
freely fix 3 of them, by using rescalings in $x$, in $y$, and an overall
SL$(2,\mathbb{Z})$ transformation. We will often make use of this freedom.

Let us unpack the main features of the geometry of $\mathcal{W}$. For $w\neq
0$, $uv=w$ parametrizes a copy of $\mathbb{C}$, with an $S^{1}$ associated to
the phase of the free complex coordinate (which can be taken to be $u$ or
$v$); this $S^{1}$ collapses to zero size at $w=0$. In turn, for fixed $w$,
the curve $P(x,y)-w=0$ defines a Riemann surface whose genus equals the number
of internal points in $\Delta$. We can look for the critical points of $P$,
i.e. for points $(x_{I},y_{I})$ such that

$\frac{\partial P}{\partial x}\Big{|}_{(x_{I},y_{I})}=\frac{\partial
P}{\partial y}\Big{|}_{(x_{I},y_{I})}=0\,.$ (2.11)

They have the meaning that on top of every $w_{I}=P(x_{I},y_{I})$, there is an
$S^{1}$ in the Riemann surface that pinches off. Thus, the segment $[0,w_{I}]$
on the $w$-plane connecting the 0 to $w_{I}$, together with the $S^{1}\times
S^{1}$ associated to the two cycles above (one on the Riemann surface and
pinching off at $w=w_{I}$ and the other one on the $uv$-plane and pinching off
at $w=0$) defines a topological $S_{I}^{3}$. The number of critical points,
and therefore of such $S^{3}_{I}$’s, is equal to the area of the toric diagram
(in units of fundamental triangles). Moreover, this collection of spheres
forms a basis of $H^{3}(\mathcal{W},\mathbb{Z})$, so the class of the $T^{3}$
toric fiber can be written as a formal linear combination of the
$S_{I}^{3}$’s.

The fiber on top of $w=0$ is of special importance, which can be understood at
an intuitive level by the fact that the $S^{3}_{I}$’s intersect on top of it.
Let us denote by $\Sigma$ the corresponding Riemann surface $P(x,y)=0$. It
contains the information of both the 5d theory as well as the corresponding
BPS quiver, via the so called _amoeba_ and _co-amoeba_ projections [33].

The amoeba projection of $\Sigma$ is defined as

$\displaystyle\mathcal{A}_{\Sigma}=\left\\{\left(\log|x|,\log|y|\right),\text{
s.t. }P(x,y)=0\right\\}\,,$ (2.12)

and it looks like a thickened version of the toric skeleton of the original
CY3 $\mathcal{M}$. Let us review how this comes about. Consider an edge of the
toric diagram composed of a single segment connecting the points with
coordinates $(a_{1},b_{1})$, $(a_{2},b_{2})$, so that the corresponding leg
is, up to a sign, $(p,q)=(b_{2}-b_{1},a_{1}-a_{2})$. The equation $P=0$ can be
written as $c_{2}x^{a_{2}}y^{b_{2}}-c_{1}x^{a_{1}}y^{b_{1}}=P^{\prime}$, where
$P^{\prime}$ contains the contribution of all dots in the toric diagram other
than the selected two. We can further write $cx^{-q}y^{p}-1=P^{\prime\prime}$,
where $P^{\prime\prime}=P^{\prime}/c_{1}x^{a_{1}}y^{b_{1}}$ and
$c=\frac{c_{2}}{c_{1}}$. Crucially, $P^{\prime\prime}$ can be regarded the
contribution of all monomials as if the origin was set in the point
$(a_{1},b_{1})$. Since the toric diagram is convex, all points lie to one side
(depending on the particular choice of edge) of $(a_{1},b_{1})$. Thus, if we
write $x=c^{\frac{1}{q}}\,t^{p}$, $y=t^{q}$, $P^{\prime\prime}$ is a
polynomial with all positive/negative (depending on the side to which the rest
of the points, as described before, lie) powers of $t$. Thus, in the
appropriate limit $t\rightarrow 0,\infty$ $P^{\prime\prime}\rightarrow 0$,
while the LHS obviously vanishes as well. Now, through the amoeba map above we
have

$\displaystyle\mathcal{A}_{\Sigma}\sim\left(p\tau+p_{0},q\tau\right),\qquad\tau=\log
t\,,\qquad p_{0}=\frac{1}{p}\log c\,;$ (2.13)

which corresponds to a infinite spike along the $(p,q)$ direction whose
position is encoded in $c$. In particular, this shows that changes in $c$ map
to changes of the positions of the external legs, that is, to mass
deformations in the 5d SCFT. More generally, the coefficients $c_{(m,n)}$ in
$P(x,y)$ are mapped to the extended Coulomb branch of the low energy theory.
This is the familiar statement that mirror symmetry exchanges complex and
Kähler moduli.

Physically, the Riemann surface $\Sigma$ also makes an appearance as the
analogue of the Seiberg-Witten curve for the 5d theory compactified on a
circle. As described originally in [28], one writes $(x_{5},x_{6})$ for the
coordinates in the plane of the web in Type IIB, and $(x_{4},x_{10})$ for the
coordinates on the $T^{2}$ where we compactify M-theory to make use of the
duality between the two setups. Then the SW curve is given precisely by
$P(x,y)=0$, where the coordinates are given by

$\displaystyle x=\exp\left[\frac{2\pi}{R}(x_{5}+ix_{4})\right]\,,\qquad
y=\exp\left[\frac{2\pi}{R}(x_{6}+ix_{10})\right]\,,$ (2.14)

with $R$ the radius of the torus.777Since we have fixed $\tau=i$, which
corresponds to the modular parameter of the M-theory torus, we are only left
with specifying the base length $R$. The various masses of BPS states of the
4d KK theory are computed using the usual SW technology, namely by integrating
the differential $\lambda=\log x\,d(\log y)$ over the corresponding cycles.

The geometry of $\mathcal{W}$ also contains the information about the BPS
quiver through the coamoeba projection of $\Sigma$. This is defined as the
projection onto the angular parts of $(x,y)$,

$\displaystyle\mathcal{A}^{\vee}_{\Sigma}=\left\\{(\arg x,\arg y)\,,\text{
s.t. }P(x,y)=0\right\\}\,.$ (2.15)

This projection lives on a $T^{2}$ and knows precisely how the $S^{3}_{I}$’s
intersect one another on top of the fiber at $w=0$. Using the fact that the
BPS quiver is identical to the fractional brane quiver for Type IIB D3-branes
probing the original geometry, the BPS quiver appears through string
dualities: the system of D3-branes probing $\mathcal{M}$ (in Type IIB)
becomes, after three T-dualities, a system of D6-branes on $\mathcal{W}$
wrapping the $S^{3}_{I}$’s (in Type IIA). On $\Sigma$, the D6-branes look like
1-cycles, each of them surrounding a puncture associated to a leg of the
amoeba, and intersecting one another according to the coamoeba. The
combinatorics of this intersection pattern is captured by the brane tiling,
which in turn specifies the $4d$ $\mathcal{N}=1$ theory living on the branes.
We refer the reader to [33, 54] for detailed discussions of this construction.

####  2.3.1 The $E_{1}$ example

Let us revisit our trusty $E_{1}$ example. The toric diagram was depicted in
Figure 1. From the associated Laurent polynomial, we find the mirror partner
to $C(\mathbb{F}_{0})$,

$\displaystyle\begin{cases}w=P(x,y)=c_{0}+c_{1}x+c_{2}y+c_{3}\frac{1}{x}+c_{4}\frac{1}{y}\\\
w=uv\end{cases}\,,$ (2.16)

where $u,v\in\mathbb{C}$ and $x,y\in\mathbb{C}^{*}$. As discussed after
(2.10), we have the freedom to fix three of the complex coefficients
$c_{i}=1$. In particular, we see that we have two complex structure moduli, as
expected.

The equation $P(x,y)=0$ is the Riemann surface above the origin of the
$w$-plane. It is straightforward to determine its amoeba and coamoeba
projections for specific values of $c_{i}$. Figure 9 shows cartoons of the
amoeba and coamoeba projections of $\Sigma$ for a choice of
coefficients.888The coefficients in this example give rise to the so-called
Phase 1 of $C(\mathbb{F}_{0})$, given by Figure 2 and (2.2). Varying these
coefficients, it is possible to obtain a qualitatively different coamoeba,
which corresponds to Phase 2. In other words, the new coamoeba gives rise to
the corresponding brane tiling or, equivalently, the quiver and
superpotential. Detailed discussions of different choices of the coefficients,
and the different toric phases (including specific analyses of
$C(\mathbb{F}_{0})$), can be found in [33, 54].

Figure 9: Amoeba and coamoeba projections for the Riemann surface for $F_{0}$.

We immediately see that the shape of the amoeba corresponds to the brane web
in Type IIB. Even though the exact map between $c$’s and (extended) Coulomb
branch parameters is relatively complicated, one can see that by varying the
$c$’s it is possible to independently tune the size of the central hole
(Coulomb branch VEV) and the horizontal distance between legs (mass parameter,
in this case corresponding to inverse squared gauge coupling).

In the coamoeba the $S^{3}_{I}$’s are projected to $S^{1}$’s on $T^{2}$. These
are wrapped by D6-branes, and we see that their pattern of intersections
reproduces the brane tiling shown in Figure 2.

##  3 Hanany-Witten transitions and polytope mutations

This section contains the main point of this paper, the implementation of the
Hanany-Witten transition of the brane web in terms of the geometry, which is
particularly easily visualized in the mirror $\mathcal{W}$. The corresponding
operation is known in the mathematical literature as a _polytope mutation_.
The connection between Hanany-Witten transitions an polytope mutations was
already noted and investigated in [39]. In this paper, however, we consider a
refined version of polytope mutation which, among other things, also specifies
the transformation of the Laurent polynomial. This prescription enables the
mapping of moduli across the transition, allowing the determination of
instances when they become frozen.

###  3.1 Hanany-Witten in the mirror

Consider a polytope $\Delta$ and its associated Laurent polynomial $P(x,y)$.
As a warm-up, let us illustrate our construction with a four-sided polytope
with two parallel sides such that the distance between them is 2 (in lattice
units). Without loss of generality, we can use an appropriate
SL$(2,\mathbb{Z})$ transformation such that one of the parallel edges is on
the $x$ axis and the other one is at $y=2$, as illustrated in Figure 10.

Picking the origin of the lattice in the interior of $\Delta$, i.e. with
$y=1$, the corresponding polynomial takes the following form

$\displaystyle P(x,y)=$
$\displaystyle\frac{1}{y}P_{-1}(x)+y^{0}P_{0}(x)+yP_{1}(x)$ (3.1)
$\displaystyle=$
$\displaystyle\frac{x^{a}}{y}\prod_{i=1}^{n_{-}}(x-x_{i})+P_{0}(x)+y\,x^{b}\prod_{j=1}^{n_{+}}(x-\widetilde{x}_{j})\,.$
(3.2)

The exponents $a$ and $b$ resulting from the factorization of $P_{\pm 1}$ can
be either positive or negative, and they depend on the precise choice of
origin along the $x$ axis. Figure 10 shows the general form of $\Delta$.

As discussed in the previous section, this polynomial specifies a geometry
$\mathcal{W}$ whose amoeba projection (of the Riemann surface $\Sigma$) has
vertical asymptotes along the bottom edge at positions $x_{i}$ and along the
top edge at positions $\widetilde{x}_{j}$. The dual IIB brane web will have
$n_{-}$ semi-infinite NS5-branes at the bottom and $n_{+}$ NS5-branes at the
top, respectively at positions

$\displaystyle x_{5,i}=\frac{R}{2\pi}\log
x_{i}\,,\qquad\widetilde{x}_{5,j}=\frac{R}{2\pi}\log\widetilde{x}_{j}\,,$
(3.3)

after using (2.14).

Figure 10: Polytope associated to the polynomial (3.1) for $a=-1$, $b=-2$,
$n_{+}=3$ and $n_{-}$=5. The origin is shown in red. We show the vectors
normal to every edge on the boundary.

Let us now consider the following change of coordinates:

$\displaystyle(x,y)\to(x,(x-x_{1})y)\,.$ (3.4)

It is immediate to see what are the consequences for the Laurent polynomial
(3.1). The $(x-x_{1})$ factor in $P_{-1}(x)$ cancels, and it appears in
$P_{1}(x)$ instead. We obtain

$\displaystyle
P(x,y)\to\frac{x^{a}}{y}\prod_{i=2}^{n_{-1}}(x-x_{i})+P_{0}(x)+y\,x^{b}\,(x-x_{1})\prod_{j=1}^{n_{1}}(x-\widetilde{x}_{j})\,.$
(3.5)

The resulting polytope is shown in Figure 11. In terms of the dual Type IIB
brane web, we have sent the external leg at position $x_{1}$ from the bottom
to the top of the brane web. Moreover, this automatically changes the slope of
the edge at the rightmost side of the polytope.999While the explicit example
in the figure was chosen such that before and after mutation both of the sides
of $\Delta$ that are not parallel to the $x$-axis consist of a single edge
(namely they do not cross over intermediate lattice points), our discussion
extends without changes to the case in which these sides contain multiple
edges. Moreover, the fact that the two lateral edges end up being parallel
after the mutation is just a non-generic feature of this specific example.
This is precisely the effect of sweeping the monodromy in the Hanany-Witten
transition resulting from moving the 7-brane at the end of the leg across the
web.

Figure 11: Mutated polytope obtained from Figure 11 by the change of variables
in (3.4). The mutation implements a Hanany-Witten transition in the dual web.

We can use the same type of coordinate transformations to implement more
general HW transitions. For example, in order to send one of the legs from the
top to the bottom of the web, we would use the transformation

$\displaystyle(x,y)\to(x,\frac{y}{x-\widetilde{x}_{j}})\,.$ (3.6)

Similarly, if we want to send several 7-branes across the web, we can employ
the transformation

$\displaystyle(x,y)\to(x,p(x)\,y)\,,$ (3.7)

with the restriction that $p(x)$ is a divisor of $P_{-1}(x)$. This requirement
ensures that the corresponding monomials cancel, and physically has the
meaning that we are picking the subset of the legs that appear as factors in
$P_{-1}(x)$.

An interesting new phenomenon arises when the length of the polytope $\Delta$
in the direction of $y$ is greater than 2. Consider, for example, an example
of height 3, and pick the origin such that the Laurent polynomial is

$\displaystyle
P(x,y)=\frac{1}{y}P_{-1}(x)+P_{0}(x)+yP_{1}(x)+y^{2}P_{2}(x)\,,$ (3.8)

and take $(x-x_{1})$ a prime factor of $P_{-1}(x)$.

Then the coordinate transformation $(x,y)\to(x,(x-x_{1})y)$ sends $P_{2}(x)\to
P_{2}(x)(x-x_{1})^{2}$. We observe that we have fewer free complex
coefficients than the naive counting from the number of monomials. In the dual
Type IIB picture, this means that two of the external legs created by the HW
transition are fixed to be together at the same position $x_{1}$. We interpret
this as the fact that two 5-branes end on the same 7-brane, and
correspondingly, that we have a white dot in the corresponding GTP.

Having a height 3 polytope, allows us to pick different vertical positions for
the origin. More generally, the distance between the origin and the edge we
want to mutate can take different values. We now discuss the important role of
the choice of origin. Returning to the example at hand, depending on the
choice of origin, we can shift the entire polytope vertically, i.e. shift the
$y$ powers in the Laurent polynomial. In particular, we can make an
alternative choice for which $P(x,y)$ becomes

$\displaystyle
P(x,y)=\frac{1}{y^{2}}P_{-2}(x)+\frac{1}{y}P_{-1}(x)+P_{0}(x)+yP_{1}(x)\,.$
(3.9)

Comparing to and (3.5) and Figure 10, we have moved the polytope downwards by
one lattice unit in the $y$ direction. Equivalently, we have picked the origin
to be one unit further from the lower edge. Now, if we want to send
$y\to(x-x_{1})y$ we also need to require that $(x-x_{1})^{2}$ divides
$P_{-2}(x)$ to ensure that the desired cancellations happen and the final
result is also a Laurent polynomial. This means that before the coordinate
transformation, we must have two legs of the web frozen together, which become
just one leg after the transition. This is the first instance where we notice
that the choice of origin of the lattice is related to whether the white dot
is present in the GTP before or after the HW transition, a point to which we
will return later.

The transformations (3.4), (3.6), (3.7) are examples of so called _mutations_
of Laurent polynomials, which in turn implies mutations of their Newton
polytopes. The key point is that the polytope mutation is implemented by a
change of variables in the corresponding Laurent polynomial. As such, it
provides a more refined description than the mere mutation of the points in
the polytope. These mutations have been discussed in the pure mathematics
literature quite extensively starting with [55]. As we will illustrate in
examples below, mutations on edges that are not horizontal can be obtained
directly by appropriate changes of variables or, equivalently, by first
applying an SL$(2,\mathbb{Z})$ transformation to turn the edge under
consideration horizontal. In the next subsection, we will review some of the
mathematical terminology and results, so that we can then import them to
Physics.

####  3.1.1 The $E_{1}$ example

Let us illustrate the previous discussion in the example of the $E_{1}$
theory. Instead of starting with the polytope in Figure 1, it is slightly more
convenient to perform a global SL$(2,\mathbb{Z})$ transformation to align one
side of the polytope with the $x$ axis, as shown in Figure 12.

Figure 12: Polytope for the $E_{1}$ theory obtained from Figure 1 via an
SL$(2,\mathbb{Z})$ transformation. The origin is indicated in red.

The corresponding Laurent polynomial reads

$P(x,y)=\frac{1}{y}(x-c_{1})+1+y\,\left(\frac{1}{x}-\frac{1}{c_{2}}\right)\,,$
(3.10)

where we have already fixed three of the coefficients, and the remaining two
are directly identified with the position of the external legs of the web. Now
we perform the following mutation,

$(x,y)\rightarrow(x,(x-c_{1})y)\,,$ (3.11)

and find

$P(x,y)\rightarrow\frac{1}{y}+1+y\,\left(\frac{1}{x}-\frac{1}{c_{2}}\right)(x-c_{1})\,,$
(3.12)

whose polytope is shown in Figure 13, which corresponds to $C(\mathbb{F}_{2})$
(up to an overall rotation, it is the one in Figure 8). In particular, note
once again that the mutation of the Laurent polynomial automatically knows
about the action due to sweeping the monodromy cut of the 7-brane across half
of the web –this is a completely generic fact. In this example, the HW
transition does not generate a white dot in the toric diagram; accordingly,
none of the moduli in (3.12) are frozen.

Figure 13: Polytope obtaines from Figure 12 by mutating it on the lower
horizontal edge.

###  3.2 Polytope mutations

As mentioned above, HW transitions translate, in the mirror, to the notion of
mutation defined in the mathematical literature [55]. The starting point is a
toric diagram in the standard sense –that is, with no decoration of white
dots– which can be regarded as a rational polytope $\Delta$ in
$\mathbb{Z}^{2}$. We assume that the toric diagram contains the origin in its
interior, which in turn implies that it has at least one internal point. This
amounts to restricting to 5d SCFTs of rank greater or equal than 1. In the
mathematical literature, it is also often required that the vertices are
primitive, which means that their coordinate vectors consist of coprime
numbers.101010Note that vertices being primitive or not depends on the choice
of origin. We will see examples of this below.. However, we will not make this
requirement here. Given one such $\Delta$, we can always write the
corresponding Laurent polynomial $P(x,y)$ as a Laurent polynomial in $y$,
whose coefficients are Laurent polynomials in $x$. Since the origin is in the
interior of $\Delta$, $P(x,y)$ contains a finite number of terms with negative
and positive powers for both $y$ and $x$. We can write

$P(x,y)=\sum_{n=-N_{-}}^{N_{+}}P_{n}(x)y^{n}\,,$ (3.13)

where $N_{-}$ and $N_{+}$ are the maximum negative and positive powers of $y$,
respectively. $P_{n}(x)$ are Laurent polynomials in $x$, whose degrees need to
satisfy that the corresponding Newton polygon is convex. A mutation is then a
birrational transformation of the form [55]

$\mu:\,(x,y)\rightarrow(x,\alpha(x)\,y)\,,$ (3.14)

where $\alpha(x)$ is a Laurent polynomial such that $\alpha^{i}(x)$ divides
$P_{-i}(x)$. Clearly, the transformation in (3.4) or its generalization (3.7)
are particular cases of (3.14). Thus, the physical avatar of the mathematical
notion of mutation is crossing 7-branes in a 5-brane web.

Note that, in order for $\alpha^{i}$ to divide $P_{-i}$, some conditions must
be met. First, there are requirements on the degree of the polynomials
$P_{i}$. Let us consider the case of $\alpha(x)$ a polynomial of degree 1. In
that case, it is clear that $P_{-i}$ must be at least a polynomial of degree
$i$ in $x$. Moreover, the coefficients of the polynomial must be tuned so as
to have the relevant number of common roots, as was discussed already around
(3.9). All in all, this selects a triangle of points ($n$ points at lattice
distance $n$ from the origin). In total, the triangle will have base $N_{-}$
and height $N_{-}$ with respect to the origin, as depicted in Figure 14.

Figure 14: The triangle selected in the mutation. The dashed line represents
the rest of the diagram, whose precise shape is not relevant, as long as it is
convex. The red dot represents the origin.

Then, the action of mutation is to insert a triangle on the the other side
with respect to the origin, only that this time with its base of length and
height are equal to $N_{+}$. Finally, if $\alpha$ is a polynomial of arbitrary
degree $d$, then we can always factor it, and a similar argument would hold
for each of the $d$ monomials. That is, we would end with $d$ triangles of
equal base and height $N_{-}$ before the mutation, that would transform into
$d$ triangles of equal base and height $N_{+}$ after it. This allows us to
translate the action of the mutation directly in terms of the polytope,
without the need to go through the Laurent polynomial as an intermediate step.

The way this is made precise in the mathematical literature is by introducing
the notion of a _primitive T-cone_. A primitive T-cone is a cone formed by the
origin and a an external side (or a collection of edges on a side) of lattice
length $N$ which is at lattice distance $N$ from the origin. The triangles in
Figure 14 and discussed above are examples of this construction. A primitive
T-cone of base $N$ is naturally identified with a 7-brane with $N$ 5-branes
ending on it. Let us stress that the definition of primitive T-cone gives a
physical meaning to the position of the origin of the polytope, as the height,
and therefore the base length of the corresponding primitive T-cone, depends
on it. This phenomenon was already discussed above in terms of the mutation of
Laurent polynomials. In pure polytope language, the mutation amounts to
removing one primitive T-cone from one side of the polytope and adding a new
primitive T-cone to the other side. Since generically the other side of the
polytope is at a different lattice height $N_{+}\neq N_{-}$, the resulting
primitive T-cone will generically be of different length $N_{+}$, which is the
manifestation of the Hanany-Witten effect. In fact, one can be fully precise
and describe the effect of the mutation combinatorially (see e.g. [56]),
precisely finding (2.7). This shows, as anticipated in [39], that the
mathematical notion of mutation precisely coincides with the HW transitions in
brane webs.

Note that, with a choice of origin such that $N_{+}$ is much bigger than
$N_{-}$ (or vice versa), the initial polytope $\Delta$ and its mutation, which
we will denote $\mu(\Delta)$, may differ a lot in size (in plain words, one of
them may have much more points than the other, precisely as a consequence of
the HW effect). This is not a problem precisely due to the fixing of the
various complex coefficients $c_{(n,m)}$ when connecting both sides of the
mutation. As observed above, in the geometry of the mirror $\mathcal{W}$, this
is understood as the freezing of some of the complex moduli, while in the
original CY3 $\mathcal{M}$ it translates as a freezing of the Kähler moduli;
precisely in such a way as to preserve the extended Coulomb branch of the 5d
SCFT.

Importantly, this freezing of moduli does not affect the functional dependence
of $P(x,y)$ on the variables $x$ and $y$, and thus the geometry of the mirror
$\mathcal{W}$. Correspondingly, we argue that the geometry $\mathcal{M}$
corresponding to a GTP is the same as for the standard toric diagram (namely,
with no white dot decoration) except that some of its Kähler moduli are
frozen. This is consistent with the proposal in [38], and we will come back to
it in Section 4.

###  3.3 Mutation invariants

In the mathematical literature, an important question is how to determine when
two polytopes can be connected via a sequence of mutations. To that end, three
invariant quantities have been defined: the _singularity content_ , the
_classical period_ and the _Hilbert series_ [57]. We will not discuss the
first invariant, except to mention that it corresponds to how each of the
edges looks as a codimension 4 singularity (see [58] for more details). The
second two, to which we now turn, can be computed from the Laurent polynomial
$P(x,y)$ associated to the polytope; we should note that the choice of origin
will be relevant in general. Lastly, one word on notation: we will often
denote either the period or the Hilbert series by the name of the CY3 variety
whose toric diagram is $\Delta$. In what follows, we will always assume that
the definition of the polytope $\Delta$ already incorporates the choice of
origin.

  * •

The (classical) period is defined in terms of the Laurent polynomial $P(x,y)$
associated to $\Delta$ as

$\pi_{\Delta}(t)=\frac{1}{(2\pi
i)^{2}}\oint_{|x|,|y|=1}\frac{dx\,dy}{x\,y}\,\frac{1}{1-t\,P(x,y)}\,,$ (3.15)

where the variables $x,y\in\mathbb{C}^{*}$. In the context at hand, it has
been argued that the classical period defined above coincides with the quantum
period and it is a generating function for Gromov-Witten invariants of the
original Calabi-Yau $\mathcal{M}$ [59, 60]. Again we remark that
$\pi_{\Delta}(t)$ generically depends on the choice of origin of the polytope.

  * •

The Hilbert series of the variety $\mathcal{M}$. Since we can label the
variety by the polytope $\Delta$, we will write ${\rm Hilb}_{\Delta}(t)$. It
turns out that this can be easily computed as the Ehrhart series of the dual
polytope $\Delta^{\circ}$ [57]. The dual polytope is defined as

$\Delta^{\circ}=\\{u\in\mathbb{Q}^{2}\,/\,u\cdot
v\geq-1\,,\forall\,v\in\Delta\\}\,.$ (3.16)

In practice, it is sufficient to consider the inequalities obtained when $v$
are the vertices of $\Delta$. Note that the dual polytope need not be
integral. Then,

${\rm Hilb}_{\Delta}(t)={\rm
Ehr}_{\Delta^{\circ}}(t)=\sum_{n=0}|n\Delta^{\circ}\cap\mathbb{Z}^{2}|\,t^{n}\,.$
(3.17)

In words, $|n\Delta^{\circ}\cap\mathbb{Z}^{2}|$ is the number of integral
lattice points in the $n$ times enlarged copy of $\Delta^{\circ}$. Once again,
note that this quantity depends on the choice of origin inside the polytope.

These quantities are invariant under mutation $\Delta\rightarrow\mu(\Delta)$,
namely

$\pi_{\Delta}(t)=\pi_{\mu(\Delta)}(t)\,,\qquad{\rm Hilb}_{\Delta}(t)={\rm
Hilb}_{\mu(\Delta)}(t)\,.$ (3.18)

An important remark is that the period $\pi_{\Delta}(t)$ depends in general on
the coefficients $c_{i}$, and therefore the fact that it is invariant under
mutation is yet another indication that for generic toric geometries the
complex moduli becoming frozen is the appropriate interpretation of the white
dot in the GTP.

Moreover, it has been conjectured [61, 62, 63] (see also [60]) that whenever
the period of two polytopes $\Delta_{1}$ and $\Delta_{2}$ coincide, then: 1)
there exists a sequence of mutations connecting the two polytopes and 2) there
exists a flat fibration of geometries interpolating between
$\mathcal{M}_{\Delta_{1}}$ and $\mathcal{M}_{\Delta_{2}}$. In physical terms,
the first result can be used to determine when two brane webs, or their
associated geometries, can be connected via a collection of HW transitions or
their corresponding mutations. On the other hand, the second result implies
that there exists a geometry (in general non-toric) that continuously
describes the HW transition.

####  3.3.1 The $E_{1}$ example

In the $E_{1}$ example, the period associated to the polytope corresponding to
$C(\mathbb{F}_{0})$ is

$\displaystyle\pi_{C(\mathbb{F}_{0})}(t)=\int\frac{dx\,dy}{x\,y}\frac{1}{1-t\left[\frac{1}{y}(x-c_{1})+1+y\left(\frac{1}{x}-\frac{1}{c_{2}}\right)\right]}\,.$
(3.19)

The integral can be computed as a series expansion in $t$ by expanding the
geometric series in the integrand. The first few orders in the series are

$\displaystyle\pi_{C(\mathbb{F}_{0})}(t)=1+t+\left(3+2\frac{c_{1}}{c_{2}}\right)t^{2}+O(t^{3})\,.$
(3.20)

The fact that the mutation (3.11) preserves the period follows trivially from
implementing it as a change of variables in the integral (3.19): the change in
the Haar measure is absorbed by the Jacobian of the change of variables. In
fact, the same argument also applies for generic mutations of Laurent
polynomials; instead, it is the reverse implication that is highly non-
trivial.

The second invariant can be computed as follows. The polytopes
$\Delta_{C(\mathbb{F}_{0})}$ and $\Delta_{C(\mathbb{F}_{2})}$ have vertices
$\\{(1,-1),(0,-1),(-1,1),(0,1)\\}$ and $\\{(0,-1),(-1,1),(1,1)\\}$,
respectively.111111Here we consider the polytopes for these geometries given
in Figures 12 and 13, which are related by mutation. Therefore, the dual
polytopes are

$\displaystyle\Delta_{C(\mathbb{F}_{0})}^{\circ}$
$\displaystyle=\\{(a,b)\in\mathbb{Q}^{2}\text{ s.t. }-1\leq b\leq
1\,,\,b-1\leq a\leq b+1\\}\,,$ (3.21)
$\displaystyle\Delta_{C(\mathbb{F}_{2})}^{\circ}$
$\displaystyle=\\{(a,b)\in\mathbb{Q}^{2}\text{ s.t. }b\leq 1\,,\,-b-1\leq
a\leq b+1\\}\,.$ (3.22)

From this, it is straightforward to scale the size of the dual polytopes and
count the number of internal integral lattice points at each $n$ as in (3.17),
finding that they are indeed the same and equal to

${\rm Hilb}_{C(\mathbb{F}_{0})}={\rm
Hilb}_{C(\mathbb{F}_{2})}=\frac{1+6t+t^{2}}{(1-t)^{3}}\,.$ (3.23)

##  4 5d SCFTs, GTPs, and the period

We have seen that there is a deep connection between the mathematical theory
of mutations and the physics of 5d SCFTs and their engineering in String
Theory. Let us make a quick summary of the discussion above:

  * •

There are two ways in which to reach a brane web corresponding to a GTP: a
Hanany-Witten transition (as in Figure 4) and a Higgs branch flow (as in
Figure 5).

  * •

The HW transition corresponds to a mutation of the geometry. One can easily
keep track of the various moduli by looking at the geometry of the mirror
$\mathcal{W}$ and, in particular, see that the extended Coulomb branch of the
5d SCFT is preserved.

  * •

The Higgs branch flow corresponds to freezing some of the moduli in M-theory
geometric engineering (complex moduli in the mirror $\mathcal{W}$, Kähler
moduli in the original $\mathcal{M}$) while keeping the geometry otherwise
identical. This process changes the extended Coulomb branch of the 5d theory,
as expected.

Now, we turn to explore some consequences of the mathematical work on
mutations in our physical setup. Here, we will discuss three items:

  1. 1.

The period $\pi_{\Delta}(t)$ can be used to efficiently classify 5d SCFTs
coming from brane webs. In particular, it allows us to determine when two _a
priori_ different looking webs can be related by a sequence of HW transitions.

  2. 2.

When two toric geometries are related by a mutation, it is possible to build a
flat family of geometries interpolating between the two. This corresponds to
the geometric version of the HW transition in the original M-theory Calabi-Yau
$\mathcal{M}$.

  3. 3.

The mirror geometry $\mathcal{W}$ automatically encodes the Seiberg-Witten
curve of the 5d theory on a circle. This can be used to extract the effect of
the frozen moduli on the low energy physical observables [44].

We should remark that in this section we will be considering simple examples
as proof of concept of our methods. It would be very interesting to
investigate these problems in more depth, which we postpone for future work.

###  4.1 The period and the classification of brane webs

Due to the constraints imposed by supersymmetry in 5 dimensions, the
exploration of the full landscape of theories appears to be an achievable
goal, at least for low rank [14, 64]. While most efforts have tackled this
problem from a geometric engineering point of view, it is also possible to
attempt a similar program in terms of brane webs [45]. In this context, the
primary challenge lies in discerning whether two brane webs correspond to
distinct 5d SCFTs or if there exists a sequence of duality transformations and
HW moves that transforms one into the other. Several invariants under these
transformations have been defined in order to tackle this problem [32]. They
are the _total monodromy_ of the brane web and the _asymptotic charge
invariant_. The total monodromy is the product of the monodromies associated
to each 7-brane of the web. In the notation of Section 2.2,

$\displaystyle M_{\rm tot}=\prod_{\ell}M_{\ell}\,.$ (4.1)

The asymptotic charge invariant $\mathcal{Q}$ is

$\displaystyle\mathcal{Q}=\text{gcd}\\{\langle\ell_{i}\,|\,\ell_{j}\rangle\,,\,\forall\,i,j\\}\,.$
(4.2)

Interestingly, the fact that these quantities are identical for two webs is
necessary but not sufficient for them to define the same low-energy 5d theory.
This raises a puzzle already at rank 2 [45], namely that the total monodromy
and asymptotic charge invariant do not completely specify the SCFT. The
specific exampled discussed in that reference were the webs labelled by (c)
and (e) in Figure 3 of [24] and shown in Figure 15. These two webs share the
same classifiers, yet they are expected to give rise to different low energy
theories.

Figure 15: The two webs to compare, taken from [24].

One immediately sees that the period $\pi_{\Delta}(t)$ seems to be precisely
the right tool to perform this task. As we have discussed above, there is
strong evidence that whenever the periods coincide there exists a sequence of
mutations connecting the corresponding Laurent polynomials, and it has been
conjectured that this should indeed be true in general [61, 62, 63]. That is,
it is to be expected that the identity of periods is a necessary and
sufficient condition for the equivalence of the corresponding polytopes under
mutation (and hence of the associated varieties). In physics terms, this would
allow to discriminate whether or not two webs describe the same 5d theory.

Indeed, one can check that the two polytopes in Figure 15 give rise to
different periods. The computation of the period as a series expansion is
straightforward, however two issues need to be taken into account: the choice
of origin (since this example has two internal points) and a possible non-
trivial map of the moduli (the coefficients $c_{(n,m)}$) from one polytope to
the other. Here, we proceed by brute force. Namely, we consider both possible
choices of origin and check that there is no possible matching of the moduli
such that the periods are equal.

Let us denote $P_{c}^{i}(x,y)$ and $P_{e}^{i}(x,y)$ the corresponding Laurent
polynomials for the two polytopes, where $i=1,2$ labels the possible choices
of origin:

$\displaystyle P_{c}^{1}(x,y)$
$\displaystyle=C_{0}+C_{1}y+C_{2}y^{2}+C_{3}xy+C_{4}\frac{1}{y}+C_{5}\frac{1}{xy}\,,$
(4.3) $\displaystyle P_{c}^{2}(x,y)$
$\displaystyle=D_{0}+D_{1}y+D_{2}x+D_{3}\frac{1}{y}+D_{4}\frac{1}{y^{2}}+D_{5}\frac{1}{xy^{2}}\,,$
(4.4) $\displaystyle P_{e}^{1}(x,y)$
$\displaystyle=E_{0}+E_{1}y+E_{2}y^{2}+E_{3}x+E_{4}xy+E_{5}\frac{1}{xy}\,,$
(4.5) $\displaystyle P_{e}^{2}(x,y)$
$\displaystyle=F_{0}+F_{1}y+F_{2}x+F_{3}\frac{1}{y}+F_{4}\frac{x}{y}+F_{5}\frac{1}{xy^{2}}\,.$
(4.6)

Note that we have chosen not to fix any values of the complex coefficients.
The first few orders of the periods for each of these polynomials read

$\displaystyle\pi_{c,1}(t)=$ $\displaystyle
1+C_{0}t+\left(C_{0}^{2}+2C_{1}C_{4}+2C_{3}C_{5}\right)t^{2}+\left(C_{0}^{3}+6C_{0}(C_{1}C_{4}+C_{3}C_{5})+3C_{1}C_{4}^{2}\right)t^{3}$
$\displaystyle+\left(C_{0}^{4}+12C_{0}^{2}(C_{1}C_{4}+C_{3}C_{5})+12C_{0}C_{1}C_{4}^{2}+6C_{1}^{2}C_{4}^{2}+24C_{1}C_{3}C_{4}C_{5}+6C_{3}^{2}C_{5}^{2}\right)t^{4}+O(t^{5})\,,$
(4.7) $\displaystyle\pi_{c,2}(t)=$ $\displaystyle
1+D_{0}t+\left(D_{0}^{2}+2D_{1}D_{2}\right)t^{2}+\left(D_{0}^{3}+6D_{0}D_{1}D_{2}+3D_{1}^{2}D_{2}\right)t^{3}$
$\displaystyle+\left(D_{0}^{4}+12D_{0}^{2}D_{1}D_{2}+12D_{0}D_{1}^{2}D_{2}+12D_{1}^{2}D_{2}D_{2}+6D_{1}^{2}D_{2}^{2}\right)t^{4}+O(t^{5})\,,$
(4.8) $\displaystyle\pi_{e,1}(t)=$ $\displaystyle
1+E_{0}t+\left(E_{0}^{2}+2E_{4}E_{5}\right)t^{2}+\left(E_{0}^{3}+6E_{0}E_{4}E_{5}+6E_{1}E_{3}E_{5}\right)t^{3}$
$\displaystyle+\left(E_{0}^{4}+12E_{0}^{2}E_{4}E_{5}+24E_{0}E_{1}E_{3}E_{5}+6E_{4}^{2}E_{5}^{2}\right)t^{4}+O(t^{5})\,,$
(4.9) $\displaystyle\pi_{e,2}(t)=$ $\displaystyle
1+F_{0}t+\left(F_{0}^{2}+2F_{1}F_{3}\right)t^{2}+\left(F_{0}^{3}+6F_{0}F_{1}F_{3}\right)t^{3}$
$\displaystyle+\left(F_{0}^{4}+12F_{0}^{2}F_{1}F_{3}+6F_{1}^{2}\left(2F_{2}F_{5}+F_{3}^{2}\right)\right)t^{4}+O(t^{5})\,.$
(4.10)

Already at fourth order one can explicitly check that there is no map between
the coefficients of $P_{c}(x,y)$ and $P_{e}(x,y)$ (allowing for both possible
choices of origin) such that their periods are equal. This shows that indeed
these two polytopes are not related by a mutation, or in other words the two
brane webs are not related by a Hanany-Witten move. Hence, they define two
different 5d SCFTs.

This example is a proof of concept of how the periods associated to the
polytopes might be used for classifying 5d SCFTs. Even though we will not
pursue this further here, one could consider proceeding in a systematic
fashion by e.g. listing the possible brane webs of a given rank, or a given
number of external legs (in the same spirit of [45]) and computing $M_{tot}$,
$\mathcal{Q}$ and $\pi(t)$ to discern if they define the same low energy
theory.

###  4.2 Hanany-Witten and geometric deformations

Having identified Hanany-Witten transitions in 5-brane webs with the
mathematical notion of mutation of a Laurent polyonomial/Newton polytope
allows us to import the results developed in the mathematical literature. In
particular, it has been proven in [61, 62] that two toric varieties
$\mathcal{M}_{0}$ and $\mathcal{M}_{\infty}$, whose toric diagrams are related
by mutation, can be regarded as deformations of one another. Technically
speaking, such deformation is q-Gorenstein (qG), meaning that some power of
the canonical bundle is principal (hence trivial as a divisor class). In this
language, since the period and the Hilbert series are invariant under qG
deformations, their identity under mutation immediately follows. In fact,
following [61], one can think of the deformation parameter as a
$\mathbb{P}^{1}$ coordinate, and regard the deformation as a flat fibration
over $\mathbb{P}^{1}$ where the fiber over 0 is $\mathcal{M}_{0}$ and the
fiber over $\infty$ is $\mathcal{M}_{\infty}$. In the particular case of
mutations relating $\mathcal{M}_{0,\infty}$ corresponding to toric diagrams
(that is, no white dot decorations), it is natural to identify this
deformation with the quiver deformation in [41].

In practice, it is not trivial to write down the explicit expression for the
flat family describing the mutation between two complicated varieties whose
brane webs are related by a HW move. Fortunately, in some examples the
deformation can be identified with the classical smoothing of singularities.
This is in fact the case of our $E_{1}$ example [61]. Focusing on the slice of
the dual of the toric fan at height 1, we can simply consider the 2d compact
spaces $\mathbb{F}_{0}$ and $\mathbb{F}_{2}$ (the construction of the complex
cone on top of the 2d compact manifold goes along for the ride in the
smoothing to be discussed below). Then, we consider the variety
$\mathcal{M}_{t}$ given by
$(y,x,t)\in\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{C}$ with the
equation [65]

$y_{0}^{2}x_{0}-y_{1}^{2}x_{1}-t\,y_{0}y_{1}x_{2}=0\,.$ (4.11)

For $t\neq 0$ one can solve this equation writing

$x_{0}=(y_{1}z_{0}+y_{1}z_{1})^{2}\,,\qquad
x_{1}=(y_{0}z_{0}-y_{0}z_{0})^{2}\,,\qquad
x_{2}=\frac{4}{t}\,z_{0}z_{1}y_{0}y_{1}\,.$ (4.12)

This provides a map from $\mathbb{P}^{1}\times\mathbb{P}^{1}$ (parametrized by
$(z,y)$) into $\mathcal{M}_{t}$, showing that for any $t\neq 0$,
$\mathcal{M}_{t}$ is isomorphic to $\mathbb{F}_{0}$. In turn, for $t=0$ the
equation boils down to $y_{0}^{2}x_{0}-y_{1}^{2}x_{1}=0$, which describes
$\mathbb{F}_{2}$. Thus we can regard $\mathcal{M}_{t}$ as a flat fibration of
deformations such that the fiber over any arbitrary non-zero $t$ is
$\mathbb{F}_{0}$ with a special fiber at $t=0$ given by $\mathbb{F}_{2}$. It
is interesting to note that the fact the generic fiber is isomorphic to
$\mathbb{F}_{0}$ is consistent with the observation, to be discussed below,
that for any non-zero quiver deformation as outlined in [41], the theory flows
to one describing $\mathbb{F}_{0}$.

It would be very interesting to make this discussion systematic and to be able
to describe the geometry interpolating between any pair of GTPs related by a
mutation explicitly. We leave this analysis for future work.

###  4.3 Seiberg-Witten curves for GTPs

As reviewed in Section 3, the vanishing locus of the Laurent polynomial
associated to a toric diagram is precisely the SW curve of the 5d SCFT. This
gives a physical meaning to the coefficients of the Laurent polynomial as VEVs
along the Coulomb branch/mass parameters. Our discussion shows that this
conclusion extends to generic GTPs, namely, the SW curve of the 5d theory on
the corresponding GTP coincides with the Laurent polynomial associated to the
GTP, a suggestion first put forward by [44]. The effect of the white dot
decoration of the GTP is to reduce the number of independent coefficients.
This can be made fully precise for GTPs which arise from mutation of a toric
diagram. We will see an explicit example in Section 6.2.

It is natural to ask how to construct the SW curve for a generic GTP with no
reference to mutation. Our general discussion instructs us to consider the GTP
with no decoration and write down its corresponding Laurent polynomial. For a
GTP with $i$ internal points and $b$ boundary points, the number of
independent coefficients is $i+b-3$, $i$ of them corresponding to Coulomb
branch VEVs and $b-3$ of them corresponding to mass deformations. The GTP
decoration freezes some of these coefficients. Through (2.5), we see that
$i-d_{C}$ Coulomb branch VEVs will be frozen. Moreover, if the GTP has $w$
white points out of the external $b$ points, $w$ masses will be frozen.
Alternatively, the number of independent coefficients in the SW for the GTP is
$d_{c}+b-w-3$.

Which specific mass or Coulomb branch parameters become frozen depends on the
locations of the white dots. Recall that that the relation between the complex
coefficients in $P(x,y)$ and the masses in the SW curve involves the circle
compactification of the brane web and the change of coordinates (2.14). For
two points in the toric diagram connected by an edge of length 1, and with
associated Laurent polynomial

$\displaystyle P(x,y)\supset
c_{1}x^{p}y^{q}+c_{2}x^{p^{\prime}}y^{q^{\prime}}\,,$ (4.13)

then the mass resulting from moving the two legs of the web apart from each
other is [28]

$\displaystyle M=\frac{R\,T_{s}}{2\pi}\log\left|\frac{c_{1}}{c_{2}}\right|\,,$
(4.14)

where $T_{s}$ is the string tension in type IIB. We see that if we constraint
the coefficients to be $c_{1}=c_{2}$, the mass is forced to be zero.

Finally, note that in order to write the Laurent polynomial for the GTP before
constraining any coefficient, an origin has to be chosen. Such choice must be
consistent with the external white dot assignation, even though at this point
we do not have a full understanding of the prescription.

##  5 BPS Quivers and the Hilbert series

So far, our discussion has focused on 5d SCFTs and their geometric realization
in M-theory. However, as we reviewed in Section 2, there is an intimate
relation between this and four-dimensional physics: the BPS quiver of the 5d
theory realized as M-theory on a CY3 $\mathcal{M}$ coincides with the 4d
theory of branes probing $\mathcal{M}$ and, in turn, the moduli space of the
4d theory gives back the CY3 $\mathcal{M}$. It is then natural to wonder
whether there is any implication of our work at the level of the BPS quiver.
The answer to this question is in the affirmative. As we discussed in Section
3.3, there is a second invariant under mutations beyond the period, namely the
Hilbert series $\text{Hilb}_{\Delta}(t)$ (3.17). It turns out that the BPS
quiver knows about this quantity: it coincides with the usual Hilbert Series
of its moduli space using the appropriate prescription.

As discussed in Section 3.3, the Hilbert series of the toric variety can be
computed through the Ehrhart series of the dual polytope. Since the Hilbert
series is a generating function for the number of holomorphic functions of a
given degree in the variety, and since those correspond to gauge-invariant
operators (GIOs) in the BPS quiver, it is natural to guess that the partition
function for holomorphic GIOs in the BPS quiver should reproduce the Hilbert
series of the variety.

However, this raises two immediate puzzles. First, the Hilbert series, being
invariant under mutation, must remain the same for two different-looking CY3s;
whereas the GIO partition function would appear to be generally different for
distinct CY3s. And second, given a particular toric diagram, one can construct
a number of different GTPs depending on the assignation of white dots on the
external lines (namely, the choice of external multiplicities in the brane web
or, equivalently, the choice of Higgs branch deformations). Indeed, in this
way of creating a GTP one finds generically inequivalent 5d theories with
different BPS quivers. However, as we have thoroughly discussed, the geometry
underlying all these cases is the same, given by regarding the GTP as a
standard toric diagram and only freezing some moduli. The first puzzle tells
us that one BPS quiver must be associated to several Hilbert series, while the
second one says that multiple BPS quivers should be associated to the same
Hilbert series of the given geometry.

The solution to both issues turns out to be the same. To compute the Hilbert
series of the variety one must assume a particular grading for the holomorphic
functions. Likewise, to compute the partition function for GIOs one must
assume a particular scaling dimension for each field. It turns out that using
the appropriate choice of scaling, the Hilbert series of the variety will
coincide with the partition function for GIOs. Moreover, different choices of
dimensions in the BPS quiver will allow us to select one GTP or the other
within the same undecorated toric diagram.

###  5.1 The $E_{1}$ example

To illustrate our discussion, let us consider the $E_{1}$ example. We have
already discussed most of the important ingredients, as it has been our
prototype example throughout the paper. As we have seen, we can engineer the
$E_{1}$ theory with M-theory on $C(\mathbb{F}_{0})$. Upon mutation, we could
equally choose the variety $C(\mathbb{F}_{2})$. In fact, we have constructed a
fibration of geometries (4.11) such that the general fiber at any $t\neq 0$ is
$\mathbb{F}_{0}$ which, at $t=0$, becomes $\mathbb{F}_{2}$.

Let us now see how this translates into the BPS quiver. The BPS quiver for
$C(\mathbb{F}_{2})$ is shown in Figure 16 [66].

Figure 16: Quiver diagram for $C(\mathbb{F}_{2})$.

The corresponding superpotential is

$\displaystyle W$ $\displaystyle=$ $\displaystyle
X_{12}X_{24}X_{41}-X_{23}X_{34}X_{42}+X_{23}Y_{12}X_{31}-X_{12}Y_{23}X_{31}$
(5.1) $\displaystyle-
X_{13}X_{41}Y_{34}+X_{42}Y_{23}Y_{34}+X_{13}X_{34}Y_{41}-X_{24}Y_{12}Y_{41}\,.$

We can introduce the following superpotential deformation [41]

$\delta W=\mu\,(X_{13}X_{31}-X_{42}X_{24})\,,$ (5.2)

which results in the quiver and superpotential for $C(\mathbb{F}_{0})$, which
were given in Figure 2 and (2.2). Note that, as anticipated, for any $\mu\neq
0$, the quiver becomes that for $C(\mathbb{F}_{0})$, in accordance with the
fact that the generic fiber of the flat fibration smoothing
$C(\mathbb{F}_{2})$ is $C(\mathbb{F}_{0})$.

Let us now turn to the Hilbert series of the BPS quivers. It is
straightforward to see that assuming equal dimensions for all fields in the
quiver of Figure 2, we reproduce the Hilbert series in (3.23), which was
computed as the Ehrhart series of the dual polytope. For convenience, we write
it again

${\rm Hilb}_{C(\mathbb{F}_{0})}=\frac{1+6t+t^{2}}{(1-t)^{3}}\,.$ (5.3)

In particular, this choice of scaling dimensions coincides with the one
arising from $a$-maximization [67] had we thought of the BPS quiver as
describing a 4d SCFT on D3-branes probing $C(\mathbb{F}_{0})$.

In turn, considering the BPS quiver for $C(\mathbb{F}_{2})$ in Figure 16, we
have to use a different choice of scaling dimensions in order for the Hilbert
series to match. If we assume that the fields
$\\{X_{42},X_{24},X_{13},X_{31}\\}$ have dimension 2 and the rest dimension 1
(in the appropriate units), we once again precisely recover (3.23). Note that
with this scaling the superpotential has dimension 4 and the deformation in
(5.2) is (classically) marginal as it has also dimension 4; this is consistent
in the sense that when performing a HW move one is not triggering an RG flow.

It is useful to identify the fields with special scaling dimensions in the
brane tiling. We show these fields with thick bars in Figure 17. We see that
the special fields are those not touching the two parallel zigzags, which
correspond to the parallel branes in the web.

Figure 17: Brane tiling for $\mathbb{C}(\mathbb{F}_{2})$ with zigzags. Fields
with different scaling dimensions are indicated with thick bars.

It is worth stressing that this choice of scaling dimension is not the one
arising from $a$-maximization of the 4d theory describing D3-branes probing
$C(\mathbb{F}_{2})$. In that case, we would have found all dimensions to be
equal, leading to

${\rm
Hilb}^{\prime}_{C(\mathbb{F}_{2})}=\frac{1-t+t^{2}+2t^{3}+t^{4}-t^{5}+t^{6}}{(1-t)^{3}\,(1+t+t^{2}+t^{3})^{2}}\,,$
(5.4)

which is different from the expected result in (5.3). It is interesting to
trace the origin of the disagreement. Since $C(\mathbb{F}_{2})$ can be
regarded as a $\mathbb{Z}_{4}$ orbifold of $\mathbb{C}^{3}$ with action
$(1,1,2)$, (5.4) can be computed by a Molien sum of the $\mathbb{C}^{3}$
Hilbert series. Indeed, one has

${\rm
Hilb}^{\prime}_{C(\mathbb{F}_{2})}=\frac{1}{4}\,\sum_{k=0}^{3}\frac{1}{(1-t\,e^{i\frac{2\pi}{4}k})\,(1-t\,e^{i\frac{2\pi}{4}k})\,(1-t\,e^{i\frac{2\pi}{4}2k})}$
(5.5)

One can check that, instead, (5.3) corresponds to

${\rm
Hilb}_{C(\mathbb{F}_{2})}=\frac{1}{4}\,\sum_{k=0}^{3}\frac{1}{(1-t\,e^{i\frac{2\pi}{4}k})\,(1-t\,e^{i\frac{2\pi}{4}k})\,(1-t^{2}\,e^{i\frac{2\pi}{4}2k})}$
(5.6)

which follows from implementing the same orbifold but on a $\mathbb{C}^{3}$
where one of the coordinates has twice the dimension of the other tow. This is
implemented at the quiver level by the chosen scaling.

##  6 Additional examples

Throughout this paper, we have discussed how mirror symmetry and the
mathematical concept of polytope mutations explain how some of the moduli in
the extended Coulomb branch of a 5d theory become frozen when the
compactification of M-theory is associated to a GTP. We also investigated the
consequences for the corresponding BPS quiver. So far, we have illustrated
these points in the context of the $E_{1}$ theory. In this section, we present
several additional examples.

###  6.1 The $E_{2}$ theory

Let us consider the $E_{2}$ theory, which can be described by the brane web in
Figure 18.

Figure 18: Toric diagram and web for $E_{2}$.

As a CY3, Figure 18 corresponds to $C(dP_{2})$. Upon crossing the brane
labelled by 4 in Figure 18, one obtains Figure 19, which corresponds to
$C(PdP_{2})$.

Figure 19: Equvalent web for $E_{2}$ after mutation.

One can easily check that the corresponding Laurent polynomials are related by
mutation. Indeed, upon SL$(2,\mathbb{Z})$ rotations to appropriately align the
toric diagram, the Laurent polynomial for Figure 18 is

$P_{C(dP_{2})}(x,y)=\frac{1}{y}\,(c_{1}+x)+(c_{2}+x)+y\,\left(c_{3}+\frac{1}{x}\right)\,.$
(6.1)

Using the following change of variables $(x,y)\rightarrow(x,y\,(c_{1}+x))$,
one finds

$P_{C(PdP_{2})}(x,y)=\frac{1}{y}+(c_{2}+x)+y\,\left(c_{1}\,\frac{1}{x}+(c_{3}c_{1}+1)+c_{3}x\right)\,,$
(6.2)

which is the Laurent polynomial associated to Figure 19. In this case, there
are six monomials before and after the mutation and no frozen complex moduli,
we have just reshuffled the coefficients.

Conversely, in terms of the polytope, it is clear that the toric diagrams in
Figures 18 and 19 are related by the inversion of a primitive T-cone, which in
both cases is of size one, thus resulting in trivial multiplicities after
mutation. It is straightforward to compute the Hilbert series associated to
the geometries, which, as expected since they are connected by mutation, are
equal

${\rm Hilb}_{C(dP_{2})}(t)={\rm
Hilb}_{C(PdP_{2})}(t)=\frac{1+5t^{5}+t^{10}}{(1-t^{5})^{3}}\,.$ (6.3)

Let us now turn to quivers. The BPS quiver for Figure 19 is shown in Figure 20
[66].

Figure 20: Quiver diagram for $PdP_{2}$.

The corresponding superpotential is

$\displaystyle W$ $\displaystyle=$ $\displaystyle
X_{45}X_{51}X_{13}X_{34}+Y_{34}X_{42}Y_{23}+X_{23}X_{35}X_{52}+X_{12}X_{24}X_{41}$
(6.4) $\displaystyle-
Y_{23}X_{35}X_{51}X_{12}-X_{41}X_{13}Y_{34}-X_{34}X_{42}X_{23}-X_{45}X_{52}X_{24}\,.$

Following [41], let us introduce the deformation

$\delta W=\mu\,(X_{24}\,X_{42}-X_{51}\,X_{13}X_{35})\,,$ (6.5)

and redefine

$X_{13}\rightarrow\frac{1}{\mu}\,X_{13}-\frac{1}{\mu}\,X_{12}Y_{23}\,,\qquad
X_{35}\rightarrow-\frac{1}{\mu}\,X_{35}+\frac{1}{\mu}\,X_{34}X_{45}\,.$ (6.6)

We obtain the quiver in Figure 21 and the superpotential

$\displaystyle W$ $\displaystyle=$ $\displaystyle
X_{35}X_{51}X_{13}-X_{13}Y_{34}X_{41}-X_{35}X_{52}Y_{23}+Y_{23}X_{34}X_{41}X_{12}$
(6.7)
$\displaystyle+X_{52}X_{23}Y_{34}X_{45}-X_{23}X_{34}X_{45}X_{51}X_{12}\,.$

This indeed corresponds to the BPS quiver of the 5d theory specified by the
geometry in Figure 18.

Figure 21: Quiver diagram for $PdP_{2}$.

Let us now turn to the partition function. Starting with the quiver in Figure
20, and assuming scaling 3 for $X_{42}$, 2 for
$\\{X_{41},X_{13},X_{35},X_{52},X_{24}\\}$ and 1 for the rest –which is not
the assignation which would come from the standard $a$-maximization– one finds
precisely (6.3). On the other hand, starting with the quiver in Figure 21 and
assuming scaling 1 for all fields except $\\{X_{41},X_{13},X_{35},X_{52}\\}$,
which have scaling 2, we again find (6.3). Once again, we see that the
deformation in (6.5), with the scaling assignations, is classically marginal
(and the re-scaling in (6.6) homogeneous).

###  6.2 The $E_{0}$ theory

Let us now consider the more involved example of $E_{0}$, whose web and toric
diagram are shown in Figure 22 (a). As a CY3, it corresponds to
$C(\mathbb{P}^{2})$, also often denoted as $C(dP_{0})$.

Figure 22: (a) Toric diagram and web for the $E_{0}$ theory. (b) GTP and web
for $E_{0}$ after the mutation, where we see that we have a white dot. (c)
Toric diagram and web for SU$(3)_{3}$ SQCD, which corresponds to the other
possible choice of origin of the polytope.

Following our general discussion, we can choose the leg corresponding to the
7-brane in yellow and cross it to the other side, obtaining Figure 22 (b).
This corresponds to a mutation of the polytope where the primitive T-cone in
yellow changes side and, more interestingly, size: in this case the resulting
brane web has non-trivial multiplicities, as shown Figure 22 (b). At the level
of the Laurent polynomial, the starting point is

$P_{\text{(a)}}(x,y)=1+c_{1}x+y+\frac{1}{xy}\,.$ (6.8)

Performing the SL$(2,\mathbb{Z})$ transformation
$(x,y)\rightarrow(\frac{x}{y},\frac{1}{y})$ to align the side to be mutated
with the $x$-axis, and performing the mutation
$(x,y)\rightarrow(x,y\,(1+c_{1}x))$, the result is

$P_{\text{(b)}}(x,y)=\frac{1}{y}+1+\big{(}2c_{1}+\frac{1}{x}+c_{1}^{2}x)\,y^{2}\,.$
(6.9)

We recognize the Laurent polynomial for the polytope in Figure 22 (b), with
three of its moduli frozen to specific values in terms of $c_{1}$ (one of them
equal to zero). Moreover, one can check that indeed the Hilbert series
–computed as the Ehrhart polynomial of the corresponding $\Delta^{\circ}$ for
both Figures 22 (a) and (b) with the given origin– agrees and reads

${\rm Hilb}_{\text{(a)}}(t)={\rm
Hilb}_{\text{(b)}}(t)=\frac{1+7t+t^{2}}{(1-t)^{3}}\,,$ (6.10)

It is also straightforward to check the invariance of the period, i.e.
$\pi_{P_{\text{(a)}}}(t)=\pi_{P_{\text{(b)}}}(t)$.

This case actually corresponds to a GTP. Imagine now changing the origin to
the other internal point as shown in Figure 22 (c). Then, the primitive cones
subtended are of length 1, which means that now the external multiplicities
are all 1 as shown in the web in Figure 22 (c) (which, as a 5d theory,
corresponds to $SU(3)_{3}$ SQCD). Computing the Hilbert series for this case
through the Ehrhart series, one finds

${\rm
Hilb}_{\text{(c)}}(t)=\frac{1+3t+10t^{2}+3t^{3}+t^{4}}{(1-t)^{3}\,(1+t)^{2}}\,.$
(6.11)

Clearly ${\rm Hilb}_{\text{(a)}}(t)\neq{\rm Hilb}_{\text{(c)}}(t)$, in
accordance with the fact that the webs (a) and (c) in Figure 22 are not
related by mutation.

Let us now turn to quivers. The original quiver for the Figure 22 (a) is that
for D3-branes probing $C(dP_{0})$, and is shown in Figure 23 [68, 69].

Figure 23: Quiver diagram for $C(dP_{0})$.

The superpotential for this theory is

$W=\epsilon_{ijk}X_{12}^{i}X_{23}^{j}X_{31}^{k}\,.$ (6.12)

It is straightforward to compute the partition function for this theory
assuming equal scaling for all fields, finding precisely (6.10).

Let us now turn to Figure 22 (b). As thoroughly argued, the corresponding
geometry is the one associated to the GTP regarding it as an ordinary toric
diagram, neglecting all white dot decorations. In this particular case, it is
the toric diagram of $\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ where
the orbifold has weights $(1,-1,0)$ and $(1,1,1)$. Its quiver is shown in
Figure 24 [70].

Figure 24: Quiver diagram for the
$\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ orbifold with weights
$(1,-1,0)$ and $(1,1,1)$.

The superpotential for this theory is

$\displaystyle W$ $\displaystyle=$ $\displaystyle
X_{56}X_{64}X_{45}+X_{63}X_{32}X_{26}+Y_{13}X_{35}Y_{51}+X_{14}Y_{45}X_{51}$
$\displaystyle+X_{21}X_{13}Y_{32}+Y_{26}Y_{64}X_{42}-X_{64}X_{42}X_{26}-Y_{13}X_{32}X_{21}$
$\displaystyle-
Y_{51}X_{14}X_{45}-Y_{64}Y_{45}X_{56}-X_{51}X_{13}X_{35}-Y_{32}Y_{26}X_{63}\,.$

One can see that assigning scaling 4 to
$\\{X_{21},X_{14},X_{42},X_{35},X_{56},X_{63}\\}$ and 1 to the rest of the
fields, the partition function of the quiver is given by (6.10). In this case,
as one side of the mutation involves a true GTP, the deformation relating the
quivers in Figures 23 and 24 is not known. Once again, it is interesting to
identify the fields with special scaling dimensions in the brane tiling [70].
We show them with thick bars in Figure 25. Once again, we see that the special
fields are those that do not intersect the two parallel zigzags, which
correspond to the parallel branes in the web.

Figure 25: Brane tiling for
$\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ with zigzags. Fields with
different scaling dimensions are indicated with thick bars.

According to our general discussion, the quiver in Figure 24 must also
correspond to Figure 22 (c). The difference must lie in the scaling
dimensions. Indeed, one can now check that, upon setting all dimensions equal,
the partition function for the quiver is precisely (6.11).

We can trace the geometric origin og the chosen dimensions. The polytope in
Figure 22 (b) and (c) corresponds to the toric diagram of
$\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ where the orbifold has
weights $(1,-1,0)$ and $(1,1,1)$. Hence, assuming equal scalings for all the
$\mathbb{C}^{3}$ coordinates one finds

${\rm
Hilb}_{\text{(c)}}(t)=\frac{1}{2}\sum_{u^{2}=1}\frac{1}{3}\sum_{v^{3}=1}\frac{1}{(1-t\,uv)\,(1-t\,\frac{v}{u})\,(1-t\,v)}\,.$
(6.14)

In turn,

${\rm
Hilb}_{\text{(b)}}(t)=\frac{1}{2}\sum_{u^{2}=1}\frac{1}{3}\sum_{v^{3}=1}\frac{1}{(1-t\,uv)\,(1-t\,\frac{v}{u})\,(1-t^{4}\,v)}\,.$
(6.15)

###  6.3 The $\tilde{E}_{1}$ theory

Let us now consider the $\tilde{E}_{1}$ theory, which can be encoded in the
toric diagram and web in Figure 26. The toric diagram corresponds to the cone
over $dP_{1}$.

Figure 26: Web for the $\tilde{E}_{1}$ theory.

Computing the Hilbert series of the geometry as the Ehrhart series of the dual
polytope one finds,

${\rm Hilb}_{C(dP_{1})}=\frac{1+6t^{4}+t^{8}}{(1-t^{4})^{3}}\,,$ (6.16)

Figure 27 shows the BPS quiver for $dP_{1}$ [68, 69].

Figure 27: Quiver diagram for $dP_{1}$.

The corresponding superpotential is

$\displaystyle W$ $\displaystyle=$ $\displaystyle
X_{13}Z_{34}Y_{41}-X_{12}X_{23}X_{34}Y_{41}+X_{12}X_{34}X_{41}Y_{23}$
$\displaystyle-Z_{34}X_{42}Y_{23}-X_{13}X_{41}Y_{34}+X_{23}X_{42}Y_{34}\,.$

One can check that, assigning scaling 2 to $\\{X_{42},X_{13}\\}$ and 1 to the
rest of the fields, one recovers (6.16).

##  7 Discussion

We have made progress towards the understanding of the geometrical engineering
of generic 5d SCFTs and its relation to brane webs, thus enlarging the class
of theories for which both descriptions are available. The key observation is
that HW transitions in brane webs correspond to simple coordinate
transformations in the geometry of the mirror, which in turn allows us to make
contact with the mutations introduced in the mathematical literature. This new
connection has very interesting consequences for the Physics of 5d SCFTs (and
may as well have implications for pure Mathematics through new insights from
Physics). Let us briefly summarize our findings.

There exists a mathematical notion of mutation, which can be expressed either
at the level of a polytope or its associated Laurent polynomial. This mutation
transforms the polytope/polynomial according to certain mutation data, which
encodes the choice of HW transition of the corresponding brane web. Regarding
the polytope as the toric diagram of a toric variety, this transformation can
be viewed as a deformation connecting two toric varieties. Moreover, it is
such that the periods and the Hilbert series remain invariant. An important
observation is that the choice of origin is crucial when implementing the
mutation. From a polytope perspective, the mutation amounts to selecting a
primitive T-cone and reversing it. In terms of the Laurent polynomial, the
choice of origin requires tuning a number of coefficients –mirroring the
primitive T-cone– in the polynomial to implement the mutation.

Our observation is that polytope mutations correspond to HW transitions in
webs of 5-branes ending on 7-branes. As usual, the graph dual to a brane web
is a GTP. Then, HW transitions correspond to mutations of the GTP upon
identifying the 7-brane to cross, along with the attached 5-branes, with the
primitive T-cone. In this identification, the length of the base of the cone
is the multiplicity of the leg, while the vector normal to the base of the
cone is the $[p,q]$ charge of the 7-brane to cross. A consequence of this
correspondence is that the GTP should be regarded as a standard toric diagram,
and that the decoration with white dots encodes the choice of origin through
the selection of primitive T-cones. For trivial external multiplicities, it
was well-known that the GTP is the toric diagram of the toric CY3 that
geometrically engineers the 5d SCFT in M-theory. In this paper, we have seen
that this extends to arbitrary GTPs. The geometry that engineers these
theories in M-theory is the one defined by the GTP interpreted as a toric
diagram, i.e. forgetting about its white dot decoration. The effect of white
dots is to freeze some of its moduli.

The correspondence between mutations and HW transitions imply that the period
and the Hilbert series are invariant. We conjecture that their inclusion
results in a complete set of classifiers of brane webs, going beyond the
proposal in [32]. Indeed, we used them to resolve a puzzle raised in [45].
Motivated by these findings, it would be very interesting to perform a
classification attempt similar to the one in [45], but based on the invariance
of the period and the Hilbert series.

We have seen that the Hilbert series can be computed from the BPS quiver,
which is formally identical to the fractional brane quiver for D3-branes
probing the associated toric geometry. Remarkably, this approach provides
further quantitative evidence supporting the conclusion that the geometry
associated with a GTP is equivalent to considering the GTP as a toric diagram.
Notably, the choice of origin, and thus the selection of the white dot
decoration, is encoded in the scaling of the fields when computing the GIO
partition function. This aspect was explicitly demonstrated in Section 6.2,
where the fractional brane quiver for the toric diagram on the right of Figure
22 reproduces, depending on the chosen field scaling, either the Hilbert
series for the toric diagram or that for the GTP in the middle of Figure 22,
corresponding respectively to the $SU(3)_{3}$ 5d SCFT or the $E_{0}$ theory.

As we have seen, white dots encode constraints on the possible resolutions of
the geometry, i.e. Coulomb branch VEVs and masses. This becomes evident when
considering the mutation of the Laurent polynomial. Indeed, starting with a
web with trivial external multiplicities, mutation typically generates a GTP.
However, it is clear that the number of coefficients in the Laurent polynomial
remains fixed under mutation. Consequently, the resulting Laurent polynomial
possesses fewer independent coefficients than a priori allowed.

Since the SW curve is the zero locus of the Laurent polynomial [28], it
follows that the effect of white dots is to freeze some of its deformations,
thus recovering the results in [44] from first principles.

###  7.1 Open questions

Our work suggests various interesting directions for future research. Below,
we summarize some of them.

While our work shed light on the importance of the choice of origin for toric
diagrams, its complete implications remain to be understood. For example, a
toric diagram with $n_{i}$ collinear edges on the $i$-th side can result in
numerous distinct GTPs, each corresponding to different possible
(supersymmetric) white dot decorations. Currently, it is unclear whether all
these possibilities can be solely encoded by the choice of origin, and if so,
how. A related question concerns the status of non-primitive vectors. In the
mathematical literature, position vectors of polytope corners are typically
required to be primitive, whereas Physics suggests the necessity of allowing
non-primitive corners. For instance, the toric diagram dual to the the web for
$SU(N)_{k}$ SQCD has $N-1$ internal collinear points. This implies that for
$N\geq 2$, no choice of origin results in primitive vectors for the corners.
In this case, it is not clear what the different choices of origin mean. Note
that for each choice, the Hilbert series computed through the Ehrhart series
of the dual polytope yields a different result. Additionally, in this case,
the choices cannot correspond to different multiplicities.

Another surprising finding of this paper is that the choice of origin is
encoded in the BPS quiver through the scaling of fields. Although we have
presented evidence for this, understanding the rationale behind the scaling
choice from first principles remains unclear. In other contexts, such as SCFTs
in 4d, 3d, and 2d, the scaling is determined by an extremization principle
arising from an RG flow (e.g., $a$-maximization for 4d SCFTs). While in the
case at hand we are dealing with a BPS quiver, which should be understood as
Matrix Quantum Mechanics, one might also anticipate a dynamical assignment of
dimensions. It would be interesting, albeit challenging, to investigate
whether some prescription can produce multiple GTPs (which arise from white
dot decorations of a single underlying toric diagram) from a unique quiver.

In all examples that we have examined where there is a mutation relating two
standard toric diagrams (i.e., without white dots), the corresponding BPS
quiver theories are connected by a superpotential deformation of the type
studied in [41], which, with our scalings, is classically marginal.121212In
our convention, this means that $\delta W$ has the same scaling as the
original superpotential. It is natural to conjecture that it is actually
marginally relevant, so that it triggers a flow to the IR geometry. This
resonates with Ilten’s geometric perspective on the deformation, which is
described in terms of a flat fibration [61]). For instance, in the $E_{1}$
example, for any non-zero deformation parameter, the generic fiber is
$\mathbb{F}_{0}$, while only at the origin do we find $\mathbb{F}_{2}$. It
would be interesting how to extend this picture to the non-toric
case.131313For instance, in the $E_{0}$ case, it would be natural to consider
a deformation of the form $\delta
W=\mu_{1}\,X_{32}X_{26}X_{64}X_{45}X_{51}X_{13}+\cdots$ of Figure 24, which is
marginal with the scalings in the text. It is unclear how this would lead to
Figure 23.

Additionally, in contrast with the HW transition, the deformation of the BPS
quiver naively seems to be an irreversible process, in harmony with the RG
flow picture.141414Allowing for massive fields to be integrated in, etc, it is
in principle possible to imagine going in both directions. The reversibility
of the process is naturally implemented in the twin perspective as advocated
in [39, 40], where the HW transition corresponds to (formal) Seiberg duality
in a (not necessarily toric) node of the twin quiver. This is also related to
the natural question of the fate of the various (toric) phases of the BPS
quiver, which from the twin point of view has been discussed in [40]. It would
be interesting to flesh out the mapping between all these different
viewpoints.

Finally, it would be interesting to explore the geometric side of the mutation
in further detail, explicitly constructing the flat family of geometries
fibered over a $\mathbb{P}^{1}$. It is natural to ask whether it is possible
to write a metric for the generic element of such interpolating fibration, and
wether extra fluxes are present. In addition, it would be nice to establish
the dictionary between the deformation of the BPS quiver and the geometric
deformation, at least in simple examples.

## Acknowledgements

We would like to thank Antoine Bourget, Tom Coates, Amihay Hanany and Mario De
Marco for useful discussions. G.A.T. is supported by the STFC Consolidated
Grants ST/T000791/1 and ST/X000575/1. S.F. is supported by the U.S. National
Science Foundation grants PHY-2112729 and DMS-1854179. D.R.G is supported in
part by the Spanish national grant MCIU-22-PID2021-123021NB-I00.

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