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# A geometric recipe for twisted superpotentials111Based on lectures given by
the first author at the Pre-StringMath Summer School, 17–21 July 2017, and at
_StringMath 2017_ , 24–28 July 2017, Hamburg University.

Lotte Hollands  Philipp Rüter  Richard J. Szabo

###### Abstract

We give a pedagogical introduction to spectral networks and abelianization, as
well as their relevance to ${\mathcal{N}}=2$ supersymmetric field theories in
four dimensions. Motivated by a conjecture of Nekrasov-Rosly-Shatashvili, we
detail a geometric recipe for computing the effective twisted superpotential
for ${\mathcal{N}}=2$ field theories of class $\mathcal{S}$ as a generating
function of the brane of opers, with respect to the spectral coordinates found
from abelianization. We present two new examples, the simplest Argyres-Douglas
theory and the pure $SU(2)$ gauge theory, while we conjecture the
$\epsilon$-expansion of the effective twisted superpotential for the $E_{6}$
Minahan-Nemeschansky theory.

EMPG–20–02

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Summary and Outline
    3. 1.3 Open Questions
  2. 2 Spectral Networks and Abelianization
    1. 2.1 WKB Spectral Networks
    2. 2.2 Higher Rank Generalization
    3. 2.3 Abelianization
    4. 2.4 Spectral Coordinates
    5. 2.5 Nonabelianization
  3. 3 Theories of Class $\mathcal{S}$ and their BPS States
    1. 3.1 Seiberg-Witten Geometry
    2. 3.2 Class $\mathcal{S}$ Geometry
    3. 3.3 BPS States from Spectral Networks
  4. 4 Twisted Superpotentials, Hitchin Systems and Opers
    1. 4.1 Effective Twisted Superpotentials
    2. 4.2 Hitchin Systems
    3. 4.3 Opers
    4. 4.4 Exact WKB Analysis
  5. 5 The Nekrasov-Rosly-Shatashvili Correspondence
    1. 5.1 The NRS Conjecture
    2. 5.2 A Geometric Recipe
    3. 5.3 AD2 Theory
    4. 5.4 Pure $SU(2)$ Theory
    5. 5.5 $E_{6}$ Minahan-Nemeschansky Theory
    6. 5.6 Derivation from Quantum Field Theory
    7. 5.7 A More General Effective Twisted Superpotential

##  1 Introduction

In these notes we give a pedagogical introduction to four-dimensional
${\mathcal{N}}=2$ supersymmetric field theories of class $\mathcal{S}$ and
certain novel geometric methods that can be used to study them.

###  1.1 Background

The origin of class $\mathcal{S}$ theories lies in six dimensions in the
${\mathcal{N}}=(2,0)$ superconformal field theory with gauge algebra
${\mathfrak{g}}$. This theory is rather mysterious and elusive: it does not
admit a Lagrangian description and it might not even exist as a classical
theory.222It is possible to find the field content and a Lagrangian
description in the specific case of the abelian gauge algebra
${\mathfrak{g}}=\mathfrak{u}(1)$ [CKVP98]. However, it can be studied
indirectly after compactification on lower-dimensional submanifolds where it
gives rise to a host of new structures in both quantum field theory and
geometry. In fact, it has broadened our fundamental understanding of what a
quantum field theory really is.

The first evidence for the six-dimensional $(2,0)$-theory traces back to the
seminal work by Nahm in his classification of the simple Lie superalgebras
which can occur as the symmetry algebra of a superconformal quantum field
theory. The highest dimensions in which a superconformal field theory can
exist was found to be six with the superconformal algebra
$\mathfrak{osp}(2,6|2k)$; the theory with the highest amount of supercharges
in six dimensions is the ${\mathcal{N}}=(2,0)$ theory [Nah78]. At the time
there was no known way to actually construct this theory, but later on
constructions of this six-dimensional superconformal field theory in terms of
string theory were realized independently by Witten [Wit96] and Strominger
[Str96].

One way to construct this theory is in type IIB string theory on an ADE
singularity. This is defined in the ten-dimensional spacetime which is the
direct product of $\mathbb{R}^{5,1}$ with the resolution of an orbifold
singularity. An orbifold singularity is a quotient $\mathbb{C}^{2}/\Gamma$ by
the natural action of a finite subgroup $\Gamma\subset SU(2)$ (with
$\mathbb{C}^{2}$ viewed as the fundamental representation of $SU(2)$). By the
McKay correspondence, $\Gamma$ can be identified with the Dynkin diagram of
type ADE of a simply-laced Lie algebra
${\mathfrak{g}}={\mathfrak{g}}(\Gamma)$. By wrapping a D3-brane on a vanishing
two-cycle of the resolution of $\mathbb{C}^{2}/\Gamma$, the localized dynamics
at the tip $0\in\mathbb{C}^{2}/\Gamma$ describes Nahm’s six-dimensional
superconformal field theory. The resulting theory on $\mathbb{R}^{5,1}$ is
labeled by the corresponding Lie algebra as
${\mathfrak{X}}={\mathfrak{X}}[{\mathfrak{g}}]$.

The theory ${\mathfrak{X}}$ also has a realization in M-theory as the
worldvolume theory of M5-branes. Consider a collection of M5-branes, whose
interactions are mediated by open M2-branes ending on them. The $(2,0)$-theory
is then realized in the limit where the M5-branes coincide; for instance, $K$
flat coincident M5-branes correspond to the $(2,0)$-theory ${\mathfrak{X}}[K]$
at an $A_{K-1}$-singularity (with ${\mathfrak{g}}=\mathfrak{su}(K)$). For
further reading about the six-dimensional $(2,0)$-theory we suggest [Moo12].

The $(2,0)$-theory can be compactified on a (possibly punctured) Riemann
surface $C$. In these notes we will study the theory
${\mathfrak{X}}[{\mathfrak{g}}]$ on the six-dimensional backgrounds

$\displaystyle C\times\mathbb{R}^{3,1}\ .$ (1.1)

After a partial topological twist of the theory, in the limit where $C$
collapses to a point this leads to a four-dimensional ${\mathcal{N}}=2$
supersymmetric field theory on $\mathbb{R}^{3,1}$ which depends only on the
complex structure of the punctured Riemann surface $C$, together with certain
specified singularity data $\mathcal{D}$ at the punctures of $C$. We denote
this theory by ${\sf T}_{{\mathfrak{g}}}[C,\mathcal{D}]$, or simply by $\sf
T$. This is called a theory of class $\mathcal{S}$ [Gai12, GMN13b].333The
$\mathcal{S}$ stands for ‘six’, or alternatively for ‘S-duality’ which acts in
a particularly nice way on the Riemann surface $C$ [Tac15].

Class $\mathcal{S}$ theories can be studied from the perspective offered by
the geometry of the Riemann surface $C$, and this is the perspective we will
take in these notes. One advantage of such a geometric approach is that it is
not only useful for understanding the conventional four-dimensional field
theories, where a Lagrangian description allows the usual methods of quantum
field theory to be employed, but also for more unconventional theories where a
Lagrangian description is not available and other methods fail. The theories
$\sf T$ of class $\mathcal{S}$ naturally encompass both types of field
theories.

###  1.2 Summary and Outline

The goal of these notes is to define and compute the effective twisted
superpotential for any four-dimensional ${\mathcal{N}}=2$ theory $\sf T$ of
class $\mathcal{S}$ in the $\frac{1}{2}\Omega$-background, following a
conjecture by Nekrasov, Rosly and Shatashvili [NRS11].444For readers more
familiar with topological string theory, we mention that this superpotential
equals the refined topological string free energy $\mathcal{F}^{\textrm{\tiny
NS}}$ in the two-dimensional Nekrasov-Shatashvili limit. The two main
ingredients are the “brane of opers”, which is a holomorphic Lagrangian
submanifold in the moduli space of complexified flat connections on $C$, and a
suitable system $\\{x_{i},y^{i}\\}$ of holomorphic Darboux coordinates on this
moduli space. With these choices the effective twisted superpotential
$\widetilde{\mathcal{W}}^{\text{eff}}$ may be extracted as the generating
function

$\displaystyle
y^{i}=\frac{1}{\epsilon}\,\frac{\partial\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)}{\partial
x_{i}}$ (1.2)

of the brane of opers with respect to the Darboux coordinates
$\\{x_{i},y^{i}\\}$, where $\epsilon$ is the deformation parameter of the
$\frac{1}{2}\Omega$-background.

These notes have grown out of a series of lectures by the first author at the
Pre-StringMath Summer School 2017. Paralleling these lectures, we have tried
to keep these notes accessible and self-contained by including extensive
background material in the first few sections. In particular, we define
spectral networks on the Riemann surface $C$ and point out how they encode the
four-dimensional BPS particle spectrum of the theory $\sf T$. We introduce the
central ideas of Seiberg-Witten theory and motivate the importance of
${\mathcal{N}}=2$ theories of class $\mathcal{S}$. We also review the
abelianization method for constructing Darboux coordinate systems on the
moduli space of complexified flat connections on $C$, which we relate to the
exact WKB method as well as TBA-like equations. Along the way we introduce the
Hitchin system and flat oper connections. All of these elements play an
important role in reaching our final goal: a geometric recipe for computing
the effective twisted superpotential $\widetilde{\mathcal{W}}^{\text{eff}}$
for any theory $\sf T$ of class $\mathcal{S}$.

These notes are naturally structured in four parts. Most of our new results
are part of Section 5, and the expert reader may wish to skip over the earlier
sections.

In Section 2 we present some of the geometry, introducing (Fenchel-Nielsen
type) spectral networks, abelianization and (Fenchel-Nielsen type) spectral
coordinates, while emphasizing their generalization to Riemann surfaces $C$
which have irregular punctures. Throughout we illustrate all of the formalism
with many explicit examples (which will be useful in Section 5).

In Section 3 we move on to the physics side of the story. After a brief
introduction to Seiberg-Witten theory, we introduce four-dimensional
${\mathcal{N}}=2$ theories of class $\mathcal{S}$ and, in particular, our main
examples. These are the simplest Argyres-Douglas theory, the pure $SU(2)$
gauge theory, the conformal $SU(2)$ gauge theory coupled to four fundamental
hypermultiplets, and the $E_{6}$ Minahan-Nemeschansky theory. As one
application of the theory of spectral networks, we review how to determine the
BPS particle spectrum of any theory of class $\mathcal{S}$.

In Section 4, we combine the geometry and physics perspectives through the
richness of the Hitchin system. We introduce the
$\frac{1}{2}\Omega$-background, along with the effective twisted
superpotential of an ${\mathcal{N}}=2$ theory of class $\mathcal{S}$ in this
background. We further motivate the brane of opers as a quantization of the
Coulomb branch of the ${\mathcal{N}}=2$ theory, while we relate the
abelianization method, applied to flat oper connections, to the exact WKB
method. This shows that the spectral coordinates from Section 2 have good WKB
asymptotics when evaluated on flat oper connections.

In Section 5, we explain the Nekrasov-Rosly-Shatashvili conjecture which
relates the generating function of opers in Fenchel-Nielsen type coordinates
to the effective twisted superpotential of an ${\mathcal{N}}=2$ theory of
class $\mathcal{S}$ in the $\frac{1}{2}\Omega$-background. We then formulate
our geometric recipe, and test it by computing the effective twisted
superpotential $\widetilde{\mathcal{W}}^{\text{eff}}(x;\epsilon)$ — in a
perturbative expansion in the coupling or ultraviolet scale, which is exact in
$\epsilon$ — in two new examples: the simplest Argyres-Douglas theory and the
pure $SU(2)$ gauge theory; for the treatment of the conformal $SU(2)$ theory
we refer to [HK18].

For the pure $SU(2)$ theory this requires computing the asymptotically small
solutions of the Mathieu equation in terms of the Mathieu functions; we are
able to do this in a perturbation expansion in the ultraviolet scale, which is
exact in $\epsilon$, following the approach of [HK18] for the Heun oper. For
the $E_{6}$ Minahan-Nemeschansky theory, we content ourselves with
conjecturing the $\epsilon$-expansion of the effective twisted superpotential.
We conclude by describing the string theory magic behind the geometric recipe,
and by giving an interpretation to the generating function of opers computed
with respect to other types of spectral coordinates.

In the time elapsed between delivering and writing up these lecture notes,
various other relevant works have appeared. We have tried to incorporate them
into these notes, either in the main text or in Section 1.3 below. Our
presentation in these notes is distinguished by deriving the familiar
effective twisted superpotential from scratch, without using any ingredients
that are not known for instrinsically strongly coupled theories (such as the
Matone relation [Mat95]). In particular, for the weakly coupled pure $SU(2)$
gauge theory we simply obtain the superpotential from the expression

$\displaystyle\widetilde{\mathcal{W}}^{\text{eff}}=\frac{1}{2}\,\int\,\mathrm{d}x\,\log\frac{\textrm{Wr}(\psi_{1}^{\prime},\psi_{1}^{\prime\prime})}{\textrm{Wr}(\psi_{1},\psi_{1}^{\prime\prime})}\,\frac{\textrm{Wr}(\psi_{1},\psi_{2}^{\prime\prime})}{\textrm{Wr}(\psi_{1}^{\prime},\psi_{2}^{\prime\prime})}\
,$ (1.3)

where ${\rm Wr}$ denotes the Wronskian and the $\psi$’s are certain solutions
to the Mathieu equation (5.24); see Section 5.4 for details.

###  1.3 Open Questions

These notes do not by any means close the door on this topic. There are
various open computational as well as conceptual questions to be addressed.
Let us mention a few here:

  * •

Can one extend the computations of the effective twisted superpotential for
the pure $SU(2)$ theory to higher order, or perhaps even to all orders?

  * •

Is it possible to find the exact form of the effective twisted superpotential
for the $E_{6}$ Minahan-Nemeshansky theory? Can one verify its relation to the
corresponding topological string partition function?

  * •

There are many more interesting examples to apply the recipe to, for instance
more general Argyres-Douglas theories. Is there some new physics encoded in
the corresponding effective twisted superpotentials?

  * •

Is it possible to find a proof of the geometric recipe (in one of the examples
or even in general), perhaps taking inspiration from [JN18]?

  * •

Can one develop a better physical framework for studying the relation between
${\mathcal{N}}=2$ boundary conditions and spectral coordinates?

  * •

Any flat connection $\nabla$ can be rewritten as an oper connection, possibly
with adjacent singularities. In these notes we only discuss the generating
function for the brane of opers without adjacent singularities. Can one define
more general generating functions? What is the physical relevance of these?
(This may compute the ${\mathcal{N}}=2$ superpotential of an ${\mathcal{N}}=2$
theory with added surface defects.)

  * •

Can one lift the geometric recipe to three dimensions, to compute the
effective twisted superpotentials of five-dimensional ${\mathcal{N}}=1$
theories in the background
$\mathbb{R}_{\epsilon}^{2}\times\mathbb{R}^{2}\times S^{1}$?

  * •

Could we also extend the recipe to four dimensions, to compute the full
Nekrasov instanton partition function of the ${\mathcal{N}}=2$ theory in the
$\Omega$-background
$\mathbb{R}_{\epsilon_{1}}^{2}\times\mathbb{R}_{\epsilon_{2}}^{2}$ (see
[Tes16a] for some initial results)? And likewise to five dimensions?

  * •

In [CPT18, CLT20] the dual topological string (or free fermion) partition
function in the self-dual $\Omega$-background
$\epsilon_{1}=-\epsilon_{2}=\epsilon$ is characterized as an isomonodromic
tau-function with respect to the same types of spectral coordinates, found
using either abelianization or the exact WKB method. Can we understand this
result from the perspective of the present paper? Aside from the difference
between the self-dual and Nekrasov-Shatashvili limits, the isomonodromic tau-
function is formulated with respect to opers with apparent singularities. A
gauge theoretic understanding of this relation has been recently found in
[Nek20, JN20], see also [Bon+17] for earlier work in this direction.

  * •

Equally intriguing are the non-perturbative approaches to topological string
theory formulated in [GHM16, CGM17, IMS19], which are centred around the
spectral determinant of the five-dimensional mirror curve. This spectral
determinant is compared to the exact WKB method for the four-dimensional pure
$SU(2)$ theory in [GGM20], and for the $SU(2)$ theory with one flavour in
[GHN21], in the $\frac{1}{2}\Omega$-background. What is the role of the
spectral determinant in our set-up?

  * •

Finally, another closely related avenue should be the mathematical programme
outlined in [Bri19, Bri20, BBS20, BM20], which studies a class of Riemann-
Hilbert problems arising in Donaldson-Thomas theory (inspired by [GMN10,
GMN13b]).555Very relevant here as well is the series of papers by Allegretti
(see for instance [All19, All21]) providing a mathematically rigorous
framework for understanding the conformal limit of the spectral coordinates
introduced by Gaiotto, Moore and Neitzke. This theory is worked out in the
example of the hypergeometric differential equation on the three-punctured
sphere in [IK20, IK21], producing the corresponding four-dimensional Nekrasov
partition function in the self-dual limit. On the physics side this programme
has inspired a better understanding of the non-perturbative topological string
partition function [Ali20, Ali21, Ali+21].

##### Acknowledgments.

LH wishes to thank Andrew Neitzke for years of discussions and collaboration
on the topics of these notes, and the organisors of the Pre-StringMath Summer
School and StringMath 2017 in Hamburg for the invitation to deliver these
lectures, and for hospitality during the school and conference. We thank
Andrew Neitzke for his Mathematica program swn-plotter [Nei20]. The work of LH
is supported by a Royal Society Dorothy Hodgkin Fellowship. The work of PR is
supported by a James Watt Scholarship from Heriot-Watt University. The work of
RJS was supported by the Consolidated Grant ST/P000363/1 from the Science and
Technology Facilities Council.

##  2 Spectral Networks and Abelianization

In this section we introduce spectral networks and explain how each spectral
network gives a coordinate system on the moduli space of flat connections.
Spectral networks were first defined in [GMN13a], where it was shown that they
are a fundamental ingredient in the story of BPS states and wall-crossing for
${\mathcal{N}}=2$ gauge theories of class $\mathcal{S}$ [Gai12, GMN13b]. In
[GMN13a] it was also observed that spectral networks can be used to construct
coordinate systems for moduli spaces of flat connections. This method, called
abelianization, was further developed in [HN16]. In this section we follow the
exposition of [HN16], while extending the abelianization method, as well as
the notion of a Fenchel-Nielsen network, to Riemann surfaces with irregular
punctures.666We have used the Mathematica program swn-plotter from Andrew
Neitzke [Nei20] for many of the spectral network plots in these notes; we
encourage the reader to play around with this her/himself.

###  2.1 WKB Spectral Networks

A spectral network is a certain kind of network that can be drawn on a
(possibly punctured) Riemann surface $C$. Suppose that $C$ has $n$ punctures
at positions $z_{i}$ and write
$C=\overline{C}\,\\!\setminus\\!\\{z_{1},\dots,z_{n}\\}$, where $\overline{C}$
is compact. Our initial input is a fixed meromorphic quadratic differential
$p_{2}$ on $C$, which can locally be written as

$p_{2}(z)=u(z)\ \mathrm{d}z\otimes\mathrm{d}z\ .$ (2.1)

The differential $p_{2}$ may be singular at the punctures. A singular point is
regular if $u(z)$ has a pole of order $2$ there; otherwise it is irregular. In
a neighbourhood of a regular puncture $z_{i}$,777 We use the terminology
regular pole, regular singularity, and regular puncture interchangeably to
mean the same thing. the function $u(z)$ may be brought to the form

$u(z)\simeq\frac{m_{i}^{2}}{(z-z_{i})^{2}}\ ,$ (2.2)

where $m_{i}$ are called the mass parameters.888We will discuss irregular
punctures later on and for the moment assume that $p_{2}$ has only regular
singularities at all punctures.

Consider the square root $\lambda=\sqrt{p_{2}}$. This is a one-form on $C$
which is locally given by $\lambda(z)=\sqrt{u(z)}\,\mathrm{d}z$. The one-form
$\lambda$ is _a priori_ multi-valued on $C$, but we will remedy this by
choosing appropriate branch cuts on $C$. Fix a phase
$\vartheta\in\mathbb{R}/2\pi\,\mathbb{Z}$. A $\vartheta$-trajectory is then a
real curve $\gamma$ on $C$ such that

${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,\sqrt{p_{2}}(v)\ \in\
\mathbb{R}^{\times}=\mathbb{R}\setminus\\{0\\}$ (2.3)

for any tangent vector $v$ to $\gamma$. This means that
${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,\sqrt{p_{2}}$ restricts to a real
and non-vanishing one-form on $\gamma$. We can write this condition more
concretely by choosing a parametrization $\gamma(t)$ of the curve and using
the local form of $p_{2}$ as in (2.1). Then (2.3) says

${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,\sqrt{u(\gamma(t))}\,\frac{\mathrm{d}\gamma(t)}{\mathrm{d}t}\
\in\ \mathbb{R}^{\times}\ .$ (2.4)

The set of all $\vartheta$-trajectories form the leaves of a foliation of
$C^{\prime}=C\\!\setminus\\!\\{z|p_{2}(z)=0\\}$. To see this, consider local
coordinates $w(z)$ defined around any point $z_{0}\in C^{\prime}$ by

$w(z)=\int_{z_{0}}^{z}\,\sqrt{u(z^{\prime})}\,\mathrm{d}z^{\prime}\ .$ (2.5)

Then $\sqrt{p_{2}(z)}=\mathrm{d}w$ and the $\vartheta$-trajectories are just
straight line segments of inclination $\vartheta$. With respect to the
coordinates $w$ we thus get a foliation whose leaves are straight lines of
inclination $\vartheta$.999Note also that $\mathrm{d}w$ is non-vanishing on
$C^{\prime}$ and thus defines a nowhere vanishing vector field on
$C^{\prime}$.

At a simple zero of $p_{2}$ the foliation by $\vartheta$-trajectories becomes
singular. In a neighbourhood of the simple zero we can choose a local
coordinate $z$ in which the zero is located at $z=0$; in this coordinate,
locally $p_{2}(z)\simeq z\,\mathrm{d}z\otimes\mathrm{d}z$. Consider a
trajectory starting at $z=0$ as
$\gamma(t)={\mathrm{e}}^{\,{\mathrm{i}}\,\alpha}\,t$ for $t>0$. Then

$\sqrt{p_{2}}(v)=\sqrt{\gamma(t)}\,\frac{\mathrm{d}\gamma}{\mathrm{d}t}={\mathrm{e}}^{\frac{3}{2}\,{\mathrm{i}}\,\alpha}\,t^{1/2}\
.$ (2.6)

The condition for $\gamma$ to be a $\vartheta$-trajectory is then

${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,{\mathrm{e}}^{\frac{3}{2}\,{\mathrm{i}}\,\alpha}\,t^{1/2}\
\in\ \mathbb{R}^{\times}$ (2.7)

which yields

$\alpha=\frac{2\,\vartheta}{3}+k\,\frac{2\pi}{3}\qquad\mbox{for}\quad k=0,1,2\
.$ (2.8)

Thus there are three $\vartheta$-trajectories emitted from every simple zero
of $p_{2}$ as shown in Figure 1.

\begin{overpic}[width=238.49231pt]{TurningPoint.png}
\put(57.0,32.0){$\frac{2}{3}\vartheta$} \end{overpic} Figure 1: Illustration
of the three $\vartheta$-trajectories emitted from a simple zero of $p_{2}$.
The orange cross depicts the position of the zero, whereas the wavy line
represents a choice of branch cut.

On the other hand, around regular poles of $p_{2}$ the behaviour of
$\vartheta$-trajectories depends on the value of the corresponding mass
parameter $m$, as depicted in Figure 2.

\begin{overpic}[width=346.89731pt]{TrajectoriesPuncture.pdf}
\put(6.0,1.0){$m\,{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\in\mathbb{R}$}
\put(26.0,1.0){$\text{Im}(m\,{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta})^{2}>0$}
\put(52.0,1.0){$m\,{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\in{\mathrm{i}}\,\mathbb{R}$}
\put(73.0,1.0){$\text{Im}(m\,{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta})^{2}<0$}
\end{overpic} Figure 2: Behaviour of $\vartheta$-trajectories near a puncture
(depicted as a solid circle) depending on the phase of
$m\,{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}$.

A generic $\vartheta$-trajectory has both endpoints at punctures of $C$. We
call a trajectory critical if one or both of its endpoints are zeroes of
$p_{2}$. We then define the critical graph to be the union of all critical
trajectories together with the zeroes of $p_{2}$. The trajectories may be
oriented as outgoing from the zeroes, and, together with a choice of branch
cuts of $\sqrt{p_{2}}$, labeled by either $12$ or by $21$ if the sign of
$\sqrt{p_{2}}(v)$ is positive or negative, respectively. We call the critical
graph together with orientations a WKB spectral network on $C$ and denote it
by $\mathscr{W}(p_{2},\vartheta)$.

######  Example 1.

Consider $C=\mathbb{P}^{1}\\!\setminus\\!\\{\infty\\}=\mathbb{C}$ with the
quadratic differential

$p_{2}(z)=(z^{2}+m)\,\mathrm{d}z\otimes\mathrm{d}z\ .$ (2.9)

This differential has an irregular singularity at $z=\infty$ of type $L=4$
(i.e. a pole of order six), since $p_{2}$ behaves as
$z^{2}\,\mathrm{d}z\otimes\mathrm{d}z$ near there. For $m>0$ and
$\vartheta=(\pi/2)^{+}$, we get the spectral network $\mathscr{W}$ shown in
Figure 3 A.101010In Figure 3 we have left out the branch cuts at infinity,
which we rectify in Example 7 below.

\begin{overpic}[width=346.89731pt]{AD2jump.png} \put(5.0,14.0){$21$}
\put(21.0,45.0){$21$} \put(3.0,40.0){$12$} \put(6.0,44.0){$12$}
\put(22.0,11.0){$12$} \put(25.0,16.0){$12$} \put(15.5,0.0){$\mathbf{A}$}
\put(42.0,14.0){$21$} \put(56.5,14.0){$12$} \put(47.0,22.0){$12$}
\put(47.0,34.0){$21$} \put(41.0,42.0){$12$} \put(57.0,42.0){$21$}
\put(50.0,0.0){$\mathbf{B}$} \put(72.0,17.0){$21$} \put(76.0,11.0){$21$}
\put(90.0,45.0){$21$} \put(94.0,39.0){$21$} \put(90.0,9.0){$12$}
\put(73.0,41.0){$12$} \put(83.0,0.0){$\mathbf{C}$} \end{overpic} Figure 3:
Spectral networks $\mathscr{W}(p_{2},\vartheta)$ for $m>0$ as $\vartheta$ is
varied.

Increasing the phase $\vartheta$ does not change the isotopy class of the
network $\mathscr{W}$, but decreasing $\vartheta$ to $\vartheta_{\rm c}=\pi/2$
does. At $\vartheta=\vartheta_{\rm c}$ the two trajectories in between the
branch points overlap and form a saddle, as shown in Figure 3 B.

By decreasing $\vartheta$ further to $\vartheta=(\pi/2)^{-}$ we end up with
the network in Figure 3 C. That is, if we vary $\vartheta$ from $(\pi/2)^{+}$
to $(\pi/2)^{-}$ the isotopy class of $\mathscr{W}$ changes, with a saddle at
the critical value $\vartheta_{\rm c}=\pi/2$. We call such a topology change a
flip. We will explain in Section 3.3 why this physically describes the
appearence of a BPS hypermultiplet of mass $m$.

######  Example 2.

Let $C$ be any punctured Riemann surface, and take $p_{2}$ to be a generic
differential with regular singularities at the punctures. For generic values
of $\vartheta$ every trajectory ends on a puncture and the foliation defined
by $p_{2}$ looks locally as in Figure 4.

\begin{overpic}[width=260.17464pt]{FoliationCell.png} \end{overpic} Figure 4:
Topology of a cell of the foliation. The cell is bounded by critical
trajectories running from a branch point to a puncture and it includes an
infinite family of generic trajectories that run between the punctures.

We call the resulting spectral network a Fock-Goncharov network (in the
terminology of [HN16]). For a Fock-Goncharov network we can use the cells of
the foliation to define an ideal triangulation
$\mathscr{T}=\mathscr{T}(p_{2},\vartheta)$ of $C$. The vertices of
$\mathscr{T}$ are the punctures of $C$ and the edges are obtained by choosing
one generic trajectory in each cell [HN16, GMN13b]. An example on the four-
punctured sphere is shown in Figure 5.

\begin{overpic}[width=260.17464pt]{Nf4SNEdges.png} \end{overpic} Figure 5: The
four-punctured sphere (viewed as the plane with the point at infinity omitted)
with a Fock-Goncharov network $\mathscr{W}$. The edges of an ideal
triangulation $\mathscr{T}$ are indicated in blue.

######  Example 3.

Consider $C=\mathbb{C}^{\times}=\mathbb{C}\setminus\\{0\\}$ with the quadratic
differential

$\displaystyle
p_{2}(z)=\Big{(}\frac{1}{z^{3}}+\frac{9}{4\,z^{2}}+\frac{1}{z}\Big{)}\,\mathrm{d}z\otimes\mathrm{d}z\
.$ (2.10)

This differential has irregular singularities at $z=0$ and $z=\infty$, since
the behaviour of $p_{2}$ is given by
$\frac{1}{\tilde{z}^{3}}\,\mathrm{d}\tilde{z}\otimes\mathrm{d}\tilde{z}$ near
there (in the local coordinate $\tilde{z}=z$ and $\tilde{z}=\frac{1}{z}$,
respectively). Near $\vartheta=\vartheta_{\rm c}=\pi/2$, something interesting
happens: Close to this critical phase, some trajectories start to wind around
the cylinder $C=\mathbb{C}^{\times}$, see Figure 6. If one studies the winding
for $\vartheta<\vartheta_{\rm c}$ (and the unwinding for
$\vartheta>\vartheta_{\rm c}$), one finds that it is really an infinite
sequence of flips; this is explained in detail in [GMN13b, Section 5.9]. At
$\vartheta=\vartheta_{\rm c}$ the network changes topology. At this phase a
family of closed trajectories emerges that surround the cylinder. This is
illustrated in Figure 6, and is called a juggle (see also [GMN13b, Section
6.6.3]).

\begin{overpic}[width=346.89731pt]{pureN=2Wjuggle.png} \end{overpic} Figure 6:
Spectral networks $\mathscr{W}(p_{2},\vartheta)$, rotated through
$-90^{\circ}$, for the quadratic differential (2.10) and phases
$\vartheta_{\rm c}-\delta$, $\vartheta_{\rm c}$, $\vartheta_{\rm c}+\delta$,
respectively, with $\vartheta_{\rm c}=\frac{\pi}{2}$ and $\delta$ small. The
transition of the spectral network from $\vartheta_{\rm c}-\delta$ to
$\vartheta_{\rm c}+\delta$ is called a juggle.

As we will see in Section 3.3, a juggle corresponds to a BPS vector multiplet
in the corresponding supersymmetric field theory. The two limits
$\vartheta\to\vartheta_{\rm c}^{\pm}$ are called the resolutions of the
spectral network at the critical phase; they are illustrated schematically in
Figure 7.

\begin{overpic}[width=346.89731pt]{pureN=2resolutions.png}
\put(13.0,0.0){$\mathscr{W}^{-}$} \put(48.5,0.0){$\mathscr{W}_{\rm c}$}
\put(83.0,0.0){$\mathscr{W}^{+}$} \put(16.0,43.0){$21$} \put(51.0,43.0){$21$}
\put(85.5,43.0){$21$} \put(16.0,20.0){$21$} \put(51.0,20.0){$21$}
\put(85.5,20.0){$21$} \put(20.0,37.0){$12$} \put(53.0,37.0){$12$}
\put(89.0,37.0){$21$} \put(17.0,33.0){$12$} \put(86.5,33.0){$21$}
\put(9.0,37.0){$12$} \put(44.0,37.0){$21$} \put(78.0,37.0){$21$}
\put(12.0,33.0){$21$} \put(81.0,33.0){$21$} \put(17.0,27.5){$12$}
\put(52.0,27.5){$12$} \put(86.5,27.5){$21$} \put(17.0,23.0){$21$}
\put(86.5,23.0){$12$} \put(10.0,27.5){$21$} \put(45.0,27.5){$21$}
\put(79.0,27.5){$12$} \put(10.0,23.0){$21$} \put(79.0,23.0){$12$}
\end{overpic} Figure 7: In the middle: The spectral network $\mathscr{W}_{\rm
c}=\mathscr{W}(p_{2},\vartheta_{\rm c})$ for the quadratic differential (2.10)
and critical phase $\vartheta_{\rm c}=\frac{\pi}{2}$. On either side: Its
resolutions $\mathscr{W}^{\pm}=\mathscr{W}(p_{2},\vartheta_{\rm c}^{\pm})$.

######  Example 4.

If $C$ is any (possibly punctured) Riemann surface with regular punctures, we
may consider a special class of quadratic differentials
${\mathrm{e}}^{-2\,{\mathrm{i}}\,\vartheta}\,p_{2}$ for which all trajectories
are compact and the critical graph is built out of saddles. These
differentials are known as Strebel differentials. In particular, given any
pants decomposition of $C$ along with a choice of “length” parameters $l_{k}$,
there is a unique Strebel differential
${\mathrm{e}}^{-2\,{\mathrm{i}}\,\vartheta}\,p_{2}$ respecting this pants
decomposition and satisfying

$\displaystyle{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,\oint_{\alpha_{k}}\,\sqrt{p_{2}}=l_{k}$
(2.11)

for all pants curves $\alpha_{k}$. The corresponding spectral network is dual
to the pants decomposition of $C$. We call such a network a Fenchel-Nielsen
network (for more details see [HN16]). See Figures 8 and 9 for examples of
Fenchel-Nielson networks on the three-punctured and four-punctured sphere,
respectively.

\begin{overpic}[width=173.44534pt]{MoleculeOne.png} \end{overpic} Figure 8:
One of two Fenchel-Nielsen network topologies on the three-punctured sphere
with the third puncture being a regular puncture at infinity. This is
equivalent to a pair of pants, if the punctures are viewed as boundaries
instead. \begin{overpic}[width=260.17464pt]{Molecules.png} \end{overpic}
Figure 9: A Fenchel-Nielsen network on the four-punctured sphere glued
together from two pairs of pants.

If the Riemann surface $C$ has irregular singularities, the Strebel condition
is too strong (there are always non-compact trajectories in this case). Yet we
would like to call a spectral network $\mathscr{W}(p_{2},\vartheta)$ of
Fenchel-Nielsen type if there is a maximal number of non-degenerate ring
domains (see also [CLT20]). This is the case, for instance, at the critical
phase $\vartheta_{\rm c}=\frac{\pi}{2}$ in Example 3. We can thus consider the
middle network in Figure 6 as another example of a Fenchel-Nielsen network.

###  2.2 Higher Rank Generalization

So far we considered spectral networks defined by a single quadratic
differential $p_{2}$. This may be generalized to a higher rank version of a
WKB spectral network by considering a tuple of differentials
$p=(p_{2},\dots,p_{K})$ on $C$, where each $p_{k}$ is a meromorphic
$k$-differential on $C$, possibly singular at the punctures $z_{i}$. In this
case we call a singularity regular if the order of the pole of each $p_{k}$ is
at most $k$, and irregular otherwise. As we will see later on, regular
punctures may be classified by Young diagrams with $K$ boxes and at most $K-1$
rows.

The tuple of differentials $(p_{2},\dots,p_{K})$ defines a (possibly branched)
$K$-fold covering $\Sigma\subset T^{*}C$ of $C$, by the equation111111In
(2.12) we have changed the sign in front of the coefficients $p_{k}$ to match
our conventions from Section 2.1. This agrees with the conventions of
[GMN13b], but unfortunately not with those of [GMN13a]. In particular, to
verify the networks below, one needs to introduce an additional minus sign in
swn-plotter [Nei20].

$\Sigma\colon\quad\lambda^{K}-\sum_{k=2}^{K}\,p_{k}\,\lambda^{K-k}=0\ ,$
(2.12)

where $\lambda=w\,\mathrm{d}z$ is the tautological one-form on the cotangent
bundle $T^{*}C$. The curve $\Sigma$ is called the spectral curve, and we will
encounter it again in Section 3 when we explain how spectral networks are
relevant to ${\mathcal{N}}=2$ field theories. In the following we assume that
the covering $\Sigma\to C$ has only simple branch points, where exactly two
sheets come together; this is the generalization of the requirement that
$p_{2}$ has only simple zeroes for $K=2$.

Let $\lambda_{i}$ be the restriction of the tautological one-form $\lambda$ to
the $i$-th sheet. We define an $ij$-trajectory on $C$ for $i\neq j$ to be a
real curve $\gamma$ on $C$ such that

${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,(\lambda_{i}-\lambda_{j})(v)\ \in\
\mathbb{R}^{\times}$ (2.13)

for any non-zero tangent vector $v$ to $\gamma$. We call such a trajectory
critical if at least one of its endpoints is a branch point. The critical
graph is again defined as the union of all critical trajectories, and we add
an orientation and a label $ij$ to each $ij$-trajectory to get a spectral
network $\mathscr{W}(p,\vartheta)$ on $C$. Sometimes we will call
$\mathscr{W}(p,\vartheta)$ a spectral network subordinate to the cover
$\Sigma\to C$.

There is a new phenomenon for critical trajectories in higher rank networks
for which $K>2$: trajectories with different labels (say $ij$ and $jk$) may
cross paths and form a junction. For instance, the crossing of a $12$ and a
$23$ trajectory will lead to a new trajectory of type $13$, see Figure 10. In
this way new trajectories can be “born”. The possibility of such crossings
vastly increases the complexity of the higher rank networks. In fact, they are
largely unexplored apart from the cases of higher rank generalizations of
Fock-Goncharov and Fenchel-Nielsen networks [GMN13a, HN17, HN16].

\begin{overpic}[width=346.89731pt]{Intersectionijk.png} \put(39.0,20.0){$ij$}
\put(39.0,4.0){$jk$} \put(57.0,20.0){$jk$} \put(57.0,4.0){$ij$}
\put(60.0,14.0){$ik$} \end{overpic} Figure 10: The crossing of walls with
labels $ij$ and $jk$ results in a new wall with label $ik$ which is “born” at
the intersection.

######  Example 5.

Let $K=3$ and
$C=\mathbb{P}^{1}_{0,1,\infty}=\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ be the
three-punctured sphere with the differentials

$\displaystyle p_{2}$
$\displaystyle=\frac{c_{\infty}\,z^{2}-(c_{0}-c_{1}+c_{\infty})\,z+c_{0}}{z^{2}\,(z-1)^{2}}\,(\mathrm{d}z)^{\otimes
2}\ ,$ $\displaystyle p_{3}$
$\displaystyle=\frac{d_{\infty}\,z^{3}+u\,z^{2}+(d_{0}+d_{1}-d_{\infty}-u)\,z-d_{0}}{z^{3}\,(z-1)^{3}}\,(\mathrm{d}z)^{\otimes
3}\ ,$ (2.14)

where $u\in\mathbb{C}$ is a free parameter, while

$\displaystyle
c_{l}=-m_{l,1}^{2}-m_{l,1}\,m_{l,2}-m_{l,2}^{2}\qquad\mbox{and}\qquad
d_{l}=m_{l,1}\,m_{l,2}\,(m_{l,1}+m_{l,2})\ ,$ (2.15)

and $m_{l,1}\neq m_{l,2}$. The residues of these differentials at the regular
punctures $z_{l}=l$ are given by $\\{m_{l,1},m_{l,2},-m_{l,1}-m_{l,2}\\}$,
respectively, and the spectral curve $\Sigma$ is a three-fold branched
covering of $C$ with six simple branch points. This implies that $\Sigma$ is a
punctured Riemann surface of genus one. In the limit $m_{l,j}\to 0$, the
branch points of the covering $\Sigma\to C$ move towards the punctures and the
topology of the corresponding spectral networks $\mathscr{W}(u,\vartheta)$
depends only on the phase of the quantity
${\mathrm{e}}^{-3\,{\mathrm{i}}\,\vartheta}\,u$. For generic phase the network
seems to be “wild”, that is, it is dense in at least some parts of $C$. For
non-generic phase the network is compact, with

$\displaystyle{\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,\oint_{\gamma}\,\lambda\
\in\ \mathbb{R}\ ,$ (2.16)

for some one-cycle $\gamma$ on $\Sigma$.

These non-generic compact networks were studied in [HN17]. They are labelled
by two coprime integers $p$ and $q$. The three simplest topologies
$\mathscr{W}_{[1,0]}$, $\mathscr{W}_{[1,2]}$ and $\mathscr{W}_{[1,3]}$ are
illustrated in Figure 11.

\begin{overpic}[width=433.62pt]{E6Networks.png} \put(14.0,-4.0){$\mathbf{A}$}
\put(50.0,-4.0){$\mathbf{B}$} \put(83.0,-4.0){$\mathbf{C}$} \end{overpic}
Figure 11: Three examples of higher rank Fenchel-Nielsen networks on the
three-punctured sphere (taken from [HN17]): A $\mathscr{W}_{[1,0]}$; B
$\mathscr{W}_{[1,2]}$; C $\mathscr{W}_{[1,3]}$. Here we have moved the three
punctures to $z=1,\omega,\omega^{2}$ with
$\omega={\mathrm{e}}^{\,2\pi\,{\mathrm{i}}/3}$.

These networks are rather degenerate: If we were to perturb $\vartheta$
slightly away from the critical value, we would see that there is infinite
winding around the saddles. For this reason we call these networks of higher
rank Fenchel-Nielsen type.

As we will learn in Section 3.3, each network $\mathscr{W}_{[q,p]}$ encodes
all BPS states of electromagnetic charge $(q,p)$ in the $E_{6}$ Minahan-
Nemeschansky theory.

###  2.3 Abelianization

In the following we fix a branched $K$-fold cover $\pi\colon\Sigma\rightarrow
C$ as in (2.12), and a spectral network $\mathscr{W}$ on $C$ subordinate to
the covering. Let $\Sigma^{\prime}$ denote $\Sigma$ with the branch points
removed. The general idea of abelianization is to lift non-abelian structures
on a given rank $K$ vector bundle $E$ over $C$ to construct corresponding
abelian structures on a line bundle $\mathcal{L}$ over the spectral cover
$\Sigma^{\prime}$. In particular, starting from a flat $SL(K,\mathbb{C})$
connection $\nabla$ on $C$, we construct a flat $GL(1,\mathbb{C})$ connection
$\nabla^{\text{ab}}$ on $\Sigma^{\prime}$, using the data of the spectral
network $\mathscr{W}$. This is illustrated in Figure 12.

\begin{overpic}[width=216.81pt]{Abelianization.png}
\put(14.0,48.0){$(\Sigma,\nabla^{\text{ab}})$}
\put(12.0,3.0){$(C,\mathscr{W},\nabla)$} \put(65.0,62.0){$\mathcal{L}$}
\put(65.0,48.0){$\mathcal{L}$} \put(65.0,10.0){$E$}
\put(8.0,24.0){abelianization} \end{overpic} Figure 12: Given a spectral
network $\mathscr{W}$ and a generic flat $SL(K,\mathbb{C})$ connection
$\nabla$ on the vector bundle $E$ over $C$, abelianization is a way of
bringing $\nabla$ into an almost diagonal form, such that it may be lifted to
a flat $GL(1,\mathbb{C})$ connection $\nabla^{\text{ab}}$ on the line bundle
$\mathcal{L}$ over the spectral cover $\Sigma^{\prime}$.

We first consider $\nabla$ on the connected components, or cells, of
$C\\!\setminus\\!\mathscr{W}$. The cells are either contractible or tubular,
and the construction works for either case [HN16]. On these we can find a
gauge which diagonalizes $\nabla$. We then glue this patchwise diagonalization
of $\nabla$ together by assigning unipotent gauge transformations to the
trajectories of $\mathscr{W}$.

Concretely, in each cell we look for a basis of sections $(s_{1},\dots,s_{K})$
of $E$ with respect to which $\nabla$ is diagonal,

$\nabla s_{i}=d_{i}\otimes s_{i}\ ,$ (2.17)

where $d_{i}$ are closed one-forms on $C$ for $i=1,\dots,K$. On crossing a
trajectory with label $ij$ we require a unipotent transformation

$\displaystyle s_{i}\longmapsto
s_{i}+c_{ij}\,s_{j}=:s_{i}^{\prime}\qquad\mbox{and}\qquad s_{k}\longmapsto
s_{k}\quad\text{for}\quad k\neq i\ ,$ (2.18)

for some function $c_{ij}$. On crossing a branch cut of type $(ij)$ we require
that the sections on either side are related by a permutation matrix as

$\displaystyle s_{i}\longmapsto s_{j}\ ,\quad s_{j}\longmapsto-
s_{i}\qquad\mbox{and}\qquad s_{k}\longmapsto s_{k}\quad\text{for}\quad k\neq
i,j\ .$ (2.19)

If we can find such a gauge, we may lift $\nabla$ on $C$ to a
$GL(1,\mathbb{C})$ connection $\nabla^{\text{ab}}$ on the spectral cover
$\Sigma^{\prime}$ as follows. On $\Sigma\\!\setminus\\!\pi^{-1}(\mathscr{W})$
we define $\nabla^{\text{ab}}$ on the $i$-th sheet by the diagonal entries
$d_{i}$ of $\nabla$:

$\nabla^{\text{ab}}(s_{i})=d_{i}\otimes s_{i}\ .$ (2.20)

To show that the unipotent gauge transformations (2.18) extend
$\nabla^{\text{ab}}$ across the trajectories to all of $\Sigma$, we need to
show that $\nabla s_{i}^{\prime}=d_{i}\otimes s_{i}^{\prime}$ for all
$i=1,\dots,K$.121212The connection $\nabla^{\text{ab}}$ can be extended to a
connection on all of $\Sigma$ which is almost flat: its holonomy around a
simple branch point is $-\mathbb{1}$. Indeed, since $\nabla$ is an
$SL(K,\mathbb{C})$ connection, on crossing a trajectory with label $ij$,
$\nabla s_{i}=d_{i}\otimes s_{i}$ is sent to

$\displaystyle\nabla s_{i}^{\prime}=d_{i}^{\prime}\otimes
s_{i}^{\prime}=d_{i}^{\prime}\otimes s_{i}+d_{i}^{\prime}\otimes
c_{ij}\,s_{j}$ (2.21)

with

$\displaystyle\nabla s_{i}^{\prime}=\nabla(s_{i}+c_{ij}\,s_{j})=\nabla
s_{i}+\mathrm{d}c_{ij}\otimes s_{j}+c_{ij}\,\nabla s_{j}=d_{i}\otimes
s_{i}+(\mathrm{d}c_{ij}+c_{ij}\,d_{j})\otimes s_{j}\ ,$ (2.22)

showing that $d_{i}^{\prime}=d_{i}$ as desired (and
$\mathrm{d}c_{ij}=c_{ij}\,(d_{i}-d_{j})$, so the functions $c_{ij}$ are
bicovariantly constant). Note that, since $\nabla$ is an $SL(K,\mathbb{C})$
connection, the connection $\nabla^{\text{ab}}$ carries some extra structure:
parallel transport of the sections $(s_{1},\dots,s_{K})$ along a path in
$C\setminus\mathscr{W}$ (not crossing any branch cuts) is given by a diagonal
matrix with determinant equal to $1$. We say that $\nabla^{\text{ab}}$ is
equivariant, see [HN16, Section 4.2] and [HK18, Section 5.3].

To help us find such a gauge, and to show it is unique, we may need some
additional discrete choices on $\nabla$; this is called a framing of $\nabla$.

######  Example 6.

Fix a Fock-Goncharov network $\mathscr{W}$ and consider a flat
$SL(2,\mathbb{C})$ connection $\nabla$ on $C$. Locally $C\setminus\mathscr{W}$
contains cells of the form shown in Figure 13 A.

\begin{overpic}[width=346.89731pt]{AbelianizationCells2.png}
\put(14.5,39.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{i},s_{j})$}}
\put(16.0,60.0){$z_{i}$} \put(16.0,8.0){$z_{j}$} \put(50.0,55.0){$z_{i}$}
\put(75.0,12.5){$z_{j}$} \put(95.0,55.0){$z_{k}$}
\put(63.0,27.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{i},s_{j})$}}
\put(80.5,27.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{k},s_{j})$}}
\put(6.0,47.5){$21$} \put(25.0,47.5){$21$} \put(6.0,21.0){$12$}
\put(25.0,21.0){$12$} \put(49.0,41.0){$21$} \put(95.0,41.0){$21$}
\put(65.0,49.0){$21$} \put(82.5,49.0){$21$} \put(61.0,19.0){$12$}
\put(88.0,19.0){$12$} \put(72.0,31.0){$12$} \put(16.0,3.0){{A}}
\put(75.0,3.0){{B}} \end{overpic} Figure 13: A: A single cell of a Fock-
Goncharov network, with basis of sections $(s_{i},s_{j})$. B: Adjacent cells
with bases related by transport over a trajectory.

For any puncture $z_{i}$, we defined the framing of $\nabla$ at $z_{i}$ to be
a choice of a $\nabla$-invariant line sub-bundle $\ell_{i}$ of $E$ in a
neighbourhood of $z_{i}$ with the following condition. If, for each cell with
two punctures $z_{i}$ and $z_{j}$, we parallel transport the respective line
bundles $\ell_{i}$ and $\ell_{j}$ to a common point $z$ in the cell, we
require that $\ell_{i}(z)\neq\ell_{j}(z)$. We call the connection $\nabla$
together with this framing data a $\mathscr{W}$-framed connection.

Note that the condition above is automatically satisfied for a generic
connection $\nabla$. Moreover, for generic $\nabla$ there are exactly $2^{n}$
possible $\mathscr{W}$-framings, where $n$ is the number of punctures. Indeed
the monodromy around each puncture has two distinct eigenspaces, and a
$\mathscr{W}$-framing corresponds to choosing just one of them.

How do we abelianize the $\mathscr{W}$-framed connection $\nabla$? Let us
locally trivialize the cover $\Sigma$ over a cell with two punctures $z_{i}$
and $z_{j}$. Suppose that puncture $z_{i}$ has incoming trajectories of type
$21$, whereas puncture $z_{j}$ has incoming trajectories of type $12$. The
basis $(s_{i},s_{j})$ is then obtained by choosing
$(s_{i}\in\ell_{i},s_{j}\in\ell_{j})$ for each cell.

The local picture for two neighbouring cells is shown in Figure 13 B. Each of
the cells has basis $(s_{i},s_{j})$ defined as above, and crossing the
trajectory between them gives a unipotent transformation that leaves one of
the sections unchanged and modifies the other one. A choice of local bases of
sections like this thus defines a $\mathscr{W}$-abelianization of $\nabla$.
This can be shown to give a canonical one-to-one correspondence between
$\mathscr{W}$-framings of $\nabla$ and $\mathscr{W}$-abelianizations of
$\nabla$ [HN16, Section 5.2].

######  Example 7.

Let us revisit Example 1 with $C=\mathbb{C}$ and the spectral network
$\mathscr{W}=\mathscr{W}(p_{2},\vartheta_{\rm c})$ with

$\displaystyle
p_{2}(z)=(z^{2}+m)\,\mathrm{d}z\otimes\mathrm{d}z\qquad\mbox{and}\qquad\vartheta_{\rm
c}=\frac{\pi}{2}\ .$ (2.23)

Since $p_{2}$ has an irregular singularity at $z=\infty$ (a pole of order
six), we consider flat $SL(2,\mathbb{C})$ connections $\nabla$ with a
corresponding irregular singularity at $z=\infty$. Any such connection
$\nabla$ experiences the Stokes phenomenon at $z=\infty$. That is, it is
impossible to find a single, well-defined section $s$ that is asymptotically
small as $z\to\infty$. Instead, for this particular singularity, we require
four sections $\tilde{s}_{1}$, $\tilde{s}_{2}$, $\tilde{s}_{3}$ and
$\tilde{s}_{4}$ that each become asymptotically small as $z\to\infty$ along
what is called a Stokes ray. (In Section 4.4 we will write down explicit
expressions for the sections $\tilde{s}_{i}$ when $\nabla$ is an oper
connection.)

To keep track of the angular information, we consider the blow-up of the
singularity at infinity and replace the puncture at $z=\infty$ with a small
circle $S^{1}$ bounding an infinitesimal disc $D_{\infty}$. We then mark four
points $\tilde{z}_{i}$ on this circle corresponding to the Stokes rays, and to
these points we assign the sections $\tilde{s}_{i}$ that become asymptotically
small along the corresponding rays. Each section $\tilde{s}_{i}$ has good
asymptotics in the angular regions adjacent to the Stokes ray labeled by
$\tilde{s}_{i}$. On the overlap of the regions where two sections are well-
defined, they are related by a Stokes matrix [Boa14].

To frame the connection $\nabla$ at infinity, we similarly replace the
puncture at $z=\infty$ by a small circle $S^{1}$ with four marked points
$z_{i}$ corresponding to the four incoming trajectories. The framing of
$\nabla$ then corresponds to a choice of $\nabla$-invariant sub-bundle
$\ell_{i}$ for every marked point $z_{i}$, again with the constraint that if
two marked points $z_{i}$ and $z_{j}$ are connected by a path which does not
cross any trajectories, and we parallel transport $\ell_{i}$ and $\ell_{j}$ to
a common point $z$ on this path, then $\ell_{i}(z)\neq\ell_{j}(z)$. More
precisely, consider the network $\mathscr{W}(p_{2},\vartheta)$ for either
$\vartheta=(\pi/2)^{+}$ or $\vartheta=(\pi/2)^{-}$; these are sometimes called
the two resolutions of the (Fenchel-Nielsen type) network $\mathscr{W}$, and
we denote them by $\mathscr{W}^{\pm}$ respectively.131313It is not possible to
abelianize precisely at the critical phase $\vartheta_{\rm c}=\pi/2$, we have
to make a choice of resolution. Then the choice of framing
$\ell_{1},\ell_{2},\ell_{3},\ell_{4}$ determines a unique
$\mathscr{W}^{\pm}$-abelianization of $\nabla$, whose basis of sections in
each cell is shown in Figure 14, with $s_{i}\in\ell_{i}$. Note that the two
bases on either side of a trajectory are indeed related by a unipotent
transformation of the type (2.18). On either side of a branch cut they are
related by a permutation matrix as in (2.19). (To trivialize the covering
$\Sigma\to C$, we choose not only the branch cuts connecting the two branch
points, but also an additional two branch cuts at $z=\infty$.) The bases of
sections for each network $\mathscr{W}^{\pm}$, as illustrated in Figure 14,
brings the connection $\nabla$ in an almost-diagonal gauge, so that it may be
lifted to a $GL(1,\mathbb{C})$ connection $\nabla^{\text{ab}}_{\pm}$ on the
spectral cover $\Sigma$ by identifying the two elements $(s_{i},s_{j})$ of the
basis in each cell with the sheets of $\Sigma$.

\begin{overpic}[width=346.89731pt]{AD2sections.png} \put(45.0,2.0){$s_{4}$}
\put(1.0,2.0){$s_{3}$} \put(1.0,67.0){$s_{2}$} \put(45.0,67.0){$s_{1}$}
\put(97.0,2.0){$s_{4}$} \put(53.0,2.0){$s_{3}$} \put(53.0,67.0){$s_{2}$}
\put(97.0,67.0){$s_{1}$} \put(21.0,68.0){$\mathscr{W}^{-}$}
\put(75.0,68.0){$\mathscr{W}^{+}$}
\put(21.0,7.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{3},s_{4})$}}
\put(11.5,45.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{2},s_{4})$}}
\put(5.0,35.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{3},s_{2})$}}
\put(34.0,35.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{1},s_{4})$}}
\put(21.0,56.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{1},s_{2})$}}
\put(71.0,7.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{3},s_{4})$}}
\put(63.0,24.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{3},s_{1})$}}
\put(87.0,35.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{1},s_{4})$}}
\put(57.0,35.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{3},s_{2})$}}
\put(71.0,56.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{1},s_{2})$}}
\end{overpic} Figure 14: Bases of sections $(s_{i},s_{j})$ in the cells of
$C\setminus\mathscr{W}^{\pm}$ defining the abelianization of $\nabla$ for the
two networks $\mathscr{W}^{\pm}$.

######  Example 8.

Let us now go back to Example 3, where $C=\mathbb{C}^{\times}$ and the
spectral network $\mathscr{W}(p_{2},\vartheta_{\rm c})$ is defined by (see
Figures 6 and 7)

$\displaystyle
p_{2}(z)=\Big{(}\frac{1}{z^{3}}+\frac{9}{4\,z^{2}}+\frac{1}{z}\Big{)}\,\mathrm{d}z\otimes\mathrm{d}z\qquad\mbox{and}\qquad\vartheta_{\rm
c}=\frac{\pi}{2}\ .$ (2.24)

Since $p_{2}$ has irregular singularities at $z=0$ and $z=\infty$, we consider
flat $SL(2,\mathbb{C})$ connections $\nabla$ with a corresponding irregular
singularity at these points. Any such connection $\nabla$ may have a monodromy
around the cylinder and is described by Stokes theory at $z=0$ and $z=\infty$.
For this particular type of singularity (the mildest case where $p_{2}$ has a
pole of order three), there is just one Stokes ray emitted from each puncture.

To frame the connection $\nabla$ we choose $\nabla$-invariant line bundles
$\ell$ and $\ell^{\prime}$ in the neighbourhood of $z=0$ and $z=\infty$,
respectively, as well as an eigenline $\ell^{\prime\prime}$ of the
counterclockwise monodromy $\sf M$ of $\nabla$ around $z=0$. With this choice
of framing data there is a unique $\mathscr{W}^{\pm}$-abelianization of
$\nabla$, for each of the two resolutions $\mathscr{W}^{\pm}$ of
$\mathscr{W}(p_{2},\vartheta_{\rm c})$. The abelianization of $\nabla$ with
respect to the resolution $\mathscr{W}^{-}$ is illustrated in Figure 15.

\begin{overpic}[width=303.53267pt]{PureSU2BritishSections.png}
\put(7.0,37.0){$21$} \put(22.0,50.0){$12$} \put(22.0,19.0){$21$}
\put(48.0,50.0){$21$} \put(48.0,19.0){$12$} \put(47.0,37.0){$21$}
\put(27.0,45.0){$21$} \put(27.0,24.0){$12$} \put(43.0,54.5){$12$}
\put(43.0,14.0){$21$}
\put(25.0,35.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1}^{\prime\prime},s_{2}^{\prime\prime}\right)$}}
\put(53.0,27.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1},s_{2}\right)$}}
\put(50.0,41.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left({\sf
M}s_{1},{\sf M}s_{2}\right)$}}
\put(5.0,26.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1}^{\prime},s_{2}^{\prime}\right)$}}
\put(54.0,51.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left({\sf
M}s_{1},s_{2}^{\prime\prime}\right)$}}
\put(54.0,17.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1},s_{2}^{\prime\prime}\right)$}}
\put(21.0,58.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1}^{\prime\prime},{\sf
M}s_{2}^{\prime}\right)$}}
\put(21.0,10.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\left(s_{1}^{\prime\prime},{\sf
M}s_{2}^{\prime}\right)$}} \end{overpic} Figure 15: Bases of sections for the
resolution $\mathscr{W}^{-}$. The dashed orange line is a monodromy cut for
the monodromy around the cylinder. Note that ${\sf M}s_{1}=s_{2}$ and ${\sf
M}s_{1}^{\prime}=s_{2}^{\prime}$.

This abelianization is characterized by the bases of sections shown in Figure
15 (in agreement with [HN20, Section 4.3]). In this figure $s_{i}$ and
$s_{i}^{\prime}$ are local bases of sections in the neighbourhood of the
incoming trajectories at $z=0$ and $z=\infty$, respectively, with
$s_{1}\in\ell$ and $s_{1}^{\prime}\in\ell^{\prime}$, and
$s_{i}^{\prime\prime}$ are local sections in some simply-connected domain of
the intermediate annulus, with $s_{2}^{\prime\prime}\in\ell^{\prime\prime}$
(in the conventions of [HN16, Sections 5.2 and 5.3]).

After possibly rescaling some of these sections, the change of basis matrices
can be brought in the desired triangular form. For instance, the matrix taking
$(s_{1},s_{2})$ to $(s_{1}^{\prime\prime},s_{2}^{\prime\prime})$ has the form

$\displaystyle\bigg{(}\begin{matrix}*&*\\\
0&*\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}*&0\\\ *&*\end{matrix}\bigg{)}$
(2.25)

relative to the basis $(s_{1},s_{2})$ in resolution $\mathscr{W}^{+}$, and

$\displaystyle\bigg{(}\begin{matrix}*&0\\\
*&*\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}*&*\\\ 0&*\end{matrix}\bigg{)}$
(2.26)

in resolution $\mathscr{W}^{-}$.

It is not always that easy (or even possible) to find a suitable framing on
$\nabla$ which makes the abelianization process one-to-one. A nice example of
this is the circular higher Fenchel-Nielsen network of Example 5. Here it
turns out that there are generically 12 abelianizations which may be
identified with the singular fibers of an auxiliary rational elliptic surface
[HN20, Section 6].

###  2.4 Spectral Coordinates

One of the applications of $\mathscr{W}$-abelianization is the construction of
a system of Darboux coordinates on the moduli space $\mathscr{M}_{\rm
flat}^{\mathscr{W}}(C,SL(K,\mathbb{C}))$ of $\mathscr{W}$-framed flat
$SL(K,\mathbb{C})$ connections on $C$. This moduli space is of importance for
example in the study of the Hitchin system, and it also shows up in a variety
of other problems in mathematical physics.

Given a basis of one-cycles on $\Sigma$, we first define a coordinate system
on the moduli space $\mathscr{M}^{\prime}_{\rm flat}(\Sigma,GL(1,\mathbb{C}))$
of equivariant $GL(1,\mathbb{C})$ connections on $\Sigma$ via the holonomies

$\mathcal{X}_{\gamma}=\text{Hol}_{\gamma}\nabla^{\text{ab}}\ \in\
\mathbb{C}^{\times}\ ,$ (2.27)

which depend only on the homology class
$[\gamma]\in\Gamma=H_{1}(\overline{\Sigma},\mathbb{Z})$ of a one-cycle
$\gamma$. Through $\mathscr{W}$-abelianization they also give a system of
coordinates $\big{\\{}\mathcal{X}^{\mathscr{W}}_{\gamma}\\}_{\gamma\in\Gamma}$
on $\mathscr{M}_{\rm flat}^{\mathscr{W}}(C,SL(K,\mathbb{C}))$, by first
abelianizing $\nabla$ into $\nabla_{\mathscr{W}}^{\rm ab}$ and then using
(2.27). We refer to these coordinates as spectral coordinates, since they are
defined in terms of spectral data, i.e. data on the spectral curve $\Sigma$.
Note that this coordinate system depends _only_ on the isotopy class of the
network $\mathscr{W}$.

The moduli space $\mathscr{M}_{\rm flat}^{\mathscr{W}}(C,SL(K,\mathbb{C}))$
has a natural holomorphic symplectic form given locally by the Atiyah-Bott-
Goldman formula [AB82, Gol84]

$\Omega=\frac{1}{2}\,\int_{C}\,\operatorname{Tr}\big{(}\delta\mathcal{A}\wedge\delta\mathcal{A}\big{)}\
,$ (2.28)

where $\mathcal{A}$ is the connection one-form of a $\mathscr{W}$-framed flat
$SL(K,\mathbb{C})$ connection $\nabla$ on $C$. Any spectral coordinate system
$\\{\mathcal{X}_{\gamma}\\}_{\gamma\in\Gamma}$ consists of (exponentiated)
holomorphic Darboux coordinates with respect to this holomorphic symplectic
form, as it brings the corresponding holomorphic Poisson structure
$\\{\cdot,\cdot\\}$ on $\mathscr{M}_{\rm
flat}^{\mathscr{W}}(C,SL(K,\mathbb{C}))$ to the form

$\\{\mathcal{X}_{\gamma},\mathcal{X}_{\gamma^{\prime}}\\}=\langle\gamma,\gamma^{\prime}\rangle\,\mathcal{X}_{\gamma}\,\mathcal{X}_{\gamma^{\prime}}=\langle\gamma,\gamma^{\prime}\rangle\,\mathcal{X}_{\gamma+\gamma^{\prime}}\
,$ (2.29)

where $\langle\cdot,\cdot\rangle$ is the intersection pairing on $\Gamma$. In
examples, we may compute the coordinates $\mathcal{X}_{\gamma}$ explicitly
through abelianization in terms of framing data [GMN13a].

######  Example 9.

Suppose $K=2$, and consider a Fock-Goncharov network $\mathscr{W}$. Given a
$\mathscr{W}$-framed $SL(2,\mathbb{C})$ connection $\nabla$ and a one-cycle
$\gamma$ on $\Sigma$, we would like to compute the holonomy
$\mathcal{X}_{\gamma}$ of the corresponding $GL(1,\mathbb{C})$ connection
$\nabla^{\text{ab}}$ along $\gamma$. The local geometry of a generic one-cycle
$\gamma$ in a Fock-Goncharov network is illustrated in Figure 16.

\begin{overpic}[width=346.89731pt]{AbelianizationCycle.png}
\put(8.0,62.0){$z_{l}$} \put(7.0,6.0){$z_{m}$} \put(89.0,6.0){$z_{n}$}
\put(89.0,62.0){$z_{k}$} \put(15.0,46.0){$21$} \put(33.0,57.0){$21$}
\put(64.0,11.0){$21$} \put(82.0,22.0){$21$} \put(17.0,10.0){$12$}
\put(80.0,58.0){$12$}
\put(29.0,46.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{l},s_{n})$}}
\put(69.0,17.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{n},s_{l})$}}
\put(11.0,26.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{l},s_{m})$}}
\put(36.0,9.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{n},s_{m})$}}
\put(81.0,30.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{n},s_{k})$}}
\put(51.0,57.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$(s_{l},s_{k})$}}
\end{overpic} Figure 16: A local Fock-Goncharov network with one-cycle
$\gamma$ indicated in blue, and the data needed to compute its holonomy.

Before starting the computation, let us recall that crossing a trajectory of
type $12$ maps the basis from $(s_{i},s_{j})$ to

$s_{j}\longmapsto s_{j}\qquad\mbox{and}\qquad s_{i}\longmapsto
s_{i}+c_{ij}\,s_{j}\ .$ (2.30)

Suppose that the basis of sections on the other side of the trajectory is
$(s_{k},s_{j})$. Then $s_{i}^{\prime}=s_{i}+c_{ij}\,s_{j}$ must be
proportional to the already chosen basis section $s_{k}$:

$\displaystyle s_{i}^{\prime}=\lambda\,s_{k}\ .$ (2.31)

Taking the exterior product on both sides with $s_{j}$, we find

$\displaystyle\lambda=\frac{s_{i}\wedge s_{j}}{s_{k}\wedge s_{j}}\ .$ (2.32)

We may thus alternatively write the mapping of $s_{i}$ as

$s_{i}\longmapsto s_{i}^{\prime}=\frac{s_{i}\wedge s_{j}}{s_{k}\wedge
s_{j}}\,s_{k}\ .$ (2.33)

In a similar manner, a trajectory of type $21$ leaves the first section in the
basis $(s_{i},s_{j})$ unchanged and maps the second section according to
(2.33). It is useful to note that the trajectory going into a puncture $z_{i}$
leaves invariant the section $s_{i}$ associated to it.

Let us now begin the computation of the holonomy of $\nabla^{\text{ab}}$ along
$\gamma$. Start at the marked point on the first sheet with section $s_{l}$.
Note that parallel transport by $\nabla^{\text{ab}}$ along $\gamma$ will
multiply the section $s_{l}$ by this holonomy:

$\displaystyle s_{l}\longmapsto\mathcal{X}_{\gamma}\,s_{l}\ .$ (2.34)

We can compute $\mathcal{X}_{\gamma}$ by following the path $\gamma$ and using
the rules (2.33) when crossing a trajectory. The first trajectory we cross
goes into $z_{l}$ and thus does not change $s_{l}$. We then cross the
trajectory going into $z_{m}$ and find

$s_{l}\longmapsto\frac{s_{l}\wedge s_{m}}{s_{n}\wedge s_{m}}\,s_{n}\ .$ (2.35)

This is left unchanged as we cross the third and fourth trajectories, as both
of these go into $z_{n}$. The next trajectory goes into $z_{k}$ and implies

$s_{n}\longmapsto\frac{s_{n}\wedge s_{k}}{s_{l}\wedge s_{k}}\,s_{l}\ ,$ (2.36)

which is again left invariant when we cross the last trajectory and close the
cycle. Hence in the end, by following the path $\gamma$ we arrive at the
section $s_{l}$ multiplied by the cross-ratio

$\mathcal{X}_{\gamma}=\frac{s_{n}\wedge s_{k}}{s_{l}\wedge
s_{k}}\,\frac{s_{l}\wedge s_{m}}{s_{n}\wedge s_{m}}\ .$ (2.37)

This is the spectral coordinate that we set out to compute. It is indeed the
familiar formula for a Fock-Goncharov coordinate on the moduli space of framed
flat $SL(2,\mathbb{C})$-connections on $C$, attached to the ideal
triangulation $\mathscr{T}$ of $C$ dual to the Fock-Goncharov network
$\mathscr{W}$ [GMN13b, GMN14]. These coordinates and their generalizations to
higher rank are for instance useful for studying the quantization of the
moduli spaces $\mathscr{M}_{\rm flat}^{\mathscr{W}}(C,SL(K,\mathbb{C}))$.

######  Example 10.

Going back to Example 7, we first need to address a subtlety. Flat
$SL(2,\mathbb{C})$ connections on this geometry may experience a monodromy
around the irregular singularity at infinity, which is usually fixed. This
implies that the moduli space of such connections is actually zero-
dimensional. However, we may consider a larger moduli space (of complex
dimension two) by considering flat $SL(2,\mathbb{C})$ connections with any
monodromy around infinity, and where we instead fix the gauge at infinity.
This latter requirement means choosing a trivialization of the bundle $E$ at
infinity.

This extended moduli space can be parameterized by two spectral coordinates
$\mathcal{X}_{A}$ and $\mathcal{X}_{B}$ corresponding to the two one-cycles
shown in Figure 17.

\begin{overpic}[width=346.89731pt]{AD2Gammas.png}
\put(8.0,44.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$B$}}
\put(53.0,44.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$B$}}
\put(20.0,35.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$A$}}
\put(65.0,35.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$A$}}
\end{overpic} Figure 17: Choice of $A$-cycle and $B$-cycle on $\Sigma$ for
$\mathscr{W}^{\pm}$. The $B$-cycle is non-compact and requires the gauge at
infinity to be fixed.

Note that the $B$-cycle begins and ends on a different sheet of the spectral
covering $\Sigma$, and $\mathcal{X}_{B}$ is only a holonomy invariant because
we fixed the gauge at infinity. The coordinates $\mathcal{X}_{A}$ and
$\mathcal{X}_{B}$ can be computed similarly to Example 9. We find that
$\mathcal{X}_{A}$ is given by the cross-ratio

$\mathcal{X}_{A}=\frac{s_{1}\wedge s_{2}}{s_{3}\wedge
s_{2}}\,\frac{s_{3}\wedge s_{4}}{s_{1}\wedge s_{4}}\ ,$ (2.38)

for both resolutions $\mathscr{W}^{+}$ and $\mathscr{W}^{-}$, while
$\mathcal{X}_{B}$ can be brought in the form

$\displaystyle\mathcal{X}_{B}^{+}=\frac{s_{1}\wedge s_{4}}{s_{2}\wedge
s_{4}}\qquad\mbox{and}\qquad\mathcal{X}_{B}^{-}=-\frac{s_{1}\wedge
s_{3}}{s_{1}\wedge s_{2}}$ (2.39)

for the networks $\mathscr{W}^{+}$ and $\mathscr{W}^{-}$, respectively. The
gauge fixing at infinity means that the sections $s_{3}$ and $s_{4}$ are
completely fixed (a $GL(1,\mathbb{C})$ gauge transformation would multiply
them by a constant), so that $\mathcal{X}_{B}^{\pm}$ are indeed invariants. It
may seem like there are various other choices to be made for the $B$-cycle,
but it actually turns out that all of them result in a spectral coordinate
that is equal to $\mathcal{X}_{B}$ up to multiplication by a power of
$\mathcal{X}_{A}$. This follows easily after realizing that the exterior
products $s_{1}\wedge s_{2}$, $s_{1}\wedge s_{4}$, $s_{3}\wedge s_{4}$ and
$s_{3}\wedge s_{2}$ are equal to one another, up to the monodromy
$\mathcal{X}_{A}$. This is because $\nabla$ is an $SL(2,\mathbb{C})$
connection and hence the exterior products $s_{i}\wedge s_{j}$ are invariant
after crossing a trajectory.141414One should exercise caution here, since
these relations only hold locally and do not account for possible monodromies
in non-simply connected regions.

######  Example 11.

Consider the Fenchel-Nielsen network from Example 8, with the choice of
$A$-cycle and $B$-cycle on $\Sigma$ as shown in Figure 18.

\begin{overpic}[width=346.89731pt]{PureSU2CyclesV2.png} \put(7.0,37.0){$21$}
\put(22.0,50.0){$12$} \put(22.0,19.0){$21$} \put(48.0,50.0){$21$}
\put(48.0,19.0){$12$} \put(47.0,37.0){$21$} \put(27.0,45.0){$21$}
\put(27.0,24.0){$12$} \put(43.6,54.0){$12$} \put(43.6,14.6){$21$}
\put(22.0,36.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$B$}}
\put(30.0,34.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$A$}}
\put(27.5,40.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$1$}}
\put(27.5,28.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$2$}}
\put(34.0,21.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$1$}}
\end{overpic} Figure 18: Choice of $A$-cycle and $B$-cycle for
$\mathscr{W}^{-}$.

The spectral coordinate $\mathcal{X}_{A}$ is simply an eigenvalue of the
monodromy matrix $\mathsf{M}$ of $\nabla$, since the one-cycle $A$ on $\Sigma$
does not cross any trajectories. (More precisely, it is the eigenvalue
corresponding to the eigenvector $s_{2}^{\prime\prime}$.) The coordinate
$\mathcal{X}_{B}$, on the other hand, is given by the cross-ratio

$\displaystyle\mathcal{X}_{B}^{-}=\mathcal{X}_{A}^{2}\,\frac{(s_{1}\wedge
s_{2}^{\prime\prime})^{2}\,(s_{1}^{\prime}\wedge
s_{1}^{\prime\prime})^{2}}{(s_{1}\wedge s_{2})\,(s_{1}^{\prime}\wedge
s_{2}^{\prime})\,(s_{1}^{\prime\prime}\wedge s_{2}^{\prime\prime})^{2}}$
(2.40)

in resolution $\mathscr{W}^{-}$, and by

$\displaystyle\mathcal{X}_{B}^{+}=\mathcal{X}_{A}^{-2}\,\frac{(s_{1}\wedge
s_{2})\,(s_{1}^{\prime}\wedge s_{2}^{\prime})\,(s_{1}^{\prime\prime}\wedge
s_{2}^{\prime\prime})^{2}}{(s_{1}\wedge
s_{1}^{\prime\prime})^{2}\,(s_{1}^{\prime}\wedge s_{2}^{\prime\prime})^{2}}$
(2.41)

in resolution $\mathscr{W}^{+}$, where the sections $s_{i}$, $s_{i}^{\prime}$
and $s_{i}^{\prime\prime}$ are all defined in Example 8 (see Figure 15).
Similarly to the previous examples, this follows by computing the parallel
transport of $\nabla^{\text{ab}}$ along the one-cycle $B$ on $\Sigma$ and
using the rules (2.33) when crossing a trajectory.

To compute $\mathcal{X}_{B}^{-}$, consider the setup in Figure 15, where we
have introduced a mondromy cut (the dashed orange line) to take care of the
monodromy that the local sections $s_{i}$, $s_{i}^{\prime}$ and
$s_{i}^{\prime\prime}$ experience under parallel transport around the
$A$-cycle. Using the identities $\mathsf{M}s_{1}=s_{2}$,
$\mathsf{M}s_{1}^{\prime}=s_{2}^{\prime}$,
$\mathsf{M}s_{1}^{\prime\prime}=\mu^{-1}\,s_{1}^{\prime\prime}$ and
$\mathsf{M}s_{2}^{\prime\prime}=\mu\,s_{2}^{\prime\prime}$ with
$\mu=\mathcal{X}_{A}$, as well as the fact that
$\mathsf{M}s\wedge\mathsf{M}s^{\prime}=s\wedge s^{\prime}$, we indeed find

$\displaystyle\mathcal{X}_{B}^{-}$ $\displaystyle=\frac{s_{2}^{\prime}\wedge
s_{1}^{\prime\prime}}{s_{2}^{\prime\prime}\wedge
s_{1}^{\prime\prime}}\,\frac{s_{2}^{\prime\prime}\wedge s_{1}}{s_{2}\wedge
s_{1}}\,\frac{s_{2}\wedge s_{2}^{\prime\prime}}{s_{1}^{\prime\prime}\wedge
s_{2}^{\prime\prime}}\,\frac{s_{1}^{\prime\prime}\wedge\mathsf{M}s_{2}^{\prime}}{s_{2}^{\prime}\wedge\mathsf{M}s_{2}^{\prime}}=\mu^{2}\,\frac{(s_{1}\wedge
s_{2}^{\prime\prime})^{2}\,(s_{1}^{\prime}\wedge
s_{1}^{\prime\prime})^{2}}{(s_{1}\wedge s_{2})\,(s_{1}^{\prime}\wedge
s_{2}^{\prime})\,(s_{1}^{\prime\prime}\wedge s_{2}^{\prime\prime})^{2}}\ .$
(2.42)

An analogous computation of $\mathcal{X}_{B}^{+}$ considers instead Figure 15
in the opposite resolution, that is, with the saddle trajectories
interchanged, and correspondingly modified bases of sections “in between” the
saddle trajectories. This yields

$\displaystyle\mathcal{X}_{B}^{+}$ $\displaystyle=\frac{s_{2}^{\prime}\wedge
s_{1}^{\prime}}{s_{2}^{\prime\prime}\wedge
s_{1}^{\prime}}\,\frac{s_{2}^{\prime\prime}\wedge
s_{1}^{\prime\prime}}{s_{2}\wedge
s_{1}^{\prime\prime}}\,\frac{s_{2}\wedge\mathsf{M}s_{2}}{s_{1}^{\prime\prime}\wedge\mathsf{M}s_{2}}\,\frac{s_{1}^{\prime\prime}\wedge
s_{2}^{\prime\prime}}{s_{2}^{\prime}\wedge
s_{2}^{\prime\prime}}=\mu^{-2}\,\frac{(s_{1}\wedge
s_{2})\,(s_{1}^{\prime}\wedge s_{2}^{\prime})\,(s_{1}^{\prime\prime}\wedge
s_{2}^{\prime\prime})^{2}}{(s_{1}\wedge
s_{1}^{\prime\prime})^{2}\,(s_{1}^{\prime}\wedge s_{2}^{\prime\prime})^{2}}\
.$ (2.43)

Following the averaging prescription of [HK18, HN20], we may thus associate
the average $B$-cycle coordinate

$\displaystyle\mathcal{X}_{B}=\sqrt{\mathcal{X}_{B}^{-}\,\mathcal{X}_{B}^{+}}=\frac{s_{1}^{\prime}\wedge
s_{1}^{\prime\prime}}{s_{1}\wedge s_{1}^{\prime\prime}}\,\frac{s_{1}\wedge
s_{2}^{\prime\prime}}{s_{1}^{\prime}\wedge s_{2}^{\prime\prime}}$ (2.44)

to the Fenchel-Nielsen network $\mathscr{W}$ at the critical phase (in
particular, this agrees with the computation in [HN20, Appendix A]).

######  Example 12.

A construction analogous to Example 11 for $K=2$ in the case of a Fenchel-
Nielson network on a surface $C$ with only regular punctures yields a
complexified version of the well-known Fenchel-Nielsen coordinates. Indeed,
one finds that there are two spectral coordinates $\mathcal{X}_{A}$ and
$\mathcal{X}_{B}$ associated to each pants cycle in the pants decomposition of
$C$, dual to the Fenchel-Nielsen network $\mathscr{W}$. Here $\mathcal{X}_{A}$
is an eigenvalue of the monodromy of $\nabla$ along the pants cycle, while
$\mathcal{X}_{B}$ is characterized by a certain property under what is called
the ‘twist flow’ on the moduli space of suitably framed flat
$SL(2,\mathbb{C})$ connections on $C$ (details are found in [HN16, Section
8.4]). These are exponentiated and complexified versions of the original
length and twist coordinates introduced in the context of hyperbolic geometry,
which for instance played an important role in Kontsevich’s proof [Kon92] of
Witten’s conjecture on the intersection numbers of the moduli space of curves
[Wit91].

The spectral coordinates obtained in Example 11 are examples of Fenchel-
Nielsen type coordinates on a Riemann surface with irregular punctures. Higher
rank generalizations of Fenchel-Nielsen length-twist coordinates have been
obtained through abelianization in [HK18],151515Another proposal for
$SL(3,\mathbb{C})$ Darboux coordinates on the four-punctured sphere is found
in [JN18]. while higher rank Fenchel-Nielsen type coordinates on the three-
punctured sphere have been introduced and studied in [HN20] by abelianization
with respect to the circular Fenchel-Nielsen network in Figure 11.161616Other
accounts and examples of abelianization may be found in e.g. [CPT18, Nik21,
Yan20].

###  2.5 Nonabelianization

It is an interesting exercise to express the monodromies of a flat
$SL(K,\mathbb{C})$ connection $\nabla$ in terms of the data of the
$GL(1,\mathbb{C})$ connection $\nabla^{\text{ab}}$. This is called
_nonabelianization_ [GMN13a]. It enables one to find the monodromy
representation of $\nabla$ in terms of spectral coordinates
$\mathcal{X}_{\gamma}$.

######  Example 13.

Returning to Example 8, let us describe the corresponding nonabelianization.
For this, we suppose instead that $\nabla^{\text{ab}}_{\pm}$ is the resulting
$GL(1,\mathbb{C})$ connection on $\Sigma$. Then we may construct the
$SL(2,\mathbb{C})$ monodromies of $\nabla$ in terms of the parallel transport
of $\nabla^{\text{ab}}_{\pm}$ as follows. Consider the path groupoid depicted
in Figure 19.

\begin{overpic}[width=346.89731pt]{PureSU2PathGroupoid.png}
\put(9.0,37.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\sf
S_{1}$}}
\put(46.0,37.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\sf
S_{4}$}}
\put(74.0,35.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\sf
S_{3}$}}
\put(88.0,34.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\sf
S_{2}$}}
\put(45.0,65.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{1}$}}
\put(45.0,5.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{2}$}}
\put(37.0,52.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{6}$}}
\put(59.0,47.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{4}$}}
\put(59.0,22.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{5}$}}
\put(81.9,33.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\varrho_{3}$}}
\end{overpic} Figure 19: A choice of paths $\varrho_{i}$ on
$C=\mathbb{C}^{\times}$ for the network $\mathscr{W}$. Each path begins and
ends at a trajectory of the network on an arrow which is labeled by a matrix
${\sf S}_{k}$.

Associate a diagonal $SL(2,\mathbb{C})$ matrix ${\sf D}_{i}$ to each path
$\varrho_{i}$ that does not cross a branch cut, and an off-diagonal
$SL(2,\mathbb{C})$ matrix $\tilde{\sf D}_{i}$ if the path $\varrho_{i}$
crosses a branch cut. We also associate a unipotent matrix ${\sf S}_{k}$ with
non-zero entry at position $ji$ to each trajectory of type $ij$.

Let the entries of the matrices

$\displaystyle{\sf D}_{i}=\bigg{(}\begin{matrix}d_{i}&0\\\
0&d_{i}^{-1}\end{matrix}\bigg{)}\qquad\mbox{and}\qquad\tilde{\sf
D}_{i}=\bigg{(}\begin{matrix}0&-\tilde{d}_{i}\\\
\tilde{d}_{i}^{-1}&0\end{matrix}\bigg{)}$ (2.45)

encode the parallel transport of the $GL(1,\mathbb{C})$ connection
$\nabla^{\text{ab}}_{\pm}$ along lifts of the paths $\varrho_{i}$ to $\Sigma$.
That is, we choose the non-zero elements $d_{i}$ and $\tilde{d}_{i}$ of the
matrices ${\sf D}_{i}$ and $\tilde{\sf D}_{i}$ as

$\displaystyle d_{1}=\frac{g_{1}}{g_{2}}\
,\quad\tilde{d}_{2}=g_{1}\,g_{2}\,\mathcal{X}_{B}\ ,\quad
d_{3}=\frac{g_{3}}{g_{2}}\ ,\quad d_{4}=\frac{g_{4}}{g_{3}}\
,\quad\tilde{d}_{5}=g_{3}\,g_{4}\ ,\quad d_{6}=\frac{1}{\mathcal{X}_{A}}\ ,$
(2.46)

for some complex numbers $g_{1}$, $g_{2}$, $g_{3}$, $g_{4}$, where
$\mathcal{X}_{A}$ and $\mathcal{X}_{B}$ are the spectral coordinates computed
in Example 11.

The unipotent transformations are of the form

$\displaystyle{\sf S}_{1}$ $\displaystyle=\bigg{(}\begin{matrix}1&f_{1}\\\
0&1\end{matrix}\bigg{)}\qquad,\qquad{\sf S}_{2}=\bigg{(}\begin{matrix}1&0\\\
f_{2}&1\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}1&f_{3}\\\
0&1\end{matrix}\bigg{)}\ ,$ $\displaystyle{\sf S}_{3}$
$\displaystyle=\bigg{(}\begin{matrix}1&0\\\
f_{4}&1\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}1&f_{5}\\\
0&1\end{matrix}\bigg{)}\qquad,\qquad{\sf
S}_{4}=\bigg{(}\begin{matrix}1&f_{6}\\\ 0&1\end{matrix}\bigg{)}\ ,$ (2.47)

or

$\displaystyle{\sf S}_{1}$
$\displaystyle=\bigg{(}\begin{matrix}1&\tilde{f}_{1}\\\
0&1\end{matrix}\bigg{)}\qquad,\qquad{\sf
S}_{2}=\bigg{(}\begin{matrix}1&\tilde{f}_{2}\\\
0&1\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}1&0\\\
\tilde{f}_{3}&1\end{matrix}\bigg{)}\ ,$ $\displaystyle{\sf S}_{3}$
$\displaystyle=\bigg{(}\begin{matrix}1&\tilde{f}_{4}\\\
0&1\end{matrix}\bigg{)}\,\bigg{(}\begin{matrix}1&0\\\
\tilde{f}_{5}&1\end{matrix}\bigg{)}\qquad,\qquad{\sf
S}_{4}=\bigg{(}\begin{matrix}1&\tilde{f}_{6}\\\ 0&1\end{matrix}\bigg{)}\ ,$
(2.48)

for the resolution $\mathscr{W}^{+}$ or $\mathscr{W}^{-}$, respectively.
Solving for the off-diagonal elements $f_{k}$ and $\tilde{f}_{k}$, by
requiring that the $SL(2,\mathbb{C})$ monodromy around the branch points on
$C$ is equal to $\mathbb{1}$, yields

$\displaystyle f_{1}$
$\displaystyle=-\frac{\mathcal{X}_{B}}{\mathcal{X}_{A}}\,\big{(}1+\mathcal{X}_{A}^{2}\big{)}\,g_{1}^{2}=\tilde{f}_{1}\
,$ $\displaystyle f_{2}$
$\displaystyle=\frac{\mathcal{X}_{A}}{\mathcal{X}_{B}\,g_{2}^{2}}\
\frac{1}{1-\mathcal{X}_{A}^{2}}\
,\qquad\tilde{f}_{2}=-\frac{1}{\mathcal{X}_{A}\,\mathcal{X}_{B}\,g_{2}^{2}}\
,$ $\displaystyle f_{3}$
$\displaystyle=-\frac{\mathcal{X}_{A}}{\mathcal{X}_{B}}\,g_{2}^{2}\
,\qquad\tilde{f}_{3}=\frac{\mathcal{X}_{A}\,\mathcal{X}_{B}\,g_{2}^{2}}{1-\mathcal{X}_{A}^{2}}\
,$ $\displaystyle f_{4}$ $\displaystyle=\frac{1}{\mathcal{X}_{A}\,g_{3}^{2}}\
,\qquad\tilde{f}_{4}=\frac{g_{3}^{2}}{\mathcal{X}_{A}}\ ,$ $\displaystyle
f_{5}$
$\displaystyle=-\frac{\mathcal{X}_{A}\,g_{3}^{2}}{1-\mathcal{X}_{A}^{2}}\
,\qquad\tilde{f}_{5}=-\frac{\mathcal{X}_{A}}{g_{3}^{2}\,(1-\mathcal{X}_{A}^{2})}\
,$ $\displaystyle f_{6}$
$\displaystyle=-\frac{g_{4}^{2}}{\mathcal{X}_{A}}\,\big{(}1+\mathcal{X}_{A}^{2}\big{)}=\tilde{f}_{6}\
.$ (2.49)

Then the monodromies of the $SL(2,\mathbb{C})$ connection $\nabla$ along any
path on $C$, written as a concatenation of the paths $\varrho_{i}$, are given
by multiplying the corresponding matrices ${\sf D}_{i}$, $\tilde{\sf D}_{i}$
and ${\sf S}_{k}$. Notice that although there are various other parameters
$g_{i}$ around (corresponding to an abelian gauge choice), the monodromy
invariants of $\nabla$ can be expressed entirely in terms of the spectral
coordinates $\mathcal{X}_{A}$ and $\mathcal{X}_{B}$.

A final point worth mentioning here is that the off-diagonal elements $f_{k}$
of the Stokes matrices ${\sf S}_{k}$ have an interpretation in terms of the
$GL(1,\mathbb{C})$ connection $\nabla^{\text{ab}}_{\pm}$ as well: They
represent the parallel transport of $\nabla^{\text{ab}}_{\pm}$ along “detour
paths” on $\Sigma$ that follow the corresponding trajectory back to the branch
point they are emitted from. Multiple terms in $f_{k}$ correspond to multiple
trajectories in the network. For example, each term in the power series
expansion

$\displaystyle
f_{2}=\frac{\mathcal{X}_{A}}{\mathcal{X}_{B}\,g_{2}^{2}}\,\big{(}1+\mathcal{X}_{A}^{2}+\mathcal{X}_{A}^{4}+\cdots\big{)}$
(2.50)

corresponds to the parallel transport of $\nabla^{\text{ab}}_{\pm}$ along a
detour path which encircles the $A$-cycle on $\Sigma$ multiple times.

A more thorough description of nonabelianization (and its relation to
abelianization) can be found in [HN16], where nonabelianization is shown to be
one-to-one for the Fock-Goncharov and Fenchel-Nielsen type networks in rank
$K=2$. However, this is not always the case, see for instance the discussion
in [HN20] on nonabelianization for the higher rank Fenchel-Nielsen networks on
the three-punctured sphere, illustrated in Figure 11. Nonabelianization is
extended to other Lie groups in [IM21].

##  3 Theories of Class $\mathcal{S}$ and their BPS States

In this section we introduce class $\mathcal{S}$ theories, which are a rich
subclass of ${\mathcal{N}}=2$ supersymmetric field theories in four
dimensions. We start with a brief introduction to four-dimensional
${\mathcal{N}}=2$ supersymmetric field theories. In Section 3.1 we go back to
the seminal work of Seiberg and Witten [SW94, SW94a], introducing Seiberg-
Witten geometry;171717However, we do not review the most physically
interesting idea of Seiberg-Witten theory, which is the relation to monopole
condensation and confinement. here we follow the reviews [Ler97, Dij97], see
also [Mar20] for a more recent account.181818We further recommend the lectures
of Witten for an introduction to Seiberg-Witten theory, which are available at
https://www.youtube.com/watch?v=EC1SvnjYWsA and
https://www.youtube.com/watch?v=9nPU3WNhH-0. In Section 3.2 we specialize to a
class of four-dimensional ${\mathcal{N}}=2$ theories known as class
$\mathcal{S}$ theories, which was first discovered by Gaiotto [Gai12]; for
more extensive reviews see e.g. [Tac14, Moo12, Tes16, Pes+17]. Finally, in
Section 3.3 we make contact with the spectral networks introduced in Section
2. We explain how the BPS spectra of class $\mathcal{S}$ theories may be
obtained from spectral networks, particularly for the example of Seiberg-
Witten theory; more details about this can be found in [GMN13a].

###  3.1 Seiberg-Witten Geometry

A four-dimensional ${\mathcal{N}}=2$ theory is a supersymmetric field theory
on $\mathbb{R}^{3,1}$ which is invariant under two supercharges in the
fundamental representation of $Spin(3,1)\cong SL(2,\mathbb{C})$, or
equivalently under eight real supercharges. One usually writes them as
$Q^{I}_{\alpha}$ and $\bar{Q}^{I}_{\dot{\alpha}}$, where $I=1,2$ counts the
supercharges (they are called R-symmetry indices: the $SU(2)_{R}\subset
SL(2,\mathbb{C})$ R-symmetry rotates them), and $\alpha,\dot{\alpha}=1,2$ are
spinor indices. There are two different particle multiplets in an
${\mathcal{N}}=2$ gauge theory. The vector multiplet consists of a gauge field
$A_{\mu}$, two spinors $\lambda^{I}_{\alpha}$ and
$\bar{\lambda}_{\dot{\alpha}}^{I}$, and a complex scalar $\phi$. All of them
transform in the adjoint representation of the gauge group $G$ (or more
precisely the gauge algebra ${\mathfrak{g}}$). The other multiplet is the
hypermultiplet. It consists of a pair of complex scalars $q^{I}$, and two
spinors $\psi_{\alpha}^{I}$ and $\bar{\psi}_{\dot{\alpha}}^{I}$. These may
transform in any representation of the gauge group $G$ (or more precisely
${\mathfrak{g}}$).191919More precisely, $q^{1}$ and $q^{2}$ live in different
representations of $G$ which are conjugate to each other, and similarly for
$\psi_{\alpha}^{I}$ and $\bar{\psi}_{\dot{\alpha}}^{I}$. The hypermultiplet
representation can sometimes be reduced to half of its degrees of freedom,
which is then called a half-hypermultiplet.

We start by studying the pure $SU(2)$ gauge theory which just consists of a
single vector multiplet in the adjoint representation of
${\mathfrak{g}}=\mathfrak{s}\mathfrak{u}(2)$. The Lagrangian includes a
potential term for the scalar field $\phi$ which is given by

$V(\phi)=\operatorname{Tr}\big{[}\phi,\phi^{\dagger}\big{]}^{2}\ .$ (3.1)

This potential vanishes if $\phi$ is restricted to the complexified maximal
torus of $SU(2)$, i.e.

$\phi=a\,\sigma_{3}\ ,$ (3.2)

where $a\in\mathbb{C}$ and $\sigma_{3}$ generates the maximal torus of
$SU(2)$. The potential $V(\phi)$ thus has flat directions or moduli.
Physically distinct vacua are parametrized by the gauge invariant Casimir
element

$u(a)=\operatorname{Tr}\phi^{2}\ ,$ (3.3)

so that the classical moduli space of vacua is $\mathscr{B}_{\rm
c}\cong\mathbb{P}^{1}$ (if we add a point at infinity). The space
$\mathscr{B}_{\rm c}$ is also called the (classical) Coulomb branch. Indeed,
if $u\neq 0$ then all fields will acquire masses by the Higgs mechanism,
except for the component of the vector multiplet in the direction of $\phi$.
After integrating out the massive fields (after all, we are interested in the
infrared physics), we are left with an effective abelian theory. This is
obviously not possible at $u=0$ and $u=\infty$, and we therefore regard these
two points as singularities of $\mathscr{B}_{\rm c}$.

The effective abelian theory is characterized by a prepotential
$\mathcal{F}_{0}(a)$. The complexified gauge coupling

$\tau(a)=\frac{\theta}{\pi}+\frac{8\pi\,{\mathrm{i}}}{g_{\textrm{\tiny
YM}}^{2}}$ (3.4)

is given as its second derivative

$\tau(a)=\frac{\partial^{2}\mathcal{F}_{0}(a)}{\partial a^{2}}\ .$ (3.5)

Furthermore, the pair

$\Omega=\bigg{(}\begin{matrix}a_{\textrm{\tiny D}}\\\ a\end{matrix}\bigg{)}\
,$ (3.6)

where

$a_{\textrm{\tiny D}}=\frac{\partial\mathcal{F}_{0}(a)}{\partial a}\ ,$ (3.7)

is known to define a holomorphic section of an $Sp(2,\mathbb{Z})\cong
SL(2,\mathbb{Z})$ vector bundle over the Coulomb branch $\mathscr{B}_{\rm c}$.
Seiberg and Witten have shown us how to find an explicit description of the
quantum moduli space $\mathscr{B}=\mathscr{B}_{\rm q}$, i.e. an explicit
description of this $SL(2,\mathbb{Z})$ bundle [SW94].

The starting point for the Seiberg-Witten solution is the behaviour of
$\mathcal{F}_{0}(a)$ around $u=\infty$, where the leading contribution can be
computed in perturbation theory. The coupling constant has an expansion

$\tau(a)=\tau_{0}+\frac{{\mathrm{i}}}{\pi}\,\log\Big{(}\frac{a}{\Lambda^{2}}\Big{)}+\sum_{n=1}^{\infty}\,c_{n}\,\Big{(}\frac{\Lambda}{a}\Big{)}^{4n}\
,$ (3.8)

where the first term $\tau_{0}$ is the classical contribution, the second term
is the one-loop contribution (this is the only perturbative correction), and
the remaining terms are instanton corrections. The ultraviolet cutoff
$\Lambda$ is exponentially small from the classical point of view, so that
quantum mechanically the single classical singularity there splits in two. We
will say more about this expansion in Section 4.

What is important here is that the logarithmic contributions determine the
local monodromy of the $SL(2,\mathbb{Z})$ bundle around $u=\infty$. If we
follow a loop around $u=\infty$ we find the transformation

$\bigg{(}\begin{matrix}a_{\textrm{\tiny D}}\\\
a\end{matrix}\bigg{)}\longmapsto\bigg{(}\begin{matrix}-a_{\textrm{\tiny
D}}+2\,a\\\ -a\end{matrix}\bigg{)}={\sf
M}_{\infty}\bigg{(}\begin{matrix}a_{\textrm{\tiny D}}\\\
a\end{matrix}\bigg{)}\qquad\mbox{with}\quad{\sf
M}_{\infty}=\bigg{(}\begin{matrix}-1&2\\\ 0&-1\end{matrix}\bigg{)}\ .$ (3.9)

The simplest solution would be to extend the $SL(2,\mathbb{Z})$ vector bundle
over $\mathbb{P}^{1}$ by having the opposite monodromy around the origin
$u=0$. This however turns out to violate the positivity constraint ${\rm
Im}(\tau)=8\pi/g_{\textrm{\tiny YM}}^{2}>0$. The Seiberg-Witten solution
assumes the minimal amount of a total of three solutions, say at $u=\infty$
and $u=\pm\,\Lambda^{2}$, with monodromy matrices generating a subgroup
$\Gamma\subset SL(2,\mathbb{Z})$. The quantum moduli space is then given by
$\mathscr{B}=\mathbb{H}/\Gamma$, where $\mathbb{H}$ is the upper complex half-
plane.

This moduli space also happens to parametrize elliptic curves of the form

$\Sigma\colon\quad
w^{2}=\frac{\Lambda^{2}}{z^{3}}-\frac{2\,u}{z^{2}}+\frac{\Lambda^{2}}{z}\ ,$
(3.10)

now known as Seiberg-Witten curves,202020This is a different formulation of
$\Sigma$ compared to the original formulation given by Seiberg and Witten. It
is the more recent class $\mathcal{S}$ theory formulation [Gai12], which is
needed to apply spectral network techniques. where the first homology of
$\overline{\Sigma}$ may be identified with the charge lattice $\Gamma$ of the
$SL(2,\mathbb{Z})$ bundle:

$\Gamma=H_{1}(\overline{\Sigma},\mathbb{Z})\ .$ (3.11)

Note that we have already encountered the family of curves $\Sigma$ (with
$\Lambda=1$ and $u=-\frac{9}{8}$) in Section 2, see Examples 3, 8, 11 and 13,
as a branched covering $\Sigma\to C$ defined by the quadratic differential
$p_{2}(z)$ from (2.10). From the Riemann-Hurwitz formula it follows that
$\Sigma$ is topologically a torus with two double punctures, see Figure 20.

\begin{overpic}[width=260.17464pt]{PureSU2SWCurveV2.png}
\put(15.0,55.0){$\Sigma$} \put(15.0,21.0){$C$} \put(25.0,15.0){$z=0$}
\put(64.0,15.0){$z=\infty$} \end{overpic} Figure 20: Seiberg-Witten geometry
for the pure $SU(2)$ theory. The ultraviolet curve $C$ is a sphere with
irregular singularities (of order three) at $z=0$ and $z=\infty$, while the
Seiberg-Witten curve $\Sigma$ is topologically a torus with two double
punctures.

Seiberg and Witten discovered that this geometry is crucial for understanding
the infrared properties of the pure $SU(2)$ gauge theory. Indeed, one can show
that the moduli $a$ and $a_{\textrm{\tiny D}}$ are given by period integrals

$a=\oint_{A}\,\lambda\qquad\mbox{and}\qquad a_{\textrm{\tiny
D}}=\oint_{B}\,\lambda\ ,$ (3.12)

where $(A,B)$ is a canonical basis of $H_{1}(\Sigma,\mathbb{Z})$ (see Figure
21) and $\lambda$ is the unique one-form on $\Sigma$ given by

$\lambda=w\,\mathrm{d}z\ ,$ (3.13)

which is called the Seiberg-Witten differential.

\begin{overpic}[width=346.89731pt]{SymplecticBasis.png}
\put(32.0,22.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$B$}}
\put(48.0,22.5){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$A$}}
\put(45.0,51.0){{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$A$}}
\put(28.5,55.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$B$}}
\put(24.0,13.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$0$}}
\put(40.0,12.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$-\Lambda$}}
\put(55.0,12.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$+\Lambda$}}
\put(73.0,15.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$\infty$}}
\end{overpic} Figure 21: Canonical basis $(A,B)$ of one-cycles on $\Sigma$
viewed as a two-sheeted branched cover of a base curve $C$.

The singularities at $u=\infty$ and $u=\pm\,\Lambda^{2}$ on $\mathscr{B}$ are
then the points in the moduli space where the Seiberg-Witten curve $\Sigma$
degenerates, and some combination of one-cycles vanishes. As we will see later
in Section 3.3, one can read off the BPS spectrum of the field theory from the
Seiberg-Witten curve. The singularities correspond to moduli where certain BPS
states become massless: $W^{\pm}$-bosons become massless at $u=\infty$
(restoring the $SU(2)$ symmetry), whereas certain dyons become massless at
$u=\pm\,\Lambda^{2}$.

The Seiberg-Witten solution was based on the premise that the abelian gauge
field was the only massless field in the theory. Thus we expect the formalism
to break down at $u=\infty$ and $u=\pm\,\Lambda^{2}$, where there are
additional massless BPS particles (that could not be integrated out).
Reinstating these massless states removes these singularities. The
singularities on $\mathscr{B}$ are therefore not physical.

Following the works of Seiberg and Witten, similar techniques have been used
to find Seiberg-Witten curves for many other Lagrangian ${\mathcal{N}}=2$
theories. More importantly, it was realized that Seiberg-Witten curves are not
just auxiliary mathematical objects which are helpful in the study of
${\mathcal{N}}=2$ gauge theories, but they can actually be identified as
physical objects appearing in various string theory constructions of
${\mathcal{N}}=2$ theories, see e.g. [Kle+96, BDS96, Wit97]. Perhaps the most
insightful construction is obtained by considering M5-branes in the M-theory
background $\mathbb{R}^{3,1}\times X\times\mathbb{R}^{3}$, with an M5-brane
wrapping $\mathbb{R}^{3,1}\times\Sigma\times\\{0\\}$, where $\Sigma\subset X$
is the Seiberg-Witten curve embedded in an ambient four-dimensional space $X$
and $\\{0\\}\subset\mathbb{R}^{3}$ is the origin of $\mathbb{R}^{3}$. This
realizes the corresponding ${\mathcal{N}}=2$ theory as the worldvolume theory
on the M5-brane.

###  3.2 Class $\mathcal{S}$ Geometry

So far we have emphasized the feature that the low energy properties of _any_
${\mathcal{N}}=2$ field theory are encoded in a geometrical object, the
Seiberg-Witten curve $\Sigma$, together with the Seiberg-Witten differential
$\lambda$. From now on we will specialize to a subclass of ${\mathcal{N}}=2$
theories for which the Seiberg-Witten curve $\Sigma$ comes with some
additional structure. Namely, $\Sigma$ is a (possibly branched) cover of
another (possibly punctured) Riemann surface $C$, where $\Sigma\subset T^{*}C$
and $\lambda$ is the tautological one-form on the cotangent bundle $T^{*}C$
restricted to $\Sigma$. In contrast to $\Sigma$, the ultraviolet curve $C$,
together with some additional data at its punctures, encodes the microscopic
properties of the ${\mathcal{N}}=2$ field theory, and it defines the
${\mathcal{N}}=2$ theory uniquely. These ${\mathcal{N}}=2$ theories are called
class $\mathcal{S}$ theories [Gai12]. Note that this is the geometric setup of
Section 2, where we considered $K$-fold coverings $\Sigma\to C$ defined by a
collection of higher differentials on $C$. Here the integer $K-1$ corresponds
to the rank of the gauge algebra ${\mathfrak{g}}$ of the ${\mathcal{N}}=2$
theory, so that $K=2$ sheeted coverings describe $SU(2)$ gauge theories.

The complex structure moduli of $C$ correspond to complexified gauge couplings
of the ${\mathcal{N}}=2$ theory, which is often a quiver gauge theory
consisting of various gauge theories coupled to one another using matter
fields. The additional data needed at the higher rank punctures is determined
by the type of singularities of the differentials. Regular singularities are
characterized by Lie algebra homomorphisms
$\varrho_{i}:\mathfrak{su}(2)\to{\mathfrak{g}}$. Roughly speaking, they encode
the flavour symmetries of the ${\mathcal{N}}=2$ theory, which are the global
symmetries acting on the matter fields of the ${\mathcal{N}}=2$ theory, with
group $G^{\varrho_{i}}$ given by the centralizer of $\varrho_{i}$ in the gauge
group $G$. We will take the gauge algebra to be
${\mathfrak{g}}=\mathfrak{su}(K)$, and we denote the corresponding
${\mathcal{N}}=2$ theory of class $\mathcal{S}$ by ${\sf
T}_{K}[C,\mathcal{D}]$, where $\mathcal{D}$ represents the additional data at
the punctures, which in this case correspond to Young diagrams with $K$ boxes.
The theory ${\sf T}_{K}[C,\mathcal{D}]$ may be realized physically as the
worldvolume theory of $K$ M5-branes wrapped on $\mathbb{R}^{3,1}\times
C\times\\{0\\}$ in the M-theory background $\mathbb{R}^{3,1}\times
T^{*}C\times\mathbb{R}^{3}$, or equivalently as a (twisted) compactification
of the six-dimensional $(2,0)$-theory ${\mathfrak{X}}[K]$ on the Riemann
surface $C$.

#### Class $\mathcal{S}$ Theories of Type $\bm{SU(2)}$

Conformal $SU(2)$ theories of class $\mathcal{S}$ can be constructed out of
just two elementary building blocks (see Figure 22):

  * •

the three-punctured sphere with an associated $SU(2)$ flavour symmetry group
at each puncture (this is only possible for $K=2$ as there is only a single
Young diagram with two boxes at each puncture); and

  * •

the cylinder with complex structure parameter

$\tau_{\textrm{\tiny
UV}}=\frac{\theta}{2\pi}+{\mathrm{i}}\,\frac{4\pi}{g_{\textrm{\tiny YM}}^{2}}\
,$ (3.14)

where $g_{\textrm{\tiny YM}}$ is real and $\theta$ is periodic with period
$2\pi$.

The cylinder corresponds to an ${\mathcal{N}}=2$ vector multiplet with gauge
group $SU(2)$ and complexified gauge coupling $\tau_{\textrm{\tiny UV}}$,
while the three-punctured sphere corresponds to a half-hypermultiplet in the
trifundamental representation of $SU(2)_{a}\times SU(2)_{b}\times SU(2)_{c}$
(see [HKS11a, Tac15] for further details about this half-hypermultiplet as
well as a Lagrangian description).

\begin{overpic}[width=346.89731pt]{BuildingBlocksSU2.png} \end{overpic} Figure
22: The three punctured sphere and the cylinder which constitute the building
blocks for $SU(2)$ class $\mathcal{S}$ theories.

All theories that are constructed using these building blocks have a
Lagrangian description and may be checked to be conformal (if all masses are
set to zero). This means in particular that the complexified coupling
$\tau_{\textrm{\tiny UV}}$ is a dimensionless parameter. (However, in contrast
to ${\mathcal{N}}=4$ supersymmetric Yang-Mills theory we should differentiate
here between the (exactly marginal) microscopic coupling $\tau_{\textrm{\tiny
UV}}$ and the infrared coupling $\tau$.)

For example, we may construct the $SU(2)$ gauge theory coupled to four
additional fundamental hypermultiplets by gluing two three-punctured spheres
using the plumbing fixture method. For this, pick a puncture on each sphere,
with the first puncture at $z=0$ in a local coordinate $z$ and the second
puncture at $w=0$ in a local coordinate $w$. Then make the identification
$z\,w=q$, where

$\displaystyle q={\mathrm{e}}^{\,2\pi\,{\mathrm{i}}\,\tau_{\textrm{\tiny
UV}}}$ (3.15)

is the exponentiated gauge coupling (see Figure 23).

\begin{overpic}[width=216.81pt]{FourPSpherePoP.png} \put(16.0,43.0){$0$}
\put(16.0,25.0){$q$} \put(87.0,43.0){$1$} \put(87.0,25.0){$\infty$}
\end{overpic} Figure 23: Plumbing fixture construction of the four-punctured
sphere.

Indeed, on the gauge theory side this means coupling the two corresponding
$SU(2)$ flavour symmetries to a dynamical $SU(2)$ gauge field with
complexified coupling $\tau_{\textrm{\tiny UV}}$. The four leftover $SU(2)$
flavour symmetries combine to give the enhanced $SO(8)$ flavour symmetry group
of the $SU(2)$ gauge theory coupled to four hypermultiplets. Note that, in
general, the Lagrangian for a theory with four hypermultiplets in a complex
representation of the gauge group is invariant under an $SU(4)$ flavour
symmetry rotating the four hypermultiplets. However, the case of $SU(2)$ gauge
group is special, since its fundamental representation is pseudoreal. This
implies that the flavour symmetry is in fact enhanced to an $SO(8)$ flavour
symmetry which rotates the eight half-hypermultiplets. We can embed the
flavour symmetries $SU(2)_{i}$ acting on the individual hypermultiplets into
this flavour symmetry group:

$SO(8)\supset SU(2)_{a}\times SU(2)_{b}\times SU(2)_{c}\times SU(2)_{d}\ .$
(3.16)

Changing the complex structure parameter of the four-punctured sphere changes
the complexified gauge coupling $\tau_{\textrm{\tiny UV}}$. But we can ask
about the effect of a $PSL(2,\mathbb{Z}$) action on $\tau_{\textrm{\tiny UV}}$
given by

$\tau_{\textrm{\tiny UV}}\longmapsto\frac{a\,\tau_{\textrm{\tiny
UV}}+b}{c\,\tau_{\textrm{\tiny
UV}}+d}\qquad\mbox{for}\quad\bigg{(}\begin{matrix}a&b\\\
c&d\end{matrix}\bigg{)}\in SL(2,\mathbb{Z})\ .$ (3.17)

This action leaves the complex structure of the four-punctured sphere
invariant and just permutes the punctures. It suggests that when the $SU(2)$
gauge theory with $N_{\rm f}=4$ flavour hypermultiplets is strongly coupled,
that is when $q\to 1$ or $q\to\infty$, we can equivalently describe it as a
weakly coupled theory with $q\to 0$ and permuted flavour symmetry groups (see
Figure 24).

\begin{overpic}[width=346.89731pt]{FourPSpheres.png} \put(19.0,58.0){$0$}
\put(19.0,46.0){$q$} \put(34.0,58.0){$1$} \put(34.0,46.0){$\infty$}
\put(58.0,56.0){$0$} \put(58.0,48.0){$q$} \put(92.0,56.0){$1$}
\put(92.0,48.0){$\infty$} \put(58.0,22.0){$0$} \put(58.0,13.0){$1$}
\put(92.0,22.0){$q$} \put(92.0,13.0){$\infty$} \put(10.0,22.0){$1$}
\put(10.0,13.0){$q$} \put(44.0,22.0){$0$} \put(44.0,13.0){$\infty$}
\end{overpic} Figure 24: The four-punctured sphere and its limits when
$q\rightarrow 0,1,\infty$.

This is indeed the S-duality found in the $SU(2)$ gauge theory coupled to four
hypermultiplets by Argyres and Seiberg [AS07]. In [Gai12] this S-duality was
generalized to any theory of class $\mathcal{S}$: any theory ${\sf
T}_{K}[C,\mathcal{D}]$ is invariant under a generalized S-duality that is
realized geometrically as the mapping class group acting on the ultraviolet
curve $C$. This is a main reason for naming them theories of class
$\mathcal{S}$.

Let us finally relate this discussion to Section 3.1 by noting that any
asymptotically free theory of class $\mathcal{S}$ may be realized in this
framework by sending some masses to infinity. For instance, the pure $SU(2)$
gauge theory is obtained by sending all four masses $m_{i}$ of the
hypermultiplets in the $N_{\rm f}=4$ $SU(2)$ theory to infinity, leaving
$q={\mathrm{e}}^{\,2\pi\,{\mathrm{i}}\,\tau_{\textrm{\tiny
UV}}}=\Lambda^{4}/m_{1}\,m_{2}\,m_{3}\,m_{4}$ finite, where $\Lambda$ is the
ultraviolet scale of the pure $SU(2)$ theory. The resulting ultraviolet curve
$C$ is a sphere with two irregular punctures. The corresponding Seiberg-Witten
geometry is described by the quadratic differential

$\displaystyle
p_{2}(z)=\Big{(}\frac{\Lambda^{2}}{z^{3}}-\frac{2\,u}{z^{2}}+\frac{\Lambda^{2}}{z}\Big{)}\,\mathrm{d}z\otimes\mathrm{d}z\
,$ (3.18)

as anticipated from (3.10).

#### Class $\mathcal{S}$ Theories of Type $\bm{SU(K)}$

As for spectral networks, the story changes quite drastically for higher rank
gauge groups $SU(K)$ with $K>2$. Conformal $SU(K)$ theories of class
$\mathcal{S}$ can again be built out of three-punctured spheres and cylinders
which are now labeled by Young diagrams with $K$ boxes. However, the
corresponding ${\mathcal{N}}=2$ field theories are often “non-Lagrangian”,
meaning that they are intrinsically stongly coupled quantum field theories
which do not admit a weakly coupled description that can be studied using
perturbation theory.212121They can be investigated physically as infrared
limits of ${\mathcal{N}}=1$ gauge theories, where supersymmetry becomes
enhanced at the fixed points, see e.g. [GRW15, MS17].

As an example, let us consider $K=3$. Class $\mathcal{S}$ theories of type
$SU(3)$ can be constructed out of three elementary building blocks (see Figure
25):

  * •

the three-punctured sphere with one “minimal” and two “maximal” punctures;

  * •

the three-punctured sphere with three “maximal” punctures; and

  * •

the cylinder with complex structure parameter $\tau_{\textrm{\tiny UV}}$
labeled by two “maximal” punctures.

Here a maximal puncture refers to a puncture labeled by a Young diagram
consisting of one row with three boxes, together with mass parameters
$(m_{1},m_{2},-m_{1},-m_{2})$ labeling the residues of the Seiberg-Witten
differential $\lambda$ at the (regular) puncture; the associated flavour
symmetry group is $SU(3)$ since the multiplicity of column heights is three. A
minimal puncture refers to a puncture labeled by a Young diagram consisting of
one column with two boxes and one column with one box, together with mass
parameters $(m,m,-2m)$; the associated flavour symmetry group is
$U(1)=S(U(1)\times U(1))$ since the multiplicities of column heights are
$(1,1)$. Similarly to the rank one case $K=2$, the cylinder corresponds to an
${\mathcal{N}}=2$ vector multiplet with gauge group $SU(3)$ and complexified
coupling $\tau_{\textrm{\tiny UV}}$, while the three-punctured sphere with two
maximal and one minimal puncture now corresponds to a free hypermultiplet in
the bifundamental representation of $SU(3)_{a}\times SU(3)_{c}$.

\begin{overpic}[width=346.89731pt]{BuildingBlocksSU3.png}
\put(15.5,5.0){$\mathbf{A}$} \put(47.0,5.0){$\mathbf{B}$}
\put(80.0,5.0){$\mathbf{C}$} \end{overpic} Figure 25: $SU(3)$ class
$\mathcal{S}$ theories are built from: $\mathbf{A}$ the sphere with two
maximal and one minimal puncture; $\mathbf{B}$ the sphere with three maximal
punctures; $\mathbf{C}$ the cylinder with two maximal punctures.

We may then construct the superconformal $SU(3)$ gauge theory coupled to six
hypermultiplets (this is the $SU(3)$ analogue of the superconformal $N_{\rm
f}=4$ $SU(2)$ theory) by gluing two such three-punctured spheres using the
plumbing fixture method; indeed, we can check that the $U(6)$ flavour symmetry
rotating the six hypermultiplets decomposes into

$U(6)\supset U(3)\times U(3)=\big{(}SU(3)_{a}\times
U(1)_{b}\big{)}\times\big{(}SU(3)_{c}\times U(1)_{d}\big{)}\ .$ (3.19)

This is illustrated in Figure 26. Suppose the punctures are positioned at
$z=0,1,q,\infty$, respectively, with $q$ close to $0$ in this weakly coupled
description of the $N_{\rm f}=6$ $SU(3)$ theory.

\begin{overpic}[width=303.53267pt]{FourPSphereSU3ab.png} \put(25.5,30.0){$a$}
\put(25.5,17.0){$b$} \put(76.5,30.0){$d$} \put(76.5,17.0){$c$} \end{overpic}
Figure 26: The ultraviolet curve for the superconformal $SU(3)$ theory as
$q\rightarrow 0$.

In the strongly coupled limit $q\to 1$, the ultraviolet curve can instead be
described by the (degenerated) four-punctured sphere, as illustrated in Figure
27 (to be precise, here we apply a mapping class group action to permute the
punctures). This decomposes into a three-punctured sphere with three maximal
punctures, and another three-punctured sphere with one maximal and two minimal
punctures.

\begin{overpic}[width=303.53267pt]{FourPSphereSU3bc.png} \put(25.5,30.0){$b$}
\put(25.5,17.0){$c$} \put(76.5,30.0){$a$} \put(76.5,17.0){$d$} \end{overpic}
Figure 27: The ultraviolet curve for the superconformal $SU(3)$ theory as
$q\rightarrow 1$.

Gaiotto interpreted this geometry in [Gai12] as the S-dual description of the
strongly coupled $N_{\rm f}=6$ $SU(3)$ theory. It has been argued by Argyres
and Seiberg [AS07] that this S-dual description is given by an intrinsically
strongly coupled interacting superconformal field theory, the $E_{6}$ Minahan-
Nemeschansky theory [MN96], coupled to a weakly coupled $SU(2)$ theory. In
particular, the field theory description of the three-punctured sphere with
three maximal punctures is given by the “non-Lagrangian” $E_{6}$ Minahan-
Nemeschansky theory [Gai12]. Microscopically, its flavour symmetry group is
given by $SU(3)_{a}\times SU(3)_{b}\times SU(3)_{c}$, which in the low energy
limit is enhanced to $E_{6}$. A classification of all the building blocks (or
“tinkertoys”) for theories of class $\mathcal{S}$ has been given in e.g.
[CD10].

To summarise, there is a large class of quantum field theories ${\sf
T}_{K}[C,\mathcal{D}]$, including most known Lagrangian222222${\mathcal{N}}=2$
complete BPS quiver gauge theories were classified in [CV13]; there are $11$
exceptional theories that are not of class $\mathcal{S}$. as well as new non-
Lagrangian ${\mathcal{N}}=2$ theories, whose microscopic data is encoded in a
(possibly punctured) Riemann surface $C$. These theories may be built out of
elementary blocks corresponding to three-punctured spheres and cylinders. The
(quantum) Coulomb branch $\mathscr{B}$ of infrared vacua of a theory ${\sf
T}_{K}[C,\mathcal{D}]$ is parameterized by $K-1$-tuples of meromorphic
$k$-differentials $(p_{2},\dots,p_{K})$ on $C$, with $2\leq k\leq K$, whose
poles at the punctures are characterized by the singularity data
$\mathcal{D}$; just as in Section 2.2, regular punctures correspond to poles
of order $k$, and wild punctures have higher order singularities. Each tuple
$(p_{2},\dots,p_{K})$ defines a spectral curve

$\displaystyle\Sigma\colon\quad w^{K}-\sum_{k=2}^{K}\,p_{k}\,w^{K-k}=0$ (3.20)

in the cotangent bundle $T^{*}C$, with local coordinates $(z,w)$ where $z\in
C$. The curve $\Sigma$ is the Seiberg-Witten curve for the theory ${\sf
T}_{K}[C,\mathcal{D}]$ at the vacuum corresponding to $(p_{2},\dots,p_{K})$.
The Seiberg-Witten differential is simply the tautological one-form
$\lambda=w\,\mathrm{d}z$ on $T^{*}C$ restricted to $\Sigma$. Explicit
descriptions of Seiberg-Witten curves and differentials for the theories
discussed above may be found in e.g. [HK18].

###  3.3 BPS States from Spectral Networks

In Section 3.2 we introduced ${\mathcal{N}}=2$ theories ${\sf
T}_{K}[C,\mathcal{D}]$ of class $\mathcal{S}$ and found that their moduli
spaces of Coulomb vacua are parametrized by tuples $u=(p_{2},\dots,p_{K})$ of
$k$-differentials on $C$, each defining a $K$-fold (possibly branched) cover
$\Sigma\to C$. We also know from Section 2 that, together with a phase
$\vartheta\in\mathbb{R}/2\pi\,\mathbb{Z}$, this data allows us to construct a
spectral network $\mathscr{W}(u,\vartheta)$ on $C$. One may wonder at this
stage whether this spectral network encodes any physical data of the theory
${\sf T}_{K}[C,\mathcal{D}]$ in the vacuum $u$. The answer is affirmative: The
spectral network $\mathscr{W}(u,\vartheta)$ encodes in a beautiful way
information about various types of BPS states [GMN13b, GMN13, GMN13a, GMN14].
In the following we will explain how spectral networks can be used to find the
spectrum of BPS particles in the theory ${\sf T}_{K}[C,\mathcal{D}]$.

Any theory with extended supersymmetry has special representations, called BPS
states, which are annihilated by certain linear combinations of supercharges.
These BPS states have masses $M$ which saturate the BPS bound

$\displaystyle M\geq|Z|\ ,$ (3.21)

where $Z$ is the central charge of the supersymmetry algebra. They are an
important tool in understanding the theory and its dualities.

Any ${\mathcal{N}}=2$ field theory has half-BPS states, which are annihilated
by half of the supercharges and form small representations of dimension two
(rather than four). In the low energy limit, when the ${\mathcal{N}}=2$ theory
is described by an abelian gauge theory with prepotential
$\mathcal{F}_{0}(a)$, the central charge $Z$ takes the form

$\displaystyle
Z=q_{i}\,a^{i}+p^{i}\,\frac{\partial\mathcal{F}_{0}(a)}{\partial a^{i}}\ .$
(3.22)

The pair of integer vectors $(\vec{q},\vec{p})$ labels the electric charges
$\vec{q}$ and the magnetic charges $\vec{p}$ of the BPS state. The central
charge can be expressed in terms of the Seiberg-Witten curve $\Sigma$ through
the period integrals

$\displaystyle a^{i}=\oint_{A^{i}}\,\lambda\qquad\mbox{and}\qquad
a_{\textrm{\tiny D},i}=\frac{\partial\mathcal{F}_{0}(a)}{\partial
a^{i}}=\oint_{B_{i}}\,\lambda\ ,$ (3.23)

where $(A^{i},B_{i})$ generate a symplectic basis232323A symplectic basis is a
homology basis for which the only non-zero intersection pairings are given by
$\langle A^{i},B_{j}\rangle=\delta^{i}_{j}$. of the first homology group
$H_{1}(\overline{\Sigma},\mathbb{Z})$ and $\lambda$ is the Seiberg-Witten
differential. (This is a generalization of the formula (3.12) for the pure
$SU(2)$ gauge theory discussed in Section 3.1.)

A four-dimensional BPS particle is a one-particle state in $\mathbb{R}^{3,1}$
that preserves half of the supersymmetry. Its electromagnetic charge
$(\vec{q},\vec{p})$ determines a one-cycle

$\displaystyle\gamma=q_{i}\,A^{i}+p^{i}\,B_{i}$ (3.24)

on $\Sigma$. Its central charge is then given by

$\displaystyle Z_{\gamma}=\oint_{\gamma}\,\lambda$ (3.25)

and its mass by $M_{\gamma}=|Z_{\gamma}|$.

Such a BPS particle in four dimensions can be regarded as a BPS string in six
dimensions [Kle+96]. It can be embedded in M-theory as an M2-brane wrapping
the product of a compact two-cycle $D$ in $T^{*}C$, which ends on $\Sigma$,
with its worldline in $\mathbb{R}^{3,1}$ (see Figure 28).

\begin{overpic}[width=346.89731pt]{M2Construction.png}
\put(25.0,5.0){$\mathbb{R}^{3}$} \put(15.5,20.0){$\mathbb{R}_{t}$}
\put(11.5,57.0){$\Sigma_{u}$}
\put(43.0,48.5){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma=\partial
D$}} \put(47.0,38.0){$\times$} \end{overpic} Figure 28: A local picture of an
M5-brane wrapping the product of the Seiberg-Witten curve $\Sigma$ with
$\mathbb{R}^{3,1}=\mathbb{R}^{3}\times\mathbb{R}_{t}$, with an M2-brane
wrapping the product of a two-cycle $D$ with the wordline $\mathbb{R}_{t}$ of
a BPS particle in $\mathbb{R}^{3,1}$.

Its electromagnetic charge $\gamma$ is encoded in the boundary $[\partial
D]=\gamma$, while the BPS condition translates into a calibration condition on
the two-cycle $D$: $D$ should be a holomorphic Lagrangian submanifold of
$T^{*}C$ with boundary $[\partial D]=\gamma$. In the simplest case, $D$ is
topologically a disc, and its projection to the ultraviolet curve $C$ is an
interval $I$ bounded on either side by a branch point of type $(ij)$ of the
covering $\Sigma\to C$ (see Figure 29).

\begin{overpic}[width=303.53267pt]{M2ProjectedDisc.png} \put(8.0,11.0){$C$}
\put(8.0,54.0){$\Sigma$} \put(51.0,54.0){$D$} \put(51.0,7.0){$I$}
\end{overpic} Figure 29: A disc $D$ bounded by $\Sigma$ which projects down to
an interval $I$ on $C$.

Let $\lambda_{i}$ be the restriction of the Seiberg-Witten differential to the
$i$-th sheet; the tension of an $(ij)$-string is given by
$\lambda_{i}-\lambda_{j}$. Then the central charge of the corresponding BPS
particle is given by

$\displaystyle
Z_{ij}=\oint_{\gamma}\,\lambda=\int_{I}\,(\lambda_{i}-\lambda_{j})\ ,$ (3.26)

while its mass is

$M_{ij}=\int_{I}\,|\lambda_{i}-\lambda_{j}|\ .$ (3.27)

The BPS condition $M_{ij}=|Z_{ij}|$ is thus satisfied if and only if the
absolute value of the integral of the one-form $\lambda_{i}-\lambda_{j}$ is
the same as the integral of its absolute value, which is true if and only if

${\mathrm{e}}^{-{\mathrm{i}}\,\vartheta}\,(\lambda_{i}-\lambda_{j})(v)\ \in\
\mathbb{R}^{\times}$ (3.28)

for any tangent vector $v$ along the projection of $\gamma$ to $C$, where
$\vartheta={\rm arg}(Z_{ij})$. (This also specifies which supercharges are
preserved by this BPS state.) Note that (3.28) is exactly the condition for a
trajectory of type $(ij)$. In other words, we can identify such a BPS particle
of charge $\gamma$ with a saddle of the spectral network
$\mathscr{W}(u,\vartheta)$ on $C$ that lifts to $\gamma$ on $\Sigma$.

More generally, any BPS particle in the vacuum $u\in\mathscr{B}$ with central
charge $Z={\mathrm{e}}^{\,{\mathrm{i}}\,\vartheta}\,M$ is encoded in the
spectral network $\mathscr{W}(u,\vartheta)$ as a finite web [GMN13a], a web
consisting of a finite number of saddles that have both ends on branch points
or junctions (see Figure 30 for some examples).

\begin{overpic}[width=346.89731pt]{Finite_Webs.png}
\put(8.0,41.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(ij)}}
\put(8.0,14.5){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(ij)}}
\put(43.0,22.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(ij)}}
\put(56.5,46.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(ij)}}
\put(90.5,46.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(jk)}}
\put(73.0,17.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}(ki)}}
\end{overpic} Figure 30: Three examples of finite webs. For class
$\mathcal{S}$ theories with $SU(2)$ gauge group, the first two webs are the
only possibilities, but for $SU(3)$ and higher rank gauge groups, more
complicated webs can form, such as the third web shown. We mark the branch
points with pairs of indices $ij$ that correspond to the sheets $\Sigma_{i}$,
$\Sigma_{j}$ that come together there, as well as the trajectories that emerge
from them, which are of type $(ij)$ or $(ji)$.

If we vary $\vartheta$ and systematically scan for such finite webs, we
uncover the spectrum of BPS particles of the class $\mathcal{S}$ theory ${\sf
T}_{K}[C,\mathcal{D}]$ in the vacuum $u$. The rays in the complex $Z$-plane
for which there exists a BPS particle in the vacuum $u$ are sometimes called
‘BPS rays’. If we vary the point $u$ on the Coulomb branch, the central
charges $Z_{\gamma}$ change and thus the BPS rays move in the $Z$-plane.

There might be real codimension one walls in the Coulomb branch $\mathscr{B}$
where the phases of some of the BPS particles coincide. Such walls are called
‘walls of marginal stability’. These are the only places where BPS bound
states can form or decay. The walls of marginal stability thus divide the
Coulomb branch into different regions where there may be a different spectrum
of BPS states.

######  Example 14.

Recall Example 1 with $C=\mathbb{C}$ and the quadratic differential

$p_{2}(z)=(z^{2}+m)\,\mathrm{d}z\otimes\mathrm{d}z\ .$ (3.29)

The corresponding four-dimensional ${\mathcal{N}}=2$ field theory is the
simplest example of an Argyres-Douglas theory [AD95], which is the theory of a
single hypermultiplet of mass $m$. It is sometimes called the AD2 theory
[GMN13b].

The Coulomb branch $\mathscr{B}$ of the AD2 theory is a single point,
corresponding to the differential $p_{2}(z)$ with fixed parameter $m$ (the
mass of the hypermultiplet), so this theory contains a unique vacuum. Its
Seiberg-Witten geometry is illustrated in Figure 31; the charge lattice
$\Gamma=H_{1}(\Sigma,\mathbb{Z})\cong\mathbb{Z}$ is generated by the one-cycle
$\gamma$.

\begin{overpic}[width=260.17464pt]{AD2SWCurve.png} \put(10.0,60.0){$\Sigma$}
\put(35.0,62.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma$}}
\put(39.0,17.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma$}}
\put(6.0,26.0){$C=\mathbb{C}$} \end{overpic} Figure 31: Seiberg-Witten
geometry of the AD2 theory.

The spectral networks for this theory were discussed in Example 1: if we vary
$\vartheta$ from $0$ to $\pi$ a saddle appears at $\vartheta=\vartheta_{\rm
c}=\pi/2$, see Figure 3. The corresponding BPS particle is of course the BPS
hypermultiplet of mass $m$. When $m=0$ the two branch points collide, the one-
cycle $\gamma$ collapses, and the hypermultiplet becomes massless.

######  Example 15.

Let us now determine the BPS spectrum of the pure $SU(2)$ gauge theory, which
we introduced in Section 3.1, using spectral networks. The Seiberg-Witten
geometry is determined by the quadratic differential

$p_{2}(z)=\Big{(}\frac{\Lambda^{2}}{z^{3}}-\frac{2\,u}{z^{2}}+\frac{\Lambda^{2}}{z}\Big{)}\,\mathrm{d}z\otimes\mathrm{d}z\
,$ (3.30)

where $\Lambda$ is the ultraviolet scale and $u$ parametrizes the Coulomb
branch $\mathscr{B}$. This differential has irregular (or wild) singularities
(of order three) at $z=0$ and $z=\infty$, and two zeroes at

$z_{\pm}=-\frac{u}{\Lambda^{2}}\pm\sqrt{\Big{(}\frac{u}{\Lambda^{2}}\Big{)}^{2}-1}\
.$ (3.31)

The Seiberg-Witten curve $\Sigma$ is defined as $w^{2}=p_{2}(z)$. As we saw in
Section 3.1, this is topologically a torus with two double punctures, see
Figure 20.

Note that there are two special points on the Coulomb branch: at
$u=\pm\,\Lambda^{2}$ both zeroes of $p_{2}(z)$ collide. These are the two
quantum singularities of $\mathscr{B}$, where the Seiberg-Witten curve
$\Sigma$ degenerates and some one-cycle $\gamma$ vanishes (see Figure 32).

\begin{overpic}[width=346.89731pt]{SWmoduli.png} \put(5.0,23.0){$\mathscr{B}$}
\put(86.0,47.0){$\Sigma_{u}$}
\put(74.0,47.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma_{\pm}$}}
\put(37.0,47.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma_{\pm}$}}
\end{overpic} Figure 32: At the singularities $u=\pm\,\Lambda^{2}$ the
Seiberg-Witten curve $\Sigma_{u}$ degenerates because a one-cycle
$\gamma_{\pm}$ vanishes. This leads to an $SL(2,\mathbb{Z})$ monodromy in the
charge lattice $\Gamma$ around these two points.

We now know that this signals the presence of massless BPS particles of
electromagnetic charge $\gamma$.

Indeed, this can be verified by examining the spectral networks
$\mathscr{W}(u,\vartheta)$ close to the singularities. It actually turns out
that the $u$-plane is divided into two regions in which the networks
$\mathscr{W}(u,\vartheta)$ behave very differently. These two regions are
known as the strong coupling region (for small $u$) and the weak coupling
region (for large $u$). They are divided by a wall of marginal stability,
which is defined by the equation ${\rm Im}(a_{\textrm{\tiny D}}/a)=0$ and can
be approximated by an ellipse passing through the singularities (see Figure
33).

\begin{overpic}[width=390.25534pt]{CouplingRegionsSU2V2.png}
\put(17.0,22.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$u=-\Lambda^{2}$}}
\put(70.0,22.0){{\color[rgb]{0.9921875,0.49609375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.9921875,0.49609375,0}$u=+\Lambda^{2}$}}
\put(39.0,24.0){strong coupling} \put(75.0,35.0){weak coupling} \end{overpic}
Figure 33: Coulomb branch, or $u$-plane, for the pure $SU(2)$ theory. The wall
of marginal stability, shown in black, separates a strongly coupled region
near $u=0$ from a weakly coupled region near $u=\infty$. Since there is an
$SL(2,\mathbb{Z})$ monodromy around the singularities at $u=\pm\,\Lambda^{2}$,
we have chosen the cut, shown in orange, to trivialize the corresponding local
system.

We choose cuts for the $SL(2,\mathbb{Z})$ monodromy as in Figure 33, and
trivialize the corresponding local system by choosing a basis of the charge
lattice $\Gamma$ as in Figure 34.

\begin{overpic}[width=346.89731pt]{PureSU2Cycles.png}
\put(52.0,30.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma_{+}$}}
\put(54.0,66.0){{\color[rgb]{0.1953125,0.1953125,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1953125,0.1953125,1}$\gamma_{-}$}}
\end{overpic} Figure 34: Choice of a basis of one-cycles $\gamma_{\pm}$ of
$\Gamma$ drawn on the ultraviolet curve $C$, where we mapped $z=\infty$ to
finite distance.

Let us consider the family of spectral networks $\mathscr{W}(u,\vartheta)$ for
varying phase $\vartheta$ at a point $u$ in the strongly coupled region first.
In Figure 35 we illustrate the topology changes that occur as $\vartheta$ is
varied from $-\frac{\pi}{4}$ to $\frac{3\pi}{4}$ at $u=0$. (The picture would
be similar at any other point in the strongly coupled region.)

\begin{overpic}[width=346.89731pt]{PureSU2FlipsStrong.png} \end{overpic}
Figure 35: The spectral networks for the pure $SU(2)$ theory as $\vartheta$ is
varied over two flips. The networks are rotated through $-90^{\circ}$.

In this family we encounter two saddles, at $\vartheta=0$ and
$\vartheta=\frac{\pi}{2}$. The first saddle corresponds to a hypermultiplet of
electromagnetic charge $\gamma_{+}$ and the second saddle to a hypermultiplet
of electromagnetic charge $\gamma_{-}$. If we continue varying $\vartheta$
from $\frac{3\pi}{4}$ to $\frac{7\pi}{4}$, we come across two saddles
corresponding to their antiparticles. The BPS spectrum in the strongly coupled
region thus consists of two hypermultiplets, traditionally called the monopole
and the dyon, and their charge conjugates.

The story in the weakly coupled region is slightly more complicated.
Fortunately we have already studied the relevant networks in Example 3.
Topology changes occur when $\vartheta$ is tuned to a critical phase
$\vartheta_{\rm c}$ (as well as $\pi+\vartheta_{\rm c}$), see Figure 6. As
$\vartheta$ increases to $\vartheta_{\rm c}$ we encounter an infinite number
of flips, which correspond to an infinite tower of hypermultiplets with
charges $(n+1)\,\gamma_{+}+n\,\gamma_{-}$ for $n\in\mathbb{Z}_{\geq 0}$. The
juggle at $\vartheta=\vartheta_{\rm c}$ encodes a vector multiplet of charge
$\gamma_{+}+\gamma_{-}$, and the infinite tower of flips as $\vartheta$
decreases to $\vartheta_{\rm c}$ correspond to another infinite number of
hypermultiplets with charges $n\,\gamma_{+}+(n+1)\,\gamma_{-}$. At the phase
$\vartheta=\pi+\vartheta_{\rm c}$ we find all of their charge conjugates.

Evidently, the spectra on either side of the wall of marginal stability are
not the same, and some BPS states have decayed. This may be captured by the
formula

$\displaystyle
K_{\gamma_{+}}\,K_{\gamma_{-}}=K_{\gamma_{-}}\,K_{\gamma_{+}+2\gamma_{-}}\,K_{2\gamma_{+}+3\gamma_{-}}\cdots
K_{\gamma_{+}+\gamma_{-}}^{2}\cdots
K_{3\gamma_{+}+2\gamma_{-}}\,K_{2\gamma_{+}+\gamma_{-}}\,K_{\gamma_{+}}\ ,$
(3.32)

where $K_{\gamma}$ is the transformation

$\displaystyle
K_{\gamma}\,:\,\mathcal{X}_{\gamma^{\prime}}\longmapsto\mathcal{X}_{\gamma^{\prime}}\,\big{(}1-(-1)^{\langle\gamma^{\prime},\gamma\rangle}\,\mathcal{X}_{\gamma}\big{)}^{\langle\gamma^{\prime},\gamma\rangle}$
(3.33)

of the spectral coordinates $\mathcal{X}_{\gamma^{\prime}}$ under a flip
[GMN13b, Section 7.6]. This is the famous Kontsevich-Soibelman wall-crossing
formula [KS08]. For a good introduction to wall-crossing in ${\mathcal{N}}=2$
field theories, we recommend [Moo10].

By now there is rich body of work on BPS states and wall-crossing in
${\mathcal{N}}=2$ theories, from many points of view. Of special note is the
BPS quiver method, which is particularly powerful when there is a region (or
chamber) in the Coulomb branch with a finite BPS spectrum. This led to the
determination of the BPS spectra of all “complete” ${\mathcal{N}}=2$ theories
[Ali+13].

The beautifully geometric spectral network method can be applied to any
${\mathcal{N}}=2$ theory of class $\mathcal{S}$ at any point $u\in\mathscr{B}$
of its Coulomb branch. It turns out to be especially valuable for conformal
${\mathcal{N}}=2$ theories with a single infinite BPS chamber. Examples
include the intrinsically strongly coupled $E_{6}$ and higher rank Minahan-
Nemeschansky theories discussed in Section 3.2, whose spectra are determined
in [HN17, HHN20], where higher rank Fenchel-Nielsen networks play a crucial
role. In these examples one finds not only saddles corresponding to
hypermultiplets or closed loops corresponding to vector multiplets, but also
more complicated topologies arise, such as the finite webs of Figure 30. The
method of spectral networks can also be extended to compute the spin content
of BPS states, with the spin identified with the writhe of paths on the
Seiberg-Witten curve $\Sigma$ [GLM15], which is related to a quantization of