2023 October 10

Approaching Argyres-Douglas theories 111This research was supported in part by
the National Science Foundation under grant PHY-22-09700.

Sriram Bharadwaj, Eric D’Hoker

Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy
University of California, Los Angeles, CA 90095, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

The Seiberg-Witten solution to four-dimensional ${\cal N}=2$ super-Yang-Mills
theory with gauge group ${\text{SU}}(N)$ and without hypermultiplets is used
to investigate the neighborhood of the maximal Argyres-Douglas points of type
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$. A convergent series expansion for the
Seiberg-Witten periods near the Argyres-Douglas points is obtained by analytic
continuation of the series expansion around the ${\mathbb{Z}}_{2N}$ symmetric
point derived in arXiv:2208.11502. Along with direct integration of the
Picard-Fuchs equations for the periods, the expansion is used to determine the
location of the walls of marginal stability for ${\text{SU}}(3)$. The
intrinsic periods and Kähler potential of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ superconformal fixed point are
computed by letting the strong coupling scale tend to infinity. We conjecture
that the resulting intrinsic Kähler potential is positive definite and convex,
with a unique minimum at the Argyres-Douglas point, provided only intrinsic
Coulomb branch operators with unitary scaling dimensions $\Delta>1$ acquire a
vacuum expectation value, and provide both analytical and numerical evidence
in support of this conjecture. In all the low rank examples considered here,
it is found that turning on moduli dual to $\Delta\leq 1$ operators spoils the
positivity and convexity of the intrinsic Kähler potential.

###### Contents

  1. 1 Introduction
  2. 2 Series expansion near a maximal AD point
    1. 2.1 Summary of the Seiberg-Witten solution
    2. 2.2 Review of the expansion around the ${\mathbb{Z}}_{2N}$ point
    3. 2.3 Expansion around a maximal AD point
    4. 2.4 The case of gauge group ${\text{SU}}(3)$
    5. 2.5 Remarks on the convergence of the ${\mathbb{Z}}_{N}$ expansion
  3. 3 Candidate walls of marginal stability revisited
    1. 3.1 Setup
    2. 3.2 Marginal stability of BPS states in ${\text{SU}}(3)$
    3. 3.3 Marginal stability of BPS states in ${\text{SU}}(4)$
    4. 3.4 Comments on the ${\text{SU}}(N)$ case for $u_{k}=0$ for $k>0$
  4. 4 The intrinsic Coulomb branch of AD theories
    1. 4.1 The $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ intrinsic Coulomb branch
    2. 4.2 The intrinsic Kähler potential
    3. 4.3 Analytical results for the periods
    4. 4.4 Analytical results for the Kähler potential
    5. 4.5 Rank-1, example 1: $(\mathfrak{a}_{1},\mathfrak{a}_{2})$
    6. 4.6 Rank-1, example 2: $(\mathfrak{a}_{1},\mathfrak{a}_{3})$
    7. 4.7 A rank-2 example: $(\mathfrak{a}_{1},\mathfrak{a}_{4})$
    8. 4.8 A rank-3 example: $(\mathfrak{a}_{1},\mathfrak{a}_{6})$
  5. 5 Conclusions and future directions
    1. 5.1 Summary of results
    2. 5.2 Future directions
  6. A Proof of Theorem 2.1
  7. B Convergence, long periods, and elliptic form for ${\text{SU}}(3)$
    1. B.1 The ${\mathbb{Z}}_{6}$ expansion
    2. B.2 The ${\mathbb{Z}}_{6}$ decomposition from the ${\mathbb{Z}}_{3}$ expansion
    3. B.3 Convergence of the ${\mathbb{Z}}_{3}$ series
    4. B.4 Elliptic expression for the $(\mathfrak{a}_{1},\mathfrak{a}_{2})$ Kähler potential

##  1 Introduction

The Seiberg-Witten (SW) solution to four dimensional ${\cal N}=2$ super-Yang-
Mills theory provides the exact low energy effective action and BPS mass
spectrum on the Coulomb branch. The vacuum expectation values (VEVs) of the
gauge scalars and their magnetic duals are encoded in terms of a family of
Riemann surfaces equipped with a meromorphic differential, referred to as the
SW curve and the SW differential, respectively. The original construction was
given for gauge group ${\text{SU}}(2)$ in [1, 2]. In the present paper, we
focus on theories with gauge group ${\text{SU}}(N)$, primarily for $N\geq 3$
and without hypermultiplets, for which the SW curve and differential were
constructed in [3, 4, 5, 6] (see also [8, 7, 9, 10, 11]).

At generic points on the Coulomb branch, the gauge group ${\text{SU}}(N)$ is
spontaneously broken to its maximal Abelian subgroup ${\text{U}}(1)^{N-1}$ and
the low energy contents of the theory consist of $N-1$ massless Abelian ${\cal
N}=2$ gauge multiplets. The spectrum of massive BPS states includes the
$N(N-1)$ gauge bosons and their magnetic counterparts. Remarkably, the SW
solution predicts the existence of isolated points in the moduli space where
the masses of one or several of these BPS state tends to zero. We now describe
two distinct scenarios where a maximal number of massive BPS states become
simultaneously massless.

There are $N$ multi-monopole points in the moduli space, at each one of which
$N-1$ mutually local (i.e. with vanishing Dirac pairing) BPS states become
simultaneously massless. At each multi-monopole point, there exists an
${\text{Sp}}(2N-2,{\mathbb{Z}})$ electric-magnetic duality frame in which all
the massless BPS states have purely magnetic charges, whence the name. In view
of their mutual locality, there exists an effective field theory description
where the massless magnetic monopoles are described by hypermultiplets.
Investigations into the behavior of the effective prepotential and periods in
the neighborhood of a multi-monopole point may be found in [12, 13, 14] as
well as in [15], where the behavior of the Kähler potential and the walls of
marginal stability are analyzed.

For gauge group ${\text{SU}}(3)$, Argyres and Douglas discovered two so-called
AD points in the moduli space, where three mutually non-local BPS states (i.e.
with non-vanishing Dirac pairing) simultaneously become massless [16]. The
corresponding AD theories are strongly interacting ${\cal N}=2$ superconformal
field theories (SCFTs). For gauge group ${\text{SU}}(N)$ with $N>3$ the
maximal number of mutually non-local BPS states become massless at two maximal
AD points, thereby generalizing the case of $N=3$, while, for $N=2$, no AD
points exist. Since the Dirac pairing is invariant under
${\text{Sp}}(2N-2,{\mathbb{Z}})$, there is no electric-magnetic duality frame
in which the massless BPS states are mutually local and simultaneously admit a
standard local field theory description.

While the absence of a standard local field theory formulation of the AD
theories presents a considerable conceptual challenge, several indirect
avenues of investigation have been explored, including the superconformal
bootstrap and brane constructions in string theory. Here, we shall investigate
the space of theories in the vicinity of the maximal AD points, first by
exploring the Coulomb branch of their embedding in ${\text{SU}}(N)$ super-
Yang-Mills and second by exploring their intrinsic Coulomb branch obtained by
sending the strong coupling scale $\Lambda$ of the ${\text{SU}}(N)$ theory to
infinity. The organization of the remainder of the paper and an overview of
the results is presented in the subsections below.

####  $\bullet$ Series expansion near the maximal Argyres-Douglas points

In section 2, we compute the SW periods in a convergent series expansion
around the maximal AD points. Our expansion provides a non-trivial analytic
continuation of the strong-coupling expansion produced in [15] around the
unique ${\mathbb{Z}}_{2N}$-symmetric point. While the latter expansion
contains the AD and the multi-monopole points on the boundary of its domain of
convergence, our expansion is centered at one or the other AD point and
thereby provides a significant extension of the domain of convergence near the
AD points. On regions where they overlap, our expansion arguably has better
convergence properties than the one given in [15]. Finally, by taking the
decoupling limit $\Lambda\rightarrow\infty$, where $\Lambda$ is the strong-
coupling scale, we obtain the intrinsic AD periods for
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ superconformal field theories in
section 4.

####  $\bullet$ Charting candidate walls of marginal stability

Two BPS states with central charges $Z_{1}$ and $Z_{2}$ and masses
$M_{1}=|Z_{1}|$ and $M_{2}=|Z_{2}|$ can form a stable bound state provided its
mass $M$ obeys $M<M_{1}+M_{2}$. The bound state is BPS when $M=|Z_{1}+Z_{2}|$,
and becomes marginally stable when the binding energy vanishes, namely when
$|Z_{1}+Z_{2}|=|Z_{1}|+|Z_{2}|$, which requires the ratio $Z_{2}/Z_{1}$ bo be
a real number. The reality of $Z_{2}/Z_{1}$ defines a real co-dimension one
sub-variety of the Coulomb branch, referred to as a candidate wall of marginal
stability. Determining this sub-variety was already undertaken in [1, 2] for
gauge group ${\text{SU}}(2)$, and discussed in more detail in [17, 18, 19].
Candidate walls of marginal stability were investigated more recently in [15]
for gauge group ${\text{SU}}(N)$ on restricted slices through the Coulomb
branch.

In section 3, we shall map out candidate walls of marginal stability beyond
the restricted slices of [15] for gauge group ${\text{SU}}(3)$, and present
partial results for $N\geq 4$. The series expansion around the AD points,
discussed in the preceding subsection, will play a key role in gaining access
to the walls of marginal stability beyond the special slices studied in [15].
In addition, we shall adapt the numerical integration methods used in [15] to
the computation of the SW periods and the central charges. These numerical
computations will allow us to reach beyond the radius of convergence of either
the ${\mathbb{Z}}_{2N}$ or the AD expansion, and to complete the charting of
candidate walls of marginal stability.

####  $\bullet$ Exploring the intrinsic Kähler potential of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ AD theories

Interest in the behavior of the Kähler potential, within the context of the SW
solution, has recently been rekindled by the role it may play in the soft
breaking of ${\cal N}=2$ super Yang-Mills theory and the renormalization group
flow of this theory to adjoint QCD [20, 14, 15, 21]. Specifically, the flow of
the mass operator $M^{2}{\rm tr}(\phi^{\dagger}\phi)$ for the gauge scalar
$\phi$ purely within the ${\cal N}=2$ super Yang-Mills theory is to the Kähler
potential of the SW solution. Motivated in part by future work on soft
supersymmetry breaking in or near AD theories, we initiate here a study of the
intrinsic Kähler potential of the maximal AD theories. Key questions concern
its positivity, convexity, and global minima properties.

In section 4, we shall study the intrinsic periods of the AD theories;
calculate the intrinsic Kähler potential; investigate the location of its
minima; and understand its positivity and convexity properties. We will find
that, in the absence of deformations, namely moduli corresponding to operators
with dimension $\Delta\leq 1$, but allowing the VEVs of genuine Coulomb branch
operators with dimension $\Delta>1$ to be non-zero, the intrinsic Kähler
potential exhibits positivity and convexity. This distinction between the
dimensions coincides precisely with the unitarity bound on scaling dimensions
of operators in any ${\cal N}=2$ SCFT: $\Delta\geq 1$, where free bosonic
fields saturate this bound. We shall gather compelling evidence that turning
on Coulomb branch operators with unitary scaling dimensions is compatible with
maintaining positivity and convexity of the Kähler potential, and its unique
global minimum being at the AD point.

### Acknowledgements

We gratefully acknowledge useful conversations with Thomas Dumitrescu and
Emily Nardoni. SB is happy to thank Lukas Lindwasser for conceptual
discussions and Amey Gaikwad for suggestions on numerical calculations. This
research was supported in part by the National Science Foundation under grant
PHY-22-09700.

##  2 Series expansion near a maximal AD point

We begin this section with a brief summary of the salient features of the
Seiberg-Witten (SW) solution for four-dimensional ${\cal N}=2$ super Yang-
Mills theory gauge group ${\text{SU}}(N)$ without hyper-multiplets [3, 4, 7].
We also review the expansion of the SW solution around the ${\mathbb{Z}}_{2N}$
symmetric point obtained in [15]. A similar set-up is then used to derive a
convergent expansion of the SW periods near one of the ${\mathbb{Z}}_{N}$
symmetric maximal AD points for $N\geq 3$, and to show that this expansion
coincides with the analytic continuation of the ${\mathbb{Z}}_{2N}$ expansion
of [15]. A key tool in matching the expansions is the Gauss-Kummer quadratic
transformation on hypergeometric functions. A detailed analysis of the domain
of convergence is undertaken and its results are presented graphically.

###  2.1 Summary of the Seiberg-Witten solution

The SW solution determines the vacuum expectation values of the gauge scalars
$a_{I}(u)$ and their magnetic duals $a_{D,I}(u)$ as locally holomorphic
functions of the gauge invariant Coulomb branch moduli $u_{n}$ for
$I=1,\cdots,N-1$ and $n=0,1,\ldots,N-2$. The SW solution is constructed from a
family of Riemann surfaces ${\cal C}(u)$ that depends holomorphically on the
moduli $u_{n}$ and is referred to as the Seiberg-Witten curve. For gauge group
${\text{SU}}(N)$ and no hyper-multiplets, the SW curve is given by,

$\displaystyle y^{2}=A(x)^{2}-\Lambda^{2N}\,,\hskip
57.81621ptA(x)=x^{N}-\sum_{n=0}^{N-2}u_{n}x^{n}\,,$ (2.1)

where $\Lambda$ is the strong-coupling scale of the non-Abelian
${\text{SU}}(N)$ super Yang-Mills theory.111 Our conventions for the curve
differ from those in [13, 14] by an $N$-dependent redefinition of the strong-
coupling scale $4\Lambda_{\text{there}}^{2N}=\Lambda_{\text{here}}^{2N}$.
Henceforth we shall set $\Lambda=1$, unless otherwise stated. Each Riemann
surface in the family is hyper-elliptic and has genus $N-1$. Choosing a basis
of homology cycles $\mathfrak{A}_{I}$ and $\mathfrak{B}_{I}$ with canonical
intersection pairing $\mathfrak{J}$,

$\displaystyle\mathfrak{J}(\mathfrak{A}_{I},\mathfrak{B}_{J})=-\mathfrak{J}(\mathfrak{B}_{I},\mathfrak{A}_{J})$
$\displaystyle=$ $\displaystyle\delta_{IJ}$
$\displaystyle\mathfrak{J}(\mathfrak{A}_{I},\mathfrak{A}_{J})\,=\,\mathfrak{J}(\mathfrak{B}_{I},\mathfrak{B}_{J})$
$\displaystyle=$ $\displaystyle 0~{},$ (2.2)

the vacuum expectation values of the gauge scalars $a_{I}(u)$ and their
magnetic duals $a_{D,I}(u)$ are obtained as the periods of a meromorphic
Abelian differential $\lambda$ as follows,

$\displaystyle 2\pi i\,a_{I}=\oint_{\mathfrak{A}_{I}}\lambda\,,\hskip
65.76596pt2\pi i\,a_{D,I}=\oint_{\mathfrak{B}_{I}}\lambda\,\hskip
65.76596pt\lambda={xA^{\prime}(x)dx\over y}\,.$ (2.3)

The matrix $\tau$ of ${\text{U}}(1)^{N-1}$ gauge couplings and mixings is
given by,

$\displaystyle\tau_{IJ}={\partial a_{D,I}\over\partial a_{J}}={\partial
a_{D,J}\over\partial a_{I}}$ (2.4)

The matrix $\tau$ is symmetric and its imaginary part is positive definite.
The symmetry of $\tau$ implies the existence of a pre-potential ${\cal F}$,
determined by $a_{D,I}=\partial{\cal F}/\partial a_{I}$, which will not be
needed in the sequel. The imaginary part of $\tau$ is the matrix of inverse
gauge couplings squared and must be positive on physical grounds. This
property is automatic in the SW solution. Indeed, the partial derivatives
$\partial\lambda/\partial u_{n}$ are holomorphic Abelian differentials on
${\cal C}(u)$, up to exact differentials of single-valued functions, so that
the partial derivatives $\partial a_{I}/\partial u_{n}$ and $\partial
a_{D,I}/\partial u_{n}$ are periods of holomorphic differentials and $\tau$ is
the period matrix of the Riemann surface ${\cal C}(u)$. The Riemann bilinear
relations automatically imply that $\tau$ has positive imaginary part.
Finally, modular transformations on the cycles $\mathfrak{A},\mathfrak{B}$
leave the canonical intersection pairing $\mathfrak{J}$ invariant and form the
duality group ${\text{Sp}}(2N-2,{\mathbb{Z}})$.

###  2.2 Review of the expansion around the ${\mathbb{Z}}_{2N}$ point

For gauge group ${\text{SU}}(2)$, the SW curve has genus one, namely it is a
torus, so that the periods may be solved in terms of elliptic functions and
modular forms [1]. For gauge group ${\text{SU}}(3)$, the periods are given by
hyper-elliptic integrals which may be reduced to linear combinations of the
Appell $F_{4}$ functions [3]. For gauge group ${\text{SU}}(N)$ with $N\geq 4$,
however, the periods are given by hyper-elliptic integrals that are no longer
tabulated special functions. Nonetheless, a relatively simple convergent
Taylor series expansion of the periods was obtained in [15] around the
${\mathbb{Z}}_{2N}$ symmetric point $u_{n}=0$ for all $n=0,1,\dots,N-2$ for
arbitrary $N$. This expansion, which we shall briefly review below, will serve
as a guide to obtaining a similar expansion around the maximal AD points.

At the ${\mathbb{Z}}_{2N}$ point we have $A(x)=x^{N}$ so that the SW curve
$y^{2}=x^{2N}-1$ manifestly exhibits the ${\mathbb{Z}}_{2N}$ symmetry
$x\to\varepsilon x$ where $\varepsilon=e^{2\pi i/2N}$. The curve contains the
two ${\mathbb{Z}}_{2N}$ symmetric points $(x,y)=(0,\pm i)$. The $2N$ branch
points are given by the $2N$-th roots of unity $(x,y)=(\varepsilon^{k},0)$ for
$k=0,\cdots,2N-1$, and are shown in the left panel of figure 1 for the case of
$N=3$. They are mapped into one another by ${\mathbb{Z}}_{2N}$. The expansion
around the ${\mathbb{Z}}_{2N}$ point is obtained by Taylor expanding the SW
periods in powers of the moduli $u_{n}$. In practice, the expansion may be
organized by setting,

$\displaystyle y^{2}=x^{2N}-1+U(x)^{2}-2x^{N}U(x)\hskip
72.26999ptU(x)=\sum_{n=0}^{N-2}u_{n}x^{n}$ (2.5)

and Taylor expanding $\lambda$ in powers of $U(x)$,

$\displaystyle\lambda=\sum_{k=0}^{\infty}{\Gamma(k+{1\over
2})\over\Gamma({1\over
2})k!}{(2x^{N}U(x)-U(x)^{2})^{k}\over(x^{2N}-1)^{k+{1\over
2}}}\Big{(}Nx^{N}-xU^{\prime}(x)\Big{)}dx$ (2.6)

The integrals of $\lambda$ along the homology cycles $\mathfrak{A}_{I}$ and
$\mathfrak{B}_{I}$, needed in the calculation of the SW periods in (2.3), may
be computed by integrating term by term in powers of $U(x)$. The homology
cycles for all terms may then be chosen along the line segments of the branch
cuts of the ${\mathbb{Z}}_{2N}$ symmetric curve $y^{2}=x^{2N}-1$, as
illustrated in figure 1 for the case $N=3$.

As shown in [15], all such integrals may be obtained by evaluating the
function $Q(\xi)$ which is defined as the Abelian integral of the SW
differential, given by (2.3),

$\displaystyle{\pi i}\,Q(\xi)=\int_{0}^{\xi}\lambda\hskip
72.26999pt\xi^{2N}=1$ (2.7)

between either ${\mathbb{Z}}_{2N}$ symmetric point $(x,y)=(0,\pm i)$, denoted
hereby $0$, and an arbitrary branch point $(x,y)=(\xi,0)$ denoted here by
$\xi$. The paths of integration are indicated in green in the left panel of
figure 1. In terms of $Q(\xi)$ the SW periods are,

$\displaystyle a_{I}$
$\displaystyle=\sum_{J=1}^{I}\Big{\\{}Q(\varepsilon^{2J-1})-Q(\varepsilon^{2J-2})\Big{\\}}$
$\displaystyle a_{D,I}$
$\displaystyle=Q(\varepsilon^{2I})-Q(\varepsilon^{2I-1})$ (2.8)

Swapping the roles of the ${\mathbb{Z}}_{2N}$ symmetric points $(0,\pm i)$
reverses the signs of $Q$ and all the periods which, in turn, is equivalent to
a modular transformation by $-I\in{\text{Sp}}(2N-2,{\mathbb{Z}})$. The Taylor
series expansion of $Q(\xi)$ in powers of the moduli $u_{n}$ is given
by,222The notation used here is related to the notation used in [15] by
letting $L+1\to L$, $M_{0}\to M$, and $Y_{M}(\xi^{N},L)\to
Y(\xi^{N},\alpha;u_{0})$, as defined in (2.10) and (2.2), and will be
convenient when matching with the expansion around the AD points in the
sequel.

$\displaystyle
Q(\xi)=\sum_{\ell_{i}=0}^{\infty}\frac{2^{\frac{M-{L}}{N}}}{2\pi^{2}N}\,\xi^{NM+{L}+N}\,\Gamma\left(\tfrac{{L}}{N}\right)Y(\xi^{N},\alpha;u_{0})\,\frac{u_{1}^{\ell_{1}}\dots
u_{N-2}^{\ell_{N-2}}}{\ell_{1}!\dots\ell_{N-2}!}$ (2.9)

where we shall use the following combinations throughout,

$\displaystyle L={1+}\sum_{j=1}^{N-2}j\ell_{j}\hskip
57.81621ptM=\sum_{j=1}^{N-2}\ell_{j}\hskip 57.81621pt\alpha=\frac{NM-{L}}{2N}$
(2.10)

The function $Y(\xi^{N},\alpha;u_{0})$ is given by the following linear
combination of Gauss hypergeometric functions $F={}_{2}F_{1}$,

$\displaystyle Y(\xi^{N},\alpha;u_{0})=$
$\displaystyle\,2u_{0}\,\xi^{N}\cos^{2}(\pi\alpha)\Gamma\left(\alpha+{\tfrac{1}{2}}\right)^{2}F\left(\alpha+{\tfrac{1}{2}},\alpha+{\tfrac{1}{2}};\tfrac{3}{2};u_{0}^{2}\right)$
$\displaystyle+\sin^{2}(\pi\alpha)\Gamma(\alpha)^{2}\,F\left(\alpha,\alpha;{\tfrac{1}{2}};u_{0}^{2}\right)$
(2.11)

Alternatively, the hypergeometric functions may themselves be expanded in
Taylor series in $u_{0}$ [15], but the above formulation will be more
pertinent to the expansion around the maximal AD points, to which we now turn.

###  2.3 Expansion around a maximal AD point

For gauge group ${\text{SU}}(N)$ with $N\geq 3$, the maximal AD points are
characterized by $u_{n}=0$ for all $n>0$ and $u_{0}=\pm 1$, recalling that we
set the strong coupling scale $\Lambda=1$. Without loss of generality we may
concentrate on the AD point with $u_{0}=1$ so that $A(x)=x^{N}-1$. The
corresponding SW curve $y^{2}=x^{N}(x^{N}-2)$ manifestly exhibits
${\mathbb{Z}}_{N}$ symmetry $x\to\varepsilon^{2}x$, recalling that
$\varepsilon=e^{2\pi i/2N}$ while the SW differential $\lambda=Nx^{N}dx/y$
transforms as $\lambda\to\varepsilon^{2}\lambda$.

####  2.3.1 Expansion of the SW differential

The neighborhood of the AD point $u_{0}=1$, inside of which we shall obtain a
convergent series expansion, may be parametrized by first taking $u_{0}$ away
from the value 1 and then turning on the moduli $u_{n}$ for $n>0$. To do so we
introduce the shifted variable $v=1-u_{0}$ keeping $u_{n}=0$ for $n>0$. In
terms of $v$ the SW curve is given by,

$\displaystyle y^{2}=(x^{N}+v)(x^{N}+v-2)$ (2.12)

Its branch points exhibit ${\mathbb{Z}}_{N}$ symmetry but, for $v\not=1$, do
not exhibit ${\mathbb{Z}}_{2N}$ symmetry. Instead, they are given as follows
for $k=0,1,\cdots,N-1$,

$\displaystyle x_{k}^{+}=(2-v)^{\frac{1}{N}}\varepsilon^{2k}\hskip
72.26999ptx_{k}^{-}=v^{\frac{1}{N}}\varepsilon^{2k+1}$ (2.13)

For sufficiently small $|v|\ll 1$ the distance from branch points to the
origin is of order $\Lambda=1$ for $x^{+}_{k}$, but of order
$|v|^{\frac{1}{N}}\ll 1$ for $x_{k}^{-}$, as illustarted in the right panel of
figure 1 for the case $N=3$. The small branch points $x_{k}^{-}$ correspond to
the intrinsic data of the AD theory, while the large branch points $x_{k}^{+}$
correspond to its embedding into ${\text{SU}}(N)$. The small branch points
dominate the dynamics of the AD theory as $\Lambda\rightarrow\infty$ since the
heavy states have masses of order $\Lambda$ and decouple. For $v\not=0,2$, the
SW curve also contains two points that are invariant under ${\mathbb{Z}}_{N}$
given by $(x,y)=(0,\pm\sqrt{v(v-2)})$ and that are referred to simply as $0$
in figure 1.

Turning on the remaining moduli $u_{n}$ for $n>0$ will modify the disposition
of the branch points from the one for the ${\mathbb{Z}}_{N}$ symmetric curve.
As long as the $u_{n}$ for $n>0$ remain sufficiently small, the branch points
will remain well-separated and a convergent Taylor series expansion should be
expected.

In this subsection we shall evaluate the periods spanned by the small branch
points, which are represented by the cycles $\mathfrak{A}_{1}^{(s)}$ and
$\mathfrak{B}_{1}^{(s)}$ in the right panel of figure 1 for ${\text{SU}}(3)$,
and will be defined for arbitrary ${\text{SU}}(N)$ in (2.3.2). The periods
involving the large branch points, which are represented by the cycles
$\mathfrak{A}_{1}^{(\ell)}$ and $\mathfrak{B}_{1}^{(\ell)}$ in the right panel
of figure 1 for ${\text{SU}}(3)$, will be defined for arbitrary
${\text{SU}}(N)$ in (2.3.5) and will be evaluated in subsection 2.3.5.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$\varepsilon^{0}$$\varepsilon^{1}$$\varepsilon^{2}$$\varepsilon^{3}$$\varepsilon^{4}$$\varepsilon^{5}$$\hat{\mathfrak{A}}_{1}$$\hat{\mathfrak{A}}_{2}$$\hat{\mathfrak{A}}_{3}$$\mathfrak{B}_{1}$$\mathfrak{B}_{2}$$\mathfrak{B}_{3}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$0$$x_{0}^{+}$$x_{0}^{-}$$x_{1}^{+}$$x_{1}^{-}$$x_{2}^{+}$$x_{2}^{-}$$\mathfrak{A}_{1}^{(\ell)}$$\mathfrak{B}_{1}^{(\ell)}$$\mathfrak{A}_{1}^{(s)}$$\mathfrak{B}_{1}^{(s)}$
Figure 1: The ${\mathbb{Z}}_{6}$ symmetric curve $y^{2}=x^{6}-1$ is shown in
the left panel, while the ${\mathbb{Z}}_{3}$ symmetric curve
$y^{2}=(x^{3}+v)(x^{3}+v-2)$ is shown in the right panel. In each case, the
branch cuts are shown in black double lines; the integration paths for
$Q(\varepsilon^{n})$ and $R(\zeta)$ are shown in green; the cycles of the
canonical homology bases $\mathfrak{A}$ and $\mathfrak{B}$ are shown in blue
and red, respectively.

To proceed with the evaluation of the small periods, we set
$A(x)=\hat{A}(x)-1$ in terms of which the SW curve and differential become,

$\displaystyle y^{2}=\hat{A}(x)(\hat{A}(x)-2)\hskip
72.26999pt\lambda={x\hat{A}^{\prime}(x)dx\over y}$ (2.14)

The AD point $u_{0}=1$ corresponds to $\hat{A}(x)=x^{N}$ and the small branch
points correspond to $|x|\ll 1$. Thus, to evaluate the periods spanned by the
small branch points, we expand the denominator in powers of $\hat{A}(x)$, as
follows,

$\displaystyle\lambda={1\over\sqrt{-2}}\sum_{k=0}^{\infty}{\Gamma(k+{1\over
2})\over 2^{k}\,\Gamma({1\over
2})\,k!}\,{x\,\hat{A}^{\prime}(x)dx\over\hat{A}(x)^{{1\over 2}-k}}$ (2.15)

By setting $u_{0}=1-v$ and rescaling the variable $x$ and the moduli $u_{n}$
as follows,

$\displaystyle x=v^{\frac{1}{N}}z\hskip
72.26999ptu_{n}=v^{1-\frac{n}{N}}v_{n}\quad\hbox{ for }n>0$ (2.16)

the function $\hat{A}(x)$ decomposes into a factor of $v$ times a factor that
only depends on the remaining rescaled moduli $v_{n}$ for $n>0$, but is
independent of $v$,

$\displaystyle\hat{A}(x)=v\Big{(}z^{N}-V(z)+1\Big{)}\hskip
72.26999ptV(z)=\sum_{n=1}^{N-2}v_{n}z^{n}$ (2.17)

Clearly, the expansion (2.14) in powers of $\hat{A}(x)$ is equivalent to an
expansion in powers of $v$. To proceed, we expand $\lambda$ in powers of the
remaining moduli $v_{n}$ with $n>0$ as follows,

$\displaystyle\lambda={v^{{1\over
2}+\frac{1}{N}}\over\sqrt{-2}}\sum_{k=0}^{\infty}{\Gamma(k+{1\over
2})\over\Gamma({1\over 2})\,k!}\,\left({v\over
2}\right)^{k}\sum_{M=0}^{\infty}{\Gamma({1\over 2}-k+M)\over\Gamma({1\over
2}-k)\,M!}{V(z)^{M}(Nz^{N}-zV^{\prime}(z))dz\over\big{(}z^{N}+1\big{)}^{{1\over
2}-k+M}}$ (2.18)

Note that all reference to the larger branch points has been translated into
analytic dependence in $z$, and the above expression for $\lambda$ can be used
only to calculate the periods spanned by the small branch points.

####  2.3.2 The short SW periods in terms of $R(\zeta)$

The short SW periods may be evaluated in terms of the function $R(\zeta)$
defined for $\zeta^{N}=-1$333We shall use the symbol $\zeta$ for the $N$-th
roots of unity satisfying $\zeta^{N}=-1$ here in order to clearly distinguish
them from the arbitrary $2N$-th roots of unity denoted by $\xi$ in the
preceding subsection. as the integral from either one of the
${\mathbb{Z}}_{N}$ symmetric points, denoted here by $z=0$, to the small
branch point $z=\zeta$ by,

$\displaystyle i\pi R(\zeta)=\int_{0}^{\zeta}\lambda\hskip
72.26999pt\zeta^{N}=-1$ (2.19)

The integral is taken along a path from $z=0$ to $z=\zeta$ that does not
intersect any of the branch cuts produced by the square root, as shown in
green in the right panel of figure 1. As in the case of the expansion around
the ${\mathbb{Z}}_{2N}$ symmetric point, swapping the sign of the
${\mathbb{Z}}_{N}$ symmetric point amounts to reversing the sign of all
periods and is equivalent to the modular transformation
$-I\in{\text{Sp}}(2N-2,{\mathbb{Z}})$. The integrals $R(\zeta)$ will soon be
related by analytic continuation to the integrals $Q(\xi)$ and hence to the
periods $a_{I}$ and $a_{D,I}$ considered in [15]. Choosing a basis for the
short homology one-cycles,

$\displaystyle\hat{\mathfrak{A}}_{j}^{(s)}$
$\displaystyle=[\varepsilon^{4j-3},\varepsilon^{4j-1}]$
$\displaystyle\mathfrak{A}_{i}^{(s)}$
$\displaystyle=\bigcup_{j=1}^{i}\hat{\mathfrak{A}}_{j}^{(s)}$
$\displaystyle\mathfrak{B}^{(s)}_{i}$
$\displaystyle=[\varepsilon^{4i-1},\varepsilon^{4i+1}]$ $\displaystyle i$
$\displaystyle=1,\cdots\left[\tfrac{N-1}{2}\right]$ (2.20)

the short periods $a_{i}^{(s)}$ and $a_{D,i}^{(s)}$ may be expressed in terms
of the periods $a_{I}$ and $a_{D,I}$ and in terms of the function $Q(\xi)$ as
follows,

$\displaystyle\hat{a}_{i}^{(s)}$
$\displaystyle=\hat{a}_{2i}+a_{D,2i-1}=R(\varepsilon^{4i-1})-R(\varepsilon^{4i-3})$
$\displaystyle a_{i}^{(s)}$ $\displaystyle=\sum_{j=1}^{i}\hat{a}^{(s)}_{j}$
$\displaystyle a_{D,i}^{(s)}$
$\displaystyle=\hat{a}_{2i+1}+a_{D,2i}=R(\varepsilon^{4i+1})-R(\varepsilon^{4i-1})$
$\displaystyle i$ $\displaystyle=1,\cdots\left[\tfrac{N-1}{2}\right]$ (2.21)

The short homology cycles $\mathfrak{A}_{1}^{(s)}$ and
$\mathfrak{B}_{1}^{(s)}$ are indicated in figure 1 for ${\text{SU}}(3)$.

####  2.3.3 Expansion of $R(\zeta)$ and the short periods

We are now ready to formulate and prove one of the fundamental results of this
paper, namely the expansion of the short periods around the maximal AD points
for arbitrary gauge group ${\text{SU}}(N)$. As shown in the preceding
subsection, the periods are given by (2.3.2) in terms of the function
$R(\zeta)$ defined in (2.19). The results below give the expansion of the
function $R(\zeta)$ around the maximal AD points.

######  Theorem 2.1.

The function $R(\zeta)$ for the small branch points $\zeta$ with
$\zeta^{N}=-1$ admits the following series expansion around $v_{n}=0$ for all
$n=1,\dots,N-2$ and $v_{0}\equiv v=1-u_{0}\neq 0$:

$\displaystyle R(\zeta)$ $\displaystyle=\frac{v^{{1\over
2}+\frac{1}{N}}}{\sqrt{2\pi}N}\sum_{{\ell_{n}=0\atop
n=0,\cdots,N-2}}^{\infty}\frac{(-)^{M+1}\zeta^{L}\Gamma(\frac{L}{N})\Gamma(\ell_{0}+{1\over
2})^{2}}{\Gamma({1\over
2})^{2}\Gamma(\frac{3}{2}-2\alpha+\ell_{0})}\,\frac{v_{0}^{\ell_{0}}\dots
v_{N-2}^{\ell_{N-2}}}{2^{\ell_{0}}\,\ell_{0}!\dots\ell_{N-2}!}$ (2.22)

where the combinations $L,M$ and $\alpha$ were defined in (2.10).

The proof proceeds from the SW differential $\lambda$ in (2.18) and is
relegated to Appendix A.

######  Corollary 2.2.

The summation over $\ell_{0}$ in the Taylor series expansion for $R(\zeta)$
for the small branch points with $\zeta^{N}=-1$ in Theorem 2.1 may be carried
out in terms of an infinite series of Gauss hypergeometric functions
${}_{2}F_{1}=F$ and the result is given by,

$\displaystyle R(\zeta)$ $\displaystyle=\frac{v^{{1\over
2}+\frac{1}{N}}}{\sqrt{2\pi}N}\sum_{{\ell_{n}=0\atop
n=1,\cdots,N-2}}^{\infty}W_{L,M}(\zeta,v)\,\frac{{v}_{1}^{\ell_{1}}\dots
v_{N-2}^{\ell_{N-2}}}{\ell_{1}!\dots\ell_{N-2}!}$ (2.23)

where $L,M$ were defined in (2.10) and the coefficient functions
$W_{L,M}(\zeta,v)$ are given by,

$\displaystyle W_{L,M}(\zeta,v)$
$\displaystyle=\frac{(-)^{M+1}\,\zeta^{L}\,\Gamma(\tfrac{L}{N})}{\Gamma(\tfrac{3}{2}-2\alpha)}F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{3}{2}-2\alpha;\tfrac{v}{2}).$
(2.24)

In the special case where ${v}_{n}=0$ for all $n\neq 0$, the function
$R(\zeta)$ reduces to,

$\displaystyle R(\zeta)=-\frac{\zeta\,v^{{1\over
2}+\frac{1}{N}}\,\Gamma(\frac{1}{N})}{\sqrt{2\pi}N\,\Gamma(\frac{3}{2}+\frac{1}{N})}F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{3}{2}+\tfrac{1}{N};\tfrac{v}{2}).$
(2.25)

The corollary readily follows from Theorem 2.1, and its proof is left to the
reader.

####  2.3.4 Evaluating $R(\zeta)$ by analytic continuation of $Q(\xi)$ for
$\xi^{N}=-1$

Before addressing the calculation of the long periods, we show that $R(\zeta)$
may be obtained from $Q(\xi)$ by analytic continuation in the variable
$v=1-u_{0}$ for the small branch points, namely $\xi=\zeta$ for which
$\zeta^{N}=-1$. Using these results, we shall then use the same analytic
continuation to obtain the long periods in the next subsection. We begin by
proving the following corollary of Theorem 2.1.

######  Corollary 2.3.

The function $R(\zeta)$ is the analytic continuation in the modulus
$v=1-u_{0}$ of $Q(\xi)$ for the small branch points specified by $\xi=\zeta$
and $\xi^{N}=-1$.

To prove the theorem, we start from the Taylor series expansion (2.9) for
$Q(\xi)$ and re-express the coefficient functions $Y$ of (2.2) using the
reflection formula for the $\Gamma$-function,

$\displaystyle\Gamma(z)\Gamma(1-z)\sin(\pi z)=\pi$ (2.26)

as well as the change of variables $v=1-u_{0}$,

$\displaystyle
Y(\xi^{N},\alpha;u_{0})=-\xi^{N}{\pi^{2}\,f_{3}(\alpha,v)\over\Gamma({\tfrac{1}{2}}-\alpha)^{2}}+{\pi^{2}\,f_{4}(\alpha,v)\over\Gamma(1-\alpha)^{2}}$
(2.27)

in terms of the functions $f_{3}$ and $f_{4}$ given by,

$\displaystyle f_{3}(\alpha,v)$ $\displaystyle=$
$\displaystyle-2\,(1-v)F\left(\alpha+{\tfrac{1}{2}},\alpha+{\tfrac{1}{2}};\tfrac{3}{2};(1-v)^{2}\right)$
$\displaystyle f_{4}(\alpha,v)$ $\displaystyle=$ $\displaystyle
F\left(\alpha,\alpha;\tfrac{1}{2};(1-v)^{2}\right)$ (2.28)

By construction, both functions admit a convergent Taylor series expansion
around the point $v=1$, which is the ${\mathbb{Z}}_{2N}$ symmetric point. Our
goal is to perform an analytic continuation to functions that admit convergent
Taylor series expansions around the point $v=0$, which is one of the AD
points. To proceed, it is readily verified that both functions $f_{3},f_{4}$
are solutions to the same hypergeometric differential equation,

$\displaystyle
v(2-v)\frac{d^{2}f}{dv^{2}}+(4\alpha+1)(1-v)\frac{df}{dv}-4\alpha^{2}f=0$
(2.29)

The solutions to this equations may alternatively be expressed in terms of
hypergeometric functions with argument $v/2$, whose normalizations are
conveniently chosen as follows,

$\displaystyle f_{1}(\alpha,v)$ $\displaystyle=$
$\displaystyle{2\,(2v)^{{1\over 2}-2\alpha}\over
1-4\alpha}F\left({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{3}{2}-2\alpha;\tfrac{v}{2}\right)$
$\displaystyle f_{2}(\alpha,v)$ $\displaystyle=$ $\displaystyle
F(2\alpha,2\alpha;2\alpha+{\tfrac{1}{2}};\tfrac{v}{2})$ (2.30)

The two bases of solutions to (2.29) are related by a matrix ${\cal
S}\in{\text{SL}}(2,{\mathbb{R}})$,

$\displaystyle\left(\begin{matrix}f_{1}(\alpha,v)\cr
f_{2}(\alpha,v)\cr\end{matrix}\right)={\cal
S}\left(\begin{matrix}f_{3}(\alpha,v)\cr
f_{4}(\alpha,v)\cr\end{matrix}\right)\hskip 72.26999pt{\cal
S}=\left(\begin{matrix}{\cal S}_{13}&{\cal S}_{14}\\\ {\cal S}_{23}&{\cal
S}_{24}\cr\end{matrix}\right)$ (2.31)

The resulting expression for $f_{2}$ is the Gauss-Kummer quadratic
transformation of hypergeometric functions. The matrix elements of ${\cal S}$
are given as follows,[22]

$\displaystyle{\cal S}_{13}(\alpha)$ $\displaystyle={\Gamma({1\over
2})\Gamma({1\over 2}-2\alpha)\over\Gamma({1\over 2}-\alpha)^{2}}$
$\displaystyle{\cal S}_{14}(\alpha)$ $\displaystyle={\Gamma({1\over
2})\Gamma({1\over 2}-2\alpha)\over\Gamma(1-\alpha)^{2}}$ $\displaystyle{\cal
S}_{23}(\alpha)$ $\displaystyle={\Gamma({1\over 2})\Gamma({1\over
2}+2\alpha)\over\Gamma(\alpha)^{2}}$ $\displaystyle{\cal S}_{24}(\alpha)$
$\displaystyle={\Gamma({1\over 2})\Gamma({1\over
2}+2\alpha)\over\Gamma({1\over 2}+\alpha)^{2}}$ (2.32)

One verifies that indeed ${\rm det\,}({\cal S})=1$ by using the reflection
relation (2.26). Inspection of the coefficients ${\cal S}_{13}$ and ${\cal
S}_{14}$ reveals that, for the special case where $\xi^{N}=-1$, the
combination $Y(-1,\alpha;u_{0})$ is proportional to the function
$f_{1}(\alpha,v)$, namely,

$\displaystyle
Y(-1,\alpha;u_{0})={\pi^{2}\,f_{1}(\alpha,v)\over\Gamma({\tfrac{1}{2}})\Gamma({1\over
2}-2\alpha)}$ (2.33)

Substituting this expression into (2.9) readily produces the expressions of
Theorem 2.1 in (2.23) and (2.24), which concludes the proof of Theorem 2.3.
Henceforth, we shall set $R(\zeta)=Q(\zeta)$ for $\zeta^{N}=-1$ and express
the short SW periods in terms of $Q(\zeta)$.

####  2.3.5 Expansion of $Q(\xi)$ for the long periods by analytic
continuation

Obtaining the expansion of the long SW periods around the maximal AD points
directly from the integral representation of the periods is considerably more
involved than the evaluation for the short periods given in Theorem 2.1.
Instead of proceeding directly here, we shall take advantage of the analytic
continuation of the expansion for $Q(\xi)$ for $\xi^{N}=1$ around the
${\mathbb{Z}}_{2N}$ symmetric point to obtain the long periods. The long
homology cycles are chosen to be,

$\displaystyle\hat{\mathfrak{A}}_{i}^{(\ell)}$
$\displaystyle=\hat{\mathfrak{A}}_{2i-1}-\mathfrak{B}_{2i-1}$
$\displaystyle\mathfrak{A}_{i}^{(\ell)}$
$\displaystyle=\bigcup_{j=1}^{i}\hat{\mathfrak{A}}_{j}^{(\ell)}$
$\displaystyle\mathfrak{B}_{i}^{(\ell)}$
$\displaystyle=-\hat{\mathfrak{A}}_{2i}+\mathfrak{B}_{2i}$ $\displaystyle i$
$\displaystyle=1,\cdots\left[\tfrac{N}{2}\right]$ (2.34)

The long periods will be denoted by $a_{i}^{(\ell)}$ and $a_{D,i}^{(\ell)}$
and may be expressed as follows,

$\displaystyle\hat{a}_{i}^{(\ell)}$ $\displaystyle=\hat{a}_{2i-1}-a_{D,2i-1}$
$\displaystyle a_{i}^{(\ell)}$
$\displaystyle=\sum_{j=1}^{i}\hat{a}_{j}^{(\ell)}$ $\displaystyle
a_{D,i}^{(\ell)}$ $\displaystyle=-\hat{a}_{2i}+a_{D,2i}$ $\displaystyle i$
$\displaystyle=1,\cdots\left[\tfrac{N}{2}\right]$ (2.35)

We alert the reader to the fact that the ranges of the index $i$ labelling the
short and long cycles (and periods) coincide for odd values of $N$ but differ
when $N$ is even, in which case there is one more pair of long periods than
short periods. Using these definitions and (2.1) one verifies that the short
and long cycles satisfy the following canonical intersection pairings,

$\displaystyle\mathfrak{J}(\mathfrak{A}^{(\ell)}_{i},\mathfrak{B}_{j}^{(\ell)})$
$\displaystyle=\delta_{i,j}$ $\displaystyle i,j$
$\displaystyle=1,\cdots\left[\tfrac{N}{2}\right]$
$\displaystyle\mathfrak{J}(\mathfrak{A}^{(s)}_{i},\mathfrak{B}_{j}^{(s)})$
$\displaystyle=\delta_{i,j}$ $\displaystyle i,j$
$\displaystyle=1,\cdots\left[\tfrac{N-1}{2}\right]$ (2.36)

while all other pairings of the cycles
$\mathfrak{A}^{(\ell)}_{i},\mathfrak{B}_{j}^{(\ell)},\mathfrak{A}^{(s)}_{i}$,
and $\mathfrak{B}_{j}^{(s)}$ vanish. The long and short homology cycles
$\mathfrak{A}_{1}^{(\ell)},\mathfrak{B}_{1}^{(\ell)},\mathfrak{A}_{1}^{(\ell)}$
and $\mathfrak{B}_{1}^{(\ell)}$ are indicated in figure 1 for
${\text{SU}}(3)$.

The Taylor series expansion of $Q(\xi)$ for $\xi^{N}=1$ at the maximal AD
point in powers of the moduli $v_{n}$ for $n>0$ is given by the following
theorem.

######  Theorem 2.4.

The Taylor series expansion near the AD point of the function $Q(\xi)$ for
$\xi^{N}=1$ is given by the following expression,

$\displaystyle Q(\xi)=\sum_{{\ell_{n}=0\atop
n=1,\cdots,N-2}}^{\infty}\frac{2^{M-\frac{L}{N}}}{2\pi^{2}N}\xi^{L}\Gamma(\tfrac{L}{N})Y(1,\alpha;u_{0})v^{\frac{NM-L+1}{N}}\frac{{v}_{1}^{\ell_{1}}\dots
v_{N-2}^{\ell_{N-2}}}{\ell_{1}!\dots\ell_{N-2}!},$ (2.37)

where $u_{0}=1-v$, the combinations $L,\alpha$ were defined in (2.10), and
$Y(1,\alpha;u_{0})$ is given by,

$\displaystyle
Y(1,\alpha;u_{0})=-\Gamma({\tfrac{1}{2}})\Gamma({\tfrac{1}{2}}+2\alpha)f_{1}(\alpha,v)+{2\pi^{2}\Gamma({\tfrac{1}{2}})\Gamma({\tfrac{1}{2}}-2\alpha)\over{\Gamma(1-\alpha)^{2}\Gamma({1\over
2}-\alpha)^{2}}}f_{2}(\alpha,v)$ (2.38)

The functions $f_{1}(\alpha,v)$ and $f_{2}(\alpha,v)$ were defined in (2.3.4).
In the special case where $v_{n}=0$ for $n>0$, only the term with $\ell_{n}=0$
for $n>0$ contributes so that $L=1$, and we have,

$\displaystyle{Q(\xi)}=\frac{2^{-\frac{1}{N}}}{2\pi^{2}N}\,\xi\,\Gamma(\tfrac{1}{N})Y(1,-\tfrac{1}{2N};u_{0})$
(2.39)

The proof of the theorem follows from using the relation (2.31) to re-express
$Q(\xi)$.

###  2.4 The case of gauge group ${\text{SU}}(3)$

The ${\cal N}=2$ super-Yang-Mills theory with gauge group ${\text{SU}}(3)$
offers one of the simplest settings in which the AD theories arise. For this
reason, and because one can at the same time obtain simplified and more
explicit formulas for the periods than in the case of arbitrary $N$, we shall
study the behavior of the ${\text{SU}}(3)$ theory in detail here. The formulas
we shall obtain may also be compared with various known results available in
the literature [3]. In terms of the moduli $u=u_{1}$ and $v=1-u_{0}$ the SW
curve and differential are given by,

$\displaystyle y^{2}=(x^{3}-ux+v)(x^{3}-ux+v-2)\hskip
72.26999pt\lambda={(3x^{3}-ux)dx\over y}$ (2.40)

Since the two factors in $y^{2}$ have no common roots, the zeros of the
discriminant of this curve obey either $4u^{3}=27v^{2}$ or
$4u^{3}=27(v-2)^{2}$.

####  2.4.1 Series expansion of the short and long periods

Using Corollary 2.2 for $N=3$ and for the case $\xi^{3}=-1$ we obtain the
following expansion for the small branch points,

$\displaystyle Q(\xi)=-{v^{\frac{5}{6}}\over
3\sqrt{2}\pi}\sum_{m=0}^{\infty}\xi^{m+1}{\Gamma(\tfrac{m+1}{3})\Gamma({1\over
2})\over\Gamma(\tfrac{11-4m}{6})\,m!}\left(-{u\over
v^{\frac{2}{3}}}\right)^{m}\,F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{11-4m}{6};\tfrac{v}{2})$
(2.41)

while using Theorem 2.4 for $N=3$ and for the case $\xi^{3}=1$ we obtain the
expansion for the large branch points,

$\displaystyle Q(\xi)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}{2^{(2m-1)/3}\over
6}\xi^{m+1}\Gamma(\tfrac{m+1}{3}){u^{m}\over
m!}\bigg{[}{\Gamma(\frac{4m-5}{6})\over\Gamma({1\over
2})^{3}}(2v)^{\frac{5-4m}{6}}F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{11-4m}{6};\tfrac{v}{2})$
(2.42) $\displaystyle\hskip 115.63243pt+{2\Gamma({1\over
2})\Gamma(\frac{5-4m}{6})\over\Gamma(\frac{2-m}{3})^{2}\Gamma(\frac{7-2m}{6})^{2}}F(\tfrac{2m-1}{3},\tfrac{2m-1}{3};\tfrac{4m+1}{6};\tfrac{v}{2})\bigg{]}$

We note that $Q(\xi)$ is analytic in $u$, but non-analytic in $v$ as it
contains powers of $v^{\frac{1}{6}}$ for all values of $\xi$. For $u=0$, its
dependence on $v$ is through a factor of $v^{\frac{5}{6}}$ times integer
powers of $v$. This scaling behavior for small $v/\Lambda^{N}$ is consistent
with the predictions of the scaling dimension $\Delta=\tfrac{6}{5}$ for the
intrinsic Coulomb branch of rank 1 AD theories [23, 11, 24, 25, 26, 27, 28,
29]. We shall return to this point in later sections.

####  2.4.2 Analyticity of the long periods

On physical grounds, the long periods, namely those associated with the
embedding of the AD theory into the ${\text{SU}}(3)$ super Yang-Mills theory,
are expected to be analytic in all moduli $u,v$ for $|u|,|v|\ll 1$. The fact
that this is the case is borne out by the following proposition.

######  Proposition 2.5.

There exists an ${\text{Sp}}(4,{\mathbb{Z}})$ electric-magnetic duality frame
such that one pair of periods is analytic in the moduli $(u,v)$, while the
other pair of periods carry the non-analyticities associated with the AD
point.

To prove the proposition, consider the following ${\text{Sp}}(4,{\mathbb{Z}})$
duality transformation, which implements the relations (2.3.2) and (2.3.5) for
the special case of $N=3$,

$\displaystyle\begin{pmatrix}a^{(\ell)}\\\ a^{(s)}\\\ a_{D}^{(\ell)}\\\
a_{D}^{(s)}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}1&0&-1&0\\\
-1&1&1&0\\\ 1&-1&0&1\\\ 0&-1&0&1\end{pmatrix}\begin{pmatrix}a_{1}\\\ a_{2}\\\
a_{D,1}\\\ a_{D,2}\end{pmatrix}$ (2.43)

where we have suppressed the sole index $i=1$. One may verify that the
corresponding cycles satisfy the canonical intersection relations of (2.3.5).
It will be convenient to use the decomposition of $Q(\xi)$ into characters of
${\mathbb{Z}}_{6}$, familiar from [15],

$\displaystyle Q(\xi)$ $\displaystyle=\sum_{n=0}^{5}\xi^{n}Q_{n}$
$\displaystyle Q_{3}$ $\displaystyle=0$ (2.44)

where we recall that $Q_{0}$ actually drops out of all SW periods (see
Appendix B). Expressing the long periods in terms of the functions $Q_{n}$, we
obtain,

$\displaystyle a^{(\ell)}$
$\displaystyle=(1+\rho)(Q_{1}-3Q_{4})-\rho(Q_{5}-3Q_{2})$ $\displaystyle
a_{D}^{(\ell)}$ $\displaystyle=(Q_{1}-3Q_{4})+(Q_{5}-3Q_{2})$ (2.45)

where $\rho=\varepsilon^{2}=e^{2\pi i/3}$. The combinations $3Q_{4}-Q_{1}$ and
$3Q_{2}-Q_{5}$ may be obtained using equations (• ‣ B.2) and (B.11) of
appendix B, and are given by,

$\displaystyle 3Q_{4}-Q_{1}$ $\displaystyle={2^{1/3}\over
2\pi^{\frac{3}{2}}}\sum_{\mu=0}^{\infty}{\Gamma(2\mu-{\frac{1}{3}})^{2}\Gamma(\mu+\tfrac{1}{3})\over\Gamma(2\mu+\frac{1}{6})\,(3\mu)!}F(2\mu-\tfrac{1}{3},2\mu-\tfrac{1}{3};2\mu+\tfrac{1}{6};\tfrac{v}{2})\,{u^{3\mu}\over
2^{2\mu}}$ $\displaystyle 3Q_{2}-Q_{5}$ $\displaystyle={2^{-1/3}\over
2\pi^{\frac{3}{2}}}\sum_{\mu=0}^{\infty}{\Gamma(2\mu+\frac{1}{3})^{2}\Gamma(\mu+\tfrac{2}{3})\over\Gamma(2\mu+\frac{5}{6})(3\mu+1)!}F(2\mu+\tfrac{1}{3},2\mu+\tfrac{1}{3};2\mu+\tfrac{5}{6};\tfrac{v}{2})\,{u^{3\mu+1}\over
2^{2\mu}}$ (2.46)

Hence, it is is clear by inspection that all non-analytic behavior of the long
periods completely cancels, as is expected on physical grounds.

####  2.4.3 ${\text{SU}}(3)$ periods via elliptic functions and modular forms

The expansion of the SW periods near the AD points for ${\text{SU}}(3)$ gauge
group may be obtained in terms of elliptic functions and modular form, in
parallel to the results of [15] for the expansion around the
${\mathbb{Z}}_{6}$ symmetric point. We shall adopt the notations and
conventions of appendix C in [15]. The derivation of these results is
analogous to the one used in section 15 of [30], and will not be presented
here.

We begin by parametrizing the genus 2 curve for $N=3$ given in (2.40) as
follows,

$\displaystyle(2\omega)^{2}x$ $\displaystyle=$ $\displaystyle\wp(z|\tau)$
$\displaystyle 4(2\omega)^{4}u$ $\displaystyle=$ $\displaystyle
g_{2}(\tau)={{{\sf E}_{4}(\tau)\over 12}}$ $\displaystyle-4(2\omega)^{6}v$
$\displaystyle=$ $\displaystyle g_{3}(\tau)=-{{{\sf E}_{6}(\tau)\over 216}}$
(2.47)

where the Weierstrass function $\wp(z|\tau)$ has periods $2\omega$ and
$2\omega\tau$, and satisfies the relation
$\wp^{\prime}(z|\tau)^{2}=4\wp(z|\tau)^{3}-g_{2}(\tau)\wp(z|\tau)-g_{3}(\tau)$,
while ${\sf E}_{4}(\tau)$ and ${\sf E}_{6}(\tau)$ are the modular forms of
weight 4 and 6 respectively, normalized to the value 1 at the cusp $i\infty$.
In terms of the parametrization (2.4.3), the SW curve (2.40) becomes,

$\displaystyle y^{2}$ $\displaystyle=$
$\displaystyle(4\omega)^{-12}(4\wp^{3}-g_{3}\wp-g_{3})(4\wp^{3}-g_{2}\wp-
g_{3}-4(2\omega)^{6})$ (2.48)

Suitably deformining the short cycles
$\mathfrak{A}=\mathfrak{A}^{(s)}_{1}=[0,2\pi i]$ and
$\mathfrak{B}=\mathfrak{B}^{(s)}_{1}=[0,2\pi i\tau]$ in order to avoid the
double-pole of $\wp(z)$ at $z=0$, the elliptic integrals defined by,

$\displaystyle A_{n}$ $\displaystyle=\frac{1}{2\pi
i}\oint_{\mathfrak{A}}dz\;\wp(z)^{n}$ $\displaystyle B_{n}$
$\displaystyle=\frac{1}{2\pi i}\oint_{\mathfrak{B}}dz\;\wp(z)^{n}$ (2.49)

satisfy the following recursion that holds for $J_{k}\in\\{A_{k},B_{k}\\}$:

$\displaystyle(8n-4)J_{n}=(2n-3)g_{2}J_{n-2}+(2n-4)g_{3}J_{n-3}$ (2.50)

with the initial conditions $A_{0}=1$ and $B_{0}=\tau$. The solution is given
by,

$\displaystyle A_{n}=K_{n}+\frac{{\sf E}_{2}}{12}L_{n-1}\hskip
72.26999ptB_{n}=\tau A_{n}+\frac{1}{2\pi i}L_{n-1}$ (2.51)

where $K_{n}$ and $L_{n}$ are modular forms of weight $2n$, determined by the
recursion relation (2.50) and the initial conditions. We have $K_{1}=L_{1}=0$
and,

$\displaystyle K_{0}$ $\displaystyle=1$ $\displaystyle K_{2}$
$\displaystyle=\tfrac{{\sf E}_{4}}{144}$ $\displaystyle K_{3}$
$\displaystyle=-\tfrac{{\sf E}_{6}}{2160}$ $\displaystyle K_{4}$
$\displaystyle=\tfrac{5\,{\sf E}_{4}^{2}}{48384}$ $\displaystyle K_{5}$
$\displaystyle=-\tfrac{{\sf E}_{4}\,{\sf E}_{6}}{77760}$ $\displaystyle L_{0}$
$\displaystyle=1$ $\displaystyle L_{2}$ $\displaystyle=\tfrac{{\sf
E}_{4}}{80}$ $\displaystyle L_{3}$ $\displaystyle=-\tfrac{{\sf E}_{6}}{1512}$
$\displaystyle L_{4}$ $\displaystyle=\tfrac{7\,{\sf E}_{4}^{2}}{48384}$
$\displaystyle L_{5}$ $\displaystyle=-\tfrac{29\,{\sf E}_{4}\,{\sf
E}_{6}}{1330560}$ (2.52)

The short periods are then given by expanding the SW differential in powers of
$1/\omega$ as given by the following theorem, which offers a non-trivial
extension of the calculation of short periods carried out to leading order in
large $\Lambda$ in [16].

######  Theorem 2.6.

The small periods $a(\tau)=a^{(s)}_{1}(\tau)$ and
$a_{D}(\tau)=a^{(s)}_{D,1}(\tau)$ admit the following Taylor series-expansion
in terms of the basis $\\{{\sf E}_{2},{\sf E}_{4},{\sf E}_{6}\\}$ of the ring
of quasi-modular forms, along with the variable $\omega$,

$\displaystyle a$ $\displaystyle=$
$\displaystyle{-i\over\sqrt{2\pi}}\sum_{k,\ell,m=0}^{\infty}{\Gamma(k+\ell+m+{1\over
2})\over
2^{k+3\ell+3m}k!\,\ell!\,m!}{(-)^{\ell+m}\,g_{2}^{\ell}\,g_{3}^{m}\over(2\omega)^{6(k+\ell+m)+5}}\left(3A_{3k+\ell+3}-{g_{2}\over
4}A_{3k+\ell+1}\right)$ $\displaystyle a_{D}$ $\displaystyle=$
$\displaystyle{-i\over\sqrt{2\pi}}\sum_{k,\ell,m=0}^{\infty}{\Gamma(k+\ell+m+{1\over
2})\over
2^{k+3\ell+3m}k!\,\ell!\,m!}{(-)^{\ell+m}\,g_{2}^{\ell}\,g_{3}^{m}\over(2\omega)^{6(k+\ell+m)+5}}\left(3B_{3k+\ell+3}-{g_{2}\over
4}B_{3k+\ell+1}\right)\qquad$ (2.53)

As a result, the following combination can be expressed in terms of modular
forms $\\{{\sf E}_{4},{\sf E}_{6}\\}$

$\displaystyle a_{D}-\tau
a={-1\over(2\pi)^{\frac{3}{2}}}\sum_{k,\ell,m=0}^{\infty}{\Gamma(k+\ell+m+{1\over
2})\over
2^{k+3\ell+3m}k!\,\ell!\,m!}{(-)^{\ell+m}\,g_{2}^{\ell}\,g_{3}^{m}\over(2\omega)^{6(k+\ell+m)+5}}\left(3L_{3k+\ell+2}-{g_{2}\over
4}L_{3k+\ell}\right)$ (2.54)

which is a locally holomorphic modular form of weight $-1$ provided $\omega$
is assigned holomorphic weight $1$ and $L_{n}$ has weight $2n$.

#### Remarks

  1. 1.

The factor $2\omega$ in the denominator of each formula in Theorem 2.6 may be
eliminated in favor of either the variables $(u,g_{2}(\tau))$ or the variables
$(v,g_{3}(\tau))$ using (2.4.3) depending on whether the expansion is sought
near the points $\tau=i$ or $\tau=\rho=e^{2\pi i/3}$ respectively.

  2. 2.

The combination $a_{D}-\tau a$ vanishes at the AD point $\tau=\rho$, since we
have $g_{2}(\rho)=0$, and the recursion relation implies $L_{3k+2}=0$ for all
$k\geq 0$. Thus, at the AD point, we have $a_{D}=\tau a$, as must be the case
for any rank-1 ${\mathcal{N}}=2$ SCFT.

  3. 3.

The ${\mathbb{Z}}_{3}$ symmetry of the AD point fixes the modulus $\tau=\rho$
and $g_{2}(\rho)=0$. In the neighborhood of the AD point, the combination
$|g_{2}(\tau)^{3}/g_{3}(\tau)^{2}|=4|u|^{3}/|v|^{2}$ is small, a condition
that coincides with the original assumption for the validity of the expansion.

  4. 4.

There exists a different potential superconformal fixed point at $\tau=i$ that
preserves ${\mathbb{Z}}_{2}$ symmetry, and where $v=g_{3}(i)=0$ and
$g_{2}(i)=(2\omega)^{4}4u$. The sum over $m$ in (2.54) then collapses to the
$m=0$ contribution, the recursion relation (2.50) for $L_{n}$ may be solved,
and the remaining dependence on $\omega$ may be eliminated in favor of $u$,

$\displaystyle a_{D}-\tau
a=-\frac{u^{\frac{5}{4}}}{9\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{2^{-10n}\,\Gamma(n+\frac{1}{4})^{2}\Gamma(n+\frac{3}{4})^{2}}{\Gamma(\frac{7}{12})\Gamma(\frac{11}{12})\Gamma(n+\frac{13}{12})\Gamma(n+\frac{17}{12})\Gamma(n+\frac{1}{2})n!}\left({u\over
3}\right)^{3n}$ (2.55)

The above series has radius of convergence $\absolutevalue{u}<3$. The scaling
dimension of the operator corresponding to the modulus $u$ is
$\Delta(u)=\frac{4}{5}<1$ is below the unitarity bound. This implies that
there is no consistent way to take $\Lambda\rightarrow\infty$ such that the
resulting theory is unitary and superconformal.

###  2.5 Remarks on the convergence of the ${\mathbb{Z}}_{N}$ expansion

In this subsection we shall discuss the convergence properties of the
expansion around the maximal AD point, briefly for the ${\text{SU}}(N)$ case,
and in more detail for ${\text{SU}}(3)$.

####  2.5.1 Heuristic analysis of the ${\text{SU}}(N)$ case

As illustrated in the right panel of figure 1 for the case of
${\text{SU}}(3)$, the branch points split into a set of small branch points of
order ${\cal O}(v^{\frac{1}{N}})$ and large branch points of order ${\cal
O}(\Lambda)$. The starting point for our expansion is a point in the moduli
space of the Coulomb branch where $v\neq 0$ and $v_{n}=0$ for all $n>0$. The
non-vanishing of $v$ guarantees that the small branch points $x^{-}_{k}$
remain well-separated. Turning on the moduli $v_{n}$ for $n\neq 0$ we observe
that the parameter entering from the Taylor expansion in corollary 2.2 is
$v_{n}$. The expansion will remain convergent as long as no two branch points
are brought to coincide with one another, which requires the parameter $u_{n}$
to remain sufficiently small with respect to $v^{(N-n)/N}$. Thus, the
conditions for convergence, derived on heuristic grounds, are as follow,

$\displaystyle|v|\ll|\Lambda|^{N}\hskip 72.26999pt{|v_{n}|\ll 1}$ (2.56)

For the case of ${\text{SU}}(3)$ we can make these bounds more precise.

####  2.5.2 Detailed analysis of the ${\text{SU}}(3)$ case

In appendix B.3, we provide a detailed derivation of the convergence
conditions for the ${\mathbb{Z}}_{3}$ series. Here, we remark on its
consequences and compare it with the ${\mathbb{Z}}_{6}$ series of [15].

  1. 1.

The ${\mathbb{Z}}_{6}$ expansion converges provided the moduli satisfy the
following inequalities [15],

$\displaystyle\tfrac{2}{\sqrt{27}}\absolutevalue{u}^{\frac{3}{2}}+\absolutevalue{1-v}<1$
(2.57)

In figure 2(a), the green translucent region shows the domain of convergence
of the ${\mathbb{Z}}_{6}$ expansion. The AD points are located on the boundary
of this domain and the three multi-monopole points are mapped to the red dot
at the peak of the conical region.

  2. 2.

The ${\mathbb{Z}}_{3}$ expansion converges provided the moduli satisfy the
inequalities,

$\displaystyle\absolutevalue{\tfrac{4u^{3}}{27}}<\absolutevalue{v}^{2}<1$
(2.58)

In figure 2(b), the red translucent cylindrical region minus the solid cone
shows the domain of convergence of the ${\mathbb{Z}}_{3}$ expansion. The AD
point is at the peak of the cone. The multi-monopole and ${\mathbb{Z}}_{6}$
points are on the boundary of the domain of convergence.

  3. 3.

Figure 3 shows the total region that we can access with the combined
expansions.

(a) Convergence region of ${\mathbb{Z}}_{6}$ expansion

(b) Convergence region of ${\mathbb{Z}}_{3}$ expansion

Figure 2: All the translucent colored regions denote convergence in the
coordinates $({\rm Re\,}v,{\rm Im\,}v,\absolutevalue{u})$, but the opaque
colored regions are excluded by convergence. Figure 3:  This plot shows the
total region of moduli-space that we can access using the two
${\mathbb{Z}}_{3}$ and ${\mathbb{Z}}_{6}$ expansions: the solid blue and red
cones are excluded from the regions of convergence of the ${\mathbb{Z}}_{3}$
expansions, and the translucent regions denote convergence.

##  3 Candidate walls of marginal stability revisited

In this section, we shall reanalyze the candidate walls of marginal stability
proposed in [15] for ${\text{SU}}(3)$, this time from the perspective of the
expansion of the periods around the AD points. We shall check agreement of the
results obtained by the two expansions on the slice with $u=0$ and map out
more walls of marginal stability beyond the $v$-plane for ${\text{SU}}(3)$,
and analyze marginal stability in the ${\text{SU}}(4)$ case on the slice
$u_{1}=u_{2}=0$.

###  3.1 Setup

In this subsection, we shall briefly summarize the setup of [15] used to
analyze the marginal stability of BPS states. At a generic point on the
Coulomb branch, the ${\text{SU}}(N)$ gauge-group is spontaneously broken to
its maximal Abelian subgroup ${\text{U}}(1)^{N-1}$. With respect to this
unbroken gauge group, the states in the theory carry both electric charges
${\bf q}=(q_{1},\cdots,q_{N-1})$ and magnetic charges ${\bf
g}=(g_{1},\cdots,g_{N-1})$, which we shall assemble into a single multiplet,

$\displaystyle\mu=({\bf q},{\bf
g})=(q_{1},\dots,q_{N-1};g_{1},\dots,g_{N-1})\in{\mathbb{Z}}^{N-1}\times{\mathbb{Z}}^{N-1}$
(3.1)

The central charge $Z[\mu]$ of the ${\cal N}=2$ supersymmetry algebra,
evaluated in a state with charge vector $\mu$, is a linear function of $\mu$
given by [1],

$\displaystyle Z[\mu]=\sum_{I=1}^{N-1}(q_{I}a_{I}+g_{I}a_{D,I})$ (3.2)

In a unitary theory the mass $M$ of any state with charge vector $\mu$
satisfies the BPS bound $\absolutevalue{Z[\mu]}\leq M$. The state is a BPS
state provided its mass $M$ saturates the BPS bound,

$\displaystyle M=\absolutevalue{Z[\mu]}$ (3.3)

Two BPS states with charges $\mu=({\bf q},{\bf g})$ and $\mu^{\prime}=({\bf
q}^{\prime},{\bf g}^{\prime})$ obey the Dirac quantization condition
$D\in{\mathbb{Z}}$ where $D$ is given by the symplectic pairing of the charges
$\mu$ and $\mu^{\prime}$,

$\displaystyle D={\bf q}\cdot{\bf g}^{\prime}-{\bf g}\cdot{\bf
q}^{\prime}=\sum_{I=1}^{N-1}\Big{(}q_{I}g_{I}^{\prime}-q_{I}^{\prime}g_{I}\big{)}$
(3.4)

When $D=0$, one may perform an ${\text{Sp}}(2N-2,{\mathbb{Z}})$ duality
transformation to new charges $\tilde{\mu}$ and $\tilde{\mu}^{\prime}$ that
have vanishing magnetic components, and are therefore mutually local. By
contrast, when $D\not=0$, the corresponding BPS states are mutually non-local.
This non-locality is, of course, familiar in the semi-classical limit where
electric charges are light and magnetic monopoles are heavy soliton states
such as the ‘t Hooft-Polyakov monopole. The novelty of the AD theories is the
presence of massless mutually non-local states and fields.

Two BPS states with charges $\mu,\mu^{\prime}$ and masses
$M=\absolutevalue{Z[\mu]},M^{\prime}=\absolutevalue{Z[\mu^{\prime}]}$,
respectively, can form a bound state of charge $\mu+\mu^{\prime}$ provided the
mass $M_{b}$ of the bound state satisfies $M_{b}<M+M^{\prime}$. The mass
$M_{b}$ satisfies the BPS bound $\absolutevalue{Z[\mu+\mu^{\prime}]}\leq
M_{b}$. In general, the inequality will be a strict one and the resulting
bound state will not be a BPS state. For special charge arrangements and for
special values of the vacuum expectation values $a_{I}$ and $a_{D,I}$,
however, two BPS states can form a BPS bound state, namely when,

$\displaystyle Z[\mu^{\prime}]=r\,Z[\mu]\hbox{ for some }r\in{\mathbb{R}}$
(3.5)

For a given pair $\mu,\mu^{\prime}$, the solutions to this equation carve out
a real co-dimension one slice of the Coulomb branch that we refer to as a
candidate wall of marginal stability. Having equality of the mass $M$ of the
bound state with its BPS bound $\absolutevalue{Z[\mu+\mu^{\prime}]}$ on the
wall does open the option of forming a stable non-BPS bound state on either
side of the wall. Whether this option is actually adopted by the theory is a
dynamical question that goes beyond the purely kinematical considerations used
here. For this reason the terminology candidate wall of marginal stability
will be used throughout.

###  3.2 Marginal stability of BPS states in ${\text{SU}}(3)$

Candidate walls of marginal stability were analyzed in [15] for
${\text{SU}}(3)$ on the slice $u=0$ for arbitrary $v$ using the
${\mathbb{Z}}_{6}$ expansion of the periods. In this subsection, we re-examine
these candidate walls of marginal stability using the ${\mathbb{Z}}_{3}$
expansion around one or the other of the AD points, first for $u=0$ and then
for arbitrary $u,v$.

####  3.2.1 The $u=0$ slice

In terms of our expansion around the AD point $u_{0}=1$, given in (2.42),
(2.41) and (2.44), the expressions for the SW periods of (2.8) for the case
$N=3$ are as follows,

$\displaystyle a_{1}$ $\displaystyle=(\rho-1)(Q_{1}-Q_{4})-(3Q_{4}-Q_{1})$
$\displaystyle a_{2}$ $\displaystyle=(1+\rho)a_{1}$ $\displaystyle a_{D,1}$
$\displaystyle=(\rho-1)(Q_{1}-Q_{4})+\rho(3Q_{4}-Q_{1})$ $\displaystyle
a_{D,2}$ $\displaystyle=\rho a_{D,1}$ (3.6)

where $\rho=e^{\frac{2\pi i}{3}}$. Consider BPS states with charge vectors
$\mu=(q_{1},q_{2};g_{1},g_{2})$ and
$\mu^{\prime}=(q_{1}^{\prime},q_{2}^{\prime};g_{1}^{\prime},g_{2}^{\prime})$
and corresponding central charges,

$\displaystyle Z[\mu^{\prime}]$ $\displaystyle=\alpha Q_{1}+\beta Q_{4}$
$\displaystyle Z[\mu]$ $\displaystyle=\gamma Q_{1}+\delta Q_{4}$ (3.7)

where we have defined the following integers of the ring ${\mathbb{Z}}[\rho]$,

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\rho\,(q_{1}^{\prime}-g_{2}^{\prime})-(q_{2}^{\prime}+g_{1}^{\prime})$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-(2+\rho)(q_{1}^{\prime}+g_{2}^{\prime})-(1+2\rho)(q_{2}^{\prime}-g_{1}^{\prime})$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\rho\,(q_{1}-g_{2})-(q_{2}+g_{1})$ $\displaystyle\delta$
$\displaystyle=$ $\displaystyle-(2+\rho)(q_{1}+g_{2})-(1+2\rho)(q_{2}-g_{1})$
(3.8)

We may parametrize the solutions to the equation (3.5) for the candidate wall
of marginal stability in terms of the real variable $r$ as follows,

$\displaystyle r$ $\displaystyle={Z[\mu^{\prime}]\over Z[\mu]}={\alpha
z+\beta\over\gamma z+\delta}$ $\displaystyle z(v)$
$\displaystyle=\frac{Q_{1}(v)}{Q_{4}(v)}$ (3.9)

Inverting the relation between $z$ and the real parameter $r$, for given
charge assignments, will make $z$ trace arcs of circles in the complex
$z$-plane that depend on the particular charge assignments of $\mu$ and
$\mu^{\prime}$. The strong-coupling spectrum of ${\text{SU}}(N)$ SW theory has
been worked out in [31], and the ${\text{SU}}(3)$ case is explained in
Appendix E of [15]. We summarize the results in the following table.

Central charges and masses of BPS dyons near the ${\text{SU}}(3)$ AD points
---
Dyon charge |  Central charge $Z[\mu_{kI}]$ |  $M(u_{0}=+1)$ |  $M(u_{0}=-1)$
$\mu_{01}=(-1,0;-1,0)$ |  $-(\rho-1)(Q_{1}+Q_{4})$ |  1.55632 |  0
$\mu_{12}=(-1,1;0,1)$ |  $-(2\rho+1)(Q_{1}+Q_{4})$ |  1.55632 |  0
$\mu_{21}=(0,1;1,1)$ |  $-(\rho+2)(Q_{1}+Q_{4})$ |  1.55632 |  0
$\mu_{02}=(0,1;0,-1)$ |  $(\rho-1)(Q_{1}-Q_{4})$ |  0 |  1.55632
$\mu_{11}=(1,0;-1,-1)$ |  $(2\rho+1)(Q_{1}-Q_{4})$ |  0 |  1.55632
$\mu_{22}=(1,-1;-1,0)$ |  $(\rho+2)(Q_{1}-Q_{4})$ |  0 |  1.55632

As indicated in the table, each AD point has three mutually non-local dyons
that become simultaneously massless. The equation (3.5) has been solved for
all possible pairs on the $u=0$ plane in [15]. We summarize these results
below, and then build on them.

There are 15 distinct pairwise ratios of the six BPS states. Two sets of three
of these ratios are between massless mutually non-local dyons. These ratios
are independent of $Q_{1}$ and $Q_{4}$ and necessarily complex, such as for
example $Z[\mu_{11}]/Z[\mu_{02}]=-\rho$. There can be no walls of marginal
stability between such pairs. The remaining nine ratios are between one
massive and one massless dyon, they do depend on $Q_{1}$ and $Q_{4}$ through
the ratio $z=Q_{1}/Q_{4}$, and can lead to candidate walls of marginal
stability. To analyze the ratios systematically, we compare the central
charges in the first triplet of mutually non-local dyons
$(Z[\mu_{01}],Z[\mu_{12}],Z[\mu_{21}])$ with cyclic permutations of the second
triplet of mutually non-local dyons $(Z[\mu_{02}],Z[\mu_{11}],Z[\mu_{22}])$,
as follows,

$\displaystyle(Z[\mu_{01}],Z[\mu_{12}],Z[\mu_{21}])$ $\displaystyle=$
$\displaystyle r_{1}\,(Z[\mu_{02}],Z[\mu_{11}],Z[\mu_{22}])$
$\displaystyle(Z[\mu_{01}],Z[\mu_{12}],Z[\mu_{21}])$ $\displaystyle=$
$\displaystyle r_{2}\,(Z[\mu_{22}],Z[\mu_{02}],Z[\mu_{11}])$ (3.10)
$\displaystyle(Z[\mu_{01}],Z[\mu_{12}],Z[\mu_{21}])$ $\displaystyle=$
$\displaystyle r_{3}\,(Z[\mu_{11}],Z[\mu_{22}],Z[\mu_{02}])$

where $z=z(v)$ and,

$\displaystyle r_{1}=-{z+1\over z-1}\hskip 57.81621ptr_{2}=-\rho{z+1\over
z-1}\hskip 57.81621ptr_{3}=\rho^{2}{z+1\over z-1}$ (3.11)

  1. 1.

The reality of $r_{1}$ parametrizes a straight line segment in the complex
$z$-plane that lies on the real-axis. We will not consider such walls because
they are non-compact.

  2. 2.

The reality of $r_{2}$ parametrizes a continuous subset of the circle
$|z+\frac{i}{\sqrt{3}}|^{2}=\frac{4}{3}$ in the complex $z$-plane for a
continuous range of values of $r_{2}\in{\mathbb{R}}$.

  3. 3.

The reality of $r_{2}$ parametrizes a continuous subset of the circle
$|z-\frac{i}{\sqrt{3}}|^{2}=\frac{4}{3}$ in the complex $z$-plane for a
continuous range of values of $r_{3}\in{\mathbb{R}}$.

In this sense, each candidate wall of marginal stability on the $v$-plane has
a three-fold degeneracy, i.e. each wall can be obtained from three distinct
pairs of dyons. Finally, we can numerically map $z(v)$ to the $u_{0}$-plane by
using the expression for $w(v)$ in terms of $v$ or $u_{0}$. This is displayed
in the right panel of figure 4, which reproduces the result of [15]

Figure 4: The candidate walls of marginal stability for ${\text{SU}}(3)$ on
the slice $u=0$ are represented by colored arcs in the $z$-plane in the left
panel and in the $u_{0}$ plane in the right panel. The contour of vanishing
Kähler potential is drawn in black [15].

####  3.2.2 Existence of walls of marginal stability away from the $v$-plane

In this sub-subsection, we give an argument for the existence of walls of
marginal stability in the ${\text{SU}}(3)$ Coulomb branch with $u\neq 0$ over
the curves of marginal stability restricted to the $v$-plane. We expect to
find a marginal stability subspace of real dimension three inside the Coulomb
branch, since we have 2 complex degrees of freedom $u$ and $v$ parametrizing
the Coulomb branch, and one real constraint $Z_{2}/Z_{1}\in{\mathbb{R}}$.
Since such a surface lives in the four-dimensional moduli space, it is rather
hard to visualize and we shall focus instead on the marginal stability
subspace of a three-dimensional slice of the Coulomb branch that we can
visualize more easily. The following proposition shows the existence of walls
in the $u$-plane for fixed values of $v$ that lie on a wall.

######  Proposition 3.1.

For any point $v_{0}$ on a curve of marginal stability in the $v$-plane, there
exists a curve of marginal stability in the $u$-plane that goes through the
point $(u,v)=(0,v_{0})$.

We hold $v=v_{0}$ fixed and consider two central charges $Z_{1}(u)$ and
$Z_{2}(u)$ evaluated at the point $v=v_{0}$, which may be regarded as a
(locally) holomorphic function of a single complex variable $u$. Then we may
define the relative phase of these two central charges as,

$\displaystyle
e^{2i\phi(u,\bar{u})}=\frac{Z_{1}(u)\bar{Z}_{2}(\bar{u})}{Z_{2}(u)\bar{Z}_{1}(\bar{u})}$
(3.12)

In what follows, we will use the fact that the phase of a holomorphic function
$f$ is harmonic for all points where the function is non-zero; this assumption
is necessary since one takes the logarithm of $f$ in the proof of this fact.
However, (potential) vanishing of the central charge $Z_{k}(u)$ does not pose
an issue since we take its complex modulus in the definition of the relative
phase. Then it follows that $\phi(u,\bar{u})$ is locally a harmonic function
on any open set that contains $u=0$. But a harmonic function on a connected
domain ${\cal D}\subset{\mathbb{C}}$ can never attain its extreme values in
the interior. Any point $u_{0}\in{\mathbb{C}}$ that lies on a wall of marginal
stability, i.e. $Z_{2}/Z_{1}\in{\mathbb{R}}$, satisfies
$\phi(u_{0},\bar{u}_{0})=0$. In particular, $\phi(0,0)=0$. This implies that
zero is neither a minimum nor a maximum of $\phi$, and that there exists a
closed subset ${\cal C}\subset\partial{\cal D}$ such that $\phi({\cal C})<0$
and $\phi({\cal C}^{c})\geq 0$. Hence, there exists a curve of marginal
stability that goes through the interior and is continuously connected to
$u=0$.

Remark. The above proposition applies locally in a neighborhood of $u=0$,
where the central charges are analytic. In particular, $Z(u)$ is not required
to be analytic globally, and hence the function $\phi$ would not be globally
harmonic due to the potential non-analyticities in $Z(u)$. Therefore, this
local existence result for walls of marginal stability does not pose an
obstruction to the compactness of the walls of marginal stability.

####  3.2.3 Numerically finding walls of marginal stability beyond the
$v$-plane

In this sub-subsection, we will find candidate walls of marginal stability
using two distinct methods: perturbation theory and numerical integration.

The first method involves first-order perturbation theory in $u$ for a fixed
value of $v$ on the orange arc in figure 4 and a mesh of values of $r$. We
note that the figure does not appreciably change even if we go to high orders
in perturbation theory, as long as we are inside the radius of convergence.

(a) Walls for the a single pair $(\mu_{01},\mu_{22})$

(b) Walls for all three pairs

(c) Full picture of the three walls beyond the radius of convergence

Figure 5: The figure above shows the curves of marginal stability on the
$u$-plane for $u_{0}\sim 0.6184i$ (the midpoint of the orange arc in figure
4). The left panel shows the walls for a single pair of dyons. The red region
is inaccessible by perturbation theory since our ${\mathbb{Z}}_{6}$-expansion
does not converge there, the blue region is where the BPS states $\mu_{01}$
and $\mu_{22}$ are stable with $\phi<0$, and the green region is where bound
states may be formed, i.e. $\phi>0$. The right panel shows a plot of the
curves of marginal stability on the $u$-plane for all three pairs of dyons
simultaneously. Again, the red region is inaccessible by perturbation theory.
The regions of mutual stability ($\phi<0$) and instability ($\phi>0$) for 2
pairs are: II and II′ for $(\mu_{01},\mu_{22})$ and $(\mu_{12},\mu_{02})$, I
and I′ for $(\mu_{12},\mu_{02})$ and $(\mu_{21},\mu_{11})$, III and III′ for
$(\mu_{21},\mu_{11})$ and $(\mu_{01},\mu_{22})$. The panel 5(c) displays the
full walls beyond the radius of convergence obtained from numerical
integration. Blue walls correspond to $(\mu_{01},\mu_{22})$, orange walls to
$(\mu_{12},\mu_{02})$, and purple walls to $(\mu_{21},\mu_{11})$.

We will focus on the segment of the curve produced by the pairs
$(\mu_{01},\mu_{22})$, $(\mu_{12},\mu_{02})$, and $(\mu_{21},\mu_{11})$ that
has positive imaginary part, i.e. the orange curve in figure 4. For any
$u_{0}$ with $\absolutevalue{u_{0}}<1$ on this arc, we have an absolutely
convergent expansion in $u$, and we can get arbitrarily close to the AD points
at $u_{0}=\pm 1$. Recall that our expansions have the following radii of
convergence in the $u$-plane.

$\displaystyle{\mathbb{Z}}_{3}\text{
points}:~{}2^{\frac{1}{3}}\absolutevalue{u}$
$\displaystyle<3\absolutevalue{\pm 1\mp u_{0}}^{\frac{2}{3}}<3\text{ for
}u_{0}=\pm 1\text{ respectively}$ $\displaystyle{\mathbb{Z}}_{6}\text{
point}:~{}2^{\frac{2}{3}}\absolutevalue{u}$
$\displaystyle<3\left(1-\absolutevalue{u_{0}}\right)^{\frac{2}{3}}\quad\text{
and }\quad\absolutevalue{u_{0}}<1.$ (3.13)

On regions of overlapping convergence, recall that
$\text{Radius}_{u}({\mathbb{Z}}_{3})\geq\text{Radius}_{u}({\mathbb{Z}}_{6})$.
Hence, it is more fruitful to apply the ${\mathbb{Z}}_{3}$ expansions near the
AD points because this inequality is significant. On the other hand, we apply
the ${\mathbb{Z}}_{6}$ expansion on the imaginary $v$-axis because that is the
boundary of convergence for the ${\mathbb{Z}}_{3}$ expansion. Precisely at the
AD points, neither expansion converges but we can get arbitrarily close.

However, this method is limited by the radius of convergence of the expansion.
To circumvent this limitation, we recall that the periods of pure
${\text{SU}}(3)$ SW theory satisfy Picard-Fuchs equations in the variables [7]

$\displaystyle(x_{1},x_{2})=\left(\frac{4u^{3}}{27},(1-v)^{2}\right)$ (3.14)

This system of second-order PDEs can be transformed into a system of first-
order ODEs which are numerically integrable (see Appendix D of [15]). We use
this method to compute the periods, central charges, and walls of marginal
stability, by scanning for points on the $u$-plane, for fixed values of $v$ on
the orange arc in figure 4 where $Z[\mu_{2}]/Z[\mu_{1}]$ is real-valued.

#### Remarks

(a) 1 pair $(\mu_{01},\mu_{22})$

(b) All $3$ pairs – degeneracy lifted

Figure 6: A three-dimensional view of the candidate walls of marginal
stability.

  1. 1.

The three pairs correspond to the three walls in figure 5, both related by
${\mathbb{Z}}_{3}$. At the fixed point $u=0$ of this ${\mathbb{Z}}_{3}$
symmetry the degeneracy among the three pairs is restored.

  2. 2.

The arcs in figure 5 with $\phi=0$ are approximately circular near the origin
because, inside the radius of convergence, equation (3.5) was truncated to
first order to give,

$\displaystyle(du-b)$ $\displaystyle\approx r(-cu+a)\Rightarrow
u(r)=\frac{ar+b}{cr+d}+\dots$

for some $a,b,c,d\in{\mathbb{C}}$ with $ad-bc\neq 0$. An equation of this form
traces out an arc of a circle inside the radius of convergence. However, the
arcs are slightly deformed from circles outside the radius of convergence, as
is clear from figure 5(c).

  3. 3.

The plots in figure 6 were created by picking 33 evenly spaced points on the
orange arc of marginal stability in figure 4 and then solving numerically for
$\phi=0$.

  4. 4.

The region of stability is a tubular neighborhood, and there are three such
compact regions corresponding to the three walls that are related by
${\mathbb{Z}}_{3}$.

###  3.3 Marginal stability of BPS states in ${\text{SU}}(4)$

On the slice that contains the AD points (i.e. $u_{k}=0$ for $k\neq 0$), we
have

$\displaystyle a_{1}$ $\displaystyle=\varepsilon Q_{-}-Q_{+}$ $\displaystyle
a_{D,1}$ $\displaystyle=\varepsilon^{2}Q_{+}-\varepsilon Q_{-}$ $\displaystyle
a_{2}$ $\displaystyle=(\varepsilon^{2}+1)a_{1}$ $\displaystyle a_{D,2}$
$\displaystyle=\varepsilon^{2}a_{D,1}$ (3.15) $\displaystyle a_{3}$
$\displaystyle=\varepsilon^{2}a_{1}$ $\displaystyle a_{D,3}$
$\displaystyle=-a_{D,1}$

The central charge can be shown to take the following form ($Q_{\pm}=Q_{1}\pm
Q_{5}$):

$\displaystyle Z[\mu]$
$\displaystyle=(m_{0}+\varepsilon^{2}m_{2})Q_{+}+\varepsilon(n_{0}+\varepsilon^{2}n_{2})Q_{-},$
(3.16)

where

$\displaystyle m_{0}$ $\displaystyle=-q_{1}-q_{2}-g_{2}$ $\displaystyle n_{0}$
$\displaystyle=q_{1}+q_{2}-g_{1}+g_{3}$ $\displaystyle m_{2}$
$\displaystyle=-q_{2}-q_{3}+g_{1}-g_{3}$ $\displaystyle n_{2}$
$\displaystyle=q_{2}+q_{3}-g_{2}$

As for ${\text{SU}}(3)$, the following coordinate will be convenient

$\displaystyle z(v)=\frac{Q_{1}}{Q_{5}}$ (3.17)

The strong-coupling spectrum of BPS dyons in pure
${\mathcal{N}}=2\;{\text{SU}}(N)$ gauge theory was worked out in [31].
Following their algorithm for the $N=4$ case, we see that there are 12 stable
BPS dyons at strong coupling which split into 2 sets of 6, each of which
becomes massless at one or the other of the two AD points corresponding to
$(u_{0},u_{1},u_{2})=(\pm 1,0,0)$. Two dyons within each set of 6 are mutually
local, while two dyons belonging to different sets of 6 are mutually non-
local. The electromagnetic charge vectors
$\mu_{kJ}=(q_{1},q_{2},q_{3};g_{1},g_{2},g_{3})$ for the IR gauge-group
${\text{U}}(1)_{1}\times{\text{U}}(1)_{2}\times{\text{U}}(1)_{3}$ are
displayed in the table that follows.

Masses and central charges of BPS dyons near the ${\text{SU}}(4)$ maximal AD
points
---
Dyon |  $Z[\mu_{kI}]$ |  $M(u_{0}=+1)$ |  $M(u_{0}=-1)$
$\mu_{01}=(-1,0,0;-1,0,0)$ |  $(1-\varepsilon^{2})Q_{+}$ |  1.2828 |  0
$\mu_{03}=(0,1,-1;0,0,-1)$ |  $-(1-\varepsilon^{2})Q_{+}$ |  1.2828 |  0
$\mu_{12}=(-1,1,0;0,1,0)$ |  $-(1+\varepsilon^{2})Q_{+}$ |  1.2828 |  0
$\mu_{32}=(0,0,1;1,1,1)$ |  $-(1+\varepsilon^{2})Q_{+}$ |  1.2828 |  0
$\mu_{21}=(0,1,0;1,1,0)$ |  $-2Q_{+}$ |  1.8142 |  0
$\mu_{23}=(-1,0,1;0,1,1)$ |  $-2\varepsilon^{2}Q_{+}$ |  1.8142 |  0
$\mu_{02}=(0,1,-1;0,-1,0)$ |  $\varepsilon(1+\varepsilon^{2})Q_{-}$ |  0 |  1.2828
$\mu_{22}=(1,0,0;-1,-1,-1)$ |  $\varepsilon(1+\varepsilon^{2})Q_{-}$ |  0 |  1.2828
$\mu_{31}=(0,0,1;0,0,-1)$ |  $-\varepsilon(1-\varepsilon^{2})Q_{-}$ |  0 |  1.2828
$\mu_{33}=(1,-1,0;-1,0,0)$ |  $\varepsilon(1-\varepsilon^{2})Q_{-}$ |  0 |  1.2828
$\mu_{11}=(1,0,-1;-1,-1,0)$ |  $2\varepsilon Q_{-}$ |  0 |  1.8142
$\mu_{13}=(0,1,0;0,-1,-1)$ |  $2\varepsilon^{3}Q_{-}$ |  0 |  1.8142

We observe a phenomenon that did not occur in ${\text{SU}}(3)$: at each AD
point, the massive BPS states belong to two different multiplets with unequal
masses $m_{0}=1.2828$ and $m_{1}=1.8142$, listed explicitly below.

  * •

IA. Massless at $u_{0}=1$ but have mass $m_{0}$ at $u_{0}=-1$:
$\\{\mu_{02},\mu_{22},\mu_{31},\mu_{33}\\}$.

  * •

IB. Massless at $u_{0}=1$ but have mass $m_{1}$ at $u_{0}=-1$:
$\\{\mu_{11},\mu_{13}\\}$.

  * •

IIA. Massless at $u_{0}=-1$ but have mass $m_{0}$ at $u_{0}=1$:
$\\{\mu_{01},\mu_{03},\mu_{12},\mu_{32}\\}$.

  * •

IIB. Massless at $u_{0}=-1$ but have mass $m_{1}$ at $u_{0}=1$:
$\\{\mu_{21},\mu_{23}\\}$.

Naïvely, there are $\binom{12}{2}$ distinct pairs. But ratios of central
charges within a single category always give rise to trivial walls of complex-
co-dimension 1 in the $z$-plane. So, it suffices to consider walls between
distinct types: $\\{$IA-IIA, IA-IIB, IB-IIA, IB-IIB$\\}$. Hence, there are
only $36$ pairs between mutually local BPS states that could give rise to
genuine walls.

  1. 1.

IA-IIA. We have the folllowing candidate walls for this case:

$\displaystyle z_{1}(r)=\frac{1+\alpha r}{1-\alpha r},\;\text{where
}\alpha\in\\{\pm\varepsilon,\pm i\varepsilon\\}.$ (3.18)

  2. 2.

IB-IIA. We have the folllowing candidate walls for this case:

$\displaystyle z_{2}(r)=\frac{1+\beta r}{1-\beta r},\;\text{where
}\beta\in\\{\pm\tfrac{1}{\sqrt{2}},\pm\tfrac{i}{\sqrt{2}}\\}.$ (3.19)

  3. 3.

IA-IIB. We have the folllowing candidate walls for this case:

$\displaystyle z_{3}(r)=\frac{1+\gamma r}{1-\gamma r},\;\text{where
}\gamma\in\\{\pm\sqrt{2},\pm i\sqrt{2}\\}.$ (3.20)

  4. 4.

IB-IIB. We have the folllowing candidate walls for this case:

$\displaystyle z_{4}(r)=\frac{1+\delta r}{1-\delta r},\;\text{where
}\delta\in\\{-\varepsilon^{\pm 1}\\}.$ (3.21)

Figure 7: The candidate walls of marginal stability for the ${\text{SU}}(4)$
case on the slice $u_{1}=u_{2}=0$ represented in the $z$-plane in the left
panel and in the $v$-plane in the right panel. The purple dots denote the AD
points, and the gray dot is the ${\mathbb{Z}}_{8}$-point. The black curve is
the contour of vanishing Kähler potential.

This set of curves is still highly degenerate: there are only three distinct
walls of marginal stability.

  1. 1.

A circle of radius $\sqrt{2}$ centred at $z=i$ with degeneracy $10$.

  2. 2.

A circle of radius $\sqrt{2}$ centred at $z=-i$ with degeneracy $10$.

  3. 3.

A circle of radius $1$ centred at $z=0$ with degeneracy $12$.

Note that the above degeneracy adds up to $32$ instead of $36$ since we do not
consider the straight lines with degeneracy $4$ corresponding to
$\beta=\pm\frac{1}{\sqrt{2}}$ and $\gamma=\pm\sqrt{2}$. The last contour to
plot is of vanishing Kähler potential:

$\displaystyle
K(v)=0\Rightarrow\absolutevalue{z}=\cot(\frac{\pi}{8})=1+\sqrt{2}.$ (3.22)

###  3.4 Comments on the ${\text{SU}}(N)$ case for $u_{k}=0$ for $k>0$

We have computed candidate walls of marginal stability up to ${\text{SU}}(7)$
on the $v$-plane with $u_{k}=0$ for $k>0$. We do not give the pictures here
explicitly since they share many features. Instead, we close this section with
some general remarks on the ${\text{SU}}(N)$ case, always assuming $u_{k}=0$
for $k>0$ in what follows. A generic wall of marginal stability on this slice
satisfies

$\displaystyle z(r)=\frac{Q_{1}}{Q_{N+1}}=\frac{\alpha r+1}{\alpha r-1},\text{
for some }\alpha\in{\mathbb{C}}\text{ and any }r\in{\mathbb{R}}.$ (3.23)

The Kähler potential on such a wall of marginal stability satisfies [15]

$\displaystyle
K(v)|_{\text{wall}}=\frac{N}{2\pi}\absolutevalue{Q_{N+1}}^{2}\tan(\tfrac{\pi}{2N})\left[\absolutevalue{\frac{\alpha
r+1}{\alpha r-1}}^{2}-\cot^{2}\left(\tfrac{\pi}{2N}\right)\right].$ (3.24)

Since $r$ can be arbitrarily rescaled, without loss of generality, we may take
$\alpha=e^{i\phi}\in S^{1}$, for $\phi\in[0,\pi]$ (we identify
$\alpha\sim-\alpha$ since they are equivalent under $\zeta\mapsto-\zeta$). For
the present discussion, we suppose that $\alpha\notin\\{0,\pi\\}$ so that we
exclude straight lines on the real axis.

Consider the special case $\alpha=i$. Then we have $\absolutevalue{z}=1$. This
is the origin-centred circle in the $z$-plane that exist for $N=2,4,6$. For
the $N=2$ case, this is precisely the $K=0$ contour found by Seiberg and
Witten [1]. However, such an origin-centred circle seems to be absent for odd
$N$ – we checked this up to ${\text{SU}}(7)$ but lack a proof.

If a wall of marginal stability that is confined to the region $K\leq 0$
contains a point where $K=0$, then we claim that it must be tangential to the
$K=0$ contour at one of $z=\pm i\cot(\frac{\pi}{2N})$. This point, together
with both the AD points, uniquely specifies a circle in the $z$-plane. This
can be seen explicitly by setting (3.24) to zero, which amounts to

$\displaystyle(1-\cot^{2}(\tfrac{\pi}{2N}))r^{2}+2\csc^{2}(\tfrac{\pi}{2N})\cos\phi
r+(1-\cot^{2}(\tfrac{\pi}{2N}))=0.$ (3.25)

This discriminant of this quadratic polynomial is

$\displaystyle
D=2\csc^{4}(\tfrac{\pi}{2N})(\cos(2\phi)-\cos(\tfrac{2\pi}{N})).$ (3.26)

This equation admits real solutions provided
$\phi\in[0,\frac{\pi}{N}]\cup[\pi-\frac{\pi}{N},\pi]$. If
$\phi\notin\\{\frac{\pi}{N},\pi-\frac{\pi}{N}\\}$, the wall intersects the
$K=0$ contour twice. Such a wall goes beyond the $K\leq 0$ region since the
circle also goes through the AD points; i.e. this is a circle through four
specified points. If $\phi\in\\{\frac{\pi}{N},\pi-\frac{\pi}{N}\\}$, then
$D=0$ and we have a unique intersection corresponding to $\alpha=e^{\pm
i\frac{\pi}{N}}=\varepsilon^{\pm 1}$. Solving the quadratic for these values
of $\alpha$ fixes $r=\pm 1$, i.e. $z=\pm i\cot(\tfrac{\pi}{2N})$. Hence, there
exist only two distinct walls of marginal stability that are confined to the
region $K\leq 0$, and are tangential to the contour $K=0$. Furthermore, such
walls exist for all $N\geq 3$. However, the above statement does not preclude
the existence of walls that are confined to the strong-coupling region but
always have $K<0$. This pair of walls is explicitly realized in the cases
$N=3,4,5,6,7$.

The contour of vanishing Kähler potential is also a universal feature, and
always has radius $\cot(\frac{\pi}{2N})$ in the $z$-plane, which scales like
$\sim N$ as $N\rightarrow\infty$. This $N$-scaling of the radius of the $K=0$
contour implies that the universal curves of marginal stability that are
tangential to the $K=0$ contour do not exist at large-$N$ since they tend to
straight lines. On the other hand, for any even $N$, we always expect to find
the contour with $\absolutevalue{z}=1$ since its radius is independent of $N$.
The fate of the other contours (in the strict region $K<0$) for even or odd
$N$ is not completely clear. We expect that the even and odd cases converge to
the same picture as $N\rightarrow\infty$. On these grounds, we suspect that
the contours that lie in the (strict) $K<0$ region for any $N$ converge to the
contour with $\absolutevalue{z}=1$ as $N\rightarrow\infty$, but we do not have
a proof. We close with an open question: is $\absolutevalue{z}=1$ the unique
contour of marginal stability as $N\rightarrow\infty$ on the $z$-plane with
$u_{k}=0$ for $k\neq 0$?

##  4 The intrinsic Coulomb branch of AD theories

In this section we shall study certain properties of the intrinsic Coulomb
branch of the AD theories by taking the $\Lambda\to\infty$ limit of the
asymptotically free embedding ${\text{SU}}(N)$ super Yang-Mills theory near
one of the maximal AD points. The resulting intrinsic AD theory is ${\cal
N}=2$ superconformal, its operators transform under representations of the
superconformal Lie algebra ${\text{SU}}(2,2|2)$, and have definite scaling
dimensions. In particular, we shall study the behavior of the Kähler potential
in this limit, and show that it is positive definite (vanishing only at the AD
point) and convex provided only genuine intrinsic Coulomb branch operators
${\cal O}_{n}$ are turned on away from the AD point, whose operator dimension
satisfies the unitarity bound $\Delta({\cal O}_{n})>1$ in a unitary
superconformal field theory.

###  4.1 The $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ intrinsic Coulomb branch

The intrinsic AD theories exist independently of their embedding into the
Coulomb branch of an asymptotically-free parent theory, and a given AD theory
may be reached from different parent theories. For example, the AD theories
obtained in the $\Lambda\to\infty$ limit of the parent theories
${\text{SU}}(3)_{N_{f}=0}$ and ${\text{SU}}(2)_{N_{f}=1}$ are the same and
referred to as the $(\mathfrak{a}_{1},\mathfrak{a}_{2})$ theory. The
nomenclature originates with yet another parent theory, namely the
$(\mathfrak{a}_{1},\mathfrak{a}_{2})$ theory may be constructed by
compactifying the six-dimensional ${\cal N}=(2,0)$ theory with gauge-algebra
$\mathfrak{a}_{1}$. The BPS quiver for such a theory is given by the product
of the $\mathfrak{a}_{1}$ and $\mathfrak{a}_{N-1}$ Dynkin diagrams, whence the
nomenclature $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ [32, 33, 34, 35].

In this section, we shall consider the intrinsic theory obtained near the
maximal AD points of the ${\cal N}=2$ super Yang-Mills theory with gauge group
${\text{SU}}(N)$ without hyper-multiplets. These theories can also be
constructed The SW curve and one-form are given by, [36]444The relation may be
derived by temporarily restoring the $\Lambda$-dependence in (2.14) to
$A(x)=\hat{A}(x)-\Lambda^{N}$ and $y^{2}=\hat{A}(x)(\hat{A}(x)-2\Lambda^{N})$
and taking the limit $-2\hat{y}^{2}=\lim_{\Lambda\to\infty}y^{2}/\Lambda^{N}$,
while keeping $x,u_{n}$ constant.

$\displaystyle\hat{y}^{2}$ $\displaystyle=x^{N}-\sum_{n=1}^{N-2}u_{n}x^{n}+v$
$\displaystyle\lambda$ $\displaystyle=\sqrt{-2}\,\hat{y}\;dx$ (4.1)

The resulting SW curve and differential are scale covariant in the following
sense. Since the SW periods are given by integrals of $\lambda$, we must
assign to $\lambda$ the scaling dimension one, which means that under a scale
transformation by a factor of $s\in{\mathbb{C}}^{*}$ we have
$\lambda\to\lambda^{\prime}=s\lambda$. This scaling relation may be derived
from the following scale transformations on $x,u_{n},v\sim u_{0}$,

$\displaystyle x\to x^{\prime}=s^{\tfrac{2}{N+2}}x\hskip
43.36243pt\hat{y}\to\hat{y}^{\prime}=s^{\frac{N}{N+2}}\hat{y}\hskip
43.36243ptu_{n}\to u_{n}^{\prime}=s^{2\frac{N-n}{N+2}}u_{n}$ (4.2)

for $n=0,1,\cdots,N-2$. The scaling dimension of the operator ${\cal O}_{n}$
whose expectation value is $u_{n}$ has the same scaling dimension as $u_{n}$
and therefore is given by,

$\displaystyle\Delta({\cal O}_{n})=2{N-n\over N+2}$ (4.3)

In a unitary superconformal field theory the dimension of every physical
operator must be larger than one in view of the unitarity bound imposed by the
representation theory of the supersymmetry algebra, which requires
$\Delta({\cal O}_{n})>1$ and thus,

$\displaystyle n+1\leq\left[\frac{N-1}{2}\right]=\mathfrak{r}$ (4.4)

The parameter $\mathfrak{r}$ is referred to as the rank of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ AD theory, and is defined to be the
dimension over ${\mathbb{C}}$ of the intrinsic Coulomb branch. By contrast,
the parameters $u_{n}$ for $n\geq\mathfrak{r}$ do not correspond to intrinsic
moduli of the AD theory.

###  4.2 The intrinsic Kähler potential

The Kähler potential of ${\text{SU}}(N)$ SW theory is defined by,

$\displaystyle
K_{{\text{SU}}(N)}=\frac{i}{4\pi}\sum_{I=1}^{N-1}\Big{(}a_{I}\,\bar{a}_{D,I}-\bar{a}_{I}\,a_{D,I}\Big{)}$
(4.5)

We may conveniently re-express $K_{{\text{SU}}(N)}$ in terms of the long and
short periods with the help of an ${\text{Sp}}(2N-2,{\mathbb{Z}})$ change of
duality frame, which results in the following expression,

$\displaystyle
K_{{\text{SU}}(N)}=\frac{i}{4\pi}\sum_{i=1}^{\mathfrak{r}}\left(a_{i}^{(s)}\bar{a}_{D,i}^{(s)}-\bar{a}_{i}^{(s)}a_{D,i}^{(s)}\right)+\frac{i}{4\pi}\sum_{i=1}^{\left[\tfrac{N}{2}\right]}\left(a_{i}^{(\ell)}\bar{a}_{D,i}^{(\ell)}-\bar{a}_{i}^{(\ell)}a_{D,i}^{(\ell)}\right)$
(4.6)

where the rank $\mathfrak{r}$ was defined in (4.4). We consider the decoupling
limit of the ${\text{SU}}(N)$ super Yang-Mills theory near the AD point as
$\Lambda\rightarrow\infty$ to obtain the intrinsic periods and the intrinsic
Kähler potential. Doing so causes the term in the Kähler potential for the
long periods to vanish, leaving the intrinsic Kähler potential
$K_{{\text{AD}}}$ of the AD theory expressed solely in terms of the short
periods,

$\displaystyle K_{\text{AD}}$
$\displaystyle=\frac{i}{4\pi}\sum_{i=1}^{\mathfrak{r}}(a_{i}^{(s)}\bar{a}_{D,i}^{(s)}-\bar{a}_{i}^{(s)}a_{D,i}^{(s)})$
(4.7)

in the limit where $v/\Lambda^{N}\to 0$. In the remainder of this section we
shall analyze the intrinsic periods and the Kähler potential using a
combination of analytical and numerical methods.

###  4.3 Analytical results for the periods

In this subsection, we obtain analytical results for the periods and the
Kähler potential, in terms of an expansion in the parameters $v$ and $v_{n}$
already encountered in (2.16),

$\displaystyle v_{n}=v^{\frac{n}{N}-1}\,u_{n}\hskip 72.26999ptn=1,\cdots N-2$
(4.8)

Note that these parameters include the moduli of the intrinsic Coulomb branch,
namely $v$ and $v_{n}$ for $n=1,2,\cdots,\mathfrak{r}-1$, but also the non-
intrinsic parameters $v_{n}$ for $n\geq\mathfrak{r}$. We consider both for the
sake of completeness, but also to contrast the difference of the behavior of
the Kähler potential under both deformations. We first turn to evaluating the
short periods.

######  Corollary 4.1.

The short periods of the $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ theory may be
expressed in terms of the function $R(\zeta)$ that admits the following
infinite series expansion in $v_{1},\dots,v_{N-2}$,

$\displaystyle R(\zeta)$ $\displaystyle=-{v^{{1\over
2}+\frac{1}{N}}\over\sqrt{2\pi}\,N}\sum_{{\ell_{n}=0\atop
n=1,\cdots,N-2}}^{\infty}{(-)^{M}}\zeta^{L}{{v}_{1}^{\ell_{1}}\cdots{v}_{N-2}^{\ell_{N-2}}\over\ell_{1}!\cdots\ell_{N-2}!}{\Gamma(\tfrac{L}{N})\over\Gamma(\frac{3}{2}+\tfrac{L}{N}-M)}$
(4.9)

where $L$ and $M$ were defined in (2.10).

This expression for $R(\zeta)$ follows from Corollary 2.2, upon restoring the
$\Lambda$-dependence and taking the limit $\Lambda\rightarrow\infty$. The
result represents a major simplification of the expressions obtained in (2.23)
and (2.24) for the embedded theory. The short periods may be expressed in
terms of the decomposition of the function $R$ in terms of characters of
${\mathbb{Z}}_{N}$ as follows,

$\displaystyle R(\zeta)$ $\displaystyle=\sum_{n=0}^{N-1}\zeta^{n}R_{n}\hskip
72.26999pt\zeta^{N}=-1$ (4.10)

Throughout, it will be convenient to use the abbreviations,

$\displaystyle s_{n}=\sin\left(\tfrac{2\pi n}{2N}\right)\hskip
72.26999ptc_{n}=\cos\left(\tfrac{2\pi n}{2N}\right)$ (4.11)

######  Corollary 4.2.

The short periods of $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ theories have the
following expressions in terms of the characters $R_{n}$ for any $N\geq 3$ and
$j=1,\dots,\mathfrak{r}$,

$\displaystyle a_{D,j}^{(s)}=2j\sum_{n=1}^{N-1}\varepsilon^{4nj}s_{n}R_{n}$
(4.12)

and,

$\displaystyle a_{j}^{(s)}$
$\displaystyle=\begin{dcases}\sum_{n=1}^{N-1}\frac{1}{2c_{n}}(\varepsilon^{4nj}-1)R_{n}&\text{
odd }N\\\ -2ijR_{\nu}+\sum_{{n=1\atop
n\neq\nu}}^{N-1}\frac{1}{2c_{n}}(\varepsilon^{4nj}-1)R_{n}&\text{ even
}N=2\nu\end{dcases}$ (4.13)

To derive $a_{D,j}^{(s)}$, we simply substitute the expansion (4.10) in terms
of characters into the expression for the dual period in (2.3.2), and apply
the definition of $s_{n}$. To derive $a_{j}^{(s)}$, we begin by substituting
the expansion (4.10) of $R(\zeta)$ into characters,

$\displaystyle a_{j}^{(s)}$
$\displaystyle=\sum_{k=0}^{j-1}\sum_{n=1}^{N-1}\varepsilon^{4nk}(\varepsilon^{3n}-\varepsilon^{n})R_{n}$
(4.14)

We then apply the following key identity

$\displaystyle\sum_{j=0}^{k-1}\varepsilon^{4jn}=\begin{dcases}k&\text{ when
}\varepsilon^{4n}=1\\\ \frac{1-\varepsilon^{4nk}}{1-\varepsilon^{4n}}&\text{
otherwise}\end{dcases}$ (4.15)

Since the range of $n$ is given by $0\leq n\leq N-1$, the instance
$\varepsilon^{4n}=1$ can occur only if $n=0$ or $n=\frac{N}{2}$. The former
does not give a contribution to the sum (4.14) thanks to the factor
$(\varepsilon^{3n}-\varepsilon^{n})$ in the summand, while the latter can only
contribute when $N$ is even. It is now clear that performing the sum over $k$
produces the different expressions for $a_{j}^{(s)}$ depending on whether $N$
is even or odd, thereby completing the proof of Corollary 4.2.

###  4.4 Analytical results for the Kähler potential

We now turn to the analytical results for the intrinsic Kähler potential, and
begin by evaluating $K_{\text{AD}}$ of (4.7) in terms of the character
coefficients $R_{n}$, using the results of Corollaries 4.1 and 4.2. The
results are given by the following theorem.

######  Theorem 4.3.

The Kähler potential $K_{\text{AD}}$ of
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ theories admits the following
decomposition in terms of the characters $R_{n}$

$\displaystyle
K_{\text{AD}}=\begin{dcases}\frac{N}{4\pi}\sum_{n=1}^{N-1}\frac{s_{n}}{c_{n}}\absolutevalue{R_{n}}^{2}&\text{
odd }N\\\ \frac{N}{4\pi}\sum_{{n=1\atop
n\not=\nu}}^{N-1}\frac{s_{n}}{c_{n}}\absolutevalue{R_{n}}^{2}+\frac{N}{8\pi}\left(R_{\nu}\sum_{{n=1\atop
n\not=\nu}}^{N-1}\frac{\varepsilon^{2n}-1}{c_{n}}\bar{R}_{n}+{\rm
c.c.}\right)&\text{ even }N=2\nu\end{dcases}$ (4.16)

$\bullet$ For odd $N$, the proof proceeds by substituting the expressions for
the periods obtained in Corollary 4.2 into the definition of the intrinsic
Kähler potential in (4.7), and we obtain,

$\displaystyle
K_{\text{AD}}=\frac{1}{4\pi}\sum_{j=1}^{\mathfrak{r}}\sum_{m,n=1}^{N-1}\frac{s_{n}}{c_{m}}R_{m}\bar{R}_{n}(\varepsilon^{4(m-n)j}-\varepsilon^{-4nj})+{\rm
c.c.}$ (4.17)

Applying the summation identity (4.15) shows that the first term receives
contributions only from $m=n$ since $m-n\neq N/2$ for any $1\leq m,n\leq N-1$
when $N$ is odd. This is a key simplification for odd $N$. After several
further elementary manipulations, we find,

$\displaystyle
K_{\text{AD}}=\frac{2r+1}{4\pi}\sum_{n=1}^{N-1}\frac{s_{n}}{c_{n}}\absolutevalue{R_{n}}^{2}-\frac{i}{8\pi}\sum_{{m,n=1\atop
m\neq
n}}^{N-1}\frac{s_{m}s_{n}}{c_{m}c_{n}c_{m-n}}\Big{(}R_{m}\bar{R}_{n}-\bar{R}_{m}{R}_{n}\Big{)}$
(4.18)

The second term on the right side cancels, since the prefactor in the summand
is symmetric under $m\leftrightarrow n$ while the combination in the
parentheses is anti-symmetric.

$\bullet$ For even $N=2\nu$, with $\nu\in{\mathbb{N}}$ and
$\mathfrak{r}=\nu-1$, the proof proceeds by separating the term $n=\nu$ in
$a_{D,i}^{(s)}$ from the other terms, and splitting the expression (4.7)
accordingly into the following four contributions,
$K_{AB}=K_{1}+K_{2}+K_{3}+K_{4}$,

$\displaystyle K_{1}$
$\displaystyle=-\frac{i}{\pi}R_{\nu}\sum_{n=1}^{N-1}s_{n}\bar{R}_{n}\sum_{j=1}^{\nu-1}j\varepsilon^{-4nj}+{\rm
c.c.}$ $\displaystyle K_{2}$
$\displaystyle=\frac{1}{4\pi}\bar{R}_{\nu}\sum_{\nu\neq
m=1}^{N-1}\frac{s_{\nu}}{c_{m}}R_{m}\sum_{j=1}^{\nu-1}(\varepsilon^{4mj}-1)+{\rm
c.c.}$ $\displaystyle K_{3}$ $\displaystyle=-\frac{1}{4\pi}\sum_{\nu\neq
m,n=1}^{N-1}\frac{s_{n}}{c_{m}}R_{m}\bar{R}_{n}\sum_{j=1}^{\nu-1}\varepsilon^{-4nj}+{\rm
c.c.}$ $\displaystyle K_{4}$ $\displaystyle=\frac{1}{4\pi}\sum_{\nu\neq
m,n=1}^{N-1}\frac{s_{n}}{c_{m}}R_{m}\bar{R}_{n}\sum_{j=1}^{\nu-1}\varepsilon^{4(m-n)j}+{\rm
c.c.}$ (4.19)

To evaluate the sum over $j$ in $K_{1}$, we use the following identity in
$x=\varepsilon^{-4n}$ with $x^{\nu}=1$,

$\displaystyle\sum_{j=0}^{\nu-1}jx^{j}=-\frac{\nu
x^{\nu}}{1-x}+\frac{x(1-x^{\nu})}{(1-x)^{2}}=-\frac{\nu}{1-x}$ (4.20)

The contribution from the $n=\nu$ term is purely real, and hence cancels. The
sums over $j$ in $K_{2}$ and $K_{3}$ are evaluated using the identity (4.15)
to give,

$\displaystyle K_{1}$ $\displaystyle=\frac{\nu}{4\pi}R_{\nu}\sum_{\nu\neq
n=1}^{N-1}\bar{R}_{n}\frac{\varepsilon^{2n}}{c_{n}}+{\rm c.c.}$ $\displaystyle
K_{2}$ $\displaystyle=-\frac{\nu}{4\pi}\bar{R}_{\nu}\sum_{\nu\neq
m=1}^{N-1}\frac{s_{\nu}}{c_{m}}R_{m}+{\rm c.c.}$ $\displaystyle K_{3}$
$\displaystyle=\frac{1}{4\pi}\sum_{\nu\neq
m,n=1}^{N-1}\frac{s_{n}}{c_{m}}R_{m}\bar{R}_{n}+{\rm c.c.}$ (4.21)

The evaluation of $K_{4}$ is a bit more subtle because,

$\displaystyle\sum_{j=1}^{N-1}\varepsilon^{4(m-n)j}=\begin{dcases}\nu-1&\text{
if }m-n\equiv 0\pmod{\nu}\\\ -1&\text{ otherwise}\end{dcases}$ (4.22)

Since the condition $m-n\equiv 0\pmod{\nu}$ can be satisfied only if
$m-n\in\\{0,\pm\nu\\}$ we have,

$\displaystyle K_{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\sum_{{m,n=1\atop m-n\neq
0,\pm\nu}}^{N-1}\frac{s_{n}}{c_{m}}\left(R_{m}\bar{R}_{n}+{\rm
c.c.}\right)+\frac{2(\nu-1)}{4\pi}\sum_{\nu\neq
m=1}^{N-1}\frac{s_{m}}{c_{m}}\absolutevalue{R_{m}}^{2}$ (4.23)
$\displaystyle+\frac{\nu-1}{4\pi}\sum_{\nu\neq
m,n=1}^{N-1}\frac{s_{n}}{c_{m}}R_{m}\bar{R}_{n}\left[\delta_{m-n,\nu}+\delta_{m-n,-\nu}\right]+{\rm
c.c.}$

Using the fact that $c_{n+\nu}s_{n+\nu}=-c_{n}s_{n}$ for any $n$, we see that
the sum on the second line above cancels. Rearranging the contributions from
$m-n=0,\pm\nu$ in the sum of $K_{1},K_{2},K_{3},K_{4}$ gives the second line
in (4.16) and completes the proof of the case when $N$ is even.

Next, we prove analytical results on the positivity and convexity of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ Kähler potential using the results of
Theorem 4.3 and Corollary 4.1, assembled in the theorem below.

######  Theorem 4.4.

The Kähler potential $K_{\text{AD}}$ of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ theories is bounded from below by zero
provided we only turn on moduli corresponding to operators with unitary
scaling dimensions, i.e. $\Delta({\cal O}_{k})>1$. Furthermore, in the absence
of deformations with non-unitary scaling dimensions, the expansion of the
Kähler potential in the rescaled moduli $v_{k}$ begins at quadratic order with
positive coefficients for $k\geq 1$ when $N$ is not divisible by 4. When $N$
is divisible by 4, an additional linear term arises.

To prove the theorem, we shall need the values of $R_{n}$ to leading orders in
$v_{1},\cdots,v_{N-2}$. This information may be read off from Corollary 4.1,

$\displaystyle R_{n}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\tfrac{n}{N})\,v^{{\tfrac{1}{2}}+\tfrac{1}{N}}}{\sqrt{2\pi}N\Gamma({\tfrac{1}{2}}+\tfrac{n}{N})}v_{n-1}+{\cal
O}(v_{i}^{2})\hskip 57.81621ptn=2,\cdots,N-1$ $\displaystyle R_{1}$
$\displaystyle=$
$\displaystyle-\frac{v^{{\tfrac{1}{2}}+\tfrac{1}{N}}}{\sqrt{2\pi}N}\frac{\Gamma(\tfrac{1}{N})}{\Gamma(\tfrac{3}{2}+\tfrac{1}{N})}\left[1+(\tfrac{1}{N^{2}}+\tfrac{1}{2N})\sum_{n=1}^{N-2}v_{n}v_{N-n}\right]+{\cal
O}(v_{i}^{3})$ (4.24)

where ${\cal O}(v_{i}^{2})$ and ${\cal O}(v_{i}^{3})$ stand for any bilinear
or trilinear terms in $v_{1},\cdots,v_{N-2}$. To confirm the absence of linear
terms in $R_{1}$, we use the fact that its contributions arise from
combinations for which $L\equiv 1\pmod{N}$. A term linear in $v_{n}$ has
$\ell_{n}=1$ and all other $\ell=0$, so that $L=1+n$. Since $n\leq N-2$, there
are no solutions to the equation $L\equiv 1~{}({\rm mod}~{}N)$, and hence no
linear terms.

To investigate the positivity and local convexity of $K_{\text{AD}}$, we
consider first the case of odd $N$, for which the Kähler potential is given by
Theorem 4.3,

$\displaystyle K_{\text{AD}}={N\over 4\pi}\sum_{n=1}^{N-1}{s_{n}\over
c_{n}}|R_{n}|^{2}$ (4.25)

Since we manifestly have the following inequalities,

$\displaystyle{s_{n}\over c_{n}}>0\hbox{ for }n=1,\cdots,\tfrac{N-1}{2}\hskip
57.81621pt{s_{n}\over c_{n}}<0\hbox{ for }n=\tfrac{N+1}{2},\cdots,N-1$ (4.26)

it is clear that the contributions from $R_{n}$ and thus $v_{n-1}$ for
$n=\tfrac{N+1}{2},\cdots,N-1$ are negative and not convex. Setting the
corresponding parameters $v_{n}=0$, we retain only those $v_{n}$ for which
$n=0,\cdots,\tfrac{N-3}{2}$. In this case, the bilinear terms in $R_{1}$
automatically vanish, and $R_{1}$ has contributions in $v_{1},\cdots,v_{N-2}$
to order zero, but no linear or bilinear contributions. Therefore, the Kähler
potential, locally near the AD point, is positive and convex. By inspection of
(4.3) and (4.4), we find, remarkably, that these values precisely correspond
to operators whose dimension is larger than 1 and thus obey the unitarity
bound, while the other values of $n$ correspond to operators whose dimension
is below the unitarity bound, thereby proving the first part of the theorem.
By contrast, turning on the deformations $v_{n-1}$ for
$n=\tfrac{N+1}{2},\cdots,N-1$ renders the Kähler potential non-positive and
non-convex.

The situation is more subtle for even $N=2\nu$. The argument for the
positivity of the diagonal part of the $K_{\text{AD}}$ is identical to the
case of odd $N$. The rest of the proof has two steps. First, we show that the
non-diagonal part of $K_{\text{AD}}$ for even $N$ gives a vanishing
contribution when all non-unitary deformations are turned off. Second, we
prove the absence of linear terms provided $N$ is not divisible by 4. To prove
the first claim, note that all contributions to $R_{\nu}$ come from the
$L\equiv\nu\pmod{N}$ sector. Parameterizing $L=\nu+Nk$ for
$k\in{\mathbb{Z}}_{\geq 0}$, the coefficients in the expansion of $R$ are
proportional to,

$\displaystyle\frac{\Gamma(\tfrac{L}{N})}{\Gamma(\tfrac{3}{2}+\tfrac{L}{N}-M)}=\frac{\Gamma(k+{\tfrac{1}{2}})}{\Gamma(2+k-M)}$
(4.27)

Furthermore, we observe that $k-M<0$. Hence, the only non-vanishing
contribution can come from $k-M=-1$. This is equivalent, for a single fixed
$j$, to $(2\nu-j)\ell_{j}=\nu+1$. Such a term contributes linearly only if
$2\nu-j\leq\nu+1$ or $j\geq\nu-1=r$. All such terms are killed if we restrict
to deformations with $\Delta>1$. To prove the second claim, we examine the
contributions to $R_{1}$ from $L\equiv 1\pmod{\nu}$ when $\nu$ is odd. This
gives rise to a linear term only if $j=\nu$, which is excluded by the
constraint $\Delta>1$. This concludes the proof of the theorem. We remark that
the Kähler potential is positive even when $N$ is divisible by 4, provided we
only turn on unitary deformation. Indeed, the non-diagonal part of the Kähler
potential then vanishes, and we are left with a sum of absolute-squares with
positive coefficients.

Remarks: To study the global structure of the Kähler potential, we use
numerics, which agree perfectly with our analytical results. At rank-1,
convexity of $K_{\text{AD}}$ over intrinsic slices follows directly from the
scaling of the periods: $a(v)\sim v^{\frac{1}{\Delta(v)}}$, which is required
by scale-invariance (cf. [11]). We numerically analyze the effect of turning
on deformations $u_{k}$ with $\Delta({\cal O}_{k})\leq 1$. Generically, such
deformations introduce points where the second-derivative test on the Kähler
potential fails to yield a unique minimum, and the determinant of the Hessian
is vanishing. In the rank-$2$ case, we primarily consider the intrinsic
Coulomb branch, which is now $2$-complex-dimensional. On the intrinsic Coulomb
branch, the Kähler potential is a positive and convex function provided there
are no deformations with $\Delta\leq 1$.

###  4.5 Rank-1, example 1: $(\mathfrak{a}_{1},\mathfrak{a}_{2})$

This is a rank-$1$ theory with $N=3$, and has an elliptic SW curve,

$\displaystyle\hat{y}^{2}=x^{3}-ux+v.$ (4.28)

The function $R(\zeta)$ for this theory is given as follows ($\zeta^{3}=-1$),

$\displaystyle R(\zeta)$
$\displaystyle=\frac{v^{\frac{5}{6}}}{3\sqrt{2\pi}}\sum_{\ell=0}^{\infty}\frac{(-\zeta)^{\ell+1}\Gamma(\frac{\ell+1}{3})\,}{\Gamma(\frac{11}{6}-\frac{2\ell}{3})\,\ell!}\,w^{\ell}\hskip
72.26999ptw=\frac{u}{v^{\frac{2}{3}}}$ (4.29)

The sum may be reorganized into hypergeometric functions of various degrees,

$\displaystyle R(\zeta)$
$\displaystyle=\frac{\,v^{\frac{5}{6}}}{3\sqrt{2\pi}}\Bigg{[}-\zeta\frac{\Gamma\left(\tfrac{1}{3}\right)}{\Gamma\left(\tfrac{11}{6}\right)}\,{}_{2}F_{1}\left(-\tfrac{5}{12},\tfrac{1}{12};\tfrac{2}{3};\tfrac{4w^{3}}{27}\right)+\zeta^{2}\frac{\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(\frac{7}{6}\right)}\,{}_{2}F_{1}\left(-\tfrac{1}{12},\tfrac{5}{12};\tfrac{4}{3};\tfrac{4w^{3}}{27}\right)w$
$\displaystyle\hskip
72.26999pt+\frac{1}{2{\Gamma({\tfrac{1}{2}})}}\,{}_{3}F_{2}\left(\tfrac{1}{4},\tfrac{3}{4},1;\tfrac{4}{3},\tfrac{5}{3};\tfrac{4w^{3}}{27}\right)w^{2}\Bigg{]}$
(4.30)

This presentation explicitly produces exact expressions for the characters
$R_{0}$, $R_{1}$ and $R_{2}$ from the last, first, and middle terms,
respectively, and allows us to compute the periods and the Kähler potential.
Before proceeding to the necessary numerical analysis, it is instructive to
evaluate the Kähler potential at low-orders in $u$, and we find,

$\displaystyle
K_{\text{AD}}=\frac{27\sqrt{3}}{50\pi^{4}}\left(\Gamma(\tfrac{1}{3})^{2}\Gamma(\tfrac{7}{6})^{2}\absolutevalue{v}^{\frac{5}{3}}-\Gamma(\tfrac{2}{3})^{2}\Gamma(\tfrac{11}{6})^{2}\absolutevalue{v}^{\frac{1}{3}}\,\absolutevalue{u}^{2}\right)+{\cal
O}(u^{3})$ (4.31)

While the Kähler potential for $u=0$ is convex and positive with a unique
vanishing point at $v=0$ by Theorem 4.4, convexity is immediately lost as soon
as we turn on $u$. Beyond the analytical result for the small $u$
approximation, numerics are required to explore the Kähler potential away from
the AD points, as shown in Figure 8.

(a) $({\rm Re\,}v,{\rm Im\,}v)$

(b) $({\rm Re\,}u,{\rm Re\,}v)$

(c) $({\rm Re\,}u,{\rm Im\,}u)$

Figure 8: A plot of the $(\mathfrak{a}_{1},\mathfrak{a}_{2})$ Kähler potential
for various slices of parameter space $u_{1},v$.

###  4.6 Rank-1, example 2: $(\mathfrak{a}_{1},\mathfrak{a}_{3})$

In this subsection, we consider the AD theory that lives in the moduli-space
of pure ${\text{SU}}(4)$ SW theory. This theory is defined by a quartic SW
curve,

$\displaystyle\hat{y}^{2}=x^{4}-u_{2}x^{2}-u_{1}x+v$ (4.32)

It is clear from Theorem 4.4 that the Kähler potential is convex and positive-
definite in the absence of any non-unitary Coulomb branch deformations with
$\Delta\leq 1$. Here, we gather further evidence that turning on such
deformations spoils both positivity and convexity.

For $v>0$ the expansion in small $u_{1}\in{\mathbb{R}}$ is given as follows,

$\displaystyle
K_{\text{AD}}=\frac{\Gamma\left(\frac{1}{4}\right)^{2}v^{\frac{3}{2}}}{32\pi^{2}\Gamma\left(\frac{7}{4}\right)^{2}}+\frac{u_{1}\Gamma\left(\frac{1}{4}\right)v^{\frac{3}{4}}}{16\sqrt{2}\pi^{3/2}\Gamma\left(\frac{7}{4}\right)}-\frac{u_{1}^{3}\Gamma\left(\frac{3}{4}\right)}{32\sqrt{2}\pi^{3/2}\Gamma\left(\frac{1}{4}\right)v^{\frac{3}{4}}}+{\cal
O}(u_{1}^{4})$ (4.33)

The expression shows that turning on $u_{1}$ spoils convexity; we find an
analogous result for $u_{2}\neq 0$. These results are qualitatively analogous
to the results we obtained for ${\text{SU}}(3)$, and we shall refrain from
presenting numerical plots for this case.

###  4.7 A rank-2 example: $(\mathfrak{a}_{1},\mathfrak{a}_{4})$

Numerical analysis confirms, here as well, that turning on any non-intrinsic
moduli, such as $u_{2}$ and $u_{3}$, spoils both convexity and positivity. We
shall now concentrate on the numerical analysis of the dependence of the
Kähler potential on the intrinsic moduli $u=u_{1},v$, with no other
deformations turned on. The SW curve is given by,

$\displaystyle\hat{y}^{2}=x^{5}-ux+v$ (4.34)

and the Taylor expansion of $R(\zeta)$ in $u$ for $v\not=0$, with
$\zeta^{5}=-1$, is given by,

$\displaystyle
R(\zeta)=\frac{v^{\frac{7}{10}}}{5\sqrt{2\pi}}\sum_{\ell=0}^{\infty}\frac{(-\zeta)^{\ell+1}\Gamma(\frac{1+\ell}{5})}{\Gamma(\frac{17}{10}-\frac{4\ell}{5})\ell!}\left(\frac{u}{v^{\frac{4}{5}}}\right)^{\ell}.$
(4.35)

This series may be summed in terms of the hypergeometric functions
${}_{5}F_{4}$ and ${}_{4}F_{3}$ with argument proportional to $u^{5}/v^{4}$.
Such a closed form for the characters $R_{n}$ is useful extract the small-$v$
behavior of $K_{\text{AD}}$ by expanding the hypergeometric functions around
$u_{1}=\infty$. We shall not produce these lengthy explicit formulas here.
Instead we concentrate on the numerical results when only the intrinsic moduli
$u=u_{1}$ and $v$ are turned on.

(a) $v$-plane

(b) $({\rm Im\,}u,{\rm Re\,}v)$-plane

(c) $u$-plane

Figure 9: Plots of $K_{\text{AD}}$ for $(\mathfrak{a}_{1},\mathfrak{a}_{4})$
on various slices of the intrinsic moduli space.

In figure 9 the Kähler potential is plotted in various slices of the intrinsic
Coulomb branch moduli $(u,v)$: versus $({\rm Re\,}(v),{\rm Im\,}(v))$ for
fixed $u$ in panel (a); versus $({\rm Re\,}(v),{\rm Im\,}(u))$ for fixed ${\rm
Re\,}(u),{\rm Im\,}(v)$ in panel (b); and $({\rm Re\,}(u),{\rm Im\,}(u))$ for
fixed $v$ in panel (c). One observes in each case that, for the domain
plotted, the Kähler potential is manifestly convex. More detailed numerical
analysis, not manifestly visible form the plots, establishes that $K_{AB}$ is
also positive definite for all values of $u,v$ studied, and vanishes only for
$u=v=0$.

In figure 10 we provide a more detailed numerical analysis of the precise
positivity and convexity properties, by taking different representative
slicing of the intrinsic moduli space. In panel (a) of figure 10 we set $u=2$,
and plot $K_{\text{AD}}$ as a function of $v=e\,e^{i\theta}$ as a function of
$r\in[-1,2]$ for a number of discrete values of $\theta$. Convexity and
positivity is observed for every such slice. In panel (b) of figure 10 we plot
$K_{\text{AD}}$ as a function of real $u\in[-2,2]$ for 10 evenly spaced
discrete values of $v\in[1,2]$. Again, we observe positivity and convexity on
each slice. Finally, in panel (c) of figure 10 we plot $K_{\text{AD}}$ as a
function of real $v\in[-2,2]$ for 10 evenly spaced real values of $u\in[1,2]$,
further confirming positivity and convexity.

(a) $v$-plane with $v=re^{i\theta}$

(b) $u$-plane for discrete $v$

(c) $v$-plane for discrete $u$

Figure 10: Plots of $K_{\text{AD}}$ for $(\mathfrak{a}_{1},\mathfrak{a}_{4})$
on discrete slices of the intrinsic moduli space.

Additional observations include the following. One verifies numerically, for
example in panel (b) of figure 10, that the minimum of the Kähler potential
over the $u$-plane always occurs at the origin $u=0$ for any $v\not=0$. From
panel (c) of figure 10, we also see that for various real (as well as) complex
values of $u$, the Kähler potential has a local minimum that is shifted from
$v=0$ for any $u\neq 0$, though still respecting positivity and convexity.
Finally, the Kähler potential is always positive in all these cases, and the
minimum of $K$ is strictly bigger than $0$ if $u\neq 0$.

###  4.8 A rank-3 example: $(\mathfrak{a}_{1},\mathfrak{a}_{6})$

We consider the dependence of the intrinsic Kähler potential on the three
intrinsic moduli $u_{2},u_{1},v$, where the SW curve is given by,

$\displaystyle\hat{y}^{2}=x^{7}-u_{2}x^{2}-u_{1}x+v$ (4.36)

The evaluation of the periods and Kähler potential proceeds very similarly to
what we have already described up to rank 2, so we will not go into that here
but rather only present the results. We study the regime of fixed $v$, and
small $u_{1,2}$ using our expansion. This allows us to produce plots in figure
11 that further support our conjecture that the intrinsic Kähler potential is
a convex and positive function with a unique minimum at $K_{\text{AD}}=0$ that
is located at the ${\mathbb{Z}}_{7}$-symmetric point.

(a) $u_{1}$-plane

(b) $u_{2}$-plane

(c) $({\rm Re\,}u_{1},{\rm Im\,}u_{2})$-plane

Figure 11: Plots of $K_{\text{AD}}$ for $(\mathfrak{a}_{1},\mathfrak{a}_{6})$
on various slices of the intrinsic moduli space for fixed value of $v$ and
different splicings of the intrinsic moduli $u_{1},u_{2}$.

##  5 Conclusions and future directions

In this paper, we have analyzed three different aspects of Argyres-Douglas
(AD) theories, and their embeddings into the moduli-spaces of asymptotically-
free gauge theories: the SW periods near the AD points; the marginal stability
of mutually local BPS states in a near the AD points; and the intrinsic
periods and Kähler potential of $(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$
theories.

###  5.1 Summary of results

  1. 1.

For gauge-group ${\text{SU}}(N)$, we evaluated SW periods near the maximal AD
points for $N\geq 3$ by a non-trivial analytic continuation of the expansion
of the periods obtained near the ${\mathbb{Z}}_{2N}$-symmetric point studied
in [15]. Since our expansion is around one or the other maximal AD point, it
allows us to access a neighborhood of the maximal AD points that includes the
intrinsic Coulomb branch of the AD theory. On regions of overlap, we showed
that our expansion has better convergence properties than the expansion
considered in [15].

  2. 2.

For gauge-group ${\text{SU}}(3)$, we revisited the structure of the walls for
marginal stability, which were analyzed in [15] only on the restricted slice
with $u=0$. Utilizing a combination of our expansion around the
${\mathbb{Z}}_{3}$ point and the numerical integration methods of [15], we
mapped out the walls of marginal stability in the 3D space $({\rm Re\,}u,{\rm
Im\,}u,\text{Arc})$, where Arc refers to a real-dimension 1 curve in the
$v$-plane. This adds a more complete understanding of the walls, and explains
the degeneracy-lifting associated with the ${\mathbb{Z}}_{3}$-symmetry of the
$v$-plane. We provide partial results for ${\text{SU}}(4)$ on the
$u_{2}=u_{1}=0$ slice, and point out some generic features for $N\geq 3$.

  3. 3.

In the last part of this paper, we explore the intrinsic Coulomb branch of
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ theories. We applied the decoupling
limit $\Lambda\rightarrow\infty$ to obtain the intrinsic periods and their
expansion around the AD point. We then apply this understanding of the periods
to exactly compute the intrinsic Kähler potential and prove its positivity and
convexity near $v_{n}=0$ on intrinsic slices, except when $4\nmid N$. We
numerically test the global positivity and convexity of the Kähler potential
over intrinsic slices in a variety of examples up to and including rank-3,
i.e. $N=3,4,5,6,7$. We find broad agreement with our analytic results, and
find that convexity and positivity are spoiled if we allow non-unitary
deformations to be turned on, namely $u_{k\geq\mathfrak{r}}\neq 0$. This is in
full agreement with Theorem 4.4.

###  5.2 Future directions

Let us close this section with some concrete directions for future work.

  1. 1.

Dynamics of wall-crossing. A key question that we have not addressed in our
considerations is whether genuine bound BPS states are formed when we follow a
trajectory in the Coulomb branch that crosses a kinematically allowed wall. It
would be interesting to explicitly determine the spectrum of BPS states after
such a wall is crossed, and determine under what conditions formation of bound
BPS states is possible. Work along these lines has been recently undertaken in
[33], and it would be interesting to apply these methods to the walls we
obtain here. It is worth emphasizing, however, that more complicated phenomena
can take place upon wall-crossing that we have also not addressed here – see
[32].

  2. 2.

Integrated correlation functions of the
$(\mathfrak{a}_{1},\mathfrak{a}_{N-1})$ stress-tensor. There is a large body
of work on integrated correlators in $3d$ ABJM and $4d$ ${\cal N}=4$ super-
Yang-Mills [37, 38, 39]. The key motivation behind such work is to provide
non-trivial non-perturbative checks of holographic duals in eleven and ten
dimensions respectively. Such works often rely on supersymmetric localization,
superconformal symmetry, and $S$-duality: all of which can be accessed for AD
theories. Recently, holographic duals have been proposed for AD theories in
M-theory [40, 41], and it would interesting to compute observables on the
field theory side to test these duals. On a related note, there are
investigations of the correlators of chiral ring operators in AD theories
using supersymmetric localization and the intrinsic SW periods [42].

In parallel to the work on ${\cal N}=4$ super-Yang-Mills, is there a
combination of supersymmetric localization and our expansion of the SW periods
that can be used to compute the stress-tensor correlators of AD theories? If
yes, what does this correspond to in the proposed holographic dual?

  3. 3.

SUSY breaking of Argyres-Douglas theories. As was already emphasized
explicitly in [20, 14, 15, 21], the convexity of the Kähler potential plays a
key role in our understanding of the IR phases of ${\text{SU}}(N)$ adjoint
$\text{QCD}_{4}$, which is obtained by deforming pure ${\cal N}=2$ gauge-
theory by ${\cal T}_{UV}\sim M^{2}{\rm tr}\bar{\phi}\phi$ near the monopole
point. On representation-theoretic grounds, we expect to find a non-
supersymmetric interacting CFT if we deform AD theories in an analogous manner
[43].

What can be said about the phases and spectrum of this IR CFT? Can this non-
supersymmetric CFT be analyzed by a Lagrangian description obtained by
deforming the ${\cal N}=1$ Lagrangian that flows to Argyres-Douglas theory
(cf. [44, 45]) in an intermediate step? If such a theory does not admit a
Lagrangian realization, can one analyze it using SW theory?

##  Appendix A Proof of Theorem 2.1

To prove Theorem 2.1 we use the SW differential given in (2.18) and the
definition of the integrals $R(\zeta)$ in (2.19). Substituting the multinomial
expansion for $V(z)^{M}$ in powers of $z$,

$\displaystyle V(z)^{M}$
$\displaystyle=\sum_{\ell_{1},\cdots,\ell_{N-2}=0}^{M}\binom{M}{\ell_{1},\cdots,\ell_{N-2}}\,v_{1}^{\ell_{1}}\cdots
v_{N-2}^{\ell_{N-2}}\,z^{L-1}$ (A.1)

where we use the combinations $L$ and $M$ familiar from (2.10), gives the
following expansion of the SW differential,

$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle{v^{{1\over
2}+\frac{1}{N}}\over\sqrt{-2}}\sum_{k=0}^{\infty}{\Gamma(k+{1\over
2})\over\Gamma({1\over 2})\,k!}\,\left({v\over
2}\right)^{k}\sum_{{\ell_{n}=0\atop n=1,\cdots,N-2}}^{\infty}{\Gamma({1\over
2}-k+M)\over\Gamma({1\over 2}-k)}{v_{1}^{\ell_{1}}\cdots
v_{N-2}^{\ell_{N-2}}\over\ell_{1}!\cdots\ell_{N-2}!}$ (A.2)
$\displaystyle\hskip
36.135pt\times\left(Nz^{N}-\sum_{n=1}^{N-2}nv_{n}z^{n}\right){z^{L-1}dz\over\big{(}z^{N}+1\big{)}^{{1\over
2}-k+M}}$

The integral over $z$ from 0 to $\zeta$ greatly simplifies for $\zeta^{N}=-1$
and we obtain,

$\displaystyle\int_{0}^{\zeta}{z^{\gamma-1}\,dz\over(z^{N}+1)^{p}}={\xi^{\gamma}\over\gamma}F(p,\tfrac{\gamma}{N};1+\tfrac{\gamma}{N};1)={\xi^{\gamma}\over
N}{\Gamma(\tfrac{\gamma}{N})\Gamma(1-p)\over\Gamma(1+\tfrac{\gamma}{N}-p)}$
(A.3)

The integrals of $\lambda$ given by the expansion in (A.2) correspond to the
values $p={\tfrac{1}{2}}-k+M$, and $\gamma=L+N$ or $\gamma=L+n$. As a result,
we obtain,

$\displaystyle i\pi R(\xi)$ $\displaystyle=$ $\displaystyle{v^{{1\over
2}+\frac{1}{N}}\over\sqrt{-2}}\sum_{k=0}^{\infty}{(-)^{M}}{\Gamma(k+{1\over
2})^{2}\over\Gamma({1\over 2})\,k!}\,\left({v\over
2}\right)^{k}\sum_{{\ell_{n}=0\atop
n=1,\cdots,N-2}}^{\infty}{v_{1}^{\ell_{1}}\cdots
v_{N-2}^{\ell_{N-2}}\over\ell_{1}!\cdots\ell_{N-2}!}$ (A.4)
$\displaystyle\times\bigg{(}{\xi^{L+N}\,\Gamma(1+\tfrac{L}{N})\over\Gamma(\frac{3}{2}+\tfrac{L}{N}+k-M)}-\sum_{n=1}^{N-2}{n\over
N}\,v_{n}\,{\xi^{n+L}\,\Gamma(\tfrac{n+L}{N})\over\Gamma({1\over
2}+\tfrac{n+L}{N}+k-M)}\bigg{)},$

where we have made use of the reflection formula for $\Gamma$-function
$\Gamma(z)\Gamma(1-z)\sin(\pi z)=\pi$. The term labelled by $n$ under the
finite sum over $n$ in the second line corresponds to shifting
$\ell_{n}\to\ell_{n}-1$ and adding these contributions simplifies the sum as
follows,

$\displaystyle i\pi R(\xi)={v^{{1\over
2}+\frac{1}{N}}\over\sqrt{-2\pi}N}\sum_{{\ell_{n}=0\atop
n=1,\cdots,N-2}}^{\infty}\\!\\!\\!\\!(-)^{M}\xi^{L}{v_{1}^{\ell_{1}}\cdots
v_{N-2}^{\ell_{N-2}}\over\ell_{1}!\cdots\ell_{N-2}!}\sum_{k=0}^{\infty}{\Gamma(k+{1\over
2})^{2}\,\Gamma(\tfrac{L}{N})\over\Gamma(\frac{3}{2}+\tfrac{L}{N}+k-M)\,k!}\,\left({v\over
2}\right)^{k}\quad$ (A.5)

Relabelling $k\to\ell_{0}$, $v\to v_{0}$ and choosing the branch
$\sqrt{-2\pi}=i\sqrt{2\pi}$ produces formula (2.22) and thereby completes the
proof of Theorem 2.1.

##  Appendix B Convergence, long periods, and elliptic form for
${\text{SU}}(3)$

We explicitly display the coefficients of the ${\mathbb{Z}}_{3}$ expansion in
this appendix, and examine the convergence criterion for the ${\text{SU}}(3)$
series. Then we demonstrate that an appropriate choice of homology basis makes
the long periods analytic.

###  B.1 The ${\mathbb{Z}}_{6}$ expansion

Explicit formulas for the periods in the case $N=3$ were obtained in [7] using
Picard-Fuchs equations. The authors expressed their results in terms of Appell
$F_{4}$ functions [46, 22], which can be defined by the following series
expansion,

$\displaystyle
F_{4}(a,b,c_{1},c_{2};x,y)=\sum_{m,n=0}^{\infty}{\Gamma(m+n+a)\Gamma(m+n+b)\Gamma(c_{1})\Gamma(c_{2})\over\Gamma(a)\Gamma(b)\Gamma(m+c_{1})\Gamma(n+c_{2}){\,m!\,n!}}\,x^{m}y^{n}$
(B.1)

The ${\text{SU}}(3)$ periods can be expressed as follows [15],

$\displaystyle a_{1}$ $\displaystyle=Q(\varepsilon^{1})-Q(\varepsilon^{0})$
$\displaystyle a_{D,1}$ $\displaystyle=Q(\varepsilon^{2})-Q(\varepsilon^{1})$
$\displaystyle a_{2}$
$\displaystyle=Q(\varepsilon^{1})-Q(\varepsilon^{0})+Q(\varepsilon^{3})-Q(\varepsilon^{2})$
$\displaystyle a_{D,2}$ $\displaystyle=Q(\varepsilon^{4})-Q(\varepsilon^{3})$
(B.2)

The function $Q(\xi)$ can be expanded in characters of ${\mathbb{Z}}_{6}$

$\displaystyle Q(\xi)$ $\displaystyle=\sum_{n=0}^{5}\xi^{n}Q_{n}$
$\displaystyle Q_{3}$ $\displaystyle=0$ (B.3)

The formula for $Q(\xi)$ given in (2.9) may be recast in terms of Appell
functions $F_{4}$ expressed as follows in terms of the variables
$x=4u_{1}^{3}/27$ and $y=u_{0}^{2}$

$\displaystyle Q_{1}$ $\displaystyle={2\pi\over 2^{{1\over 3}}\,3^{{3\over
2}}\,\Gamma(\tfrac{2}{3})^{3}}\,v\,F_{4}(\tfrac{1}{3},\tfrac{1}{3},\tfrac{2}{3},\tfrac{3}{2};x,y)$
$\displaystyle Q_{2}$ $\displaystyle={2\pi\over 2^{{1\over
3}}\,3^{2}\,\Gamma(\tfrac{2}{3})^{3}}\,u\,F_{4}(\tfrac{1}{6},\tfrac{1}{6},\tfrac{4}{3},\tfrac{1}{2};x,y)$
$\displaystyle Q_{4}$ $\displaystyle={2^{{1\over 3}}\,3^{{3\over
2}}\,\Gamma(\tfrac{2}{3})^{3}\over
4\pi^{2}}\,F_{4}(-\tfrac{1}{6},-\tfrac{1}{6},\tfrac{2}{3},\tfrac{1}{2};x,y)$
$\displaystyle Q_{5}$ $\displaystyle={2^{{1\over
3}}\,\Gamma(\tfrac{2}{3})^{3}\over
4\pi^{2}}\,uv\,F_{4}(\tfrac{2}{3},\tfrac{2}{3},\tfrac{4}{3},\tfrac{3}{2};x,y)$
(B.4)

Additionally, $Q_{3}=0$, while $Q_{0}$ cancels out of all periods. Note that
the double infinite series for the Appell function is absolutely convergent
for $\sqrt{|x|}+\sqrt{|y|}<1$ which gives the following region of absolute
convergence in terms of $u$ and $v$,

$\displaystyle\tfrac{2}{\sqrt{27}}\,|u|^{{3\over 2}}+|v|<1\,.$ (B.5)

Beyond this region, partial analytic continuation formulas are known for
$F_{4}$,555 These are obtained by expressing $F_{4}$ as an infinite sum of
hypergeometric functions, such as  $\displaystyle
F_{4}(a,b,c_{1},c_{2};x,y)=\sum_{n=0}^{\infty}{\Gamma(n+a)\Gamma(n+b)\Gamma(c_{2})\over\Gamma(a)\Gamma(b)\Gamma(n+c_{2}){\,n!}}\,y^{n}\,F(n+a,n+b;c_{1};x)\,,$
(B.6) and applying inversion formulas for the hypergeometric functions.

$\displaystyle F_{4}(a,b,c_{1},c_{2};x,y)$
$\displaystyle={\Gamma(c_{1})\Gamma(b-a)\over\Gamma(b)\Gamma(c_{1}-a)}\,(-x)^{-a}F_{4}(a,a+1-c_{1},a+1-b,c_{2};\tfrac{1}{x},\tfrac{y}{x})$
(B.7) $\displaystyle\
+{\Gamma(c_{1})\Gamma(a-b)\over\Gamma(a)\Gamma(c_{1}-b)}\,(-x)^{-b}F_{4}(b,b+1-c_{1},b+1-a,c_{2};\tfrac{1}{x},\tfrac{y}{x})\qquad$

which gives the following region in terms of $u_{1}$ and $u_{0}$,

$\displaystyle 1+|u_{0}|<\tfrac{2}{\sqrt{27}}\,|u_{1}|^{{3\over 2}}$ (B.8)

allowing us to explore the region of large $|u_{1}|$ and small $|u_{0}|$.
Recent progress on the analytic continuation of $F_{4}$ may be found in [47].

###  B.2 The ${\mathbb{Z}}_{6}$ decomposition from the ${\mathbb{Z}}_{3}$
expansion

The above-mentioned decomposition in terms of the characters of
${\mathbb{Z}}_{6}$ reduces to

$\displaystyle Q(\xi)$
$\displaystyle=(Q_{0}+Q_{3})\xi^{0}+(Q_{1}+Q_{4})\xi^{1}+(Q_{2}+Q_{5})\xi^{2}\qquad\text{
for }\xi^{3}=+1$ $\displaystyle Q(\xi)$
$\displaystyle=(Q_{0}-Q_{3})\xi^{0}+(Q_{1}-Q_{4})\xi^{1}+(Q_{2}-Q_{5})\xi^{2}\qquad\text{
for }\xi^{3}=-1$ (B.9)

These coefficients may be extracted from the expression for $Q(\xi)$ in (2.42)
by parametrizing $m=3\mu+\nu$ where $\nu=0,1,2$ and $\mu\geq 0$.

  * •

For $\xi^{3}=-1$ we obtain,

$\displaystyle Q_{0}-Q_{3}$ $\displaystyle=$ $\displaystyle{u^{2}\,v^{-{1\over
2}}\over 3\sqrt{2}\,\pi}\sum_{\mu=0}^{\infty}{\mu!\,\Gamma({1\over
2})\over\Gamma({\tfrac{1}{2}}-2\mu)\,(3\mu+2)!}\,F({\tfrac{1}{2}},{\tfrac{1}{2}};{\tfrac{1}{2}}-2\mu;\tfrac{v}{2})\left({u^{3}\over
v^{2}}\right)^{\mu}$ $\displaystyle Q_{1}-Q_{4}$ $\displaystyle=$
$\displaystyle-{v^{\frac{5}{6}}\over
3\sqrt{2}\,\pi}\sum_{\mu=0}^{\infty}{\Gamma(\tfrac{1}{3}+\mu)\Gamma({1\over
2})\over\Gamma(\tfrac{11}{6}-2\mu)\,(3\mu)!}\,F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{11}{6}-2\mu;\tfrac{v}{2})\left({u^{3}\over
v^{2}}\right)^{\mu}$ $\displaystyle Q_{2}-Q_{5}$ $\displaystyle=$
$\displaystyle{u\,v^{\frac{1}{6}}\over
3\sqrt{2}\,\pi}\sum_{\mu=0}^{\infty}{\Gamma(\tfrac{2}{3}+\mu)\Gamma({1\over
2})\over\Gamma(\tfrac{7}{6}-2\mu)\,(3\mu+1)!}\,F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{7}{6}-2\mu;\tfrac{v}{2})\left({u^{3}\over
v^{2}}\right)^{\mu}$ (B.10)

  * •

For $\xi^{3}=1$ we obtain,

$\displaystyle Q_{0}+Q_{3}$ $\displaystyle=$ $\displaystyle{u^{2}\ v^{-{1\over
2}}\over 3\sqrt{2}\,\pi}\sum_{\mu=0}^{\infty}{\mu!\,\Gamma({1\over
2}+2\mu)\over\Gamma({1\over
2})(3\mu+2)!}F({\tfrac{1}{2}},{\tfrac{1}{2}};{\tfrac{1}{2}}-2\mu;\tfrac{v}{2})\left({u^{3}\over
v^{2}}\right)^{\mu}$ (B.11) $\displaystyle Q_{1}+Q_{4}$ $\displaystyle=$
$\displaystyle\sum_{\mu=0}^{\infty}{\Gamma(\tfrac{1}{3}+\mu)\over(3\mu)!}\bigg{[}{v^{\frac{5}{6}}\over
3\sqrt{2}\,\pi}\,{\Gamma(-\frac{5}{6}+2\mu)\over\Gamma({1\over
2})}F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{11}{6}-2\mu;\tfrac{v}{2})\left({u^{3}\over
v^{2}}\right)^{\mu}$ $\displaystyle\hskip 79.49744pt{+}{2^{2/3}\Gamma({1\over
2})\Gamma(\frac{5}{6}-2\mu)\,(4u^{3})^{\mu}\over
6\,\Gamma(\frac{2}{3}-\mu)^{2}\Gamma(\frac{7}{6}-\mu)^{2}}F(2\mu-\tfrac{1}{3},2\mu-\tfrac{1}{3};\tfrac{1}{6}+2\mu;\tfrac{v}{2})\bigg{]}$
$\displaystyle Q_{2}+Q_{5}$ $\displaystyle=$ $\displaystyle
u\sum_{\mu=0}^{\infty}{\Gamma(\tfrac{2}{3}+\mu)\over(3\mu+1)!}\bigg{[}2^{1/3}{\Gamma(-\frac{1}{6}+2\mu)\,(4u^{3})^{\mu}\over
6\,\Gamma({1\over
2})^{3}}(2v)^{\frac{1}{6}-2\mu}F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{7}{6}-2\mu;\tfrac{v}{2})$
$\displaystyle\hskip 79.49744pt{+}{2^{4/3}\Gamma({1\over
2})\Gamma(\frac{1}{6}-2\mu)\,(4u^{3})^{\mu}\over
6\,\Gamma(\frac{1}{3}-\mu)^{2}\Gamma(\frac{5}{6}-\mu)^{2}}F(\tfrac{1}{3}+2\mu,\tfrac{1}{3}+2\mu;\tfrac{5}{6}+2\mu;\tfrac{v}{2})\bigg{]}$

Using the reflection formula, $Q_{0}+Q_{3}=Q_{0}-Q_{3}$ so that $Q_{3}=0$,
consistent with [15].

###  B.3 Convergence of the ${\mathbb{Z}}_{3}$ series

To study the convergence properties of the series for $Q(\xi)$ when
$\xi^{3}=-1$ we use the reflection and multiplication formulas for
$\Gamma$-functions to obtain,

$\displaystyle Q(\xi)$ $\displaystyle=$ $\displaystyle{v^{\frac{5}{6}}\over
3\pi}\sum_{\nu=0,1,2}(-)^{\nu+1}\xi^{\nu+1}\sin\pi(\tfrac{4\nu-5}{6}){\cal
Q}_{\nu}$ $\displaystyle{\cal Q}_{\nu}$ $\displaystyle=$ $\displaystyle{1\over
2^{\frac{4}{3}}3^{1\over
2}}\sum_{\mu=0}^{\infty}{\Gamma(\mu+\frac{4\nu-5}{12})\Gamma(\mu+\frac{4\nu+1}{12})\over\Gamma(\mu+\frac{\nu+2}{3})\Gamma(\mu+\frac{\nu+3}{3})}\,F({\tfrac{1}{2}},{\tfrac{1}{2}};\tfrac{11-4\nu}{6}-2\mu;\tfrac{v}{2})\left({4u^{3}\over
27v^{2}}\right)^{\mu+\frac{\nu}{3}}$ (B.1)

We begin by considering the $\lambda\to+\infty$ behavior of the hypergeometric