# Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Nikita Nekrasov  IHES, France & ITEP, Russia & SCGP, USA
<EMAIL_ADDRESS>and  Vasily Pestun  IAS, USA<EMAIL_ADDRESS>
###### Abstract.

Seiberg-Witten geometry of mass deformed ${\mathcal{N}}=2$ superconformal ADE
quiver gauge theories in four dimensions is determined. We solve the limit
shape equations derived from the gauge theory and identify the space
${{\mathfrak{M}}}$ of vacua of the theory with the moduli space of the genus
zero holomorphic (quasi)maps to the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$G^{\mathbb{C}}$-bundles on a (possibly degenerate) elliptic curve
$\mathscr{E}$ defined in terms of the microscopic gauge couplings, for the
corresponding simple ADE Lie group $G$. The integrable systems
${\mathfrak{P}}$ underlying, or, rather, overlooking the special geometry of
${{\mathfrak{M}}}$ are identified. The moduli spaces of framed $G$-instantons
on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, of $G$-monopoles with
singularities on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$, the Hitchin systems
on curves with punctures, as well as various spin chains play an important
rôle in our story. We also comment on the higher dimensional theories. In the
companion paper the quantum integrable systems and their connections to the
representation theory of quantum affine algebras will be discussed.

### Acknowledgments

We are grateful to S. Cherkis, R. Donagi, E. Frenkel, A. Kapustin, J. Morgan,
H. Nakajima, A. Okounkov, S. Shatashvili, and E. Witten for discussions and
useful remarks.

Research of NN was supported in part by RFBR grants 12-02-00594, 12-01-00525,
by Agence Nationale de Recherche via the grant ANR 12 BS05 003 02, by Simons
Foundation, and by Stony Brook Foundation.

VP gratefully acknowledges support from Institute for Advanced Study, NFS
grant PHY-0969448, Rogen Dashen Membership, MSERF 14.740.11.0081, NSh
3349.2012.2 and RFBR grant 10-02-00499

The work was reported at the conferences and invited talks [conferences]. We
thank the organizers of these conferences and the institutions for the
invitations and the participants for the questions and the comments.

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the material
    2. 1.2 Notations
  2. 2 Supersymmetric quiver theories
    1. 2.1 Gauge group, matter fields, couplings, parameters
  3. 3 The ADE classifications of superconformal ${\mathcal{N}}=2$ theories
    1. 3.1 Beta functions and Cartan matrix
    2. 3.2 Class I theories
    3. 3.3 Class II theories
    4. 3.4 Class II* theories
  4. 4 Low-energy effective theory
    1. 4.1 Special Kähler geometry
    2. 4.2 Extended moduli space
    3. 4.3 Finite size effects
    4. 4.4 The appearance of an integrable system
    5. 4.5 Integrable systems from classical gauge theories
    6. 4.6 Extended moduli space as a complex integrable system
  5. 5 The limit shape equations
    1. 5.1 The amplitude functions
    2. 5.2 The limit shape prepotential
    3. 5.3 The iWeyl group.
    4. 5.4 Moduli of vacua and mass parameters
  6. 6 Solution of the limit shape equations
    1. 6.1 From invariants to the curve
    2. 6.2 The cameral curve as a modular object
    3. 6.3 Spectral curves
    4. 6.4 Obscured curve
  7. 7 The Seiberg-Witten curves in some detail
    1. 7.1 Class I theories of $A$ type
    2. 7.2 Class I theories of $D$ type
    3. 7.3 Class I theories of $E$ type
    4. 7.4 Class II theories of $A$ type and class II* theories
    5. 7.5 Class II theories of $D$ type
    6. 7.6 Class II theories of $E$ type
  8. 8 The integrable systems of monopoles and instantons
    1. 8.1 Periodic Monopoles and the phase space of Class I theories
    2. 8.2 Double-periodic instantons and Class II theories
    3. 8.3 Noncommutative instantons and Class II* theories
  9. 9 Higher dimensional theories
  10. A McKay correspondence, D-branes, and M-theory
    1. A.1 From finite groups to Lie groups
    2. A.2 Platonic symmetries
    3. A.3 D-branes at singularities
  11. B Partitions and free fermions
  12. C Lie groups and Lie algebras
    1. C.1 Finite dimensional Lie algebras
    2. C.2 Affine Lie algebras
    3. C.3 ADE Cartan matrices, roots and weights
    4. C.4 Affine Weyl group
    5. C.5 Conjugacy classes and moduli of bundles
    6. C.6 Infinite matrices and their Weyl group
    7. C.7 The representation theory

##  Chapter 1 Introduction

In this work a class of quiver ${\mathcal{N}}=2$ supersymmetric theories in
four dimensions is analyzed. The first problem of this sort was solved in
[Seiberg:1994rs, Seiberg:1994aj] for the $SU(2)$ gauge theory in four
dimensions with eight supercharges.

We study mass perturbed ${\mathcal{N}}=2$ superconformal theories, and compute
the exact metric

$ds_{{\mathfrak{M}}}^{2}=G_{{\mathcal{I}}{\bar{\mathcal{J}}}}du^{\mathcal{I}}d{\bar{u}}^{\bar{\mathcal{J}}}$

on the moduli space ${{\mathfrak{M}}}$ of vacua of the low-energy effective
theory. We also compute the vacuum expectation values

$\langle{\mathcal{O}}_{i,n}\rangle_{u}$

of all gauge invariant ${\mathcal{N}}=2$ chiral operators.

Our theories have the gauge group $G_{\text{g}}$ which is a product of a
finite number of special unitary groups. The technique we use is the saddle
point approach to the calculation of the supersymmetric partition function of
the theory in $\Omega$-background [Nekrasov:2002qd]. The partition function is
given by the sum over special instanton configurations. In the limit, where
the $\Omega$-deformation is removed so that the theory approaches the original
flat space theory, the sum over the special instantons is dominated by the
contribution of one particular special instanton configuration, of a very
large instanton charge (with the expected small effective density of instanton
charge). This configuration, the so-called limit shape, is found in this work
using a novel approach, built on the analytic techniques of [Nekrasov:2003rj].
Namely, we interpret the limit shape equations as the conditions defining the
analytic continuations of the generating functions

${\mathscr{Y}}_{i}(x)={\exp}\ \langle\,{\operatorname{tr}}\,{\rm
log}\left(x-{\Phi}_{i}\right)\,\rangle_{u},$ (1.1)

where $i$ labels the simple factors in the gauge group $G_{\text{g}}$, and
${\Phi}_{i}$ is the corresponding complex adjoint Higgs scalar field. We get
the system of (algebraic) equations determining these functions by fixing the
set of basic invariants of the monodromy of the analytic continuation.

Recall that a complex Lie group ${\mathbf{G}}_{\text{q}}$ is naturally
associated with the quiver gauge theory. This group is different from the
original gauge group $G_{\text{g}}$ of the theory. Roughly speaking the Dynkin
diagram of ${\mathbf{G}}_{\text{q}}$ is the universal cover of the quiver of
the gauge theory. The group ${\mathbf{G}}_{\text{q}}$ may be infinite-
dimensional. In fact, for the ${\mathcal{N}}=2$ superconformal theories the
corresponding Lie algebra $\mathfrak{g_{\text{q}}}$ is the finite dimensional
simple Lie algebra of the ADE type $\mathfrak{g}$, or its affine version
$\widehat{\mathfrak{g}}$, or the algebra $\widehat{\mathfrak{gl}}_{\infty}$.

Our main construct is the $x$-dependent element $g(x)$ of the maximal torus
${\mathbf{T}}^{\text{ad}}_{\text{q}}$ of
${\mathbf{G}}^{\text{ad}}_{\text{q}}$, which can also be viewed as the multi-
valued ${\bf T}_{\text{q}}$-valued function. The group element is locally
analytic in $x$:

$g(x)=\prod_{i\in\widetilde{\mathrm{Vert}_{\gamma}}}{\mathscr{P}}_{i}(x+{\mu}_{i})^{-{\lambda}^{\vee}_{i}}{\mathscr{Y}}_{i}(x+{\mu}_{i})^{{\alpha}_{i}^{\vee}}$
(1.2)

where $i$ runs over the set $\widetilde{\mathrm{Vert}_{\gamma}}$ of vertices
of the universal cover of the quiver graph $\gamma$, the polynomials
${\mathscr{P}}_{i}(x)$ and the complex parameters ${\mu}_{i}$ are determined
by the gauge couplings and the masses of matter hypermultiplets, and
${\alpha}_{i}^{\vee}$ and ${\lambda}_{i}^{\vee}$ are the simple coroots and
the fundamental coweights of $\mathfrak{g_{\text{q}}}$. It is also convenient
to introduce another group element:

$g_{\infty}(x)=\prod_{i\in\widetilde{\mathrm{Vert}_{\gamma}}}{\mathscr{P}}_{i}(x+{\mu}_{i})^{-{\lambda}^{\vee}_{i}}$
(1.3)

Our main claim is that the conjugacy class
$[g(x)]\in{{\mathbf{T}}^{\text{ad}}_{\text{q}}}/W({\mathfrak{g_{\text{q}}}})$
is holomorphic in $x$, so that the basic adjoint invariants of
${\mathbf{G}}_{\text{q}}$, evaluated on $[g(x)]$ (up to some twist discussed
further) are polynomials of $x$, leading to a system of equations relating
${\mathscr{Y}}$ and $x$:

${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=T_{i}(x)=T_{i,0}x^{{\mathbf{v}}_{i}}+T_{i,1}x^{{\mathbf{v}}_{i}-1}+\ldots+T_{i,{\mathbf{v}}_{i}}$
(1.4)

which define what we call the _cameral curve_

${\mathcal{C}}_{u}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times\left({\mathbb{C}}^{\times}\right)^{\mathrm{Vert}_{\gamma}}$
(1.5)

The invariants ${{\mathscr{X}}}_{i}$ are normalized characters of $g(x)$ in
the fundamental representations ${\mathbf{R}}_{i}$ of
${\mathbf{G}}_{\text{q}}$, of the highest weight $\lambda_{i}$:

${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=g_{\infty}(x)^{-\lambda_{i}}\,{\operatorname{Tr}}_{R_{i}}g(x)$

Moreover, from the work of Steinberg [Steinberg:1965] (see also
[Morgan:2002math]) we know that for the finite-dimensional
${\mathbf{G}}_{\text{q}}$ one can conjugate $g(x)$ in
${\mathbf{G}}_{\text{q}}$ to obtain a smooth ${\mathbf{G}}_{\text{q}}$-valued
function ${\bf g}(x)$ of $x$. Further inspection shows that ${\bf g}(x)$ is a
quasi-classical limit of an element of the Yangian algebra
$Y({\mathfrak{g_{\text{q}}}})$, built on the Lie algebra
$\mathfrak{g_{\text{q}}}$ of ${\mathbf{G}}_{\text{q}}$. Hopefully the
analogous statements hold for all ${\mathbf{G}}_{\text{q}}$’s we encounter.

In this way one recovers all known results about the Seiberg-Witten geometries
of the ${\mathcal{N}}=2$ theories in four dimensions (we do not review all of
them in this work) as well as finds new results. In particular, we find the
families of curves describing the geometry of the moduli space of vacua for
the theories which were previously believed not to have such description. We
also find that the special geometry of the quiver theories is captured in
general by a polylogarithmic system of differentials on these curves.

##### Higher dimensions

The gauge theories we discuss can be also lifted to five dimensional theories
compactified on a circle $\mathbb{S}^{1}_{\left\langle\beta\right\rangle}$ of
circumference $\beta$, or even to the six dimensional theories, compactified
on a two-torus, of the area $\beta^{2}$. In the limit $\beta\to 0$ one
recovers the original four dimensional theory. In the five dimensional case
the polynomials $T_{i}(x)$ in Eq. (1.4) are replaced by Laurent polynomials in
$e^{\mathrm{i}\beta x}$, while in the six dimensional case the functions
$T_{i}(x)$ become elliptic.

##### Defreezing

One of the initial questions which led us to the subject of this work was the
following. Consider the $SU(2)$ theory with $N_{f}=4$ hypermultiplets in the
fundamental representation, with the coupling $\mathfrak{q}$. By now there is
an overwhelming evidence for the connection of this theory to Liouville
conformal blocks on a sphere with four punctures. The momenta of Liouville
vertex operators at the punctures are related to the masses of the
hypermultiplets, the locations of the vertex operators are, e.g.
$0,1,{\mathfrak{q}},{\infty}$, and the momentum at the intermediate channel is
the Coulomb parameter $a$.

Let us view this theory as a $U(2)$ theory, and let us single out the maximal
torus $U(1)^{4}$ of the $Spin(8)$ flavor symmetry group. Let us gauge these
$U(1)$ groups. This gauging is possible in the noncommutative geometry setup.
One acquires four additional coupling constants. What will happen to the
Liouville theory?

Upon some reflection one concludes that the resulting theory is a particular
case of the ${\widehat{D}}_{4}$ theory, with
${\mathbf{v}}_{0}={\mathbf{v}}_{1}={\mathbf{v}}_{3}={\mathbf{v}}_{4}=1$, and
${\mathbf{v}}_{2}=2$. We then decided to solve the general quiver
superconformal theory which led us to discover many other interesting things.

Figure 1.1. The three major ways to construct $\mathcal{N}=2$ theories

##### Classification

Another motivation was the question whether Hitchin’s system exhaust the list
of all reasonable Seiberg-Witten integrable systems. From the early discovery
[Donagi:1995cf] that the ${\mathcal{N}}=2^{*}$ theory with
${G_{\text{g}}}=SU(N)$ is governed by the $SU(N)$ Hitchin system on a one-
punctured torus (which is nothing but the elliptic Calogero-Moser system, as
shown previously in [Gorsky:1994dj]), proposals in [Martinec:1996by], and
subsequent developments culminating in the introduction of the ‘‘S-class’’
theories [Witten:1997sc, Gaiotto:2009we, Gaiotto:2009hg, Alday:2009aq] there
was a lot of activity with experimental evidence suggesting that
${\mathcal{N}}=2$ theories can be described by some version of Hitchin’s
system. The underlying construction in these approaches is the
compactification of the six dimensional superconformal $(0,2)$-theory on some
Riemann surface embedded as a supersymmetric cycle in some ambient geometry,
and it is believed that the global features of the embedding should play
virtually no rôle in the effective gauge theory dynamics.

Another way of engineering ${\mathcal{N}}=2$ theories, using string theory, is
the so-called geometric engineering [Katz:1996fh, Lerche:1996xu], which is the
study of the gravity-decoupled limit of the IIA compactification on a Calabi-
Yau threefold, with the Calabi-Yau becoming effectively non-compact. A large
class of models comes from toric Calabi-Yau’s. One then employs the local
mirror symmetry to generate curves with differentials, whose periods capture
the special geometry of the ${\mathcal{N}}=2$ theory.

In our work we presented another characterization of the integrable systems
underlying the special geometry of the ${\mathcal{N}}=2$ theories with the
superconformal ultraviolet limit. Namely, we identify these systems with the
moduli spaces of some gauge/Higgs configurations, such as monopoles or
instantons, with the gauge group ${\mathbf{G}}_{\text{q}}$ corresponding to
the quiver diagram encoding, among other things, the matter sector of the
theory. Unlike all previous approaches, see fig. 1.1, which involved some
reference to the non-perturbative dualities, or even embedding of the gauge
theory to string theory and M-theory, we derive these statements within the
quantum field theory, by analyzing the instanton contributions to the low-
energy effective action.

In some cases (e.g. in a simple fashion for the $A_{r}$ type theories, in a
more subtle way for the $D_{r}$ type theories) our phase spaces can be
identified with the phase spaces of Hitchin systems on the low genus curves
with punctures, using some version of Nahm-Fourier-Mukai transform, but in
general we don’t have such a duality. Provided a complete description of the
Seiberg-Witten curves and algebraic integrable systems for the $\mathcal{N}=2$
ADE quiver theories it would be interesting to further investigate this ADE
quiver class, along the lines of [Gaiotto:2009hg, Gaiotto:2010be] or
[Alim:2011ae] for the ‘‘S-class’’.

In the other cases, e.g. the class II $E_{r}$ type theories we can use the
relation between the moduli of del Pezzo surfaces and the moduli of
$E$-bundles on elliptic curve to assign to our version of the Seiberg-Witten
curve a one-parametric family of del Pezzo surfaces, which can be viewed as an
example of the mirror noncompact threefold of [Katz:1997eq].

##### Outlook

|  |  |  | classical | quantum | double quantum
---|---|---|---|---|---|---
|  |  |  | $\begin{array}[]{c}\epsilon_{1}=0\\\ \epsilon_{2}=0\end{array}$ | $\begin{array}[]{c}\epsilon_{1}\neq 0\\\ \epsilon_{2}=0\end{array}$ | $\begin{array}[]{c}\epsilon_{1}\neq 0\\\ \epsilon_{2}\neq 0\end{array}$
4d | $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}$ | XXX | rational | ${\mathbf{G}}_{\text{q}}(\mathbb{C})$ | $Y(\mathfrak{g_{\text{q}}})$ | …
5d | $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}^{\times}$ | XXZ | trigonometric | ${\mathbf{G}}_{\text{q}}(\mathbb{C}^{\times})$ | $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ | …
6d | $\mathbf{C}_{\left\langle x\right\rangle}=E_{T}$ | XYZ | elliptic | ${\mathbf{G}}_{\text{q}}(E)$ | $E_{T,\eta}(\mathfrak{g_{\text{q}}})$ | …
Table 1.1. The (rational/trigonometric/elliptic) by (classical/quantum/double
quantum)

In the companion paper [NP2012b] we study in details the connection between
the class of ADE quiver gauge theories and _quantum_ integrable ADE
spinchains. In particular we explain there that the five dimensional version
of the ADE quiver gauge theory on the twisted bundle
$\mathbb{R}^{4}{\tilde{\times}}\mathbb{S}^{1}_{\left\langle\epsilon_{1},\epsilon_{2};\beta\right\rangle}$
[Nekrasov:2003af] with the equivariant parameters set to
$\epsilon_{1}=\epsilon$, $\epsilon_{2}=0$ as in [Nekrasov:2009ui] is
associated with the XXZ spin chain $\mathfrak{g_{\text{q}}}$. The theory is
solved by the quantum version of the master equation (1.4): the group
${\mathbf{G}}_{\text{q}}$ is replaced by the _quantum affine algebra_
$U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ with the quantum parameter
$q=e^{\mathrm{i}\beta\epsilon}$, while the characters ${\mathscr{X}}_{i}$ are
promoted to the $q$-characters of Frenkel-Reshetikhin [Frenkel:1998]. (If
${\mathbf{G}}_{\text{q}}$ is itself affine Kac-Moody group
${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ then
$U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ is naturally _quantum toroidal
algebra_). In the four dimensional limit the XXZ $\mathfrak{g_{\text{q}}}$
spin chain turns into the XXX $\mathfrak{g_{\text{q}}}$ spin chain, the
quantum affine algebra $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ degenerates
into Yangian $Y(\mathfrak{g_{\text{q}}})$, and the gauge theory on the twisted
bundle becomes the four dimensional theory subject to a two dimensional
$\Omega$-background. Finally, the six dimensional theory compactified on a
torus $E_{T}$ corresponds to the XYZ $\mathfrak{g_{\text{q}}}$ spinchain, with
the quantum affine group $U_{q}(\widehat{\mathfrak{g_{\text{q}}}})$ elevated
to the quantum elliptic group $E_{T,{\eta}}(\mathfrak{g_{\text{q}}})$
[Drinfeld_1986, Felder_1995, Felder_1996], with
${\eta}={\beta}{\epsilon}/2\pi$.

It is clear that there is an even larger picture in which the algebraic
integrable systems we encountered in this work are in a additionally
quantized, or deformed, this time with the with parameter $\epsilon_{2}$, c.f.
table 1.1, with the rational/trigonometric/elliptic trichotomy in the vertical
direction established in [Faddeev:1982, Sklyanin:1979, Belavin:1982] and
connected with the gauge theories in [Nekrasov:1996cz]. It would be exciting
to explore the connection with H. Nakajima’s work [Nakajima:2001q] on quiver
varieties and quantum affine algebras as well as the connection with elliptic
cohomology [Ginzburg_1995, Grojnowski_2007, Lurie_2009] of moduli spaces.
Notice that the quantum or double quantum exploration of the ADE quiver world
is in a sense orthogonal to the approach of [Alday:2009aq] dealing with the
‘‘S-class’’ world in the Fig. 1.1. Classically, on the overlap, the relation
between the corresponding algebraic integrable systems comes from the Nahm-
Fourier-Mukai/Corrigan-Goddard/ADHM reciprocity relating the moduli space of
$\mathbf{G}$-bundles and Hitchin systems. The (doubly) quantum version of this
Nahm transform, if it exists, seems to cover the ‘‘quantum’’ geometric
Langlands duality, Separation of variables for quantum systems
[Sklyanin:1987ih, Feigin:1994in, Frenkel:1995zp, Sklyanin:1995bm,
Enriquez:1996xc] and the quasi-particle or ‘‘free field’’ Dotsenko-Fateev
[Dotsenko:1984nm, Gerasimov:1990fi, Feigin:1990qn] representation of conformal
blocks. The new ingredient [Nekrasov:2009rc] in this relatively classic field
of research are the supersymmetric gauge theories in four dimensions.

The task to fill the table 1.1 with all glory (double) quantum details is left
for future work.

###  1.1. Organization of the material

Chapter 2 introduces the quiver supersymmetric gauge theories which we shall
study.

Chapter 3 presents the classification of the gauge theories which are
${\mathcal{N}}=2$ superconformal in the ultraviolet. We distinguish three
classes of such theories, I, II, and II*. The I and II classes have an ADE
classification so that for class I ${\mathbf{G}}_{\text{q}}=\mathbf{G}$ and
for class II ${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ where $\mathbf{G}$
is ADE group, the II* theories correspond to $\widehat{GL}_{\infty}$ group.

Chapter 4 reviews the special Kähler geometry of the vectormultiplet moduli
spaces ${\mathfrak{M}}$ of vacua of ${\mathcal{N}}=2$ theories. We also recall
the relation of ${\mathfrak{M}}$ to the algebraic integrable systems and the
hyperKähler manifolds. We give some examples to be used later.

Chapter 5 introduces our main tool: the limit shape equations, which summarize
the microscopic gauge theory calculation leading to the effective low-energy
action, i.e. the prepotential ${{\mathscr{F}}}$.

Chapter 6 presents the solution of the limit shape equation. We reformulate
the equations as the Riemann-Hilbert problem for the set of functions
${\mathscr{Y}}(x)$ and solve it by equating the invariants
${{\mathscr{X}}}({\mathscr{Y}}(x))$ of the monodromy group, the _iWeyl group_
which we attach to every ${\mathcal{N}}=2$ gauge theory, to some polynomials
$T(x)$. In this manner we find an (algebraic) curve ${\mathcal{C}}$ and a
system of differentials, whose periods give the special coordinates
$\mathfrak{a}$ and the derivatives
${\partial}{{\mathscr{F}}}/{\partial}{\mathfrak{a}}$ of the prepotential
${\mathscr{F}}$.

Chapter 7 analyzes the solution in some detail. We interpret the data for the
solution of the class I theories as describing a holomorphic map with
prescribed singularities of ${\mathbb{C}\mathbb{P}}^{1}$ to the space of
conjugacy classes ${\mathbf{T}}/W({\mathbf{g}})$ in a complex Lie group
$\mathbf{G}$, which can be also viewed as the moduli space of holomorphic
$\mathbf{G}$-bundles on a degenerate elliptic curve. For the class II theories
the analogous data parametrizes (quasi)maps to the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$\mathbf{G}$-bundles on elliptic curve. In some cases we relate the curve
$\mathcal{C}$ to the more familiar Seiberg-Witten curves. For the theories
corresponding to the $A$ series we manage to relate our curves $\mathcal{C}$
to the spectral curves of rational and elliptic Gaudin models (the Hitchin
system on the genus zero and one curves with punctures), and also reproduce
the results of [Witten:1997sc], [Shadchin:2005cc]. For the class II $D$ type
theories we reproduce the results of [Kapustin:1998fa]. For the class II $E$
type theories we find yet another interpretation of our solution, in terms of
families of del Pezzo surfaces. In this way we get a field theory
understanding of some of the local mirror symmetry predictions [Katz:1997eq]
and brane construction [Kapustin:1998pb, Kapustin:1998xn].

Chapter 8 discusses the moduli spaces ${\mathfrak{P}}$ of vacua of the gauge
theory compactified on a circle ${\mathbb{S}}^{1}$. We don’t present the full
analysis of the hyperKähler metric on ${{\mathfrak{P}}}$ in this work.
Instead, we focus on the geometry of ${{\mathfrak{P}}}$ in the complex
structure inherited from four dimensions (this complex structure is sometimes
called the complex structure $\bf I$), in which it presents itself as an
(algebraic) integrable system. Our solution of the four dimensional theory
comes in a form which leads to a natural guess for the phase space
${\mathfrak{P}}$ of the integrable systems corresponding to our theories. For
the class I theories it is the moduli space of $G$-monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with singularities, for the class II
theories it is the moduli space of $G$-instantons on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, and for the class II*
${\widehat{A}}_{r}$ theories it is the moduli space of noncommutative $U(r+1)$
instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$. Of course these spaces
have a natural hyperkähler structure which depends in the expected fashion on
all the parameters of the theory and its compactification. Although our
motivation comes from the field theory analysis in the previous chapters, our
results confirm the conjectures of [Chalmers:1996xh], [Hanany:1996ie],
[Kapustin:1998fa],[Kapustin:1998pb], [Kapustin:1998xn], [Cherkis:1997aa],
[Cherkis:2000cj], [Cherkis:2001gm] which are motivated by the string theory
analysis, and in particular by the brane constructions.

Chapter 9 discusses the modifications of our solutions in the five and six
dimensional cases.

In the Appendix A we review the affine ADE graphs, the McKay correspondence
and the M-theory/D-brane picture for the present work; in the Appendix B we
put our conventions on the partitions and representations by free fermions; in
the Appendix C we review some standard material on Lie groups and Lie algebras
which we use in solving our theories. We recall the notions of the (co)root
and the (co)weight lattices, Weyl groups, and the integrable highest weight
representations; in the Appendix LABEL:se:elliptic we collect our conventions
for elliptic functions; in the Appendix LABEL:se:appendixE we give some
technical details on spectral curves of affine E-series.

###  1.2. Notations

###### Quivers

chapter 2

$\displaystyle{\mathrm{Vert}_{\gamma}}$ $\displaystyle{\rm set\ of\ vertices}$
$\displaystyle{\mathrm{Edge}_{\gamma}}$ $\displaystyle{\rm set\ of\ edges}$
$\displaystyle s(e)\in\mathrm{Vert}_{\gamma}$ $\displaystyle{\rm the\ source\
of\ the\ edge\ }e\in\mathrm{Edge}_{\gamma}$ $\displaystyle
t(e)\in\mathrm{Vert}_{\gamma}$ $\displaystyle{\rm the\ target\ of\ the\ edge\
}e\in\mathrm{Edge}_{\gamma}$
$\displaystyle\mathfrak{q}_{i}=e^{2\pi\mathrm{i}\tau_{i}}$ gauge coupling
constants $\displaystyle{\tilde{\mathfrak{q}}}$
$\displaystyle({\mathfrak{q}}_{i})_{i\in\mathrm{Vert}_{\gamma}}$
$\displaystyle\mathbf{v}_{i}$ number of colors for $i$-th node gauge group
$SU(\mathbf{v}_{i})$ $\displaystyle\mathbf{w}_{i}$ number of flavors for
$i$-th node fundamental matter $\displaystyle\mathfrak{a}_{i,\mathbf{a}}$
eigenvalues of the complex scalars
$\displaystyle\mathfrak{a}_{i,\mathbf{a}}\leftrightarrow\mathfrak{a}^{\mathcal{I}}$
the special coordinates on Coulomb moduli space $\displaystyle C_{ij}$ Cartan
matrix associated to the quiver by its Dynkin graph $\displaystyle
r=\operatorname{rk}(C)$ $|\mathrm{Vert}_{\gamma}|$ if $\gamma$ is finite ADE
or $|\mathrm{Vert}_{\gamma}|-1$ if $\gamma$ is affine ADE

###### Lie groups

$\displaystyle\mathrm{i}$ $\displaystyle\sqrt{-1}$ $\displaystyle
G_{\text{g}}$ Gauge group of the four-dimensional theory
$\displaystyle{\mathbf{G}}_{\text{q}}$ Kac-Moody group associated with quiver
Dynkin diagram $\displaystyle{G_{M}}$ the flavor group
$\displaystyle{\mathbf{G}}=G^{\mathbb{C}}$ $\displaystyle{\rm finite\
dimensional\ complex\ Lie\ group}\quad$ $\displaystyle{\widehat{\mathbf{G}}}$
affine Kac-Moody group for $\mathbf{G}$ $\displaystyle{G}$ maximal compact
subgroup of $\mathbf{G}$ $\displaystyle{\mathbf{T}}$ $\displaystyle{\rm
maximal\ torus\ of}\ {\mathbf{G}}\quad$ $\displaystyle T$ $\displaystyle{\rm
maximal\ torus\ of}\ G$ $\displaystyle Z$ the center of both $G$ and
$\mathbf{G}$ $\displaystyle{{\mathbf{G}}^{\text{ad}}}={\mathbf{G}}/Z$
$\displaystyle{\rm adjoint\ form\ of\ the\ complex\ Lie\ group}\
{\mathbf{G}}\quad$ $\displaystyle{{\mathbf{T}}^{\text{ad}}}={\mathbf{T}}/Z$
the maximal torus of ${\mathbf{G}}^{\text{ad}}$
$\displaystyle{{\mathbf{G}}^{\text{ad}}_{\text{q}}}={{\mathbf{G}}_{\text{q}}}/Z$
$\displaystyle{\rm adjoint\ form\ of\ the\ complex\ Lie\ group}\
{{\mathbf{G}}_{\text{q}}}\quad$
$\displaystyle{{\mathbf{T}}^{\text{ad}}_{\text{q}}}={{\bf T}_{\text{q}}}/Z$
$\displaystyle{\rm its\ maximal\ torus}\quad$

###### Lie algebras

$\displaystyle\mathfrak{g_{\text{q}}}$ Kac-Moody Lie algebra associated with
quiver Dynkin diagram $\displaystyle{\mathfrak{g}}$ $\displaystyle\ {\rm
Lie}({\mathbf{G}})$ $\displaystyle{\mathfrak{h}}$ $\displaystyle\ {\rm
Lie}({\mathbf{T}})$

###### Representation theory

appendix C

$\displaystyle a_{i}$ Kac-Dynkin marks $\displaystyle{{\rm
Q}},{\Lambda}^{\vee}$ $\displaystyle{\rm root\ lattice},\ {\rm coweight\
lattice}$ $\displaystyle{\Lambda},{{\rm Q}}^{\vee}$ $\displaystyle{\rm weight\
lattice},\ {\rm coroot\ lattice}$ $\displaystyle R_{i}$ $\displaystyle
i^{\prime}{\rm th\ fundamental\ representation\ of}\ {\mathbf{G}}$
$\displaystyle{\widehat{R}}_{i}$ $\displaystyle i^{\prime}{\rm th\
fundamental\ representation\ of}\ {\widehat{\mathbf{G}}}$
$\displaystyle{\mathcal{R}}_{i}$ $\displaystyle i^{\prime}{\rm th\
fundamental\ representation\ of}\ {\widehat{GL}_{\infty}}$

###### Spaces

$\displaystyle{B({\mathfrak{g}})}$
$\displaystyle{\mathbf{T}}/W({\mathfrak{g}})\text{ the space of conjugacy
classes in $\mathbf{G}$ }$ $\displaystyle{B({\mathfrak{g_{\text{q}}}})}$
$\displaystyle{{\bf T}_{\text{q}}}/W({\mathfrak{g_{\text{q}}}})\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}_{\text{q}}}$
$\displaystyle{B^{\text{ad}}({\mathfrak{g}})}$
$\displaystyle{\mathbf{T}}/\left(Z\times W({\mathfrak{g}})\right)\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}^{\text{ad}}}$
$\displaystyle{B^{\text{ad}}({\mathfrak{g_{\text{q}}}})}$ $\displaystyle{{\bf
T}_{\text{q}}}/\left(Z\times W({\mathfrak{g_{\text{q}}}})\right)\ {\rm the\
space\ of\ conjugacy\ classes\ in}\ {{\mathbf{G}}^{\text{ad}}_{\text{q}}}$
$\displaystyle{\mathbf{C}_{\left\langle x\right\rangle}}$ complex plane
$\mathbb{C}$ in 4d, cylinder $\mathbb{C}^{\times}$ in 5d, torus $E$ in 6d
$\displaystyle{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$
$\displaystyle={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\,\infty\,\\}$
$\displaystyle{\mathscr{E}}(\mathfrak{q})$ elliptic curve
$\mathbb{C}^{\times}/\mathfrak{q}^{\mathbb{Z}}$ $\displaystyle\mathfrak{q}$
$\prod_{i\in\mathrm{Vert}_{\gamma}}\mathfrak{q}_{i}^{a_{i}}$ for class II
theories $\displaystyle\mathrm{Bun}_{\mathbf{G}}(\mathscr{E})$ coarse moduli
space of semistable holomorphic $\mathbf{G}$-bundles on $\mathscr{E}$
$\displaystyle{\mathfrak{M}}$ the Couloumb moduli space of the 4d $\times
SU(\mathbf{v}_{i})$ gauge theory $\displaystyle{\mathfrak{M}}^{\mathrm{ext}}$
the Couloumb moduli space of the 4d $\times U(\mathbf{v}_{i})$ gauge theory
$\displaystyle{\mathfrak{P}}\to{\mathfrak{M}}$ the algebraic integrable system
$\dim_{\mathbb{C}}{\mathfrak{P}}=2\dim_{\mathbb{C}}{\mathfrak{M}}$
$\displaystyle{\mathfrak{P}}^{\mathrm{ext}}\to{\mathfrak{M}}^{\mathrm{ext}}$
the complex integrable system
$\dim_{\mathbb{C}}{\mathfrak{P}}^{\mathrm{ext}}=2\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}$

###### Seiberg-Witten curves

$\displaystyle\mathcal{C}_{u}$ cameral curve: section 6.2 $\displaystyle
C_{u}$ spectral curve: section 6.3 $\displaystyle\mathscr{C}$ obscure curve:
section 6.4 $\displaystyle x$ flat coordinate on $\mathbf{C}_{\left\langle
x\right\rangle}$ $\displaystyle\mathscr{Y}_{i}(x)$ amplitudes (the solution of
the theory): section 5.1 $\displaystyle T_{i}(x)$ gauge polynomials of degree
$\mathbf{v}_{i}$ $\displaystyle\mathscr{P}_{i}(x)$ matter polynomials of
degree $\mathbf{w}_{i}$ $\displaystyle g(x)$
${\mathbf{T}}^{\text{ad}}_{\text{q}}$ valued analytic function on
$\mathbf{C}_{\left\langle x\right\rangle}$ $\displaystyle\mathbf{g}(x)$
${\mathbf{G}}_{\text{q}}$ valued analytic function on
$\mathbf{C}_{\left\langle x\right\rangle}$ $\displaystyle{\mathscr{X}}_{j}$
${\mathbf{G}}_{\text{q}}$ character (or Weyl invariant) for $i$-th fundamental
weight of ${\mathbf{G}}_{\text{q}}$

###### Partitions

$\displaystyle{\lambda}$ $\displaystyle{\rm partition}\
({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots{\lambda}_{{\ell}({\lambda})}>0),\qquad{\lambda}_{i}\in{\mathbb{Z}}_{\geq
0}$ $\displaystyle{\ell}({\lambda})$ $\displaystyle{\rm the\ length\ of\ the\
partition\ }{\lambda}$ $\displaystyle|{\lambda}|$ the size of the partition
$|\lambda|=\sum_{i=1}^{{\ell}({\lambda})}{\lambda}_{i}$

Let $x\in R^{\mathbb{Z}}$ be a sequence $(x_{i})_{i\in\mathbb{Z}}$, with
$x_{i}$ in some ring $R$.

###### Consecutive products

$\displaystyle x^{[i]}_{j}\equiv$
$\displaystyle\,\frac{\prod_{k=-\infty}^{k=j+i-1}x_{k}}{\prod_{k=-\infty}^{j-1}x_{k}}\,,$
$\displaystyle x^{[i]}=$ $\displaystyle\,x^{[i]}_{1},$ $\displaystyle
x^{[i+1]}=$ $\displaystyle\,x^{[i]}x_{i+1}$

for example

$\displaystyle x^{[0]}=$ $\displaystyle\,1,$ $\displaystyle x^{[2]}_{4}=$
$\displaystyle\,x_{4}x_{5},$ $\displaystyle x^{[-1]}=$
$\displaystyle\,x_{0}^{-1},$ $\displaystyle x_{5}^{[-4]}=$
$\displaystyle\,x_{1}^{-1}x_{2}^{-1}x_{3}^{-1}x_{4}^{-1}$

###### Consecutive sums

$\displaystyle x_{(i)}=$
$\displaystyle\,\sum_{j=-\infty}^{i}x_{j}-\sum_{j=-\infty}^{0}x_{j}$
$\displaystyle x_{(i+1)}=$ $\displaystyle\,x_{(i)}+x_{i+1}$

for example

$\displaystyle x_{(0)}=$ $\displaystyle\,0,$ $\displaystyle x_{(3)}=$
$\displaystyle\,x_{1}+x_{2}+x_{3},$ $\displaystyle x_{(-2)}=$
$\displaystyle\,-x_{0}-x_{-1},$
$\displaystyle{\lambda}_{({\ell}({\lambda}))}=$ $\displaystyle\,|{\lambda}|$

##  Chapter 2 Supersymmetric quiver theories

Consider any ${\mathcal{N}}=2$ supersymmetric field theory in four dimensions
whose gauge group is a product of special unitary groups, while the matter
hypermultiplets are in the fundamental, bi-fundamental, and adjoint
representations. The field content, the parameters of the Lagrangian, and the
choice of the vacuum are conveniently encoded in the quiver data, which is:

  1. (1)

An oriented graph $\gamma$ with the set $\mathrm{Vert}_{\gamma}$ of vertices
and the set
$\mathrm{Edge}_{\gamma}\subset\mathrm{Vert}_{\gamma}\times\mathrm{Vert}_{\gamma}$
of oriented edges. Let
$s,t:\mathrm{Vert}_{\gamma}\times\mathrm{Vert}_{\gamma}\to\mathrm{Vert}_{\gamma}$
by the projections onto the first and the second factors. They define the two
maps $s,t:\mathrm{Edge}_{\gamma}\to\mathrm{Vert}_{\gamma}$ which assign to an
oriented edge its source and the target, respectively. In what follows we
shall use the notation

$|\mathrm{Vert}_{\gamma}|=\\#\mathrm{Vert}_{\gamma}$

for the number of vertices in the quiver.

  2. (2)

An assignment of polynomials to the vertices:

${\mathscr{T}},{\mathscr{P}}:\mathrm{Vert}_{\gamma}\to{\mathbb{C}}[x],\qquad
i\mapsto{\mathscr{T}}_{i}(x),{\mathscr{P}}_{i}(x)$ (2.1)

and ${\mathbf{v}},{\mathbf{w}}:\mathrm{Vert}_{\gamma}\to{\mathbb{Z}}_{+}$,
where

${\mathbf{v}}_{i}={\deg}\,{\mathscr{T}}_{i},\
{\mathbf{w}}_{i}={\deg}\,{\mathscr{P}}_{i},\qquad i\in\mathrm{Vert}_{\gamma}$

The polynomials ${\mathscr{T}}_{i}(x)$ are monic, the highest order term
coefficients ${\mathfrak{q}}_{i}$ of the polynomials ${\mathscr{P}}_{i}(x)$
are required to obey: $|{\mathfrak{q}}_{i}|<1$.

  3. (3)

A $1$-cocycle ${\bf m}\in{\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$, in other
words an assignment

$e\in\mathrm{Edge}_{\gamma}\mapsto m_{e}\in{\mathbb{C}}\ .$ (2.2)

We now proceed with the explanation of the rôles of the polynomials
${\mathscr{T}},{\mathscr{P}}$, as well as that of the cocycle $\bf m$.

###  2.1. Gauge group, matter fields, couplings, parameters

####  2.1.1. The gauge group

We denote the gauge group by $G_{\text{g}}$. It is the product:

${G_{\text{g}}}=\prod_{i\in\mathrm{Vert}_{\gamma}}SU({\mathbf{v}}_{i})$ (2.3)

The vector multiplet therefore splits into a collection of vector multiplets
for the $SU({\mathbf{v}}_{i})$ gauge factors:

${\Phi}_{\text{g}}=\left({\Phi}_{i}\right)_{i\in\mathrm{Vert}_{\gamma}}$

We have a gauge coupling $e_{i}$ and the theta angle $\vartheta_{i}$ for each
$i\in\mathrm{Vert}_{\gamma}$. As usual, we combine them into the complexified
gauge couplings,

${\tau}_{i}=\frac{1}{2\pi\mathrm{i}}{\log}({\mathfrak{q}}_{i})=\frac{{\vartheta}_{i}}{2\pi}+\frac{4\pi\mathrm{i}}{e_{i}^{2}}$
(2.4)

The bosonic part of the action for gauge fields is given by:

${\mathcal{L}}_{\text{YM}}=\sum_{i\in{\mathrm{Vert}_{\gamma}}}\left(\frac{1}{e_{i}^{2}}\int{\operatorname{tr}}_{{\mathbf{v}}_{i}}F_{A_{i}}\wedge\star
F_{A_{i}}+\frac{{\mathrm{i}}{\vartheta}_{i}}{8\pi^{2}}\int{\operatorname{tr}}_{{\mathbf{v}}_{i}}F_{A_{i}}\wedge
F_{A_{i}}\right)$ (2.5)

where ${\operatorname{tr}}_{v}$ denotes the trace of a $v\times v$ matrix. The
exponentiated coupling

${\mathfrak{q}}_{i}=e^{2\pi\mathrm{i}\tau_{i}}$

enters the path integral measure. The perturbative effects do not depend on
$\vartheta_{i}$, while the non-perturbative effects, which are the
contributions of the gauge fields with non-trivial instanton charge, depend on
${\mathfrak{q}}_{i},{\bar{\mathfrak{q}}}_{i}$. In other words, the partition
function is expected to be invariant under the shifts

${\tau}_{i}\longrightarrow{\tau}_{i}+1\ .$ (2.6)

For $i\in\mathrm{Vert}_{\gamma}$ let ${\Phi}_{i}$ denote the corresponding
complex scalar in the adjoint representation. The bosonic potential of the
vector multiplet field $\Phi_{i}$ contains a universal term

${\operatorname{tr}}_{{\mathbf{v}}_{i}}\,[{\Phi}_{i},{\Phi}_{i}^{\dagger}]^{2}$

plus some possible non-negative terms coming from interactions with matter
fields. If the matter fields are massive then this term alone forces
${\Phi}_{i}$ to commute with its conjugate at low energies. Therefore, at low
energy the field $\Phi_{i}$ can be diagonalized:

${\Phi}_{i}\longrightarrow{\rm
diag}(\mathfrak{a}_{i,{\alpha}})_{{\alpha}=1}^{{\mathbf{v}}_{i}},$ (2.7)

The gauge invariant order parameters are the vacuum expectation values of the
coefficients of the characteristic polynomial ${\mathscr{T}}_{i}$ of
${\Phi}_{i}$:

${\mathscr{T}}_{i}(x)=\left\langle{\det}_{{\mathbf{v}}_{i}}(x-{\Phi}_{i})\right\rangle_{u}$
(2.8)

where we assume the normalization $\langle 1\rangle_{u}=1$, and ${\det}_{v}$
denotes the determinant of a $v\times v$-matrix. Therefore the polynomials
${\mathscr{T}}_{i}(x)$ in (2.8) are monic.

Thus, a collection of polynomials ${\mathscr{T}}_{i}(x)$,
$i\in{\mathrm{Vert}_{\gamma}}$ fixes the choice of the vacuum
$u=(u_{i,{\mathbf{a}}})\in{{\mathfrak{M}}}$:

${\mathscr{T}}_{i}(x)=x^{{\mathbf{v}}_{i}}+\sum_{{\mathbf{a}}=2}^{{\mathbf{v}}_{i}}u_{i,{\mathbf{a}}}x^{{\mathbf{v}}_{i}-{\mathbf{a}}}$
(2.9)

Because of the non-perturbative (instanton) effects the relation between
$u_{i,{\mathbf{a}}}$ and $\mathfrak{a}_{i,{\mathbf{a}}}$ is not polynomial,
and for the same reason ${\mathscr{T}}_{i}(x)\neq{\mathscr{Y}}_{i}(x)$.

####  2.1.2. The hypermultiplets in the bi-fundamental, or adjoint
representations

The bifundamental or adjoint hypermultiplet $H_{e}$,
$e\in{\mathrm{Edge}_{\gamma}}$ transforms in the following ${G_{\text{g}}}$
representation:

$\left({\overline{\mathbf{v}}_{s(e)}},{\mathbf{v}}_{t(e)}\right)\oplus\left({\overline{\mathbf{v}}_{t(e)}},{\mathbf{v}}_{s(e)}\right),\qquad{\rm
for}\quad s(e)\neq t(e)$ ${\rm Adj}(su({\mathbf{v}}_{i})),\qquad{\rm for}\quad
s(e)=t(e)=i$

The masses of the bi-fundamental hypermultiplets are conveniently represented
by the $1$-cocycle: ${\bf m}\in{\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$:

$e\mapsto{\bf m}_{e}$

Let $[{\bf m}]\in H^{1}({\gamma},{\mathbb{C}})$ be the corresponding
cohomology class. If we denote by $m^{*}$ a particular representative of
$[{\bf m}]$ in ${\mathcal{C}}^{1}({\gamma},{\mathbb{C}})$, then

${\bf
m}=m^{*}+{\delta}{\mu}\,,\qquad{\mu}\in{\mathcal{C}}^{0}({\gamma},{\mathbb{C}})$
(2.10)

or, in components,

${\bf m}_{e}=m^{*}_{e}+{\mu}_{t(e)}-{\mu}_{s(e)}$

####  2.1.3. The hypermultiplets in the fundamental representations

These are assigned to the vertices $i\in\mathrm{Vert}_{\gamma}$. We have
${\mathbf{w}}_{i}$ such multiplets. Write

${\mathscr{P}}_{i}(x)={\mathfrak{q}}_{i}\,\prod_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}(x-m_{i,{\mathfrak{f}}})$
(2.11)

Then $m_{i,{\mathfrak{f}}}$ are the masses of the fundamental hypermultiplets,
charged under $SU({\mathbf{v}}_{i})$. A ${\mathbf{w}}_{i}$-tuplet of
fundamental hypermultiplets can be thought as a bifundamental
$(\mathbf{v}_{i},\bar{\mathbf{w}}_{i})$ for $SU({\mathbf{v}}_{i})$ and an
auxiliary frozen $U({\mathbf{w}}_{i})$, so that $m_{i,{\mathfrak{f}}}$ can be
interpreted as the values of the frozen scalar field in the vector multiplet
of $U({\mathbf{w}}_{i})$.

##  Chapter 3 The ADE classifications of superconformal ${\mathcal{N}}=2$
theories

In quantum gauge theory the coupling constants $e_{i}$ are subject to the
renormalization which leads to their dependence on the energy scale at which
one measures the interaction between the charged particles. The consistent
theories have the gauge couplings which tend to zero as the scale approaches
ultraviolet, or approach some fixed values. These theories are called
asymptotically free and asymptotically conformal, respectively. Moreover,
starting with the asymptotically conformal theory, one can perturb it by the
mass terms. Then, by tuning the masses and the bare couplings, one arrives at
the asymptotically free theory. All asymptotically free quiver theories arise
in this way. Therefore it suffices to solve the asymptotically conformal
theories.

From the perspective of geometrical engineering the ADE quiver theories were
studied in [Katz:1997eq], and three dimensional ADE quiver theories were
studied in [Gaiotto:2008ak].

###  3.1. Beta functions and Cartan matrix

The running of the gauge coupling ${\tau}_{i}$ is described by the Gell-Mann-
Low equations which are one-loop exact for the ${\mathcal{N}}=2$
supersymmetric theories, the result of [Novikov:1983uc]. The actual
contributions of the matter and gauge multiplets to the gauge couplings are:

${\beta}_{i}={2\pi\mathrm{i}}\frac{d{\tau}_{i}}{d{\log}{\Lambda}}={\mathbf{w}}_{i}-2{\mathbf{v}}_{i}+\sum_{e:\
t(e)=i}{\mathbf{v}}_{s(e)}+\sum_{e:\ s(e)=i}{\mathbf{v}}_{t(e)}$ (3.1)

where ${\Lambda}$ is the energy scale. The theory is asymptotically conformal
if ${\beta}_{i}=0$ vanishes for all $i\in\mathrm{Vert}_{\gamma}$. The theory
is asymptotically free if ${\beta}_{i}<0$.

Let us define the incidence of the pair of vertices $I_{ij}$ to be the number
of edges $e$ connecting the vertices $i$ and $j$:

$I_{ij}=\\#\\{e\,|\,s(e)=i,\,t(e)=j,\ {\rm or}\ s(e)=j,\,t(e)=i\\}$ (3.2)

with the understanding that if the vertex $i\in\mathrm{Vert}_{\gamma}$ is
connected to itself by a loop, then the corresponding edge contributes $2$ to
the incidence matrix element $I_{ii}$. Define, for all quivers, the Cartan
matrix of size $|\mathrm{Vert}_{\gamma}|\times|\mathrm{Vert}_{\gamma}|$

$C_{ij}=2{\delta}_{ij}-I_{ij}$ (3.3)

Then

${\beta}_{i}\propto({\mathbf{w}}-C{\mathbf{v}})_{i}$

where

$(C{\mathbf{v}})_{i}=\sum_{j\in\mathrm{Vert}_{\gamma}}C_{ij}{\mathbf{v}}_{j}\
.$

Let us solve the ${\beta}_{i}=0$ conditions (cf. [Howe:1983wj], [Katz:1997eq],
[Lawrence:1998ja]). It is convenient to separate the solutions into three
cases, which we shall call the theories of Class I, the theories of Class II
and the theories of Class II*. By $r$ we shall denote the rank of the Cartan
matrix $C$

$r=\operatorname{rk}(C)$ (3.4)

The main difference between the Class I and Class II,II* theories is that the
Cartan matrix of Class I theories has the maximal rank

$r(C_{\text{$\gamma$ of Class I}})=|\mathrm{Vert}_{\gamma}|$ (3.5)

while for the theories of Class II and Class II* the Cartan matrix has one-
dimensional kernel,

$r(C_{\text{$\gamma$ of Class II}})=|\mathrm{Vert}_{\gamma}|-1$ (3.6)

###  3.2. Class I theories

The solutions to the equations ${\beta}_{i}=0$ with ${\mathbf{w}}\neq 0$ are
the theories of Class I. It is well-known that the graph $\gamma$ is in this
case a Dynkin diagram of a finite dimensional simple simply-laced Lie algebra
$\mathfrak{g_{\text{q}}}$, of the ADE type, with $\mathrm{Vert}_{\gamma}$
labeling the simple roots of $\mathfrak{g_{\text{q}}}$:

$i\in\mathrm{Vert}_{\gamma}\mapsto{\alpha}_{i}$

To solve the equation ${\beta}_{i}=0$ is equivalent to finding two vectors

${\mathbf{v}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{v}}_{i}{\alpha}_{i}^{\vee},\qquad{\mathbf{w}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{w}}_{i}{\alpha}_{i}^{\vee}$
(3.7)

with non-negative components
${\mathbf{v}}_{i},{\mathbf{w}}_{i}\in{\mathbb{Z}}_{\geq 0}$, such that (cf.
(3.3)

${\mathbf{w}}=C^{\mathfrak{g_{\text{q}}}}{\mathbf{v}}$ (3.8)

where $C^{\mathfrak{g_{\text{q}}}}$ is the Cartan matrix of the corresponding
finite dimensional Lie algebra $\mathfrak{g_{\text{q}}}$ of the ADE type.
Equivalently

${\mathbf{v}}=\sum_{i\in\mathrm{Vert}_{\gamma}}{\mathbf{w}}_{i}{\lambda}_{i}^{\vee}$

where ${\lambda}_{i}^{\vee}$ are the fundamental coweights of
$\mathfrak{g_{\text{q}}}$.

For class I theories we set ${\mathbf{G}}_{\text{q}}=\mathbf{G}$ where
$\mathbf{G}$ is finite dimensional complex ADE group.

######  Remark.

In the case of $\mathfrak{g_{\text{q}}}$ of the $A_{r}$ type the dimensions
${\mathbf{v}}_{i}$ must be a convex function of $i$. In particular, they grow
with $i$, for $i=1,\ldots,i_{*}$, and then decrease:

${\mathbf{v}}_{1}\leq{\mathbf{v}}_{2}\leq\ldots\leq{\mathbf{v}}_{i_{*}-1}\leq{\mathbf{v}}_{i_{*}}\geq{\mathbf{v}}_{i_{*}+1}\geq\ldots\geq{\mathbf{v}}_{r}$
(3.9)

######  Remark.

The graphs of the $D_{r}$ and $E_{r}$ Dynkin type have a single tri-valent
vertex, let us call it $i_{*}$. One can easily show using the ${\beta}_{i}=0$
equations that ${\mathbf{v}}_{i_{*}}$ is the maximal value of
${\mathbf{v}}_{i}$ on $i\in\mathrm{Vert}_{\gamma}$, and that
${\mathbf{v}}_{i}$ decrease along each leg emanating from the tri-valent
vertex $i_{*}$.

###  3.3. Class II theories

The Class II theories have ${\mathbf{w}}\equiv 0$, and $[{\bf m}]=0$. It is
well-known that the graphs $\gamma$, such that the corresponding Cartan matrix
$C$ has a zero eigenvector with positive integer entries are in one-to-one
correspondence with the simply laced affine Dynkin diagrams (see Appendix A
for our conventions on ADE graphs and McKay correspondence):

  1. (1)

${\widehat{A}}_{r}$, $r\geq 2$,

  2. (2)

${\widehat{D}}_{r}$, $r\geq 5$,

  3. (3)

${\widehat{E}}_{r}$, $r=7,8,9$

These Dynkin diagrams correspond to the affine Lie algebras
$\widehat{\mathfrak{g}}$ associated to finite dimensional Lie algebras
$\mathfrak{g}$ of rank $r$. We set
${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$ and
$\mathfrak{g_{\text{q}}}=\widehat{\mathfrak{g}}$. We discuss the relevant
aspects of the theory of affine Kac-Moody algebras in the following
subsections.

Note that the ${\widehat{A}}_{0}$ case (its quiver has one vertex and one edge
connecting it to itself), given our constraint $[{\bf m}]=0$ for the Class II
theory, corresponds to the ${\mathcal{N}}=4$ superconformal theory. It is well
known that the classical moduli space of vacua gets no quantum corrections in
this theory.

The dimensions ${\mathbf{v}}$ are uniquely specified, up to a single multiple:

${\mathbf{v}}_{i}=Na_{i}$ (3.10)

where $a_{i}$ are the so-called Dynkin labels. We shall recall several
interpretations of these numbers below.

###  3.4. Class II* theories

The Class II* theories have ${\mathbf{w}}=0$ and $[{\bf m}]\neq 0$. The first
condition reduces our choice of $\gamma$ to the affine Dynkin diagrams
(including the ${\widehat{A}}_{0}$ case of the quiver with one vertex and one
loop connecting this vertex with itself). The second condition implies that
$\gamma$ is the Dynkin diagram of the ${\widehat{A}}_{r}$ type for some $r\geq
1$. Indeed, only in this the affine Dynkin diagram has
$H^{1}({\gamma},{\mathbb{C}})={\mathbb{C}}$, the diagram being a regular
$r+1$-gon. The dimensions ${\mathbf{v}}_{i}$ are all equal to $N$, a non-
negative integer.

In particular, the Class II* $r=0$, ${\widehat{A}}_{0}$-theory with $[{\bf
m}]\neq 0$ is the celebrated ${\mathcal{N}}=2^{*}$ theory, the $SU(N)$ theory
with massive adjoint hypermultiplet.

We shall see that the Kac-Moody Lie algebra which corresponds to the theories
of Class II* is the $\widehat{{\mathfrak{g}\mathfrak{l}}({\infty})}$ algebra,
which contains $\widehat{\mathfrak{u}(r)}$ as a subalgebra of $r$-periodic
matrices.

##  Chapter 4 Low-energy effective theory

We now can proceed with the main subject of our study. Our goal is to
determine the two-derivative/four-fermions terms in the low-energy effective
action of our theory.

The low-energy effective theory of the ${\mathcal{N}}=2$ supersymmetric quiver
theory with generic masses $(m_{e}),(m_{i,{\mathfrak{f}}})$ is the abelian
${\mathcal{N}}=2$ theory of ${\mathbf{r}}$ vector multiplets,

${\mathbf{r}}=\sum_{i\in{\mathrm{Vert}_{\gamma}}}({\mathbf{v}}_{i}-1)$ (4.1)

For generic masses the theory has the manifold ${\mathfrak{M}}$ of vacua,
which is a complex variety of complex dimension $\mathbf{r}$:

${\rm dim}_{\mathbb{C}}\,{{\mathfrak{M}}}={\mathbf{r}}\,,$

The effective theory is a sigma model on ${\mathfrak{M}}$, interacting with
$\mathbf{r}$ abelian gauge fields $A_{\mu}^{\mathcal{I}}$,
${\mathcal{I}}=1,\ldots,{\mathbf{r}}$, and some fermionic fields. Our goal is
to determine the metric on ${\mathfrak{M}}$, the effective gauge couplings
${\rm Im}{\tau}_{\mathcal{I}\mathcal{J}}$ and the effective theta-angles ${\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}$ of these gauge fields.

###  4.1. Special Kähler geometry

One can interpret the eigen-values (2.7) obeying

$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}=0$
(4.2)

as the _special coordinates_ on the moduli space ${\mathfrak{M}}$ of vacua. As
is well known, ${\mathfrak{M}}$ is a Kähler manifold, with a peculiar metric,
and a rigid system of local coordinate systems. The corresponding geometry is
called the _rigid special geometry_ , and it is a limit [Seiberg:1994rs] of
the _special geometry_ of ${\mathcal{N}}=2$ supergravity, studied in
[Cremmer:1984hj].

Let us label the effective abelian vector multiplets
${\mathcal{A}}^{\mathcal{I}}$ by ${\mathcal{I}}=(i,{\mathbf{a}})$,
$i\in\mathrm{Vert}_{\gamma}$, ${\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}$. In
components:

${\mathcal{A}}^{\mathcal{I}}={\mathfrak{a}}^{\mathcal{I}}+{\vartheta}{\psi}^{\mathcal{I}}+{\vartheta}{\vartheta}(F_{A}^{\mathcal{I}})^{-}+\ldots$

where

$(F_{A}^{\mathcal{I}})^{\pm}=\frac{1}{2}\left(F_{A}^{\mathcal{I}}\pm\star
F_{A}^{\mathcal{I}}\right)$ (4.3)

The scalar components
${\mathfrak{a}}^{\mathcal{I}}\leftrightarrow{\mathfrak{a}}_{i,{\mathbf{a}}}$,
${\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}$, more precisely their vacuum
expectation values are the local _special coordinates_. Globally they are
subject to monodromy transformations, unlike the global coordinates
$(u_{i,{\mathbf{a}}})$ in Eq. (2.9), which are defined via the expectation
values of the gauge invariant local operators of the microscopic theory.

The monodromy transformations act by symplectic transformations mixing the
special coordinates ${\mathfrak{a}}^{\mathcal{I}}$ and their duals
${\mathfrak{a}}_{\mathcal{I}}^{D}$ together with the masses $(m_{e})$,
$e\in\mathrm{Edge}_{\gamma}$ and $(m_{i,{\mathfrak{f}}})$,
$i\in\mathrm{Vert}_{\gamma}$, ${\mathfrak{f}}=1,\ldots,{\mathbf{w}}_{i}$. The
dual coordinates are the derivatives of the prepotential ${{\mathscr{F}}}$,

${\mathfrak{a}}^{D}_{\mathcal{I}}=\frac{{\partial}{{\mathscr{F}}}}{{\partial}{\mathfrak{a}}^{\mathcal{I}}}$
(4.4)

The prepotential is a multi-valued analytic function of
${\mathfrak{a}}^{\mathcal{I}}$, it is the superspace action which determines
the low-energy effective action in the approximation we are working:

${\mathcal{L}}^{\rm
eff}=\frac{\mathrm{i}}{4\pi}\int{\tau}_{\mathcal{I}\mathcal{J}}(F_{A}^{\mathcal{I}})^{-}\wedge(F_{A}^{\mathcal{J}})^{-}+{\bar{\tau}}_{\mathcal{I}\mathcal{J}}(F_{A}^{\mathcal{I}})^{+}\wedge(F_{A}^{\mathcal{J}})^{+}-\mathrm{i}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\star
d{\bar{\mathfrak{a}}}^{\mathcal{J}}$ (4.5)

where

${\tau}_{\mathcal{I}\mathcal{J}}=\frac{{\partial}^{2}{{\mathscr{F}}}}{{\partial}{\mathfrak{a}}^{\mathcal{I}}{\partial}{\mathfrak{a}}^{\mathcal{J}}}$
(4.6)

The invariant formulation of Eq. (4.4) is that the two-form

$\sum_{\mathcal{I}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}^{D}_{\mathcal{I}}$

identically vanishes on ${\mathfrak{M}}$. The proper formulation of this
condition uses the additional structure which we review below.

###  4.2. Extended moduli space

In our solution of the theories of class I and class II it would be sometimes
convenient to trade the bifundamental masses formally with the $U(1)$ factors
as explained in (2.10) if one considers
$\times_{i\in\mathrm{Vert}_{\gamma}}U(\mathbf{v}_{i})$ gauge group instead of
$\times_{i\in\mathrm{Vert}_{\gamma}}SU(\mathbf{v}_{i})$. The
$|\mathrm{Vert}_{\gamma}|-1$ bifundamental masses111Recall that in the class
II $\widehat{A}_{r}$ theory there are $r$ mass parameters that can be traded
for the $U(1)$ scalars and $1$ additional “twist” mass parameter $m^{*}$
promoting the class II to class II* and one overall $U(1)$ factor add
$|\mathrm{Vert}_{\gamma}|$ parameters to ${\mathfrak{M}}$. We set

${\mathfrak{M}}^{\mathrm{ext}}={\mathfrak{M}}\times\mathbb{C}^{\mathrm{Vert}_{\gamma}}$
(4.7)

with

$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=\dim_{\mathbb{C}}{\mathfrak{M}}+|\mathrm{Vert}_{\gamma}|=\sum_{i\in\mathrm{Vert}_{\gamma}}\mathbf{v}_{i}$
(4.8)

For the class II theories

$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=\sum_{i\in\mathrm{Vert}_{\gamma}}\mathbf{v}_{i}=Nh$
(4.9)

where $h=\sum_{i\in\mathrm{Vert}_{\gamma}}a_{i}$ is the Coxeter number of
$\mathbf{G}$. Recall that we only encounter simply-laced Lie algebras for
which $h=h^{\vee}$.

We should emphasize that only the true moduli space ${\mathfrak{M}}$ of vacua
has the special geometry with (4.6) defining a positive (outside the loci of
singularities which signal the appearance of massless BPS particles) metric in
the appropriate duality frame. On the extended moduli space
${\mathfrak{M}}^{\mathrm{ext}}$ the prepotential ${\mathscr{F}}$ still defines
some kind of metric, but it cannot be positive everywhere throughout the
variety of masses. This is because the dependence on masses is purely
perturbative. Once we gauge the flavor symmetry (an example of such gauging,
promoting the $A_{1}$ class I theory to the $\widehat{D}_{4}$ class II theory,
will be discussed in section 7.5.1), we correct the metric by the instanton
contributions.

###  4.3. Finite size effects

Subjecting the gauge theory to some boundary conditions reveals more
structure.

For example, we can compactify the four dimensional ${\mathcal{N}}=2$ gauge
theory on a circle ${\mathbb{S}}^{1}$ of radius $R$. The resulting theory
looks like a three dimensional sigma model with the target space
${{\mathfrak{P}}}$ which is a hyperkähler manifold of real dimension
$4{\mathbf{r}}$. The hyperkähler metric on ${\mathfrak{P}}$ contains a lot of
interesting information about the particle content of the original four
dimensional theory.

The hyperkähler structure on ${\mathfrak{P}}$ is a triplet of integrable
complex structures, ${\bf I},{\bf J},{\bf K}$, such that every linear
combination $(a{\bf I}+b{\bf J}+c{\bf K})$ for $a^{2}+b^{2}+c^{2}=1$ is also
an integrable complex structure, and a triplet of the corresponding symplectic
forms ${\omega}_{\bf I},{\omega}_{\bf J},{\omega}_{\bf K}$ which are the
Kähler forms for the metric $g$ on ${\mathfrak{P}}$ in the corresponding
complex structures.

Among the two-sphere of complex structures, one complex structure, which is
usually called $\bf I$, plays a special rôle. This complex structure and the
corresponding $(2,0)$ symplectic form ${\Omega}_{\bf I}={\omega}_{\bf
J}+\mathrm{i}{\omega}_{\bf K}$ are visible in the limit $R\to\infty$, where
${\mathfrak{P}}$ as a metric space collapses to ${\mathfrak{M}}$. For very
large but finite $R$ the manifold ${\mathfrak{P}}$ looks like a fibration over
${\mathfrak{M}}$ whose fibers ${{\mathscr{A}}_{u}}$, $u\in{{\mathfrak{M}}}$
are the abelian varieties (complex tori, which we describe in more detail
momentarily) of diameter which scales like $R^{-1}$.

These fibers ${{\mathscr{A}}_{u}}$ parametrize the ${\mathbb{S}}^{1}$ holonomy
of the abelian gauge fields $A^{\mathcal{I}}$ and their duals
$A^{D}_{\mathcal{I}}$. The reduction of the action (4.5) on ${\mathbb{S}}^{1}$
gives:

$\displaystyle{\mathcal{L}}^{\rm eff3d}=\int\mathrm{i}{\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\alpha}^{\mathcal{I}}\wedge
B^{\mathcal{J}}+$ (4.10) $\displaystyle\qquad\frac{1}{2}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,\left(Rd{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+R^{-1}d{\alpha}^{\mathcal{I}}\wedge\ast
d{\alpha}^{\mathcal{J}}+RB^{\mathcal{I}}\wedge\ast B^{\mathcal{J}}\right)$

where we denote by $\ast$ the three dimensional Hodge star, and by
$B^{\mathcal{I}}$ the curvature of the three dimensional gauge field
$B^{\mathcal{I}}=dA^{\mathcal{I}}_{\rm 3d}$ which is obtained by decomposing
$A^{\mathcal{I}}_{\rm 4d}={\alpha}^{\mathcal{I}}d{\theta}+A^{\mathcal{I}}_{\rm
3d}$. The scalar ${\alpha}^{\mathcal{I}}$ is actually circle-valued, since the
gauge transformations $e^{2\pi\mathrm{i}n_{\mathcal{I}}{\theta}}$ shift it by
$2\pi\mathrm{i}n^{\mathcal{I}}$, $n^{\mathcal{I}}\in{\mathbb{Z}}$. Next we
dualize the three dimensional abelian gauge field, by promoting
$B^{\mathcal{I}}$ to the independent $2$-form, and coupling it to the dual
scalar ${\beta}_{\mathcal{I}}$, which is also circle-valued, in order to
ensure the flux quantization of the original gauge curvature
$B_{\mathcal{I}}$:

$\displaystyle{\mathcal{L}}^{\rm
eff3dd}=\int\mathrm{i}\left(d{\beta}_{\mathcal{J}}+{\rm
Re}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\alpha}^{\mathcal{I}}\right)\wedge
B^{\mathcal{J}}+$ (4.11) $\displaystyle\qquad\frac{1}{2}{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,\left(R\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+R^{-1}d{\alpha}^{\mathcal{I}}\wedge\ast
d{\alpha}^{\mathcal{J}}+RB^{\mathcal{I}}\wedge\ast B^{\mathcal{J}}\right)$
$\displaystyle\qquad\longrightarrow\frac{R}{2}\int{\rm
Im}{\tau}_{\mathcal{I}\mathcal{J}}\,d{\mathfrak{a}}^{\mathcal{I}}\wedge\ast
d{\bar{\mathfrak{a}}}^{\mathcal{J}}+\frac{1}{2R}\int({\rm
Im}{\tau}^{-1})^{\mathcal{I}\mathcal{J}}dz_{\mathcal{I}}\wedge\ast
d{\bar{z}}_{\mathcal{J}}$

In the last line we have integrated out the unconstrained Gaussian field
$B_{\mathcal{I}}$. We also introduced the holomorphic coordinates

$z_{\mathcal{I}}={\beta}_{\mathcal{I}}+{\tau}_{\mathcal{I}\mathcal{J}}{\alpha}^{\mathcal{J}},\qquad
I=1,\ldots,{\mathbf{r}}$ (4.12)

on the fibers ${{\mathscr{A}}_{u}}$ of the fibration
${{\mathfrak{P}}}\to{{\mathfrak{M}}}$. Both ${\mathfrak{a}}_{\mathcal{I}}$ and
$z_{\mathcal{I}}$ are the ${\bf I}$-holomorphic coordinates on
${\mathfrak{P}}$. By construction, the coordinates $z_{\mathcal{I}}$ are
subject to the periodic identifications:

$z_{\mathcal{I}}\to
z_{\mathcal{I}}+2\pi\mathrm{i}\left(n_{\mathcal{I}}+{\tau}_{\mathcal{I}\mathcal{J}}m^{\mathcal{J}}\right),\qquad
n_{\mathcal{I}},m^{\mathcal{I}}\in{\mathbb{Z}}$ (4.13)

which confirm our assertion that the fibers ${{\mathscr{A}}_{u}}$ of the map
${\mathfrak{P}}\to{\mathfrak{M}}$ are abelian varieties (recall that the
metric ${\rm Im}{\tau}d{\mathfrak{a}}\otimes d{\bar{\mathfrak{a}}}$ is
positive definite, the unitarity requirement). The coordinates
${\mathfrak{a}}_{\mathcal{I}},z_{\mathcal{I}}$ are the Darboux coordinates for
the $(2,0)$ form ${\Omega}_{\bf I}$:

${\Omega}_{\bf
I}=\sum_{{\mathcal{I}}=1}^{{\mathbf{r}}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
dz_{\mathcal{I}}=\sum_{{\mathcal{I}}=1}^{{\mathbf{r}}}d{\mathfrak{a}}^{D}_{\mathcal{I}}\wedge
dz_{D}^{\mathcal{I}}$ (4.14)

as well as the electric-magnetic duals ${\mathfrak{a}}^{D}_{\mathcal{I}}$ and
$z_{D}^{\mathcal{I}}=({\tau}^{-1})^{\mathcal{I}\mathcal{J}}{\beta}_{\mathcal{J}}+{\alpha}^{\mathcal{I}}$.
The fibers ${{\mathscr{A}}_{u}}$ are Lagrangian with respect to $\Omega_{\bf
I}$.

The metric on ${\mathfrak{P}}$, which enters the kinetic term in the Eq.
(4.11) is actually not the correct hyperkähler metric on ${\mathfrak{P}}$ for
finite $R$. It receives corrections which are exponentially small with $R$,

$\sim e^{-M({\mathfrak{a}})R}$ (4.15)

where $M({\mathfrak{a}})$ is the mass of a BPS particle in the Hilbert space
of the theory in four dimensions built over the vacuum $u\in{\mathfrak{M}}$.
As is well-known, the masses of some BPS particles vanish along some loci in
${\mathfrak{M}}$, where the corrections (4.15) become significant. One can
show, however, that $\Omega_{\bf I}$ does not get corrected by the finite size
effects of these BPS particles.

One can also compactify the theory on a two-dimensional Riemann surface
$\Sigma$ (with a partial twist along $\Sigma$, to preserve some
supersymmetry). For $\Sigma$ other then two-torus this leads to the two
dimensional theory with ${\mathcal{N}}=2$ supersymmetry. One has various
sectors labeled by the electric and magnetic fluxes ${\bf
e}=(e_{\mathcal{I}}),{\bf m}=(m^{\mathcal{I}})$ through $\Sigma$. In the
sector where $({\bf e},{\bf m})\neq(0,0)$ one gets an effective superpotential
[Losev:1997tp]:

$W_{({\bf e},{\bf
m})}=\sum_{\mathcal{I}=1}^{\mathbf{r}}e_{\mathcal{I}}\mathfrak{a}^{\mathcal{I}}+m^{\mathcal{I}}{\mathfrak{a}}^{D}_{\mathcal{I}}$

which in four dimensional theory is the central charge of the
${\mathcal{N}}=2$ superalgebra. It is also equal to one of the _action
variables_ of the Seiberg-Witten integrable system [Gorsky:1995zq],
[Donagi:1995cf], [Donagi:1997sr].

If one compactifies on a two-torus, then the resulting two dimensional theory
is the ${\mathcal{N}}=4$ supersymmetric sigma model whose target space
${\mathfrak{P}}$ is the hyperkähler manifold.

It turns out to be quite useful to interpret the ${\mathcal{N}}=2$ theory on a
four dimensional manifold $X$ which can be viewed as a two-torus fibration
over some base $B$, as an effective sigma model with $B$ as a world sheet. In
case where the fibration has singularities of real codimension one (for
example, if $X$ is a product of a disk and a cylinder), then $B$ has a
boundary, and the smoothness of the four dimensional field configurations
translates to particular boundary conditions in the two dimensional sigma
model [Nekrasov:2010ka]. An interesting class of such boundary conditions come
from the so-called canonical coisotropic branes [Kapustin:2001ij,
Kapustin:2006pk, Nekrasov:2010ka]. The algebra of the open string vertex
operators corresponding to such a brane turns out to be the deformation
quantization [MR2062626] of the algebra of holomorphic (in the appropriate
complex structure) functions on ${\mathfrak{P}}$. Remarkably, when
${\mathfrak{P}}$ is an algebraic integrable system in one of the complex
structures, one can apply the fiberwise T-duality along the Liouville fibers,
leading to the mirror perspective on the quantization procedure. First of all,
in the case of the Hitchin system the mirror manifold turns out to be the
Hitchin system for the Langlands dual group. In the general case the mirror
${{\mathfrak{P}}}^{\vee}$ of the original hyperkähler manifold
${\mathfrak{P}}$ is also expected to be an integrable system. The mirror of
the canonical coisotropic brane is believed to to be a holomorphic (in an
appropriate complex structure) Lagrangian brane. In the case of Hitchin system
this brane is the so-called brane ${\mathcal{B}}_{\mathcal{O}}$ of opers
[Kapustin:2006pk].

This approach to quantization of the integrable systems from gauge theories
will be elaborated upon in [NP2012b].

###  4.4. The appearance of an integrable system

The complex symplectic manifold $({{\mathfrak{P}}},{\Omega}_{\bf I})$, its
projection ${\pi}:{\mathfrak{P}}\to{\mathfrak{M}}$ with Lagrangian fibers
${{\mathscr{A}}_{u}}={\pi}^{-1}(u)$, $u\in{\mathfrak{M}}$, which are
principally polarized abelian varieties (the principal polarization comes from
the restriction of ${\omega}_{\bf I}$ onto the fibers) define what is known as
the _algebraic integrable system_ [Donagi:1995am], [Donagi:1995cf],
[Donagi:1997sr]. It is one of the possible complexifications of the familiar
notion of the completely integrable system in the classical mechanics.

The other possibility, namely a complex symplectic manifold with the
Lagrangian fibration whose fibers are the complex tori
$\left({\mathbb{C}}^{\times}\right)^{\mathbf{r}}$, is also realized in the
context of gauge theories. However, the base of such a system typically
parametrizes the space of mass parameters of the gauge theory.

The fibers ${{\mathscr{A}}_{u}}$ are the Liouville tori, while
$({\mathfrak{a}}_{\mathcal{I}},z_{\mathcal{I}})$ are the action-angle
variables. The novelty of the complex case is the doubling of the possible
choices of the action-angle variables with fixed Liouville fibration. Indeed,
the fibers ${{\mathscr{A}}_{u}}$ are the $2{\mathbf{r}}$-real dimensional
tori, therefore in producing the action variables as in the Arnol’d-Liouville
theorem one has a choice of $\mathbf{r}$ out of $2\mathbf{r}$ cycles in
$H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$. The lattice
$H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$ has a symplectic form $\varpi$,
which comes from the polarization, i.e. a properly normalized class of the
restriction ${\omega}_{\bf I}|_{{{\mathscr{A}}_{u}}}$. It turns out that any
Lagrangian sublattice $L$ in $H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$ defines
a system of local coordinates $({\mathfrak{a}}_{I})$ on the base
${\mathfrak{M}}$ near the point $u\in{\mathfrak{M}}$, as well as the conjugate
angle-like coordinates $(z_{\mathcal{I}})$ on the fiber ${{\mathscr{A}}_{u}}$
itself. Let ${\mathcal{A}}_{\mathcal{I}}$ be the integral basis of this
sublattice $L\subset H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$. Then

$d{\mathfrak{a}}^{\mathcal{I}}=\oint_{{\mathcal{A}}_{\mathcal{I}}}{\Omega}_{\bf
I}$ (4.16)

One can also define

$d{\mathfrak{a}}^{D}_{\mathcal{I}}=\oint_{{\mathcal{B}}^{\mathcal{I}}}{\Omega}_{\bf
I}$ (4.17)

where ${\mathcal{B}}^{\mathcal{I}}$ is the basis in the dual sublattice
$L^{\vee}\subset H_{1}({{\mathscr{A}}_{u}},{\mathbb{Z}})$, such that

${\varpi}({\mathcal{A}}_{\mathcal{I}},{\mathcal{A}}_{\mathcal{J}})={\varpi}({\mathcal{B}}^{\mathcal{I}},{\mathcal{B}}^{\mathcal{J}})=0,\qquad{\varpi}({\mathcal{A}}_{\mathcal{I}},{\mathcal{B}}^{\mathcal{J}})={\delta}_{\mathcal{I}}^{\mathcal{J}}$
(4.18)

One then shows that

$\sum_{{\mathcal{I}}=1}^{\mathbf{r}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}_{\mathcal{I}}^{D}\equiv 0$ (4.19)

on ${\mathfrak{M}}$, which, in turn, implies (4.4). The coordinates
$z_{\mathcal{I}}$ along ${{\mathscr{A}}_{u}}$ are defined using (4.14) with
the normalization (4.13) that half of the periods of $z_{\mathcal{I}}$ are in
$2\pi\mathrm{i}{\mathbb{Z}}$.

The integrable systems which one encounters in the classical mechanics are
rarely given in the form of the action-angle variables. Usually one has the
phase space ${\mathfrak{P}}$, the symplectic form ${\Omega}_{\bf I}$, perhaps
some Darboux coordinates

${\Omega}_{\bf I}=\sum_{{\mathcal{I}}=1}^{\mathbf{r}}dp_{\mathcal{I}}\wedge
dq^{\mathcal{I}}$

and the collection of Poisson-commuting functionally independent Hamiltonians
$U_{1}(p,q),\ldots,U_{\mathbf{r}}(p,q)$. One then looks for the action-angle
coordinates, i.e. the Darboux coordinates $({\mathfrak{a}},z)$, such that the
Hamiltonians $U_{\mathcal{I}}(p,q)=u_{\mathcal{I}}({\mathfrak{a}})$ depend
only on ${\mathfrak{a}}$, the action variables. The Hamiltonian evolution then
linearizes on the fibers ${{\mathscr{A}}_{u}}$, which are the level sets of
the Hamiltonians. The motion is a constant velocity motion in the $z$
coordinates:

$z_{\mathcal{I}}(t)=z_{\mathcal{I}}(0)+\sum_{\mathcal{J}=1}^{\bf
r}\,t_{\mathcal{J}}\,\frac{{\partial}u_{\mathcal{J}}}{{\partial}{\mathfrak{a}}_{\mathcal{I}}}$

It is interesting to study the level sets ${{\mathscr{A}}_{u}}$ of the
Hamiltonians, the Liouville tori. The algebraic integrable systems are such,
that the fibers can be compactified to become the polarized abelian varieties.
Where do the polarized abelian varieties come from?

###  4.5. Integrable systems from classical gauge theories

One source of the polarized abelian varieties are the Jacobians of the
algebraic curves. The Liouville tori of algebraic integrable systems can be
often found inside the Jacobians of the algebraic curves, constructed while
solving some classical gauge field equations.

####  4.5.1. Hitchin system

There is an interesting class of algebraic integrable systems for which the
Liouville tori are precisely these Jacobians. Take the $U(N)$ Hitchin system
on a genus $g$ Riemann surface. The phase space ${\mathfrak{P}}$ is the
cotangent bundle (up to a birational transformation) to the moduli space
$M_{N,c}$ of holomorphic rank $N$ vector bundles $E$ over $\Sigma$ with fixed
first Chern class $c=c_{1}(E)$. It is convenient to take $(c,N)=1$ to avoid
complications coming from the reducible connections.

In the complex structure $\bf I$ the holomorphic coordinates on
${\mathfrak{P}}$ are $({\bar{A}},{\Phi})$, where ${\bar{\partial}}+{\bar{A}}$
is the $(0,1)$-connection on the smooth vector bundle $E$ which endows it with
the complex structure, and $\Phi\in End(E)\otimes{\Omega}^{1,0}({\Sigma})$ is
the holomorphic Higgs field:

${\bar{\partial}}{\Phi}+[{\bar{A}},{\Phi}]=0$ (4.20)

The symplectic form on ${\mathfrak{P}}$ comes from the $(2,0)$ symplectic form
on the space of all smooth pairs $({\bar{A}},{\Phi})$

${\Omega}_{\bf
I}=\int_{\Sigma}{\operatorname{tr}}\,{\delta}{\Phi}\wedge{\delta}{\bar{A}}$
(4.21)

by the symplectic reduction with respect to the action of the gauge group:

$g:\,({\bar{A}},{\Phi})\longrightarrow(g^{-1}{\bar{A}}g+g^{-1}{\bar{\partial}}g,g^{-1}{\Phi}g)$

The set of Poisson-commuting Hamiltonians is given by:

$U_{i,{\mathbf{a}}}=\int_{\Sigma}{\nu}_{i,{\mathbf{a}}}{\operatorname{tr}}\,{\Phi}^{i},\qquad
i=1,\ldots,N$ (4.22)

where ${\nu}_{i,{\mathbf{a}}}\in H^{0,1}({\Sigma},K_{\Sigma}^{\otimes(1-i)})$,
${\mathbf{a}}=1,\ldots,(2i-1)(g-1)+{\delta}_{i,1}$ form a basis in the space
of holomorphic $(1-i,1)$-differentials. Fixing the values $u_{i,{\mathbf{a}}}$
of all the Hamiltonians $U_{i,{\mathbf{a}}}$ gives us a point
$u\in{{\mathfrak{M}}}$ in the vector space

${{\mathfrak{M}}}=\bigoplus_{i}H^{0,1}({\Sigma},K_{\Sigma}^{\otimes(1-i)})$
(4.23)

One defines the _spectral curve_ $C_{u}\subset T^{*}{\Sigma}$ as the zero
locus of the characteristic polynomial of $\Phi$:

${\rm Det}({\Phi}-{\lambda})=0$ (4.24)

It is a holomorphic curve thanks to (4.20), which is invariant under the
Hamiltonian flows generated by the Hamiltonians (4.22). The curve $C$ is an
$N$-sheeted cover of $\Sigma$

${\pi}:C_{u}\to\Sigma$ (4.25)

Its genus can be computed using the Riemann-Hurwitz formula:

$2-2g_{C_{u}}=N(2-2g_{\Sigma})-{\delta}$

where ${\delta}=2N(N-1)(g_{\Sigma}-1)$ is the number of branch points. The
latter is the number of zeroes of the discriminant of the polynomial (4.24),
which is a holomorphic $N(N-1)$-differential on $\Sigma$. Thus:

$g_{C_{u}}=N^{2}(g_{\Sigma}-1)+1$

The Jacobian of $C$ is thus an abelian variety of dimension

$g_{C_{u}}=g_{\Sigma}+\sum_{j=2}^{N}(2j-1)(g_{\Sigma}-1)$ (4.26)

which is equal to the dimension of the base ${\mathfrak{M}}$ of the Hitchin
fibration. The fibers ${\mathscr{A}}_{u}$ of the Hitchin fibration are thus
the Jacobians of the corresponding spectral curves.

One generalization is to study the $SL(N)$ Hitchin system. In this case the
corresponding rank $N$ vector bundles have the trivial determinant, and the
corresponding Higgs field is traceless. The base of the Hitchin fibration now
has the dimension $(N^{2}-1)(g_{\Sigma}-1)$, the equation (4.24) has vanishing
$\propto{\lambda}^{N-1}$ term, and the fibers ${\mathscr{A}}_{u}$ are not the
full Jacobians of the spectral curve $C_{u}$, which still has the genus
$g_{C_{u}}$ (4.26) but the kernel $J_{0}$ of the map ${\pi}_{*}:Jac(C_{u})\to
Jac({\Sigma})$, which sends the degree zero line bundle $L$ on $C_{u}$ to the
line bundle ${\mathscr{L}}={\rm Det}{\pi}_{*}L$ on $\Sigma$, whose fiber
${\mathscr{L}}_{z}$ over the point $z\in\Sigma$ is the tensor product of the
fibers $L_{y}$ of $L$ over all preimages of $z$:

${\mathscr{L}}_{z}=\bigotimes_{y\in{\pi}^{-1}(z)}L_{y}$ (4.27)

The Hitchin system can be defined [Hitchin:1987mz] for any algebraic Lie group
$G$, with the maximal torus $T$. Let $\mathfrak{g}=Lie(G)$,
$\mathfrak{h}=Lie(T)$. The Hitchin space is the moduli space of stable pairs
$({\mathcal{P}},{\Phi})$, where ${\mathcal{P}}$ is a holomorphic $G$-bundle
over $\Sigma$, and $\Phi$ is a holomorphic $(1,0)$-form on $\Sigma$, valued in
the bundle of Lie algebras $\mathfrak{g}$, associated with $\mathcal{P}$ via
the adjoint representation:

${\Phi}\in H^{0}({\Sigma},K_{\Sigma}\otimes{\rm ad}({\mathcal{P}}))\ .$

The Hitchin fibration is again defined by fixing the gauge-invariant
polynomials $P_{j}({\Phi})\in H^{0}\left({\Sigma},K_{\Sigma}^{\otimes
d_{j}}\right)$ of the $\mathfrak{g}$-valued Higgs field $\Phi$:

$u=\left(P_{j}({\Phi})\right)_{j=1}^{r}\in\bigoplus_{j=1}^{r}{\mathbb{C}}^{(2d_{j}-1)(g_{\Sigma}-1)}={{\mathfrak{M}}}$

where $d_{j}$’s are the degrees of basic Ad-invariant polynomials on
$\mathfrak{g}$.

The fibers of the Hitchin fibration are now trickier to define. First of all,
there is no preferred notion of the spectral curve. For some gauge groups one
can use the minuscule representation, but this is not always available.

One option is to consider the so-called _cameral curve_ ${\mathcal{C}}_{u}$,
which is a $W({\mathfrak{g}})$-cover of the base curve $\Sigma$. The points of
the cameral curve ${\mathcal{C}}_{u}$ are, over generic $z$, the pairs
$({\varphi},z)$, where $z\in\Sigma$ and $\varphi\in\mathfrak{h}$ is the
element of the fixed Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ which
is conjugate to the Higgs field $\Phi(z)$. This definition makes sense for the
points $z\in\Sigma$ for which $\Phi(z)$ is semi-simple, i.e. belongs to the
$ad(G)$-orbit of an element in $\mathfrak{h}$. If this is not the case (e.g.
$\Phi(z)$ is conjugate to a Jordan block in the $GL(N)$ case), one can find an
appropriate representative in $\mathfrak{h}$ by modifying the equivalence
relation (e.g. two matrices are equivalent if their characteristic polynomials
coincide). To stress the fact that ${\mathcal{C}}_{u}$ depends on $u$ which is
the set of holomorphic $d_{j}$-differentials $P_{j}({\Phi})$ .

Over ${\mathcal{C}}_{u}$ so defined one has $r$ line bundles,
${\mathcal{L}}_{i}$, $i=1,\ldots,r$, which correspond to the fundamental
weights $\lambda_{i}\in\mathfrak{h}^{*}$. The line bundle ${\mathcal{L}}_{i}$
is a subbundle in the holomorphic vector bundle ${\bf
R}_{i}=R_{i}\times_{G}{\mathcal{P}}$, associated with $\mathcal{P}$ via the
$i$’th fundamental representation $R_{i}$ of $G$. The fiber of
${\mathcal{L}}_{i}\subset{\bf R}_{i}$ over $({\varphi},z)$ is the eigenspace
corresponding to the eigenvalue ${\lambda}_{i}({\varphi})$.

To any weight vector ${\lambda}\in\Lambda$ a line bundle
${\mathcal{L}}_{\lambda}$ over $\mathcal{C}_{u}$ can be associated:

${\lambda}=\sum_{i=1}^{r}n_{i}{\lambda}_{i}\
\mapsto{\mathcal{L}}_{\lambda}=\bigotimes_{i=1}^{r}{\mathcal{L}}_{i}^{\otimes\,n_{i}}$
(4.28)

In a more physical language, the Hitchin moduli space is the quotient of the
space of pairs $({\bar{A}},{\Phi})$, where ${\bar{A}}$ is a $(0,1)$-connection
on smooth principal $G$-bundle ${\mathcal{P}}$ over $\Sigma$, and $\Phi$ is a
$(1,0)$ $\mathfrak{g}$-valued form, which are compatible, i.e. solve the Eq.
(4.20), and are considered up to the $G$-gauge transformations:

$g:({\bar{A}},{\Phi})\mapsto(g^{-1}{\bar{\partial}}g+Ad_{g}{\bar{A}},Ad_{g}{\Phi})$

By fixing the partial gauge ${\Phi}={\varphi}\in\mathfrak{h}$ for fixed
$\mathfrak{h}\subset\mathfrak{g}$, one reduces the gauge invariance from $G$
to $N(T)$. The Eq. (4.20) imply that in this gauge $\bar{A}$ is a
$T$-connection $\bar{A}=\bar{a}$, with the $T$ subgroup of $N(T)$ acting by
the $T$-gauge transformations $\bar{a}\mapsto\bar{a}+{\bar{\partial}}{\chi}$,
$e^{\chi}\in T$. On $\Sigma$ the $T$-valued gauge field and the
$\mathfrak{h}$-valued Higgs field $\varphi$ are not well-defined, since there
are the $W({\mathfrak{g}})=N(T)/T$ remaining gauge transformations. On
$\mathcal{C}$, however, both $\varphi$ and $\bar{a}$ are well-defined. In
fact, ${\bar{a}}$ defines on $\mathcal{C}_{u}$ a holomorphic principal
$T$-bundle $\mathcal{T}$, so that $L_{i}={\mathcal{T}}^{{\lambda}_{i}}$. The
$T$-bundle $\mathcal{T}$ is $W({\mathfrak{g}})$-equivariant. This is the
translation of the fact that the Weyl group $W({\mathfrak{g}})$ acts
simultaneously on $\varphi$ and $\bar{a}$. The isomorphism (properly
understood at the ramification points)

Holomorphic principal $G$-bundles $\mathcal{P}$ on $\Sigma$, holomorphic Higgs
fields $\Phi\in H^{0}({\Sigma},K_{\Sigma}\otimes{\rm ad}({\mathcal{P}}))$
$\Leftrightarrow$ $W({\mathfrak{g}})$-covers $\mathcal{C}$ of $\Sigma$,
$\mathcal{C}\subset T^{*}{\Sigma}\otimes\mathfrak{h}$, holomorphic
$W({\mathfrak{g}})$-equivariant principal $T$-bundles on $\mathcal{C}$

allows to represent the Hitchin moduli space as a fibration over the vector
space ${\mathfrak{M}}$, whose points are the $W({\mathfrak{g}})$-invariant
curves $\mathcal{C}_{u}$ sitting in the tensor product
$T^{*}\Sigma\otimes\mathfrak{h}$ (this is almost a tautology: a
$W({\mathfrak{g}})$-invariant curve in $T^{*}\Sigma\otimes\mathfrak{h}$ is a
curve in $T^{*}\Sigma\otimes\mathfrak{h}/W({\mathfrak{g}})$, i.e. a
holomorphic section of the vector bundle
$T^{*}{\Sigma}\otimes{\mathbb{C}}[{\mathfrak{h}}]^{W({\mathfrak{g}})}$).

The fiber ${\mathscr{A}}_{u}$ of the Liouville fibration (which is called
Hitchin’s fibration in this case) is a generalized Prym variety, which is,
roughly speaking,

${{\mathscr{A}}_{u}}\approx{\operatorname{Hom}}_{W({\mathfrak{g}})}\left({\Lambda},{\rm
Pic}({\mathcal{C}}_{u})\right)={\mathrm{Bun}}_{T}({\mathcal{C}}_{u})^{W({\mathfrak{g}})}$
(4.29)

The papers [Donagi:1995alg], [Donagi:2000dr], [Donagi:1998vx] correct the Eq.
( 4.29 ) in a couple of subtle points as well as provide the additional
theory.

####  4.5.2. Instanton moduli spaces as integrable systems

Hitchin’s equations (4.20), for flat $\Sigma$, are the dimensional reduction
of the instanton (or anti-self-duality) equations from four dimensions. It
turns out that one can get an integrable system directly from the moduli
spaces of four dimensional instantons, or three dimensional monopoles.

We only briefly sketch the constructions here.

Let $S$ be an elliptic K3 manifold, i.e. an algebraic surface, with the
holomorphic ${\omega}_{S}^{2,0}$ form, and with the projection
${\pi}:S\to{\mathbb{C}\mathbb{P}}^{1}$ whose fibers ${\pi}^{-1}(z)$,
$z\in{\mathbb{C}\mathbb{P}}^{1}$ are the elliptic curves ${\mathscr{E}}_{z}$
(generically nonsingular). One can endow $S$ with the hyperkähler metric.
Consider the moduli space ${{\mathfrak{P}}}={\mathcal{M}}_{N}(G)$ of charge
$N$ $G$-instantons on $S$, i.e. the solutions to the system of partial
differential equations

$\displaystyle F_{A}\wedge{\omega}_{\bf I}=F_{A}$
$\displaystyle\wedge{\omega}_{\bf J}=F_{A}\wedge{\omega}_{\bf K}=0$ (4.30)
$\displaystyle F^{0,2}_{A}=0$

(the last equation is a linear combination of the $\bf J$ and $\bf K$
equations from the first line) of fixed instanton charge $N\geq 0$:

$-\frac{1}{8\pi^{2}}\int_{S}{\operatorname{tr}}F_{A}\wedge F_{A}=N$ (4.31)

Here $G$ is some compact simply-connected simple Lie group, which has a
simply-laced Lie algebra $\mathfrak{g}$. The moduli space
${{\mathfrak{P}}}={\mathcal{M}}_{N}(G)$ is also hyperkähler, in particular it
is holomorphic symplectic, with the $(2,0)$-form given by:

${\Omega}_{\bf
I}^{N,G}=\int_{S}{\omega}_{S}^{2,0}\wedge\,{\operatorname{tr}}\,{\delta}{\bar{A}}\wedge{\delta}{\bar{A}}$
(4.32)

The integrable system structure is obtained by studying the restriction of the
instanton gauge field on the elliptic fibers, where generically they define a
point in the coarse moduli space $\mathrm{Bun}_{\bf G}({\mathscr{E}}_{z})$ of
semi-stable principal holomorphic $\bf G$-bundles on the fiber, see C.5.
Thanks to E. Loojienga’s theorem, this moduli space is a weighted projective
space, which can be identified for different non-singular fibers. One gets
thus a section of the locally trivial bundle of

${\Pi}:{\mathscr{P}}=\bigcup_{z\in{\mathbb{C}\mathbb{P}}^{1}}\mathrm{Bun}_{\bf
G}({\mathscr{E}}_{z})\longrightarrow{\mathbb{C}\mathbb{P}}^{1}$

One has to be careful at the singular fibers. The base ${\mathfrak{M}}$ of the
integrable systems is the properly compactified moduli space of the
holomorphic sections
${\sigma}:{\mathbb{C}\mathbb{P}}^{1}\longrightarrow{\mathscr{P}}$ of
appropriate degree with some ramification conditions at the discriminant locus
of the original elliptic fibration $\pi$.

In this work we shall not encounter these difficulties.

In fact, as we shall explain in more detail in the section 8, the moduli
spaces of vacua of the quiver gauge theories we study lead to the integrable
systems which arise from the the moduli spaces of $G$-monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ for class I theories with
${\mathbf{G}}_{\text{q}}=\mathbf{G}$, or from the moduli spaces of
$G$-instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ for class II
theories with ${\mathbf{G}}_{\text{q}}=\widehat{\mathbf{G}}$. Here $G$ is a
compact Lie group, whose complexification is the complex simple Lie group
$\mathbf{G}$.

The moduli space ${\mathfrak{P}}$ of $G$-instantons, viewed in the complex
structure where
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}={\mathbb{C}}^{1}\times{\mathscr{E}}$,
is birational to the moduli space of semi-stable holomorphic
$\mathbf{G}$-bundles on ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$, with fixed trivialization at
${\infty}\times{\mathscr{E}}$. The moduli space ${{\mathfrak{P}}}$ projects
down to the moduli space ${\mathfrak{M}}$ of quasimaps from
${\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}$ to the moduli space
of semi-stable holomorphic bundles $\mathrm{Bun}_{\mathbf{G}}(\mathscr{E})$ on
a fixed elliptic curve $\mathscr{E}$. The moduli space of monopoles maps to
the moduli space of quasimaps with prescribed singularities on
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$ to
$B({\mathfrak{g}})={\mathbf{G}}/Ad({\mathbf{G}})={\mathbf{T}}/W({\mathfrak{g}})$.

###  4.6. Extended moduli space as a complex integrable system

The extended moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ is a base of a
complex, but not algebraic, integrable system
${{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$. The
Liouville tori of this integrable system are acted on by an algebraic torus
$({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$, so that the quotients are
the compact abelian varieties, the Liouville tori fibered over
${\mathfrak{M}}$. The symplectic quotient of ${\mathfrak{P}}^{\mathrm{ext}}$
with respect to $({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$ at some
level of the moment map, which is linearly determined by the values of the bi-
fundamental masses, gives ${\mathfrak{P}}$. Recall that Duistermaat-Heckmann
theorem [Duistermaat:1982vw] then implies that the cohomology class
$[{\Omega}_{\bf I}]$ of the $(2,0)$-symplectic form ${\Omega}_{\bf I}$ on
${\mathfrak{M}}$ is linear with masses.

##  Chapter 5 The limit shape equations

In this section we return to the microscopic analysis of our gauge theory.
Recall that the ${\mathcal{N}}=2$ supersymmetry algebra is generated by four
supercharges ${\mathcal{Q}}_{{\alpha}i}$, ${\alpha}=1,2$, $i=1,2$ of the left
and by four supercharges ${\bar{\mathcal{Q}}}_{\dot{\alpha}}^{i}$,
${\dot{\alpha}}=1,2$ of the right chirality. The prepotential
${\mathcal{F}}({\mathcal{A}})$ of the theory is a function of the superfield
${\mathcal{A}}$ which is annihilated by ${\mathcal{Q}}_{{\alpha}i}$’s. We
shall now focus on the observables which are in the cohomology of one of the
${\mathcal{Q}}_{{\alpha}i}$ supercharges, which we shall call simply
$\mathcal{Q}$.

###  5.1. The amplitude functions

The basic such observable is the scalar ${\Phi}_{i}$ in the vector multiplet.
More precisely, any gauge invariant functional, in particular the local
operator $P({\Phi}_{i}({\bf x}))$, where $P$ is some invariant polynomial on
the Lie algebra of $SU(\mathbb{B}_{i})$, and $\bf x$ is a point in space-time,
is annihilated by $\mathcal{Q}$. Moreover, the observables $P({\Phi}_{i}({\bf
x}))$ and $P({\Phi}_{i}({\bf x}^{\prime}))$ for two different points ${\bf x}$
and ${\bf x}^{\prime}$ are in the same $\mathcal{Q}$-cohomology class.
Therefore, one may talk about the vacuum expectation value of $P({\Phi}_{i})$
without specifying the point $\bf x$.

Consider the observables
${\mathcal{O}}_{n,i}={\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\Phi}_{i}^{n}$.
Form the generating function:

${\mathscr{Y}}_{i}(x)=x^{{\mathbf{v}}_{i}}\,{\exp}\,\left(-\sum_{n=1}^{\infty}\frac{\langle{\mathcal{O}}_{n,i}\rangle_{u}}{n}\,x^{-n}\right)$
(5.1)

which turns out to be well-behaved for sufficiently large $x$. We shall denote
the $x$-plane where ${\mathscr{Y}}_{i}(x)$ are defined, by
$\mathbf{C}_{\left\langle x\right\rangle}$. Actually, the analytic
continuation in the $x$ variable gives us the set ${\bf\mathscr{Y}}(x)$ of
multi-valued analytic functions on $\mathbf{C}_{\left\langle x\right\rangle}$,

${\bf\mathscr{Y}}(x)=({\mathscr{Y}}_{i}(x))_{i\in\mathrm{Vert}_{\gamma}},$

This set of multi-valued functions captures the vacuum expectation values of
all the local gauge invariant observables commuting with the supercharge
${\mathcal{Q}}$.

######  Remark.

The general relation between the amplitude functions ${\mathscr{Y}}_{i}(x)$
and the polynomials ${\mathscr{T}}_{i}(x)$ generating the first non-trivial
Casimirs of the gauge group $G_{\text{g}}$ is

${\mathscr{T}}_{i}(x)=\left({\mathscr{Y}}_{i}(x)\right)_{+}$

where $(\ldots)_{+}$ denotes the polynomial part.

In what follows we shall use another set of
$(T_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ polynomials, ${\rm
deg}T_{i}(x)={\mathbf{v}}_{i}$, which are not monic. The coefficients of
$T_{i}(x)$ are related to the coefficients of ${\mathscr{T}}_{i}(x)$ by a
‘‘mirror map’’ change of variables, which will become clear in the course of
our exposition.

######  Remark.

In the way we defined these functions, the information about the vacuum
expectation values of these observables is contained in the expansion of
${\mathscr{Y}}_{i}(x)$ near $x=\infty$ on the physical sheet of these
functions. It would be interesting to see whether the expansion at $x=\infty$
of the branches of ${\mathscr{Y}}_{i}(x)$ contains information about the vevs
of the chiral observables of the theories, related to the one we started with
via some version of $S$-duality.

The functions ${\mathscr{Y}}_{i}(x)$ are the integral transforms of the
densities

${\rho}_{i}({\mathrm{x}}),\qquad{\mathrm{x}}\in{\mathbb{R}},i\in\mathrm{Vert}_{\gamma},$

which describe the combinatorics of the set of fixed points of the symmetry
group action on the instanton moduli space used in the localization approach
to the calculation of the supersymmetric partition function of the gauge
theory. We do not review the formalism [Nekrasov:2003rj] in this work. For the
prior work on the subject see [Moore:1997dj, Lossev:1997bz, Losev:1997wp,
Moore:1998et, Losev:1999nt, Losev:1999tu, Losev:2000bf, Nekrasov:2002dd,
Nekrasov:2002qd, Nekrasov:2003zj, Losev:2003py] and for the novel applications
and refinements see [Pestun:2007rz, Alday:2009aq].

Instead we write down the equations obeyed by the amplitude functions, the so-
called _limit shape equations_ , generalizing the limit shape equations
studied in [Nekrasov:2003rj, Nekrasov:2004vw, Shadchin:2005mx,
Shadchin:2005cc, Shadchin:2005hp]. We shall solve the limit shape equations
using the analytic properties of the amplitude functions. One finds that the
analytic continuation of these functions is governed by the monodromy group,
which we shall call the iWeyl group (the _instanton Weyl group_).

The iWeyl group is the Weyl group $W(\mathfrak{g_{\text{q}}})$. For the Class
I theories $W(\mathfrak{g_{\text{q}}})$ is the finite Weyl group
$W({\mathfrak{g}})$ of the corresponding ADE simple Lie algebra
$\mathfrak{g}$, for the Class II theories the iWeyl group turns out to be the
affine Weyl group $W(\widehat{\mathfrak{g}})$ of the corresponding affine Lie
algebra $\widehat{\mathfrak{g}}=\mathfrak{g_{\text{q}}}$. The Weyl group of
$\widehat{GL}_{\infty}$ shows up in the Class II* theories.

We solve the limit shape equations by constructing the iWeyl invariants
${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))$ of ${\mathscr{Y}}_{i}(x+{\mu}_{i})$,
for the appropriate shifts ${\mu}_{i}$, and showing that these invariants are
polynomials in $x$,

${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))=T_{j}(x),\qquad
j\in\mathrm{Vert}_{\gamma}$ (5.2)

For the Class II theories the invariants ${{\mathscr{X}}}_{j}$ are convergent
power series in ${\mathfrak{q}}_{j}$. Moreover, in each order in expansion in
${\mathfrak{q}}=\prod_{i}{\mathfrak{q}}_{i}^{a_{i}}$ they are finite Laurent
polynomials in ${\mathscr{Y}}_{j}$’s. For the Class II* theories the
invariants ${{\mathscr{X}}}_{j}$ are convergent power series in
${\mathfrak{q}}_{j}$, and finite Laurent polynomials in
${\mathscr{Y}}_{j}(x+{\mu}_{j}+lm^{*})$, for a finite collection of integers
$l\in{\mathbb{Z}}$, again in every order in $\mathfrak{q}$ expansion. For the
Class I theories the functions ${{\mathscr{X}}}_{j}$ are polynomials in
${\mathscr{P}}_{j}(x)$ and Laurent polynomials in ${\mathscr{Y}}_{j}(x)$.

####  5.1.1. The densities and the amplitude functions

The amplitude functions ${\mathscr{Y}}_{i}(x)$ are the multi-valued analytic
functions, which we defined, for large $x$, via Eq. (5.1):

${\mathscr{Y}}_{i}(x)={\exp}\
\langle{\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\rm
log}\left(x-{\Phi}_{i}\right)\rangle_{u}$ (5.3)

One shows, using the fixed point techniques that

${\mathscr{Y}}_{i}(x)={\exp}\
\int_{\mathbb{R}}\,d{\mathrm{x}}\,{\rho}_{i}({\mathrm{x}}){\log}(x-{\mathrm{x}})$
(5.4)

where the _density_ function ${\rho}_{i}(x)$ has compact support which
consists of ${\mathbf{v}}_{i}$ intervals

${\rm supp}\,{\rho}_{i}=\bigcup_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\qquad
I_{i,{\mathbf{a}}}$

The intervals $I_{i,{\mathbf{a}}}$ should be thought of as the ‘‘fattened’’
versions of the eigen-values ${\mathfrak{a}}_{i,{\mathbf{a}}}$. More
precisely:

$1=\int_{I_{i,{\mathbf{a}}}}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}},\qquad{\mathfrak{a}}_{i,{\mathbf{a}}}=\int_{I_{i,{\mathbf{a}}}}{\mathrm{x}}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}}$
(5.5)

The functions ${\mathscr{Y}}_{i}(x)$ have, therefore, the cuts at the
intervals $I_{i,{\mathbf{a}}}$, with the limit values
${\mathscr{Y}}^{\pm}_{i}$ of the ${\mathscr{Y}}_{i}$ function at the top and
the bottom banks of the interval $I_{i,{\mathbf{a}}}$ being related via:

${\mathscr{Y}}_{i}^{+}(x)/{\mathscr{Y}}_{i}^{-}(x)={\exp}\
2\pi\mathrm{i}\int^{x}_{-\infty}{\rho}_{i}({\mathrm{x}})d{\mathrm{x}}$

One then analytically continues ${\mathscr{Y}}_{i}(x)$ across the cuts, which
leads to the set of the multi-valued analytic functions. We shall describe
this analytic continuation in detail in the coming section.

####  5.1.2. The special coordinates

From the Eqs. (5.3) and (5.5) one derives:

${\mathfrak{a}}_{i,{\mathbf{a}}}=\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d\,{\log}{\mathscr{Y}}_{i}(x)$
(5.6)

where $A_{i{\mathbf{a}}}$ is a small loop surrounding the cut
$I_{i,{\mathbf{a}}}$, see Fig. 5.1.

Figure 5.1. Support intervals $I_{i,\mathbf{a}}$ of the densities $\rho_{i}$
and the cycles $A_{i\mathbf{a}}$

###  5.2. The limit shape prepotential

The prepotential of the low-energy theory is expressed in terms of the
densities as follows:

$\displaystyle{\mathcal{F}}({\mathfrak{a}};m;{\tau})$ $\displaystyle\ =\
-\int\int_{{\mathbb{R}}^{2}}d{\mathrm{x}}^{\prime}d{\mathrm{x}}^{\prime\prime}\sum_{i\in\mathrm{Vert}_{\gamma}}{\rho}_{i}({\mathrm{x}}^{\prime}){\rho}_{i}({\mathrm{x}}^{\prime\prime}){\mathcal{K}}({\mathrm{x}}^{\prime}-{\mathrm{x}}^{\prime\prime})$
$\displaystyle+\int\int_{{\mathbb{R}}^{2}}d{\mathrm{x}}^{\prime}d{\mathrm{x}}^{\prime\prime}\sum_{e\in\mathrm{Edge}_{\gamma}}{\rho}_{t(e)}({\mathrm{x}}^{\prime}){\rho}_{s(e)}({\mathrm{x}}^{\prime\prime}){\mathcal{K}}({\mathrm{x}}^{\prime}-{\mathrm{x}}^{\prime\prime}+m_{e})$
$\displaystyle+\sum_{i\in\mathrm{Vert}_{\gamma}}\int_{{\mathbb{R}}}d{\mathrm{x}}\,{\rho}_{i}({\mathrm{x}})\left(({\log}{\mathfrak{q}}_{i})\frac{{\mathrm{x}}^{2}}{2}+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}({\mathrm{x}}-m_{i,{\mathfrak{f}}})\right)$
$\displaystyle\qquad\qquad\qquad+\sum_{i,\mathbf{a}}\mathfrak{b}_{i,\mathbf{a}}\left(1-\int_{I_{i,\mathbf{a}}}\rho_{i}({\mathrm{x}})d{\mathrm{x}}\right)$
$\displaystyle+\sum_{i,\mathbf{a}}\mathfrak{a}_{i,\mathbf{a}}^{D}\left(\mathfrak{a}_{i,\mathbf{a}}-\int_{I_{i,\mathbf{a}}}{\mathrm{x}}\rho_{i}({\mathrm{x}})\,d{\mathrm{x}}\right)$

where the constraints (5.5) are incorporated by the last two lines via
Lagrangian multipliers
$\mathfrak{b}_{i,\mathbf{a}},\mathfrak{a}_{i,\mathbf{a}}^{D}$,

${\mathcal{K}}(x)=\frac{x^{2}}{2}\left(\log\left(\frac{x}{\Lambda_{UV}}\right)-\frac{3}{2}\right)$
(5.7)

and $\Lambda_{UV}$ is the UV cutoff scale. In fact, the
$\Lambda_{UV}$-dependence drops out for the theories solving the
${\beta}_{i}=0$ equations. However, in the intermediate formulae we keep the
explicit $\Lambda_{UV}$-dependence.

####  5.2.1. The limit shape equations.

Originally the limit shape equations were derived as the variational equations
on $\mathcal{F}$ (see [Nekrasov:2003rj], [Nekrasov:2004vw]). These are linear
integral equations on the densities ${\rho}_{i}(x)$: for any $x\in
I_{i,{\mathbf{a}}}$ the following should hold

$\displaystyle-2\int_{\mathbb{R}}{\rho}_{i}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}})d{\mathrm{x}}+$
(5.8) $\displaystyle\qquad+\sum_{e:\
t(e)=i}\int_{\mathbb{R}}{\rho}_{s(e)}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}}+m_{e})d{\mathrm{x}}$
$\displaystyle\qquad\quad+\sum_{e:\
s(e)=i}\int_{\mathbb{R}}{\rho}_{t(e)}({\mathrm{x}}){\mathcal{K}}(x-{\mathrm{x}}-m_{e})d{\mathrm{x}}$
$\displaystyle\qquad\qquad\qquad+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}(x-m_{i,\mathfrak{f}})+\frac{x^{2}}{2}{\log}({\mathfrak{q}}_{i})=x\,{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}+{\mathfrak{b}}_{i,{\mathbf{a}}}$

where ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$, ${\mathfrak{b}}_{i,{\mathbf{a}}}$
are some constants, the Lagrange multipliers for the conditions (5.5) which
are determined from the solution. Actually,
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ _is_ the dual special coordinate, cf.
(4.4)

$\frac{{\partial}{\mathcal{F}}}{{\partial}{\mathfrak{a}}_{i,{\mathbf{a}}}}={\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$
(5.9)

We find it useful to rewrite the second derivative with respect to $x$ of the
linear integral equations (5.8) on $\rho_{i}({\mathrm{x}})$ as the non-linear
polynomial difference equations on the amplitudes ${\mathscr{Y}}_{i}(x)$:

${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathscr{P}}_{i}(x)\prod_{e:\
t(e)=i}{\mathscr{Y}}_{s(e)}(x+m_{e})\prod_{e:\
s(e)=i}{\mathscr{Y}}_{t(e)}(x-m_{e})$ (5.10)

for $x\in I_{i,{\mathbf{a}}}$, ${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}$, where
we used the notation:

${\mathscr{Y}}^{\pm}_{i}(x)={\mathscr{Y}}_{i}(x\pm\mathrm{i}0),\qquad x\in
I_{i,{\mathbf{a}}}$ (5.11)

####  5.2.2. The mass cocycles.

In what follows we shall redefine the amplitude functions and the
${\mathscr{P}}$-polynomials

${\mathscr{Y}}_{i}(x)\to{\mathscr{Y}}_{i}(x+{\mu}_{i}),\qquad{\mathscr{P}}_{i}(x)\to{\mathscr{P}}_{i}(x+{\mu}_{i})$
(5.12)

so as to simplify the shifts of the arguments by the masses $m_{e}$ of the bi-
fundamental hypermultiplets:

$m_{e}\longrightarrow m_{e}+{\mu}_{t(e)}-{\mu}_{s(e)}$

The equations (5.10) are the main equations which determine the low-energy
effective action as well as the expectation values of all gauge invariant
chiral observables. One can view the Eqs. (5.10) as a Riemann-Hilbert problem.
They are also similar, but not identical, to the so-called Y-systems and
discrete Hirota equations.

For the class I and class II theories the shift (5.12) maps the equations
(5.10) to

${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathscr{P}}_{i}(x)\prod_{j\in\mathrm{Vert}_{\gamma},j\neq
i}{\mathscr{Y}}_{j}(x)^{I_{ij}}$ (5.13)

where for the class II theories ${\mathscr{P}}_{i}(x)={\mathfrak{q}}_{i}$. For
the class II* theory $\widehat{A}_{r}^{*}$, with the clockwise, say,
orientation of the quiver, we can make all masses $m_{e}$ to be equal,
$m_{e}=\frac{1}{r+1}{\mathfrak{m}}$, by using the shift (5.12). More
precisely, in writing (5.14) we chose the representative $m^{*}$ such that if
all the edges are oriented so that $t(e)=(s(e)+1)\,{\rm mod}\,(r+1)$, then
$m^{*}_{e}=\frac{1}{r+1}{\mathfrak{m}}$. Then,

${\mathscr{Y}}^{+}_{i}(x){\mathscr{Y}}^{-}_{i}(x)={\mathfrak{q}}_{i}\,{\mathscr{Y}}_{i-1}\left(x-\frac{\mathfrak{m}}{r+1}\right){\mathscr{Y}}_{i+1}\left(x+\frac{\mathfrak{m}}{r+1}\right)$
(5.14)

where ${\mathscr{Y}}_{i+r+1}(x)={\mathscr{Y}}_{i}(x)$.

####  5.2.3. Analytic continuation

We use Eq. (5.3) to analytically continue the functions ${\mathscr{Y}}_{i}(x)$
through the cuts $I_{i,{\mathbf{a}}}$: for the Class I and II theories, after
the redefinition (5.12) which eliminates $m_{e}$:

$r_{i}:{\mathscr{Y}}_{i}(x)\longrightarrow{\mathscr{Y}}_{i}(x){\mathscr{P}}_{i}(x)\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathscr{Y}}_{j}(x)^{-C_{ij}}$
(5.15)

For the Class II* theories after the redefinition (5.12) we have the following
analogue of (5.15):

$r_{i}:{\mathscr{Y}}_{i}(x)\longrightarrow{\mathfrak{q}}_{i}\frac{{\mathscr{Y}}_{i-1}\left(x-\frac{\mathfrak{m}}{r+1}\right){\mathscr{Y}}_{i+1}\left(x+\frac{\mathfrak{m}}{r+1}\right)}{{\mathscr{Y}}_{i}(x)}$
(5.16)

###  5.3. The iWeyl group.

The transformations (5.15, 5.16) generate a group, which we shall call the
_instanton Weyl group_ , or _iWeyl group_ , ${{}^{i}{\mathcal{W}}}$, for
short. This group can be defined for a much larger class of ${\mathcal{N}}=2$
theories, not necessarily of the superconformal quiver type we study in this
work.

It is clear that the transformations $r_{i}$ are reflections $r_{i}\circ
r_{i}=Id$, so the iWeyl group is the group, generated by reflections.

Now, by comparing Eq. (5.15) and Eqs. (C.31), (C.58) we see that for the Class
I theories the iWeyl group is the finite Weyl group $W({\mathfrak{g}})$.
Similarly, for the Class II theories the iWeyl group coincides with the affine
Weyl group $W(\widehat{\mathfrak{g}})$. For the Class II* theory the iWeyl
group is the Weyl group $W({\widehat{\mathfrak{gl}}_{\infty}})$ of the group
$\widehat{GL}_{\infty}$.

The groups $\mathbf{G}$, $\widehat{\mathbf{G}}$, $\widehat{GL}_{\infty}$ and
their Weyl groups are discussed in Appendix.

###  5.4. Moduli of vacua and mass parameters

After all the redefinitions (5.12) the original mass parameters $m_{e}$, the
moduli $({\mathfrak{a}}^{\mathcal{I}})_{{\mathcal{I}}=1}^{\bf r}$ of the vacua
$u\in{\mathfrak{M}}$, and the derivatives of the prepotential
$({\partial}{\mathcal{F}}/{\partial}{\mathfrak{a}}^{\mathcal{I}})$ are
recovered from the study of periods of certain differentials on the curve
${\mathcal{C}}_{u}$ defined as follows.

####  5.4.1. The first glimpses of the cameral curve

The functions ${\mathscr{Y}}_{i}(x)$, $i\in\mathrm{Vert}_{\gamma}$, after the
maximal analytic continuation through the cuts $I_{i,{\mathbf{a}}}$ form a
local ${{}^{i}{\mathcal{W}}}$-system. It is easy to see that, as long as
$|{\mathfrak{q}}_{j}|\ll 1$ for all $j\in\mathrm{Vert}_{\gamma}$, there is
exactly one branch of ${\mathscr{Y}}_{i}(x)$ as $x\to\infty$ which behaves as:

${\mathscr{Y}}_{i}^{\rm phys}(x)\sim x^{{\mathbf{v}}_{i}}+\ldots$ (5.17)

The other branches behave as

${\mathscr{Y}}_{i}^{\rm
unphys}(x)\sim\left(\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathfrak{q}}_{j}^{n_{ji}}\right)\,x^{{\mathbf{v}}_{i}}+\ldots$
(5.18)

with $n_{ji}\geq 0$, and

$\sum_{j}n_{ji}>0\ .$

Now, the branches meet at the cuts

$\bigcup_{i\in\mathrm{Vert}_{\gamma};\,{\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}}I_{i,{\mathbf{a}}}$
(5.19)

for the Class I and II theories, and at the cuts

$\bigcup_{i\in\mathrm{Vert}_{\gamma};\,{\mathbf{a}}=1,\ldots,N}\,I_{i,{\mathbf{a}}}+\frac{\mathfrak{m}}{r+1}\,{\mathbb{Z}}$
(5.20)

for the Class II* theories.

The collection of these branches defines a curve ${\mathcal{C}}_{u}$ which we
shall describe explicitly in the next section. The curve ${\mathcal{C}}_{u}$
is a ${{}^{i}{\mathcal{W}}}$-cover of the $x$-plane $\mathbf{C}_{\left\langle
x\right\rangle}$, with the branch points at the ends of the cuts
$I_{i,{\mathbf{a}}}$. Because of the ${{}^{i}{\mathcal{W}}}$-action on
${\mathcal{C}}_{u}$ and the relation to the Weyl groups which permute Weyl
cameras, the curve ${\mathcal{C}}_{u}$ will be called the _cameral curve_ ,
following [Donagi:1995alg]

####  5.4.2. The special coordinates and the mass parameters

Figure 5.2. The cycle at $x=\infty$ surrounding the branch cuts of
$\mathscr{Y}_{i}^{\text{phys}}(x)$

Take the physical branch and expand it at $x=\infty$. Then the next-to-leading
term gives ${\mu}_{i}$, the mass shift which determines (or partially
determines, in the II* case) the bi-fundamental masses:

${\mathscr{Y}}^{\rm phys}_{i}(x)\sim
x^{{\mathbf{v}}_{i}}+{\mathbf{v}}_{i}{\mu}_{i}x^{{\mathbf{v}}_{i}-1}+\ldots$
(5.21)

The Eq. (5.6) is modified by the ${\mu}_{i}$-shift:

${\mathfrak{a}}_{i,{\mathbf{a}}}+{\mu}_{i}=\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d{\log}{\mathscr{Y}}_{i}(x)$
(5.22)

where $A_{i{\mathbf{a}}}$ is a loop on the physical sheet of
${\mathcal{C}}_{u}$ which surrounds the cut $I_{i,{\mathbf{a}}}$. Note that
the only singularities of ${\mathscr{Y}}_{i}(x)$ on the physical sheet are at
$x=\infty$ and at the cuts. Therefore (see fig. 5.2):

$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\frac{1}{2\pi\mathrm{i}}\oint_{A_{i{\mathbf{a}}}}\,x\,d{\log}{\mathscr{Y}}_{i}(x)=\frac{1}{2\pi\mathrm{i}}\oint_{\infty}\,x\,d{\log}{\mathscr{Y}}_{i}(x)={\mathbf{v}}_{i}{\mu}_{i}$
(5.23)

which is consistent with Eqs. (5.21), (5.22) thanks to (2.3), i.e.

$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}=0$
(5.24)

######  Remark.

The residues

$U_{ji}=\frac{1}{2\pi\mathrm{i}}\oint_{\infty}\,x^{j}\,d{\log}{\mathscr{Y}}_{i}(x)$
(5.25)

determine the ‘‘mirror map’’, the change of variables
$\left(T_{i}(x)\right)_{i\in\mathrm{Vert}_{\gamma}}\longrightarrow\left({\mathscr{T}}_{i}(x)\right)_{i\in\mathrm{Vert}_{\gamma}}$
we talked about earlier.

####  5.4.3. The dual special coordinates

Now let us discuss the dual coordinates ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$.
First of all, the Eq. (5.9) does not quite make sense in view of (5.24).
Suppose we relax (5.24) by absorbing ${\mu}_{i}$ into the definition of
${\mathfrak{a}}_{i,{\mathbf{a}}}$. We can then analyze the limit shape problem
in the usual fashion. We should keep in mind, however, that only the
$SU({\mathbf{v}}_{i})$ part of the gauge group is dynamical. The trace part of
the dual special coordinates ${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$,

${\mu}^{D}_{i}=\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$

is ambiguous. The traceless part, i.e.

$\sum_{{\mathbf{a}}=1}^{{\mathbf{v}}_{i}}\lambda_{\mathbf{a}}{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$

for any weight vector $({\lambda}_{\mathbf{a}})$,

$\sum_{\mathbf{a}}{\lambda}_{\mathbf{a}}=0\ ,$

should be well-defined. Let us now see how this works in detail.

By differentiating (5.8) with respect to $x$ we find

$\displaystyle-2\int_{\mathbb{R}}{\rho}_{i}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}})d{\tilde{x}}+$
(5.26)
$\displaystyle\qquad+\sum_{e:t(e)=i}\int_{\mathbb{R}}{\rho}_{s(e)}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}}+m_{e})d{\tilde{x}}+\sum_{e:s(e)=i}\int_{\mathbb{R}}{\rho}_{t(e)}({\tilde{x}}){\mathcal{K}}^{\prime}(x-{\tilde{x}}-m_{e})d{\tilde{x}}$
$\displaystyle\qquad\qquad\qquad+\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}{\mathcal{K}}^{\prime}(x-m_{i,\mathfrak{f}})+x\,{\log}{\mathfrak{q}}_{i}={\mathfrak{a}}_{i,{\mathbf{a}}}^{D},\qquad\qquad
x\in I_{i,{\mathbf{a}}}$

where

${\mathcal{K}}^{\prime}(x)=x\,\log\left(\frac{x}{\tilde{\Lambda}_{UV}}\right)\equiv\int_{\tilde{\Lambda}_{UV}}^{x}\mathcal{K}^{\prime\prime}({\tilde{x}})d{\tilde{x}}$

with

$\mathcal{K}^{\prime\prime}(x)=\log\left(\frac{x}{\Lambda_{UV}}\right)$ (5.27)

and $\tilde{\Lambda}_{UV}=\exp(1)\Lambda_{UV}$. Using the definition (5.4) of
$\mathscr{Y}_{i}(x)$ functions we find

$\displaystyle{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\tilde{\Lambda}_{UV}\log\mathfrak{q}_{i}-\int_{\tilde{\Lambda}_{UV}}^{x}\,d{\tilde{x}}\,\log\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,+$
(5.28)
$\displaystyle\qquad\int_{\tilde{\Lambda}_{UV}}^{x}d{\tilde{x}}\left(-\log\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})+\sum_{e:\,t(e)=i}\log\mathscr{Y}^{\mathrm{phys}}_{s(e)}({\tilde{x}}+m_{e})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\sum_{e:\,s(e)=i}\log\mathscr{Y}^{\mathrm{phys}}_{t(e)}({\tilde{x}}-m_{e})+\log{\mathscr{P}}_{i}({\tilde{x}})\right)$

The integration contour in the above formula runs over a physical sheet from a
marked point $p_{*}\in{\mathcal{C}}_{u}$ which sits over the point
$\tilde{\Lambda}_{UV}\in\mathbf{C}_{\left\langle x\right\rangle}$ to a point
$x\in I_{i,\mathbf{a}}$ which we view as sitting on $\mathcal{C}_{u}$. The
choice of $x$ is is irrelevant111Physically the meaning of the integrand in
the effective electrostatic problem is the force acting on elementary charge,
and the integral is the chemical potential for the charge, or the energy
required to move an elementary charge from the density support to infinity.
The force vanishes on the support of the charge in the stationary charge
distribution. as long as $x\in I_{i,\mathbf{a}}$ precisely due to the critical
point equations (5.10). The above expression for
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ can be converted into much nicer form by
noting that the integral on the second line in (5.28) is in fact

$\int_{\tilde{\Lambda}_{UV}}^{x}\log\left(\,{}^{r_{i}}\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)d{\tilde{x}}$
(5.29)

where $r_{i}$ is the $i$’th reflection (5.15). In other words, (5.29) is the
integral of the analytic continuation of the function
$\mathscr{Y}_{i}({\tilde{x}})$ onto the mirror sheet of ${\mathcal{C}}_{u}$
obtained from the physical sheet by the ${{}^{i}{\mathcal{W}}}$-reflection
$r_{i}$, i.e. by continuing across any of the cuts $I_{i,\mathbf{a}}$,
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}$, supporting the density
$\rho_{i}(x)$. Thus the integral of the expression in the brackets on the last
two lines in (5.28) is equal to the integral

$\int_{r_{i}(p_{*})}^{x}\,\log{\mathscr{Y}}_{i}({\tilde{x}})d{\tilde{x}}$
(5.30)

Thus we conclude

${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\tilde{\Lambda}_{UV}\log\mathfrak{q}_{i}-\int_{B_{i\mathbf{a}}}\,\log\mathscr{Y}_{i}({\tilde{x}})d{\tilde{x}}$
(5.31)

where the contour $B_{i\mathbf{a}}$ starts at the point $p_{*}$ which sits
over $x=\tilde{\Lambda}_{UV}$ on the physical sheet, runs through the cut
$I_{i,\mathbf{a}}$ to the mirror sheet $r_{i}(\mathrm{phys})$ and terminates
at the point $r_{i}(p_{*})$, which sits over $x=\tilde{\Lambda}_{UV}$ on the
mirror sheet.

It is tempting to send $\Lambda_{UV}$ to infinity. However, there is a
subtlety which we already discussed in the beginning of this section. The
integral (5.31) diverges for ${\Lambda}_{UV}\to\infty$. The linear divergence
is canceled by the constant term
${\tilde{\Lambda}}_{UV}{\log}{\mathfrak{q}}_{i}$ due to

$\log\left(\,{}^{r_{i}}\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)-\log\left(\,\mathscr{Y}_{i}^{\mathrm{phys}}({\tilde{x}})\,\right)=\log\mathfrak{q}_{i},\quad{\tilde{x}}\to\infty$
(5.32)

However the subleading logarithmic divergence does not, in general, cancel.
The simplest way to calculate it is to compute the logarithmic derivative
${\Lambda}_{UV}d{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}/d{\Lambda}_{UV}$ and then
send $\Lambda_{UV}\to\infty$:

${\Lambda}_{UV}\frac{d{\mathfrak{a}}_{i,{\mathbf{a}}}^{D}}{d{\Lambda}_{UV}}=-\sum_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}m_{i,{\mathfrak{f}}}+\sum_{e:\,t(e)=i}{\mathbf{v}}_{s(e)}m_{e}-\sum_{e:\,s(e)=i}{\mathbf{v}}_{t(e)}m_{e}$
(5.33)

Luckily the right hand side of (5.33) does not depend on ${\mathbf{a}}$.

We can use the formal expression

${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}=\int_{B_{i\mathbf{a}}}\,x\,d\,{\log}\,\mathscr{Y}_{i}$
(5.34)

where the contour

$B_{i\mathbf{a}}:\infty_{\mathrm{phys}}\to\ \stackrel{{\scriptstyle
I_{i,\mathbf{a}}}}{{\qquad}}\to r_{i}(\infty_{\mathrm{phys}})$ (5.35)

starts at the point $x=\infty_{\mathrm{phys}}$ on the physical sheet, then
runs through the cut $I_{i,\mathbf{a}}$ to the mirror sheet
$r_{i}(\mathrm{phys})$ and finishes at the point
$r_{i}(\infty_{\mathrm{phys}})$ on this mirror sheet.

The canonical contour $B_{i\mathbf{a}}$ computing
$\mathfrak{a}_{i\mathbf{a}}^{D}$ is an open contour, and, as we said above,
the integral of $xd{\log}{\mathscr{Y}}_{i}$ is divergent. However the
variation of Coulomb parameters in $SU({\mathbf{v}}_{i})$ concerns only the
differences

$\mathfrak{a}_{i,\mathbf{a}^{\prime}}^{D}-\mathfrak{a}_{i,\mathbf{a}^{\prime\prime}}^{D}=\int_{B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}}\,x\,d\,\log\mathscr{Y}_{i}$
(5.36)

computed by the closed contour
$B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}=B_{i\mathbf{a}^{\prime}}-B_{i\mathbf{a}^{\prime\prime}}$
running on physical sheet through the cut $I_{i\mathbf{a}^{\prime}}$ to the
mirror sheet $r_{i}(\mathrm{phys})$ and then through the cut
$I_{i\mathbf{a}^{\prime\prime}}$ back to the physical sheet. The divergence
(5.33) cancels in the integration over
$B_{i{\mathbf{a}}^{\prime}}-B_{i{\mathbf{a}}^{\prime\prime}}$.

In fact, the difference (5.36) is represented as the integral over the closed
contour $B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}$ without any
divergent quantities, as follows immediately from the (5.26), by connecting
two points $x^{\prime}\in I_{i,\mathbf{a}^{\prime}}$ and $x^{\prime\prime}\in
I_{i,\mathbf{a}^{\prime\prime}}$ on the physical sheet and replacing the
integrand as in the second integral of (5.28) over the physical sheet by an
integral of $-\log\mathscr{Y}(x^{\prime})dx^{\prime}$ over the return segment
from $x^{\prime\prime}\in I_{i,\mathbf{a}^{\prime\prime}}$ to $x^{\prime}\in
I_{i,\mathbf{a}^{\prime}}$ on the mirror sheet $r_{i}(\mathrm{phys})$, see
Fig. 5.3

Figure 5.3. The cycles $B_{i;\mathbf{a}^{\prime}}^{\mathbf{a}^{\prime\prime}}$

In the weakly coupled regime we have the following BPS particles in the gauge
theory: for each gauge group factor $i$ the $W$-bosons associated with the
breaking $SU({\mathbf{v}}_{i})\to U(1)^{{\mathbf{v}}_{i}-1}$, which correspond
to the roots of the $SU({\mathbf{v}}_{i})$, and magnetic monopoles, which
correspond to the fundamental weights. Accordingly, it seems natural to define
the following cycles on the cameral curve: the $\mathcal{A}$\- and
$\mathcal{B}$-cycles, more precisely ${\mathcal{A}}_{\mathcal{I}}$,
${\mathcal{B}}^{\mathcal{I}}$, labelled by ${\mathcal{I}}=(i,{\mathbf{a}})$,
with $i\in\mathrm{Vert}_{\gamma}$, and
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}-1$:

${\mathcal{A}}_{\mathcal{I}}=A_{i{\mathbf{a}}}-A_{i({\mathbf{a}}+1)}\,,\qquad{\mathcal{B}}^{\mathcal{I}}=\sum_{{\mathbf{a}}^{\prime}=1}^{\mathbf{a}}B_{i{\mathbf{a}}^{\prime}}-\frac{\mathbf{a}}{{\mathbf{v}}_{i}}\sum_{{\mathbf{a}}^{\prime\prime}=1}^{{\mathbf{v}}_{i}}B_{i{\mathbf{a}}^{\prime\prime}}$
(5.37)

The cycles ${\mathcal{A}}_{\mathcal{I}},{\mathcal{B}}^{\mathcal{I}}$ determine
the special coordinates

${\mathfrak{a}}^{\mathcal{I}}=\oint_{{\mathcal{A}}_{\mathcal{I}}}\,x\,d{\log}\,{\mathscr{Y}}_{i},\qquad{\mathfrak{a}}_{\mathcal{I}}^{D}=\oint_{{\mathcal{B}}^{\mathcal{I}}}\,x\,d{\log}\,{\mathscr{Y}}_{i}$
(5.38)

In the weak coupling regime the pairing (5.38) between the cycles and the
differentials is non-zero only for ${\mathcal{I}}=(i,{\mathbf{a}})$, for some
${\mathbf{a}}=1,\ldots,{\mathbf{v}}_{i}-1$. The main property of these cycles
is the vanishing of the following two-form on the space ${\mathfrak{M}}$ of
$u$-parameters:

$0=\sum_{\mathcal{I}}d{\mathfrak{a}}^{\mathcal{I}}\wedge
d{\mathfrak{a}}_{\mathcal{I}}^{D}$ (5.39)

which follows simply from the Eq. (5.9).

##  Chapter 6 Solution of the limit shape equations

In this section we solve the Eqs (5.10), (5.13), (5.14), and get an explicit
formula for the curve ${\mathcal{C}}_{u}$.

###  6.1. From invariants to the curve

Our strategy is to define a set of basic invariants
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))$ of the ${{}^{i}{\mathcal{W}}}$ group.
We shall find the basic invariants which are power series in
${\mathfrak{q}}_{j}$’s, and which are normalized in such a way that, for the
class I and II theories:

$\displaystyle{{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}(x)+\sum_{{\nu},\,|{\nu}|>0}\prod_{j\in\mathrm{Vert}_{\gamma}}{\mathscr{P}}_{j}^{{\nu}_{j}}(x)\
{\Psi}_{i\nu}({\mathscr{Y}}_{1}(x),\ldots,{\mathscr{Y}}_{r}(x))$ (6.1)
$\displaystyle\qquad{\nu}=({\nu}_{1},\ldots,{\nu}_{r}),\,\qquad|{\nu}|=\sum_{j\in\mathrm{Vert}_{\gamma}}{\nu}_{j}$

where
${\Psi}_{\nu}({\mathscr{Y}}(x))\in{\mathbb{C}}[{\mathscr{Y}}_{j}(x),{\mathscr{Y}}_{j}(x)^{-1}]$
are quasi-homogeneous Laurent polynomials:

$\sum_{j\in\mathrm{Vert}_{\gamma}}\left({\mathbf{v}}_{j}{\mathscr{Y}}_{j}\frac{\partial}{{\partial}{\mathscr{Y}}_{j}}+{\mathbf{w}}_{j}{\nu}_{j}\right){\Psi}_{i\nu}={\mathbf{v}}_{i}{\Psi}_{i\nu}$
(6.2)

For the class II* theories there is one modification:

$\displaystyle{{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}(x)+\sum_{{\nu},\,|{\nu}|>0}\
{\tilde{\mathfrak{q}}}^{\nu}\,{\tilde{\Psi}}_{i\nu}({\mathscr{Y}}(x))$ (6.3)
$\displaystyle\qquad{\nu}=({\nu}_{0},\ldots,{\nu}_{r}),\,\qquad|{\nu}|=\sum_{j=0}^{r}{\nu}_{j}$

where

${\tilde{\mathfrak{q}}}^{\nu}=\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{{\nu}_{j}}$

and
${\tilde{\Psi}}_{i\nu}({\mathscr{Y}}(x))\in{\mathbb{C}}\biggl{[}{\mathscr{Y}}_{j}(x+\frac{n}{r+1}{\mathfrak{m}}),{\mathscr{Y}}_{j}^{-1}(x+\frac{n}{r+1}{\mathfrak{m}})\biggr{]}$
with $-|{\nu}|\leq n\leq|{\nu}|$. The functions $\tilde{\Psi}_{i\nu}$ are
quasi-homogeneous:

$\sum_{j\in\mathrm{Vert}_{\gamma}}a_{j}{\mathscr{Y}}_{j}\frac{\partial}{{\partial}{\mathscr{Y}}_{j}}{\tilde{\Psi}}_{i\nu}=a_{i}{\tilde{\Psi}}_{i\nu}$
(6.4)

####  6.1.1. Master equations

Now, the ${{}^{i}{\mathcal{W}}}$-invariance of
${{\mathscr{X}}}_{j}({\mathscr{Y}}(x))$ implies that they are continuous
across all the cuts, that is they are single-valued analytic functions of $x$.
Given their large $x$ asymptotics, they are polynomials in $x$:

$\boxed{{{\mathscr{X}}}_{j}({\mathscr{Y}}(x))=T_{j}(x)}$ (6.5)

where

$T_{j}(x)=T_{j,0}({\tilde{\mathfrak{q}}})x^{{\mathbf{v}}_{j}}+T_{j,1}({\tilde{\mathfrak{q}}};m)x^{{\mathbf{v}}_{j}-1}+\ldots+T_{j,{\mathbf{v}}_{j}}$
(6.6)

The coefficients $T_{j,0}({\tilde{\mathfrak{q}}})$ are determined by the gauge
couplings ${\mathfrak{q}}_{k}$

$T_{j,0}({\tilde{\mathfrak{q}}})=1+\sum_{{\nu},\,|{\nu}|>0}{\tilde{\mathfrak{q}}}^{\nu}{\Psi}_{j\nu}(1,\ldots,1)$

The coefficients $T_{j,1}({\tilde{\mathfrak{q}}};m)$ are determined by the
masses $m_{e}$ from (6.1),(6.3),(5.21).

The rest of the coefficients

$\left(T_{j,{\mathbf{a}}}({\tilde{\mathfrak{q}}};m,u)\right)^{j\in\mathrm{Vert}_{\gamma}}_{{\mathbf{a}}=2,\ldots,{\mathbf{v}}_{j}}\
$ (6.7)

is determined by
$u=(u_{i,{\mathbf{a}}})_{i\in\mathrm{Vert}_{\gamma}}^{{\mathbf{a}}=2,\ldots,{\mathbf{v}}_{i}}$.
The Eqs. (6.5,6.6) define an analytic curve

${\mathcal{C}}_{u}^{\circ}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}\,,$

which can be compactified to the _cameral curve_ ${\mathcal{C}}_{u}$ which is
the ${{}^{i}{\mathcal{W}}}$-cover of
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\infty\\}$:

${\mathcal{C}}_{u}\longrightarrow{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$

unramified (for generic $\tilde{\mathfrak{q}}$) over $x=\infty$. The details
of the compactification of ${\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}}$ and
${\mathcal{C}}_{u}^{\circ}\subset{\mathcal{C}}_{u}$ will be discussed
elsewhere. In what follows we drop the superscript $\circ$ in the definition
of $\mathcal{C}_{u}$.

####  6.1.2. The periods

The cameral curve ${\mathcal{C}}_{u}$ depends on $u$, forming a family
${\mathcal{C}}$ of curves parametrized by the ‘‘u-plane’’ ${\mathfrak{M}}$.
The family $\mathcal{C}$ depends on the microscopic couplings
${\mathfrak{q}}_{i}$ and on the mass parameters. Let us keep the masses fixed.
When $|{\mathfrak{q}}_{i}|\ll 1$ for all $i\in\mathrm{Vert}_{\gamma}$ we have
a well-separated system of cycles ${\mathcal{A}}_{\mathcal{I}}$ and
${\mathcal{B}}^{\mathcal{I}}$, which we defined in the section 5.4. We
transport this system of cycles throughout the moduli space of gauge couplings
via Gauß-Manin connection.

####  6.1.3. Vector-valued Seiberg-Witten differential

Let us introduce the following vector-valued $1$-differential which is
schematically given by:

$d{\mathbb{S}}=x\sum_{i\in\mathrm{Vert}_{\gamma}}\left(d\,{\log}\,{\mathscr{Y}}_{i}{\alpha}^{\vee}_{i}-d\,{\log}\,{\mathscr{P}}_{i}(x){\lambda}^{\vee}_{i}\right)$
(6.8)

where ${\alpha}_{i}^{\vee}$ and ${\lambda}_{i}^{\vee}$ are the simple coroots
and the fundamental coweights of $\mathfrak{g_{\text{q}}}$. We shall have more
specific notations for each class.

The differential $d{\mathbb{S}}$ takes values in the vector space
${\mathbb{C}}^{\mathrm{Vert}_{\gamma}}$ which is acted upon by the
${{}^{i}{\mathcal{W}}}$-group. The group ${{}^{i}{\mathcal{W}}}$ also acts on
the curve $\mathcal{C}_{u}$. It is clear from our construction that
$d{\mathbb{S}}$ is ${{}^{i}{\mathcal{W}}}$-equivariant:

$w^{*}d{\mathbb{S}}=w\cdot d{\mathbb{S}}$ (6.9)

for any $w\in{{}^{i}{\mathcal{W}}}$.

####  6.1.4. Degeneration and filtration

In this section we consider the theories of class $I$ and class $II$, and the
extended Coulomb moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ (which includes
the masses of bifundamental hypermultiplets, recall eq. 4.7). Consider
conformal quiver with assigned dimensions $(\gamma,\mathbf{v},\mathbf{w})$ at
vertices and recall that they satisfy $\mathbf{w}=C\mathbf{v}$,
$\mathbf{w}_{i}\geq 0,\mathbf{v}_{i}\geq 0$ where $C$ is the Cartan matrix of
$\gamma$. We say that the theory $(\gamma,\mathbf{v},\mathbf{w})$ strictly
contains the theory $(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ if
$0\leq\mathbf{v}^{\prime}\leq\mathbf{v},0\leq\mathbf{w}^{\prime}\leq\mathbf{w}$
and $C\mathbf{v}^{\prime}=\mathbf{w}^{\prime}$, and
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})\neq(\gamma,\mathbf{v},\mathbf{w})$.
The extended Coulomb moduli space ${\mathfrak{M}}^{\mathrm{ext}}$ of theory
$(\gamma,\mathbf{v},\mathbf{w})$ contains a locus related to the Coulomb
moduli ${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}$ space of
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ as follows. Suppose that
given a point $\mathfrak{m}_{f}\in\mathbf{C}_{\left\langle x\right\rangle}$
the polynomials $T_{i}(x),\mathscr{P}_{i}(x),i\in\mathrm{Vert}_{\gamma}$
factorize as follows

$\displaystyle T_{i}(x)$
$\displaystyle=T^{\prime}_{i}(x)(x-\mathfrak{m}_{f})^{\mathbf{v}_{i}-\mathbf{v}^{\prime}_{i}}$
(6.10) $\displaystyle\mathscr{P}_{i}(x)$
$\displaystyle=\mathscr{P}^{\prime}_{i}(x)(x-\mathfrak{m}_{f})^{\mathbf{w}_{i}-\mathbf{w}^{\prime}_{i}}$

Then it is clear that the character equations (6.5) factorize as well, and the
$\mathscr{Y}_{i}(x)$ functions solving the theory
$(\gamma,\mathbf{v},\mathbf{w})$ are expressed in terms of
$\mathscr{Y}_{i}^{\prime}(x)$ functions solving the theory
$(\gamma,\mathbf{v}^{\prime},\mathbf{w}^{\prime})$ as

$\mathscr{Y}_{i}(x)=(x-\mathfrak{m}_{f})^{(\mathbf{v}_{i}-\mathbf{v}^{\prime}_{i})}\mathscr{Y}_{i}^{\prime}(x)$
(6.11)

Suppose that the degeneration eq. 6.10 is minimal, i.e. there is no
intermediate and different
$(\gamma,\mathbf{v}^{\prime\prime},\mathbf{w}^{\prime\prime})$ such that
$\mathbf{v}^{\prime}\leq\mathbf{v}^{\prime\prime}\leq\mathbf{v}$ and
$\mathbf{w}^{\prime}\leq\mathbf{w}^{\prime\prime}\leq\mathbf{w}$. Then we see
that ${\mathfrak{M}}^{\mathrm{ext}}$ includes the loci
${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}\times\mathbf{C}_{\left\langle
x\right\rangle}$

${{\mathfrak{M}}^{\mathrm{ext}}}^{\prime}\times\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathfrak{M}}^{\mathrm{ext}}$ (6.12)

where $\mathbf{C}_{\left\langle x\right\rangle}$ parameterizes the location of
$\mathfrak{m}_{f}$.

In the monopole picture of 8.1 such degeneration corresponds to the complete
screening of the point-like non-abelian monopoles by several Dirac monopoles.

Recall that dimensions $\mathbf{v}$ for the theories of class II parametrized
by a single integer $N$ such that $\mathbf{v}_{i}=Na_{i}$ where $a_{i}$ are
Dynkin marks. Therefore, the above inclusion is

${\mathfrak{M}}^{\mathrm{ext}}_{N-1}\times\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathfrak{M}}^{\mathrm{ext}}_{N}$ (6.13)

Geometrically, such inclusion for theories of class II is associated with
freckled (point) instantons described in more details after eq. 6.30.

###  6.2. The cameral curve as a modular object

In this section we give the modular interpretation of the curve
$\mathcal{C}_{u}$.

####  6.2.1. The Class I theories

Let $\mathbf{G}={\mathbf{G}}_{\text{q}}$ be the simple complex Lie group
corresponding to the quiver of the class I theory. Let $Z$ be its center, and
let C$\mathbf{G}$ be the conformal extension of $\mathbf{G}$. Let
${\check{\lambda}}_{i}^{\vee},{\check{\alpha}}_{i}^{\vee}$, $i=1,\ldots,r$ be
the fundamental coweights and the simple coroots in C$\mathfrak{h}\subset$
C$\mathfrak{g}$. Let

$g(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\check{\lambda}}^{\vee}_{i}}{\mathscr{Y}}_{i}(x)^{{\check{\alpha}}_{i}^{\vee}}\in{\rm
C}{\mathbf{T}}\subset{\rm C}{\mathbf{G}}$ (6.14)

In the notations of (C.56):

$g(x)={\mathbf{g}}_{{\mathscr{P}}(x),{\mathscr{Y}}_{1}(x),\ldots,{\mathscr{Y}}_{r}(x)}$

We also use

$g_{\infty}(x)={\mathbf{g}}_{{\mathscr{P}}(x),1,\ldots 1}$

The importance of $g(x)$ is that it transforms by the reflection in the Weyl
group $W(\mathfrak{g})$ when crossing the cuts $I_{i,{\mathbf{a}}}$, cf.
(C.58),

$g_{+}(x)\longrightarrow g_{-}(x)=\ ^{r_{i}}g_{+}(x)$

which implies that for the class I theories the iWeyl group
${{}^{i}{\mathcal{W}}}$ is the Weyl group $W(\mathfrak{g})$ of the
corresponding simple Lie group $\mathbf{G}$. In order to construct the
${{}^{i}{\mathcal{W}}}$-invariants one could take any C$\mathbf{G}$-invariant
function on C$\mathbf{G}$. In fact, cf. (C.56)

${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=g_{\infty}(x)^{-{\lambda}_{i}}\,{\chi}_{i}\left(g(x)\right)$
(6.15)

Using the formulae (LABEL:eq:wmult),(LABEL:eq:poswe) we write:

${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))={\mathscr{Y}}_{i}\sum_{{\nu}=({\nu}_{1},\ldots,{\nu}_{r})}c_{\nu}^{i}\,\prod_{j=1}^{r}\left({\mathscr{P}}_{j}\prod_{k=1}^{r}{\mathscr{Y}}_{k}^{-C_{kj}^{\mathfrak{g_{\text{q}}}}}\right)^{{\nu}_{j}}$
(6.16)

where

$c_{\nu}^{i}={\chi}_{R_{i},{\lambda}_{i}-\sum_{j=1}^{r}{\nu}_{j}{\alpha}_{j}},\qquad{\nu}\in{\mathbb{Z}}_{+}^{r}$
(6.17)

In fact, the sum in (6.16) is finite, i.e. only for a finite number of vectors
$\nu$’s the multiplicity $c_{\nu}^{i}$ is non-zero.

We thus obtain the following geometric picture. The solution of the Class I
theory is a $\mathbf{C}_{\left\langle x\right\rangle}$-parametrized family
$[g(x)]$ of conjugacy classes in C${\mathbf{G}}$, which vary with $x$
polynomially, in the appropriate sense, and such that the value of the
$D$-homomorphism on $[g(x)]$ is fixed for the theory:

$D(g(x))=\left(\prod_{j=1}^{r}{\mathscr{P}}_{j}(x)^{-l_{j{\xi}}}\right)_{{\xi}=1}^{z_{\mathfrak{g}}}$
(6.18)

Actually, as we explain in the section C.1.8, the coweights
${\check{\lambda}}_{i}^{\vee}$ are not uniquely specified. The group element
$g(x)$ in (6.14) defines a well-defined conjugacy class
$[g(x)]\in{B^{\text{ad}}({\mathfrak{g_{\text{q}}}})}$ in
${{\mathbf{G}}_{\text{q}}}/Z$. Its lift to $C\mathbf{G}$ can be twisted by any
C-valued (meromorphic) function of $x$. We shall use this freedom in our
manipulations with spectral curves.

The cameral curve ${\mathcal{C}}_{u}$ can be viewed, geometrically, as the
lift to $C\mathbf{T}$ of the parametrized rational curve in
$C\mathbf{T}/W({\mathfrak{g}})\approx({\mathbb{C}}^{\times})^{z_{\mathfrak{g}}}\times{\mathbb{C}}^{r}$:

$x\mapsto D(g(x))\times\left(T_{1}(x),\ldots,T_{r}(x)\right)$ (6.19)

####  6.2.2. The class II theories

As we mentioned above, the quivers of the class II theories correspond to the
simply laced affine Kac-Moody algebras, i.e.
$\mathfrak{g_{\text{q}}}=\widehat{\mathfrak{g}}$. Let $\widehat{\mathbf{G}}$
be the corresponding Kac-Moody group. Let ${\widehat{\lambda}}_{i}^{\vee}$,
${\widehat{\alpha}}_{i}^{\vee}$ be the corresponding affine coweights and
coroots, $i=0,1,\ldots,r$ (see Appendix C.2). Define:

$g(x)=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}(x)^{{\widehat{\alpha}}_{i}^{\vee}}\in{\widehat{\mathbf{T}}}\subset{\widehat{\mathbf{G}}}$
(6.20)

Again, strictly speaking $g(x)$ takes values in ${\widehat{\mathbf{G}}}/Z$ and
so we should consider the modification of $\widehat{\mathbf{G}}$ corresponding
to the conformal extension $C\widehat{\mathbf{G}}$, but since the subtlety
with the center $Z\subset\mathbf{G}$ only involves the $x$-independent factor

${\widehat{g}}_{\infty}=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}$
(6.21)

it will not affect the $x$-dependence of the invariants. The limit shape
equations, as in the class I case, translate to the jump conditions

$g_{+}(x)\longrightarrow g_{-}(x)=\,^{r_{i}}g_{+}(x)$

for $x\in I_{i,{\mathbf{a}}}$, with $r_{i}$ being the simple reflections
generating the affine Weyl group $W({\widehat{\mathfrak{g}}})$, which is the
${{}^{i}{\mathcal{W}}}$ group for the class II theories.

The invariants of $W({\widehat{\mathfrak{g}}})$ are constructed using the
characters $\widehat{\chi}_{i}$ of the fundamental representations
${\widehat{R}}_{i}$ of $\widehat{\mathbf{G}}$:

${{\mathscr{X}}}_{i}({\mathscr{Y}}(x))=\left({\widehat{g}}_{\infty}^{{\widehat{\lambda}}_{i}}\right)^{-1}\,{\widehat{\chi}}_{i}(g(x))$
(6.22)

They can also be obtained by starting with ${\mathscr{Y}}_{i}(x)$ and
averaging with respect to the $W({\widehat{\mathfrak{g}}})$-action. The
$W({\widehat{\mathfrak{g}}})$-action consists of the translations by the
coroot lattice ${{\rm Q}}^{\vee}$ and the $W({\mathfrak{g}})$-transformations.
The ${{\rm Q}}^{\vee}$-averaging produces the lattice theta-functions of
various characteristics, of the schematic form (the details are given in the
Appendix LABEL:se:lattice-theta):

${\Theta}({\xi},{\mathfrak{q}})=\sum_{{\nu}\in{{\rm
Q}}^{\vee}}{\mathfrak{q}}^{\frac{1}{2}\langle{\nu},{\nu}\rangle}e^{i\langle{\nu},{\xi}\rangle}$
(6.23)

where

${\mathfrak{q}}=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{a_{i}}$ (6.24)

The affine analogue of the formula (6.16) is an infinite sum, however, it is a
power series in ${\mathfrak{q}}$. Using the fact that the weights
$\widehat{\lambda}$ of the fundamental representation ${\widehat{R}}_{i}$