differ from the highest weight $\widehat{\lambda}_{i}$ by a positive linear
combination of simple roots,
${\widehat{\lambda}}={\widehat{\lambda}}_{i}-{\widehat{\nu}}$,

${\widehat{\nu}}=\sum_{j=0}^{r}{\nu}_{j}{\widehat{\alpha}}_{j},\qquad{\nu}_{j}\in{\mathbb{Z}}_{+}$

we can write, with

${\tilde{\mathfrak{q}}}^{\widehat{\nu}}=\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{\nu_{j}}$
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x);{\tilde{\mathfrak{q}}})={\mathscr{Y}}_{i}\sum_{\widehat{\nu}}{\widehat{c}}_{\widehat{\nu}}^{i}{\tilde{\mathfrak{q}}}^{\widehat{\nu}}\,\prod_{k,j=0}^{r}{\mathscr{Y}}_{k}(x)^{-C_{kj}^{\widehat{\mathfrak{g}}}{\nu}_{j}}$
(6.25)

where we made the
${\tilde{\mathfrak{q}}}=({\mathfrak{q}}_{0},\ldots,{\mathfrak{q}}_{r})$
dependence explicit, and

${\widehat{c}}_{\widehat{\nu}}^{i}={\chi}_{{\widehat{R}}_{i},{\widehat{\lambda}}_{i}-{\widehat{\nu}}},\qquad$
(6.26)

Write ${\widehat{\nu}}=n{\delta}+{\nu}$, where $n\in{\mathbb{Z}}_{+}$, and
${\nu}\in{{\rm Q}}$ belongs to the root lattice of $\mathfrak{g}$. Notice that
the factor ${\tilde{\mathfrak{q}}}^{\widehat{\nu}}$ in (6.25) depends on $n$
only via the ${\mathfrak{q}}^{n}$ factor. For fixed $n$ the number of
${\nu}\in{{\rm Q}}$ such that ${\widehat{c}}_{n{\delta}+{\nu}}^{i}\neq 0$ is
finite.

The characters of $\widehat{\mathbf{G}}$ are well-studied [Kac:1984].
Physically they are the torus
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$ conformal
blocks of the WZW theories with the group $G$, and levels $k=a_{i}$,
$i=0,1,\ldots,r$ (see [Dolan:2007eh] for recent developments). The argument of
the characters can be viewed as the background $\mathbf{G}$ $(0,1)$-gauge
field $\bar{\bf A}$, which couples to the holomorphic current ${\bf
J}=g^{-1}{\partial}g$:

$Z_{k}({\tau};{\bar{\bf A}})=\int Dg\,{\exp}\,k\left(S_{\rm
WZW}(g)+\int_{\mathscr{E}}\left\langle{\bf J},{\bar{\bf
A}}\right\rangle\right)=\sum_{\widehat{\lambda}\text{ at level
$k$}}c_{\widehat{\lambda}}\cdot{\widehat{\chi}}_{\widehat{\lambda}}({\widehat{t}};{\mathfrak{q}})$
(6.27)

The background gauge field has only $r$ moduli. In practice, one chooses the
gauge ${\bar{\bf A}}=\frac{\pi}{{\rm Im}{\tau}}{\xi}$, where ${\xi}={\rm
const}\in{\mathfrak{h}}$.

Technically, it is more convenient to built the characters using the free
fermion theory, at least for the $A_{r}$, $D_{r}$ cases, and for the groups
$E_{6},E_{7},E_{8}$ at level $1$. We review this approach in the appendix.

The master equations (6.5)
${{\mathscr{X}}}_{i}({\mathscr{Y}}(x);{\tilde{\mathfrak{q}}})=T_{i}(x)$
describe a curve ${\mathcal{C}}_{u}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times({\mathbb{C}}^{\times})^{r+1}$ which is a
$W({\widehat{\mathfrak{g}}})$-cover of the $x$-parametrized rational curve
$\Sigma_{u}$ in ${\mathbb{C}}^{r+1}={\rm
Spec}{\mathbb{C}}[{\widehat{\chi}}_{0},\ldots,{\widehat{\chi}}_{r}]$, cf.
(6.22):

${\widehat{\chi}}_{i}=\prod_{j}{\mathfrak{q}}_{j}^{-{\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee})}\
T_{i}(x),\qquad i=0,\ldots,r$ (6.28)

Now, as we recall in the section C.5, the characters ${\widehat{\chi}}_{i}$,
$i=0,\ldots,r$ are the sections of the line (orbi)bundle ${\mathcal{O}}(1)$
over the coarse moduli space ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of
holomorphic principal semi-stable $\mathbf{G}$-bundles over the elliptic curve
$\mathscr{E}$. Therefore (6.28),(6.5) define for each $u$ a quasimap $U$ of
the compactified $x$-plane ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ to ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$, which is
actually an honest map near $x=\infty$, whose image approaches the fixed
$\mathbf{G}$-bundle ${\mathcal{P}}_{\tilde{\mathfrak{q}}}$. This bundle can be
described, e.g. by the transition function $g_{\infty}$, which is one of the
$\mathbf{T}$ lifts of

${\tilde{g}}_{\infty}=\prod_{i=1}^{r}{\mathfrak{q}}_{i}^{-{\lambda}^{\vee}_{i}}\in{\mathbf{T}}/Z$
(6.29)

By definition, the local holomorphic sections of
${\mathcal{P}}_{\tilde{\mathfrak{q}}}$ are the ${\mathbf{G}}$-valued functions
${\Psi}(z)$, defined in some domain in ${\mathbb{C}}^{\times}$ such that

${\Psi}({\mathfrak{q}}z)=g_{\infty}{\Psi}(z)$

The complex dimension of the space of quasimaps $U$ with fixed $U({\infty})$
is the number of coefficients in the polynomials
$(T_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ excluding the highest coefficients,
that is (cf. Eq.(4.7)),

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

We say that $U$ is a quasimap, and not just a holomorphic map
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$ for two
reasons. Technically, a collection of ${\widehat{\chi}}_{i}$ in (6.28) defines
a point in $\mathbb{W}\mathbb{P}^{a_{0},a_{1},\ldots,a_{r}}$ only if the
polynomials $T_{i}(x)$ don’t have common weighted factors. If, however, for
some ${\mathfrak{m}}_{f}\in{\mathbf{C}_{\left\langle x\right\rangle}}$:

$T_{i}(x)={\tilde{T}}_{i}(x)(x-{\mathfrak{m}}_{f})^{a_{i}},{\rm for\ all\
}i=0,\ldots,r$ (6.30)

then the map ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}\to{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ is not well-
defined at $x={\mathfrak{m}}_{f}$. It is trivial to extend the map there by
removing the $(x-{\mathfrak{m}}_{f})^{a_{i}}$ factors. This operation lowers
$N\to N-1$. In a way, the point ${\mathfrak{m}}_{f}$ carries a unit of the
instanton charge. Such a configuration is called a freckled instanton
[Losev:1999tu]. Thus, the extended moduli space eq. 4.7 of vacua
${{\mathfrak{M}}^{\mathrm{ext}}}_{N}$ of the gauge theory with
${G_{\text{g}}}=\times_{i}SU(Na_{i})$, contains the locus
${{\mathfrak{M}}^{\mathrm{ext}}}_{N-1}\times{\mathbf{C}_{\left\langle
x\right\rangle}}$. Allowing for several freckles at the unordered points
${\mathfrak{m}}_{f1},{\mathfrak{m}}_{f2},\ldots,{\mathfrak{m}}_{fi}$ we arrive
at the hierarchy of embeddings of the moduli spaces of vacua of the gauge
theories with different gauge groups $G_{\text{g}}$:

$\displaystyle{{\mathfrak{M}}^{\mathrm{ext}}}_{N}=$
$\displaystyle\,\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-1}\times{\mathbf{C}_{\left\langle
x\right\rangle}}\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-2}\times
Sym^{2}{\mathbf{C}_{\left\langle x\right\rangle}}$ (6.31)
$\displaystyle\qquad\qquad\cup\ldots\cup\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N-i}\times
Sym^{i}{\mathbf{C}_{\left\langle x\right\rangle}}\cup\ldots\cup
Sym^{N}{\mathbf{C}_{\left\langle x\right\rangle}}$

where $\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$ stands for the space of
degree $N$ rational maps $U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$.

This hierarchy of gauge theories is more familiar in the context of class I
theories. Presently, the freckled instantons to
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ correspond to the loci in
${\mathfrak{M}}$ where a Higgs branch of the gauge theory can open. Indeed, if
(6.30) holds, then we can solve the master equation (6.5) by writing

${\mathscr{Y}}_{j}(x)=(x-{\mathfrak{m}}_{f})^{a_{j}}\
{\tilde{\mathscr{Y}}}_{j}(x)$ (6.32)

with ${\tilde{\mathscr{Y}}}_{j}(x)$ solving the master equation (6.5) of the

$\times_{i\in\mathrm{Vert}_{\gamma}}\ SU\left(\left({N-1}\right)a_{i}\right)$

gauge theory. In the IIB string theory picture A.3 the full collection of
fractional branes in the amount of $a_{i}$ for the $i$’th type recombine, and
detach themselves from the fixed locus, moving away at the position
${\mathfrak{m}}_{f}$ on the transverse
${\mathbb{R}}^{2}={\mathbf{C}_{\left\langle x\right\rangle}}$.

Now let us take $u\in\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$. The
corresponding map $U$ defines a rational curve $\Sigma_{u}$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of degree $N$.

######  Remark.

Actually, there is another compactification of
$\mathring{{\mathfrak{M}}^{\mathrm{ext}}}_{N}$, via genus zero Kontsevich
stable maps of bi-degree $(1,N)$ to
${\mathbb{C}\mathbb{P}}^{1}\times{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$
(see [Givental:1997] where the space of quasimaps is called the toric map
spaces). It would be interesting to study its gauge theoretic meaning.

######  Remark.

The highest order coefficients $T_{i,0}({\tilde{\mathfrak{q}}})$ of the
polynomials $T_{i}(x)$ depend only on the gauge coupling constants, and
determine the limit $U(x)$, $x\to\infty$

$U({\infty})=[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]\in{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$
(6.33)

The next-to-leading terms $T_{i,1}({\tilde{\mathfrak{q}}},m)$ depend only on
the gauge couplings and the bi-fundamental masses. These define the first jet
${\mathscr{T}}_{[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]}{\Sigma}_{u}$ of the
rational curve $\Sigma_{u}$ at $x=\infty$.

Summarizing, _the moduli space ${{\mathfrak{M}}}_{N}$ of vacua of the class II
theory with the gauge group_

${G_{\text{g}}}=\times_{i\in\mathrm{Vert}_{\gamma}}\ SU(Na_{i})$

_is the moduli space of degree $N$ finely framed at infinity quasimaps_

$U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}\approx{\mathbb{W}\mathbb{P}}^{a_{0},a_{1},\ldots,a_{r}}$
(6.34)

_where the fine framing is the condition that $U$ is the honest map near
$x=\infty$, and the first jet (the value and the tangent vector) at $x=\infty$
are fixed:_

$\left(U({\infty}),U^{\prime}({\infty})\right)\leftrightarrow\left({\tilde{\mathfrak{q}}},m\right)$
(6.35)

_We also have the identification of the extended moduli space
${{\mathfrak{M}}^{\mathrm{ext}}}$ with the space of framed quasimaps_

####  6.2.3. The class II* theories

The theories with the affine quiver of the ${\widehat{A}}_{r}$ type can be
solved uniformly in both class II and class II* cases. This is related to the
fact that the current algebra ${\widehat{u(r+1)}}$, the affine Kac-Moody
algebra based on $U(r+1)$ is a subalagebra of
$\widehat{\mathfrak{gl}}_{\infty}$, consisting of the $(r+1)$-periodic
infinite matrices.

Let $\gamma$ be the affine Dynkin graph of the ${\widehat{A}}_{r}$ algebra. We
have, $\mathrm{Vert}_{\gamma}=\mathrm{Edge}_{\gamma}=\\{0,1,\ldots,r\\}$.
Choose such an orientation of the graph $\gamma$ that for any
$e\in\mathrm{Edge}_{\gamma}$: $s(e)=e$, $t(e)=(e+1)$ mod $r+1$. Let $m_{e}$,
$e=0,\ldots,r$ be the corresponding bi-fundamental multiplet masses, and

${\mathfrak{m}}=\sum_{e=0}^{r}m_{e}$ (6.36)

We are in the class II* theory iff $\mathfrak{m}\neq 0$.

It is convenient to extend the definition of $m_{e}$ to the universal cover of
$\gamma$. Thus, we define

$\displaystyle{\mathfrak{m}}_{i}=m_{i\,\mathrm{mod}\,(r+1)},$ (6.37)
$\displaystyle
Y_{i}(x)={\mathscr{Y}}_{i\,\mathrm{mod}\,(r+1)}(x-{\mathfrak{m}}_{(i)})\
,\qquad i\in\mathbb{Z}$

The extended amplitudes $Y_{i}(x)$ obey

$Y_{i+r+1}(x)=Y_{i}\left(x-{\mathfrak{m}}\right)$ (6.38)

Define

$t_{j}(x)={\check{t}}_{j}\,\frac{Y_{j}(x)}{Y_{j-1}(x)}$ (6.39)

where

$\displaystyle{\check{t}}_{j+1}={\mathfrak{q}}_{j\,\mathrm{mod}\,(r+1)}\,{\check{t}}_{j}$
(6.40) $\displaystyle\prod_{j=0}^{r}{\check{t}}_{j}=1\,,\ $
$\displaystyle{\check{t}}_{j+r+1}={\mathfrak{q}}\,{\check{t}}_{j}$

Then

$t_{j+r+1}(x)={\mathfrak{q}}\,t_{j}(x-{\mathfrak{m}})$

where for the $\widehat{A}_{r}$-series,

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

Now, consider the following element of $\widehat{GL}_{\infty}$:

$g(x)={\mathscr{Y}}_{0}(x)^{K}\times\prod_{i\in\mathbb{Z}}t_{i}(x)^{E_{i,i}}$
(6.41)

with $t_{i}(x)$ from (6.39), and $E_{i,j}$ denoting the matrix with all
entries zero except $1$ at the $i$’th row and $j$’th column. A closer
inspection shows (6.41) is the direct generalization of (6.20) with the
$r+1$-periodic matrix $g_{\infty}$, and
$({\mathscr{Y}}_{i}(x))_{i\in\mathrm{Vert}_{\gamma}}$ replaced by the infinite
array $(Y_{i}(x))_{i\in\mathbb{Z}}$. Indeed, the simple coroots of
$\widehat{GL}_{\infty}$ are the diagonal matrices, shifted in the central
direction

${\alpha}_{i}^{\vee}=K{\delta}_{i,0}+E_{i,i}-E_{i+1,i+1},\qquad
i\in\mathbb{Z}$ (6.42)

so that the analogue of (C.64) holds

$K=\sum_{i\in\mathbb{Z}}{\alpha}_{i}^{\vee}$

if we drop the telescopic sum $\sum_{i\in\mathbb{Z}}E_{i,i}-E_{i+1,i+1}\sim
0$.

We do not need to deal with all the coweights of $\widehat{GL}_{\infty}$, only
with the $r+1$-periodic ones, defined via:

$\prod_{j=0}^{r}{\mathfrak{q}}_{j}^{-{\tilde{\lambda}}_{j}^{\vee}}=\prod_{b\in\mathbb{Z}}\prod_{j=1}^{r+1}\left({\mathfrak{q}}^{b}{\check{t}}_{j}\right)^{E_{i+b(r+1),i+b(r+1)}}$

These coweights are the coweights of the ${\widehat{A}}_{r}$ Kac-Moody
algebra, embedded into $\mathfrak{gl}_{\infty}$ as the subalgebra of
$r+1$-periodic matrices

$\sum_{i,j\in\mathbb{Z}}a_{i,j}E_{i,j},\qquad a_{i+r+1,j+r+1}=a_{i,j}$

We shall describe the solution of this theory in detail in the next section.

###  6.3. Spectral curves

The cameral curve captures all the information about the limit shape, the
special coordinates, the vevs of the chiral operators, and the prepotential.
Its definition is universal.

However, the cameral curve is not very convenient to work with. In many cases
one can extract the same information from a ‘‘smaller’’ curve, the so-called
_spectral curve_. In fact, there are several notions of the spectral curve in
the literature.

Suppose ${\lambda}\in{\rm
Hom}(({\mathbb{C}}^{\times})^{\mathrm{Vert}_{\gamma}},{\mathbb{C}}^{\times})$
is a dominant weight, i.e. ${\lambda}({\alpha}_{i}^{\vee})\geq 0$ for all
$i\in\mathrm{Vert}_{\gamma}$. Let $R_{\lambda}$ be the irreducible highest
weight module of ${\mathbf{G}}_{\text{q}}$ with the highest weight $\lambda$,
and ${\pi}_{\lambda}:{{\mathbf{G}}_{\text{q}}}\longrightarrow{\rm
End}(R_{\lambda})$ the corresponding homomorphism. Then the spectral curve
$C^{R_{\lambda}}_{u}$ in $\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbf{C}_{\left\langle t\right\rangle}$ is

${\det}_{R_{\lambda}}\left(1-t^{-1}{\zeta}(x)^{-1}{\pi}_{\lambda}(g(x))\right)=0$
(6.43)

where

  1. (1)

for the class I theories we introduce the factor

$\zeta(x)=g_{\infty}(x)^{\lambda}\times{\rm\ a\ rational\ function\ of\ }x\,,$

having to do with the lift of the conjugacy class $[g(x)]$ from
${{\mathbf{G}}^{\text{ad}}}$ to ${\rm C}{\mathbf{G}}$. The rational function
is chosen so as to minimize the degree of the curve $C^{R_{\lambda}}_{u}$, as
we explain in the examples below.

  2. (2)

for the class II, II* theories the factor $\zeta(x)$ is a constant.

Generally, the curve $C^{R_{\lambda}}_{u}$ defined by (6.43) is not
irreducible. The equation (6.43) factorizes into a product of components, one
component for each Weyl orbit in the set of weights $\Lambda_{R_{\lambda}}$
for the module $R_{\lambda}$. Each Weyl orbit intersects dominant chamber at
one point and therefore can be parametrized by dominant weights $\mu$.
Therefore

$C^{R_{\lambda}}_{u}=\bigcup_{\mu\in\Lambda_{R_{\lambda}}\cap\Lambda^{+}}{\mathrm{mult}(\lambda:\mu)}\cdot\left(C_{u}^{\mu}\right)$
(6.44)

where $\mathrm{mult}(\lambda:\mu)$ denotes multiplicity of weight $\mu$ in the
module $R_{\lambda}$. If $R_{\lambda}$ is minuscule module, then, by
definition, the curve $C^{R_{\lambda}}_{u}$ is irreducible.

######  Example.

Consider the $A_{1}$ theory and take $\lambda=3\lambda_{1}$, i.e. the spin
$\frac{3}{2}$ representation. If
$T_{1}(x)=\operatorname{tr}_{R_{1}}g(x)=t(x)+t(x)^{-1}$ one finds that

$\displaystyle C^{R_{\lambda_{1}}}:$ $\displaystyle\ 0=1-T_{1}(x)t+t^{2}$
(6.45) $\displaystyle C^{R_{3\lambda_{1}}}:$ $\displaystyle\
0=(1-T_{1}(x)t+t^{2})(1+3T_{1}(x)t-T_{1}(x)^{3}t+t^{2})$

Let ${{}^{i}{\mathcal{W}}}_{\mu}\subset{{}^{i}{\mathcal{W}}}$ be the
stabilizer of $\mu$ in ${{}^{i}{\mathcal{W}}}$, a subgroup of
${{}^{i}{\mathcal{W}}}$. Consider the map:

$p_{\mu}:{\mathbf{C}_{\left\langle
x\right\rangle}}\times\left({\mathbb{C}}^{\times}\right)^{\mathrm{Vert}_{\gamma}}\longrightarrow{\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}$

given by:

$\displaystyle
p_{\mu}:(x,({\mathscr{Y}}_{i})_{i\in\mathrm{Vert}_{\gamma}})\mapsto$
$\displaystyle\,(x,t(x)),$ (6.46) $\displaystyle\qquad
t(x)=g(x)^{\mu}/g_{\infty}(x)^{\mu}=\prod_{i\in\mathrm{Vert}_{\gamma}}{\mathscr{Y}}_{i}^{{\mu}({\alpha}_{i}^{\vee})}$

Under the map $p_{\mu}$ the curve ${\mathcal{C}}_{u}$ maps to
$C_{u}^{\mu}={\mathcal{C}}_{u}/{{{}^{i}{\mathcal{W}}}}_{\mu}\subset{\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}$, the
irreducible $\mu$-component of the spectral curve. This curve comes with the
canonical differential, which is the restriction of the differential on
${\mathbf{C}_{\left\langle x\right\rangle}}\times{\mathbf{C}_{\left\langle
t\right\rangle}}^{\times}$:

$dS=x\frac{dt}{t}$ (6.47)

Actually, in the case of the class II, II* theories the commonly used notion
of the spectral curve differs from the one in (6.43).

Although we suspect the study of spectral curves associated with the
integrable highest weight representations of affine Kac-Moody algebras may be
quite interesting, in this paper for the analysis of the class II and II*
theories we use the conventional notion of the spectral curve used for the
study of families of $\mathbf{G}$-bundles.

To define it, let us fix an irreducible representation $R$ of $\mathbf{G}$,
${\pi}_{R}:{\mathbf{G}}\to{\rm End}(R)$, and let us study the theory of a
complex chiral fermion valued in $R$, more precisely, an $(1,0)$ $bc$ system
in the representations $(R^{*},R)$:

${\mathscr{L}}_{bc}=\sum_{i=1}^{{\rm dim}R}\int b_{i}{\bar{\partial}}c^{i}$
(6.48)

coupled to a background ${\mathbf{G}}\times{\mathbb{C}}^{\times}$ gauge field
$\bar{\bf A}\oplus\bar{A}$, and compute its partition function on the torus
$\mathscr{E}$:

$Z({\bf t},t,q)={\operatorname{Tr}}_{{\mathcal{H}}_{R}}\left((-t)^{J}_{0}{\bf
t}^{{\bf J}_{0}}q^{L_{0}}\right)$ (6.49)

Mathematically, we consider the space

$H_{R}=R[z,z^{-1}]=H_{R}^{+}\oplus H_{R}^{-}$ (6.50)

of $R$-valued functions on the circle ${\mathbb{S}}^{1}$. In (6.50) we took
Laurent polynomials in $z\in{\mathbb{C}}^{\times}$, which correspond to
Fourier polynomials on the circle. We may take some completion of $H_{R}$ but
we shall not do this in the definition of the spectral determinant below.
Consider an element ${\widehat{g}}\in{\widehat{\mathbf{G}}}$ of the affine
Kac-Moody group, i.e. the central extension of
${\widetilde{L\mathbf{G}}}=L{\mathbf{G}}\ltimes{\mathbb{C}}^{\times}$, the
loop group $L{\mathbf{G}}$ extended by the $\mathbb{C}^{\times}$ acting by
loop rotations. We have the canonical homomorphism-projection
$f:{\widehat{\mathbf{G}}}\longrightarrow{\widetilde{L\mathbf{G}}}$ with the
fiber ${\mathbb{C}}^{\times}$, the center of the central extension:

$f:{\widehat{g}}\mapsto g(z)q^{z{\partial}_{z}}$ (6.51)

The projection is topologically non-trivial.

Now, ${\widetilde{L\mathbf{G}}}$ acts in $H_{R}$ via rotation and evaluation,
and so does $\widehat{\mathbf{G}}$ thanks to (6.51) : for ${\Psi}\in H_{R}$:

$\left(f({\widehat{g}})\cdot{\Psi}\right)(z)={\pi}_{R}(g(z))\cdot{\Psi}(qz)$
(6.52)

We would like to define the spectral determinant of $f({\widehat{g}})$ in the
representation $H_{R}$. The eigenvalues of $f({\widehat{g}})$ are easy to
compute

${\rm Eigen}(f({\widehat{g}}))=\\{\,{\bf t}^{\mu}q^{n}\,|\
{\mu}\in{\Lambda}_{R},\,n\in\mathbb{Z}\,\\}$ (6.53)

where we transformed $g(z)$ to a constant ${\bf t}\in\mathbf{T}$ by means of a
$z$-dependent $\mathbf{G}$-gauge transformation:

$g(z)\mapsto h^{-1}(z)g(z)h(qz)={\bf t}$ (6.54)

The fibration $f:{\widehat{\mathbf{G}}}\to{\widetilde{L{\mathbf{G}}}}$,
restricted onto
${\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}\subset{\widetilde{L{\mathbf{G}}}}$
becomes trivial,
$f^{-1}\left({\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}\right)\approx{\mathbb{C}}^{\times}_{c}\times{\mathbb{C}}^{\times}_{q}\times{\mathbf{T}}$.
Let us denote by $c$ the coordinate on the first factor.

The eigenvalues (6.53) concentrate both near $0$ and $\infty$, so we define:

$\displaystyle{\det}_{H_{R}}\,\left(1-t^{-1}{\widehat{g}}\right):={\det}_{H_{R}^{+}}(1-t^{-1}{\widehat{g}}){\det}_{H_{R}^{-}}(1-t{\widehat{g}}^{-1})=$
(6.55) $\displaystyle\qquad\qquad
c^{{\kappa}_{R}}\prod_{{\mu}\in{\Lambda}_{R}}\prod_{n=0}^{\infty}\left(1-q^{n}t^{-1}{\bf
t}^{\mu}\right)\left(1-q^{n+1}t\,{\bf t}^{-{\mu}}\right)$

The expression (6.55) is $W({\widehat{\mathfrak{g}}})$-invariant. The shifts
by ${\rm Q}$ act as follows, cf. (C.104):

$({\bf t},c)\mapsto\left(q^{\beta}\cdot{\bf t},\,{\bf
t}^{\beta}q^{\frac{1}{2}\left\langle\beta,\beta\right\rangle}\cdot c\right)$
(6.56)

where we view $\beta\in{\rm Q}$ both as a vector in the root lattice and as a
vector in the coroot lattice, and $\left\langle,\right\rangle$ is the Killing
metric. The level ${\kappa}_{R}$ in (6.55) is defined as follows:

$\sum_{{\mu}\in{\Lambda}_{R}}{\mu}\left\langle\mu,\beta\right\rangle={\kappa}_{R}{\beta}$
(6.57)

for any vector $\beta\in{\rm Q}$. Geometrically the spectral curve
corresponding to $R$ is obtained as follows: consider the universal principal
$\mathbf{G}$-bundle ${\mathcal{U}}$ over
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})\times\mathscr{E}$, and associate
the vector bundle $\mathscr{R}$ with the fiber $R$:

${\mathscr{R}}={\mathcal{U}}\times_{\mathbf{G}}R$

Now restrict it onto the rational curve
$\Sigma_{u}\subset{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$. We get the
$R$-bundle over $\Sigma_{u}\times{\mathscr{E}}$.

For generic point $x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ over the corresponding point $U(x)\in\Sigma_{u}$ we get the
vector bundle ${\mathscr{R}}_{x}$ over $\mathscr{E}$, which is semi-stable,
and splits as a direct sum of line bundles

${\mathscr{R}}_{x}=\bigoplus_{{\mu}\in{\Lambda}_{R}}{\mathscr{L}}_{{\mu},x}$
(6.58)

where the summands are the degree zero line bundles on $\mathscr{E}$. Under
the identification $Pic_{0}({\mathscr{E}})$ with $\mathscr{E}$ the line bundle
${\mathscr{L}}_{{\mu},x}$ corresponds to the point ${\bf t}(x)^{\mu}$
mod$\,{\mathfrak{q}}^{\mathbb{Z}}$ for some ${\bf
t}(x)\in{\mathbf{T}}/{\mathfrak{q}}^{{{\rm Q}}^{\vee}}$. The closure of the
union

$\bigcup_{x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}}\,\left\\{\ {\bf t}(x)^{\mu}\ |\ {\mu}\in{\Lambda}_{R}\
\right\\}\ \subset{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$ (6.59)

is the spectral curve
$C^{R}_{u}\subset{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$. It is given by the vanishing locus of
the regularized determinant (6.55):

$c(x)^{{\kappa}_{R}}\prod_{{\mu}\in{\Lambda}_{R}}{\theta}(t^{-1}{\bf
t}(x)^{\mu};q)=0$ (6.60)

the choice of the $x$-dependence of $c(x)$ seems immaterial at this point, as
long as $c(x)\in{\mathbb{C}}^{\times}$.

##### Degree of the spectral curve

The $x$-degree of the spectral curve for class II theories in representation
$R$ is $N\kappa_{R}$ where $\kappa_{R}$ is given by (6.57). The $\kappa_{R}$
is the proportionality constant for the second Casimir in representation $R$
$\operatorname{tr}_{R}(\cdot,\cdot)=\kappa_{R}(\cdot,\cdot)_{2}$ where the
$(\cdot,\cdot)_{2}$ is the canonical Killing form in which the long roots have
length square equal to $2$. The standard computations leads to

$\kappa_{R}=\frac{\dim_{R}}{\dim_{\mathfrak{g}}}(\lambda_{R},\lambda_{R}+2\rho)_{2}$
(6.61)

where $\rho=\frac{1}{2}\sum_{\alpha>0}\alpha$ is the Weyl vector. For
fundamental representations $R_{1}$ we find for all cases

$\displaystyle\kappa_{R_{1}}(A_{r})=1$ (6.62)
$\displaystyle\kappa_{R_{1}}(D_{r})=2$ $\displaystyle\kappa_{R_{1}}(E_{6})=6$
$\displaystyle\kappa_{R_{1}}(E_{7})=12$
$\displaystyle\kappa_{R_{1}}(E_{8})=60$

###  6.4. Obscured curve

In the previous construction, in view of the identification
${\mathscr{L}}_{{\mu},x}\leftrightarrow{\bf t}(x)^{\mu}$ we can decompose, for
each weight

${\mu}=\sum_{i=1}^{r}{\mu}_{i}{\lambda}_{i}\in{\Lambda}_{R}$
${\mathscr{L}}_{{\mu},x}=\bigotimes_{i=1}^{r}{\mathbb{L}}_{i,x}^{\otimes{\mu}_{i}}$

for some "basic" line bundles ${\mathbb{L}}_{i,x}$ corresponding to the
fundamental weights. These basic line bundles are ordered, so they define a
point

$\\{\ {\mathbb{L}}_{1,x},\ldots,{\mathbb{L}}_{r,x}\ \\}\in
Pic_{0}({\mathscr{E}})^{r}\approx{\mathscr{E}}^{r}\ ,$

the Cartesian product of $r$ copies of the elliptic curve. Taking the whole
family and including the parametrization we obtain the _obscured curve_
${\mathscr{C}}_{u}$:

${\mathscr{C}}_{u}=\\{\ (x;{\mathbb{L}}_{1,x},\ldots,{\mathbb{L}}_{r,x})\ |\
x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}\
\\}\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}^{r}$ (6.63)

Let us present another simple construction of ${\mathscr{C}}_{u}$. Namely, let
us use the fact [Friedman:1997yq, Friedman:1997ih, Donagi:1997dp,
Friedman:1998si, Friedman:2000ze], that

${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}=({\mathscr{E}}\otimes{{\rm
Q}})/W({\mathfrak{g}})$ (6.64)

where the tensor product is understood in the category of abelian groups. At
the level of manifolds, (6.64) simply says that

${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}={\mathscr{E}}^{r}/W({\mathfrak{g}})$
(6.65)

for some natural action of the Weyl group $W({\mathfrak{g}})$ on the Cartesian
product of $r$ copies of $\mathscr{E}$. Let us denote by ${\pi}_{W}$ the
projection

${\pi}_{W}:{\mathscr{E}}^{r}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}={\mathscr{E}}^{r}/W({\mathfrak{g}})$
(6.66)

The rational curve $\Sigma_{u}$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ lifts to a curve in
${\mathscr{E}}^{r}$, and the graph of the parametrized curve
$\Sigma_{u}\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$ lifts to
the graph in ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}^{r}$ which is our friend _obscured curve_
${\mathscr{C}}_{u}$. It is the quotient of the cameral curve by the lattice
${\rm Q}^{\vee}$:

${\mathscr{C}}_{u}={\mathcal{C}}_{u}/{{\rm Q}}^{\vee}$ (6.67)

In the section 8.2 we shall present yet another construction of
${\mathbb{L}}_{i,x}$’s, using gauge theory.

There is the so-called determinant line bundle $L$ over the moduli space
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$, whose sections are the
fundamental characters $\widehat{\chi}_{i}$, $i=0,1,\ldots,r$. In the E.
Loojienga’s identification
${{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}\approx{\mathbb{W}\mathbb{P}}^{a_{0},a_{1},\ldots,a_{r}}$
this line bundle is just the ${\mathcal{O}}(1)$ orbibundle over the weighted
projective space.

We have then the line bundle ${\mathscr{L}}$ over ${\mathscr{E}}^{r}$:

${\mathscr{L}}={\pi}_{W}^{*}L$ (6.68)

Let us call this line bundle _the abelianized determinant line bundle_.

##  Chapter 7 The Seiberg-Witten curves in some detail

In this section we shall discuss the geometry of curves describing the limit
shape configurations and the special geometry of the gauge theories under
consideration. When possible we identify the cameral or the spectral curves
with the analogous curves of some algebraic integrable systems, namely the
Hitchin systems on the genus zero (i.e. Gaudin model) or genus one (i.e. spin
elliptic Calogero-Moser system) curves with punctures. These identifications
are less universal then the identification with the spectral curves of the
spin chains based on the Yangian algebra built on $\mathfrak{g}$,
$\widehat{\mathfrak{g}}$, or $\widehat{GL}_{\infty}$, respectively. The latter
identification is a subject of a separate venue of research which touches upon
various advances in geometric representation theory, study of the symplectic
geometry of moduli spaces of instantons and monopoles, quantum cohomology of
quiver varieties, to name just a few. We shall only mention the relation to
spin chains in a few examples, in this work.

Throughout this section we shall use the notation

$g_{\lambda}(x)={\zeta}(x)^{-1}{\pi}_{\lambda}(g(x))$ (7.1)

for the projectively modified operator in the representation
$(R_{\lambda},{\pi}_{\lambda})$ of ${\mathbf{G}}_{\text{q}}$, corresponding to
the group element $g(x)\in{\mathbf{G}}_{\text{q}}$.

###  7.1. Class I theories of $A$ type

This is the so-called linear quiver theory. The set of vertices
$\mathrm{Vert}_{\gamma}=\\{1,\ldots,r\\}$, the set of edges
$\mathrm{Edge}_{\gamma}=\\{1,\ldots,r-1\\}$, the maps $s,t$ for a particular
orientation are given by: $s(e)=e$, $t(e)=e+1$. The bi-fundamental masses are
a trivial cocycle:

$m_{e}={\mu}_{e+1}-{\mu}_{e}$ (7.2)

The corresponding conformal group $C\mathbf{G}=GL(r+1,{\mathbb{C}})$, the
fundamental characters ${\chi}_{i}$ are the characters of the representations
${\Lambda}^{i}{\mathbb{C}}^{r+1}$. We shall now describe the spectral curve in
the representation $R_{\lambda_{1}}\approx{\mathbb{C}}^{r+1}$. The
corresponding group element $g_{\lambda_{1}}(x)$ in (6.14) is the diagonal
matrix

$g_{\lambda_{1}}(x)={\rm diag}(t_{1}(x),\ldots,t_{r+1}(x))$

with

$\displaystyle t_{1}(x)={\zeta}(x){\mathscr{Y}}_{1}(x),\quad
t_{r+1}(x)={\zeta}(x){\mathscr{P}}^{[r]}(x)\,{\mathscr{Y}}_{r}(x)^{-1}$ (7.3)
$\displaystyle\quad
t_{i}(x)={\zeta}(x){\mathscr{P}}^{[i-1]}(x)\,{\mathscr{Y}}_{i}(x){\mathscr{Y}}_{i-1}(x)^{-1},\qquad
i=2,\ldots,r$

with some normalization factor ${\zeta}(x)$ which we choose shortly, and the
explicit formula for the invariants ${\mathcal{X}}_{i}({\mathscr{Y}}(x))$ is
(we omit the $x$-dependence in the right hand side):

$\displaystyle{\mathcal{X}}_{i}({\mathscr{Y}}(x))=\prod_{j=1}^{i-1}{\mathscr{P}}_{j}^{j-i}\times$
(7.4) $\displaystyle\qquad
e_{i}\left({\mathscr{Y}}_{1},{\mathscr{Y}}_{2}{\mathscr{Y}}_{1}^{-1}{\mathscr{P}}^{[1]},\ldots,{\mathscr{Y}}_{i}{\mathscr{Y}}_{i-1}^{-1}{\mathscr{P}}^{[i-1]},\ldots,{\mathscr{Y}}_{r}^{-1}{\mathscr{P}}^{[r]}\right)$

where $e_{i}$ are the elementary symmetric polynomials in $r+1$ variables. Our
master equations (6.5) equate the right hand side of (7.4) with the degree
${\mathbf{v}}_{i}$ polynomial $T_{i}(x)$ in $x$, cf. (6.6).

It is convenient to organize the invariants (7.4) into a generating
polynomial, which is nothing but the characteristic polynomial of the group
element $g(x)$ in some representation of $C{\mathbf{G}}$. The most economical
is, of course, the defining fundamental representation ${\mathbb{C}}^{r+1}$
with the highest weight ${\lambda}_{1}$:

$\displaystyle{\det}\left(t\cdot 1_{r+1}-g_{\lambda_{1}}(x)\right)=$ (7.5)
$\displaystyle\qquad
t^{r+1}+\sum_{i=1}^{r}(-1)^{i}t^{r+1-i}{\zeta}(x)^{i}\prod_{j=1}^{i-1}{\mathscr{P}}_{j}^{i-j}(x)\,{\mathcal{X}}_{i}({\mathscr{Y}}(x))$
$\displaystyle\qquad\qquad\qquad+(-{\zeta}(x))^{r+1}\prod_{j=1}^{r}{\mathscr{P}}_{j}^{r+1-j}(x)$

The group ${{}^{i}{\mathcal{W}}}$ is the symmetric group
${\mathcal{S}}_{r+1}$, which acts by permuting the eigenvalues of $g(x)$ in
(7.3). The cameral curve ${\mathcal{C}}_{u}$ is the $(r+1)!$-fold ramified
cover of the compactified $x$-plane ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}$. The points in the fiber are the _ordered_ sets of roots
$(t_{1}(x),\ldots,t_{r+1}(x))$ of the polynomial (7.5).

The curve ${\mathcal{C}}_{u}$ covers the _spectral curve_ $C_{u}$. The latter
is defined as the zero locus of the characteristic polynomial (7.5). The cover
${\mathcal{C}}_{u}\to C_{u}$ is $r!:1$, it sends the ordered $r+1$-tuple of
roots $(t_{1},\ldots,t_{r+1})$ to the first root $t_{1}$. The cover
$C_{u}\to\mathbf{C}_{\left\langle x\right\rangle}$ is $(r+1):1$.

Explicitly, the curve $C_{u}$ is given by:

$0={\mathcal{P}}(t,x)=\sum_{i=0}^{r+1}(-1)^{i}t^{r+1-i}{\zeta}(x)^{i}\,{\prod_{j=1}^{i-1}{\mathscr{P}}_{j}(x)^{i-j}}\
T_{i}(x)$ (7.6)

####  7.1.1. Relation to Gaudin model

Figure 7.1. Degree profile example for $A_{4}$ theory and
$(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4})=(4,7,8,5)$. For
convenience one can set boundary conditions
$\mathbf{v}_{0}=\mathbf{v}_{r+1}=\mathbf{w}_{0}=\mathbf{w}_{r+1}=0$.

It is easy to see, using the Eqs. (3.8), (3.9) and Fig. 7.1 that
${\mathbf{w}}_{i_{*}}=w_{+}+w_{-}$, where

$\displaystyle w_{+}={\mathbf{v}}_{i_{*}}-{\mathbf{v}}_{i_{*}+1}\geq 0$ (7.7)
$\displaystyle w_{-}={\mathbf{v}}_{i_{*}}-{\mathbf{v}}_{i_{*}-1}\geq 0$

and it useful to record

$\displaystyle\mathbf{v}_{1}$
$\displaystyle=\mathbf{w}_{1}+\dots+\mathbf{w}_{i_{*}-1}+\mathbf{w}_{-}$ (7.8)
$\displaystyle\mathbf{v}_{r}$
$\displaystyle=\mathbf{w}_{r}+\dots+\mathbf{w}_{i_{*}+1}+\mathbf{w}_{+}$
$\displaystyle\mathbf{v}_{*}$
$\displaystyle=\sum_{i=1}^{i_{*}-1}i\mathbf{w}_{i}+i_{*}\mathbf{w}_{-}$
$\displaystyle\mathbf{v}_{*}$
$\displaystyle=\sum_{i=i_{*}+1}^{r}(r+1-i)\mathbf{w}_{i}+(r+1-i_{*})\mathbf{w}_{+}$

Accordingly, we can factorize the polynomial ${\mathscr{P}}_{i_{*}}(x)$ as:

${\mathscr{P}}_{i_{*}}(x)={\mathfrak{q}}_{i_{*}}{\mathscr{P}}^{+}(x){\mathscr{P}}^{-}(x)$
(7.9)

where ${\mathscr{P}}^{\pm}$ are monic polynomials of degrees

${\rm deg}{\mathscr{P}}^{\pm}=w_{\pm}$

We can actually transform (7.6) into something nice, by adjusting
${\zeta}(x)$:

${\zeta}(x)^{-1}={\mathscr{P}}^{-}(x){\mathscr{P}}^{[i_{*}-1]}(x)$ (7.10)

Then $D(g(x))$ is given by:

$\displaystyle D(g(x))=\frac{P_{0}(x)}{P_{\infty}(x)}$ (7.11) $\displaystyle
P_{0}(x)={\mathscr{P}}^{+}(x)^{r+1-i_{*}}\prod_{j=i_{*}+1}^{r}{\mathscr{P}}_{j}(x)^{r+1-j}$
$\displaystyle
P_{\infty}(x)={\mathscr{P}}^{-}(x)^{i_{*}}\prod_{j=1}^{i_{*}-1}{\mathscr{P}}_{j}(x)^{j}$

Then ${\mathcal{P}}(t,x)$ can be written as:

${\mathcal{P}}(t,x)=\prod_{j=1}^{i_{*}-1}{\mathfrak{q}}_{j}^{j}\cdot\frac{P(t,x)}{P_{\infty}(x)}$

where $P(t,x)$ is a degree $N={\mathbf{v}}_{i_{*}}$ polynomial in $x$, and the
degree $r+1$ polynomial in $t$, which is straightforward to calculate:

$\displaystyle(-1)^{i_{*}}\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}^{j}\ P(t,x)=$
(7.12)
$\displaystyle\qquad(-{\mathfrak{q}}_{i_{*}})^{i_{*}}t^{r+1}P_{\infty}(x)\,+$
$\displaystyle\qquad\qquad+\sum_{i=1}^{i_{*}}t^{r+1-i}\,T_{i}(x){\mathfrak{q}}_{i_{*}}^{i_{*}-i}(-{\mathscr{P}}_{*}^{-}(x))^{i_{*}-i}\prod_{j=i}^{i_{*}-1}{\mathscr{P}}_{j}^{j-i}(x)$
$\displaystyle\qquad\qquad\qquad+\sum_{i=i_{*}+1}^{r}t^{r+1-i}\,T_{i}(x)(-{\mathscr{P}}_{*}^{+}(x))^{i-i_{*}}\prod_{j=i_{*}+1}^{i-1}{\mathscr{P}}_{j}^{i-j}(x)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(-1)^{r+1-i_{*}}P_{0}(x)$

Now, recall that $T_{j,0}$ is fixed by the couplings ${\mathfrak{q}}$:

$T_{j,0}({\mathfrak{q}})=\prod_{j=1}^{i-1}{\mathfrak{q}}_{j}^{j-i}\,e_{i}(1,{\mathfrak{q}}_{1},{\mathfrak{q}}_{1}{\mathfrak{q}}_{2},\ldots,{\mathfrak{q}}_{1}{\mathfrak{q}}_{2}\ldots{\mathfrak{q}}_{i},\ldots{\mathfrak{q}}_{1}\ldots{\mathfrak{q}}_{r})$
(7.13)

and the coefficient $T_{j,1}$ is fixed by the masses $m_{i,{\mathfrak{f}}}$
and $m_{e}$.

Therefore, the coefficient of $x^{N}$ in $P(t,x)$ can be computed explicitly:

$\displaystyle\sum_{i=0}^{r+1}(-1)^{i}t^{r+1-i}\,\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}^{-i}\,e_{i}(1,{\mathfrak{q}}_{1},{\mathfrak{q}}_{1}{\mathfrak{q}}_{2},\ldots,{\mathfrak{q}}_{1}{\mathfrak{q}}_{2}\ldots{\mathfrak{q}}_{i},\ldots,{\mathfrak{q}}_{1}\ldots{\mathfrak{q}}_{r})=$
(7.14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad=\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)$

where

${\check{t}}_{i}=\frac{\prod_{j=1}^{i}{\mathfrak{q}}_{j}}{\prod_{j=1}^{i_{*}}{\mathfrak{q}}_{j}},\qquad\qquad
i=0,\ldots,r$ (7.15)

We thus rewrite the curve $C_{u}$ in the $(x,t)$-space, defined by the
equation

$\displaystyle
0={\mathcal{R}}_{A_{r}}(t,x)=\frac{P(t,x)}{\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)}=\prod_{l=1}^{N}\left(x-x_{l}(t)\right)=$
(7.16)
$\displaystyle\qquad\qquad\qquad\qquad=x^{N}+\frac{1}{\prod_{i=0}^{r}\left(t-{\check{t}}_{i}\right)}\sum_{j=1}^{N}p_{j}(t)x^{N-j}$

where

$N={\mathbf{v}}_{i_{*}}$ (7.17)

It is clear from the Eq. (7.16) that as $t\to{\check{t}}_{i}$ one of the roots
$x_{l}(t)$ has a pole, while the other $N-1$ roots are finite. Near $t=0$ the
polynomial $P(t,x)$ approaches:

$P(0,x)=(-1)^{r+1-i_{*}}P_{0}(x)$ (7.18)

while near $t=\infty$

$P(t,x)t^{-r-1}\to(-{\mathfrak{q}}_{i_{*}})^{i_{*}}P_{\infty}(x)$ (7.19)

Let

$dS=x\frac{dt}{t}$ (7.20)

Then our discussion above implies that the differential $dS$ has the first
order poles on $C_{u}$: at one of the $N$ preimages of the points
${\check{t}}_{i}$, $i=0,1,\ldots,r$, and at all preimages of the points $t=0$
and $t=\infty$. The residues of $dS$ are linear combinations of the masses of
the hypermultiplets, in agreement with the observations in
[Seiberg:1994aj],[Donagi:1995cf].

Remarkably, we can identify $C_{u}$ with the spectral curve of the meromorphic
Higgs field ${\Phi}$:

${\Phi}={\Phi}(t)dt=\sum_{j=-1}^{r+1}\,{\Phi}_{j}\frac{dt}{t-{\check{t}}_{j}}$
(7.21)

where ${\check{t}}_{-1}=0$, ${\check{t}}_{r+1}={\infty}$, and ${\Phi}_{j}$ are
$N\times N$ matrices, which have rank one for $j=0,1,\ldots,r$, and have the
maximal rank for $j=-1,r+1$. Moreover, the eigenvalues of ${\Phi}_{j}$ are all
fixed in terms of the masses. The spectra of ${\Phi}_{j}$, $j=-1,\ldots,r+1$
have specified multiplicity:

  1. (1)

the matrix ${\Phi}_{-1}$ has $w_{+}$ eigenvalues of multiplicity $r+1-i_{*}$,
and ${\mathbf{w}}_{r+1-j}$ eigenvalues of multiplicity $j$, for
$j=1,\ldots,r-i_{*}$;
the eigenvalues are fixed by the masses

  2. (2)

the matrices ${\Phi}_{j}$, $j=0,1,\ldots,r$ has one non-vanishing eigenvalue
each, and $N-1$ vanishing eigenvalues; We can write

$({\Phi}_{j})_{a}^{b}=u_{a}^{j}v_{j}^{b},\qquad a,b=1,\ldots,N$

for some vectors $u^{j}$, $v_{j}\in{\mathbb{C}}^{N}$, obeying

$\sum_{a=1}^{N}u^{j}_{a}v_{j}^{a}=M_{j}$ (7.22)

and considered up to an obvious ${\mathbb{C}}^{\times}$-action, for some
$M_{j}$ which is linear in the bi-fundamental and fundamental masses.

  3. (3)

the matrix ${\Phi}_{r+1}$ has $w_{-}$ eigenvalues of multiplicity $i_{*}$, and
${\mathbf{w}}_{j}$ eigenvalues of multiplicity $j$, for $j=1,\ldots,i_{*}-1$.

Then:

$\left(\frac{dt}{t}\right)^{N}\,{\mathcal{R}}_{A_{r}}(t,x)={\det}\left(\,x\frac{dt}{t}-{\Phi}\right)$
(7.23)

We can make an $SL(N)$ Higgs field out of $\Phi$ by shifting it by the scalar
meromorphic one-form $\frac{1}{N}{\operatorname{Tr}}_{N}{\Phi}$, which is
independent of the moduli $u$ of the curve $C_{u}$.

The moduli space of $r+3$-ples of matrices ${\Phi}_{j}$, obeying

$\sum_{j=-1}^{r+1}{\Phi}_{j}=0$ (7.24)

with fixed eigenvalues of the above mentioned multiplicity, considered up to
the simultaneous $SL(N)$-similarity transformation, is the phase space
${{\mathfrak{P}}}^{H}_{0,r+3}$ of the genus zero version of $SL(N)$ Hitchin
system, the classical Gaudin model on $r+3$ sites. The general Gaudin model
has the residues ${\Phi}_{j}$ belonging to arbitrary conjugacy classes.

See [Kronheimer:1990a, Kronheimer:1990] for the geometry of complex coadjoint
orbits. The Hitchin system with singularities was studied in [Gorsky:1994dj,
Nekrasov:1995nq, Donagi:1995am, Gukov:2006jk, Witten:2007td, Gukov:2008sn]. In
[Gaiotto:2009we, Nanopoulos:2009uw, Nanopoulos:2010zb, Nanopoulos:2010ga,
Nanopoulos:2009xe] this Hitchin system with singularities was discussed from
the point of view of brane constructions such as [Witten:1997sc,
Gaiotto:2009we].

######  Remark.

The curve $C_{u}$ is much more economical then ${\mathcal{C}}_{u}$. However,
the price we pay is the complexity of the relation between the special
coordinates ${\mathfrak{a}}_{i{\mathbf{a}}}$,
${\mathfrak{a}}_{i{\mathbf{a}}}^{D}$ and the moduli $u$ of the curve $C_{u}$.
Roughly speaking all special coordinates are linear combinations of the
periods of the differential

$x\frac{dt}{t}$

and the masses. The coordinates ${\mathfrak{a}}_{1{\mathbf{a}}}$ come from the
periods

$\oint x\,d{\log}g_{1}(x)\sim\oint xdt/t$

the coordinates ${\mathfrak{a}}_{2{\mathbf{a}}}$ come from the periods

$\oint x\,d{\log}(g_{1}(x)g_{2}(x))\sim\oint xdt/t+\oint xdt/t$

the coordinates ${\mathfrak{a}}_{i{\mathbf{a}}}$ come from the periods

$\oint x\,d{\log}(g_{1}(x)\ldots g_{i}(x))\sim\oint xdt/t+\ldots+\oint xdt/t$

etc.

######  Remark.

In the $A_{2}$ case our solution matches the one found in [Shadchin:2005cc].

######  Remark.

We can connect the cameral curve ${\mathcal{C}}_{u}$ to the spectral curve
$C_{u}$ via a tower of ramified covers:

${\mathcal{C}}_{u}\to C_{u}^{(r)}\to C_{u}^{(r-1)}\to\dots\to
C_{u}^{(1)}=C_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ (7.25)

which we can call the _Gelfand-Zeitlin_ tower of curves. The curve
$C_{u}^{(i)}$ is the quotient of ${\mathcal{C}}_{u}$ by the subgroup
$W(A_{r-i})$ of the Weyl group $W(A_{r})$, which acts on the amplitudes
$({\mathscr{Y}}_{i+1},\ldots,{\mathscr{Y}}_{r})$ while preserving
$({\mathscr{Y}}_{1},\ldots,{\mathscr{Y}}_{i})$.

######  Remark.

We should warn the reader that our cameral curves need not be the cameral
curves of Hitchin systems [Donagi:1995alg]. We mapped the spectral curve of
the family of conjugacy classes $[g(x)]$ corresponding to the fundamental
representation $R_{1}$ to the spectral curve of the $GL(N)$-Gaudin system,
i.e. the genus zero Hitchin system, corresponding to the $N$-dimensional
representation. One could then build the cameral curve for the $GL(N)$-Gaudin
system. This curve has all the reasons to differ from our cameral curve
${\mathcal{C}}_{u}$.

However, the identification of ${\mathfrak{M}}$ with the moduli spaces of
curves describing the spectrum of the transfer matrix in the quasi classical
limit of the $Y(A_{r})$ spin chain is more natural, and carries over to the
level of cameral curves. This statement will be elaborated upon below and in
[NP2012b].

######  Remark.

In view of [Gaiotto:2009we] it is natural to identify the space of couplings
${\tilde{\mathfrak{q}}}=({\mathfrak{q}}_{1},\ldots,{\mathfrak{q}}_{r})$ with a
coordinate patch in the moduli space $\overline{\mathcal{M}}_{0,r+3}$ of
stable genus zero curves with $r+3$ punctures. In this fashion the linear
quiver theories (the class I type $A_{r}$ theories) can be analytically
continued to other weakly coupled regions (weak coupling corresponds to the
maximal degeneration of the stable curve). Most of these regions do not have a
satisfactory Lagrangian description. Nevertheless, it would be interesting to
try to generalize the limit shape equations even without knowing their
microscopic origin. What would the iWeyl group look like in this case?

####  7.1.2. Quiver description

We have thus found that a particular subset of Gaudin-Hitchin models, with all
but two residues of the minimal type, are the Seiberg-Witten integrable
systems of the class I $A_{r}$ type theories. As a check, let us compute the
dimension of the moduli space ${{\mathfrak{P}}}^{H}_{0,r+3}$ of solutions to
the (traceless part of the) moment map equation (7.24) divided by the
$SL(N,{\mathbb{C}})$-action is equal to:

$\displaystyle 2(r+1)(N-1)-2(N^{2}-1)+$ (7.26)
$\displaystyle\qquad+\left(N^{2}-\sum_{j=1}^{i_{*}-1}j^{2}{\mathbf{w}}_{j}-i_{*}^{2}w_{-}\right)+$
$\displaystyle\qquad+\left(N^{2}-\sum_{j=i_{*}+1}^{r}(r+1-j)^{2}{\mathbf{w}}_{j}-(r+1-i_{*})^{2}w_{+}\right)=$
$\displaystyle\qquad\qquad\qquad=2\sum_{i=1}^{r}({\mathbf{v}}_{i}-1)=2\,{\dim}{{\mathfrak{M}}}$

Actually, the moduli space ${{\mathfrak{P}}}^{H}_{0,r+3}$ can be described as
a quiver variety. Its graph is an $r+3$-pointed star, with $r+1$ legs of
length $1$, and two long legs, of the lengths $l_{-1}={\mathbf{v}}_{r}-1$ and
$l_{r+1}={\mathbf{v}}_{1}-1$, respectively. The dimensions of the vector
spaces assigned to vertices are: the $r+3$-valent vertex (the star) has
dimension $N$, the tails of the short legs all have dimension $1$, the
dimensions along the long legs start at $1$ at the tails, then grow with the
step $1$ for the first ${\mathbf{w}}_{1}$ (respectively, ${\mathbf{w}}_{r}$)
vertices, then grow with the step $2$ for the next ${\mathbf{w}}_{2}$
(respectively, ${\mathbf{w}}_{r-1}$) and so on. (See example in figure 7.2).

Figure 7.2. The example quiver variety for $A_{4}$ quiver at
$\mathbf{v}=(7,10,8,5)$ and $\mathbf{w}=(4,5,1,2)$ with $i_{*}=2$ and
$\mathbf{w}_{*}=\mathbf{w}_{-}+\mathbf{w}_{+}$ with $\mathbf{w}_{-}=3$ and
$\mathbf{w}_{+}=2$. The labels at vertices denote the dimensions in the
pattern as explained

The extended phase space ${\mathfrak{P}}^{\mathrm{ext}}$ for the class I
$A_{r}$ type theories is easy to describe. One just need to relax the
${\mathbb{C}}^{\times}$ moment map constraints (7.22) as well as the analogous
${\mathbb{C}}^{\times}$ constraints for the ${\Phi}_{-1}$, ${\Phi}_{r+1}$
residues. In the quiver description we make the quiver gauge group the product
of the special unitary groups as opposed to the product of unitary groups.

####  7.1.3. Reduction to the spin chain

The simplest example of the Class I theory of the $A$ type is, of course, the
$A_{1}$ theory. This is the celebrated $N_{f}=2N_{c}$ theory, with
${\mathbf{w}}_{1}=N_{f}$, ${\mathbf{v}}_{1}=N_{c}=N$, in our notation. Let
${\mathfrak{q}}={\mathfrak{q}}_{1}$ and let $T(x)=T_{1,0}^{-1}T_{1}(x)$ denote
the monic degree $N$ polynomial.

The reduced curve (7.12) assumes a very simple form:

${\mathfrak{q}}{\mathscr{P}}^{-}(x)t+{\mathscr{P}}^{+}(x)t^{-1}=(1+{\mathfrak{q}})T(x)$
(7.27)

It is not difficult to recognize in this formula the quasiclassical limit of
Baxter’s $T-Q$ equation [Baxter:1985] for the $XXX$ $sl_{2}$ spin chain. In
fact, it was observed already in [Gorsky:1996hs, Gorsky:1996qp] that the
Seiberg-Witten curve of the ${\mathcal{N}}=2$ supersymmetric QCD can be
interpreted using the integrable spin chain, albeit in a somewhat different
fashion. Note that a possible lift of $[g(x)]$ to
$C{\mathbf{G}}=GL(2,{\mathbb{C}})$ in this case is given by the diagonal
matrix:

$g(x)=\left(\begin{matrix}{\mathfrak{q}}t{\mathscr{P}}^{-}(x)&0\\\
0&t^{-1}{\mathscr{P}}^{+}(x)\end{matrix}\right)$ (7.28)

where $t$ solves (7.27). However this choice of $g(x)$ is not continuous in
$x$. As we cross the cuts $I_{1,{\mathbf{a}}}$ the matrix $g(x)$ will have its
diagonal entries exchanged. We can conjugate $g(x)\to h^{-1}(x)g(x)h(x)$ into
a form, e.g.

$\mathbf{g}(x)=\left(\begin{matrix}{\mathfrak{q}}T(x)&1\\\
{\mathfrak{q}}\left(T^{2}(x)-{\mathscr{P}}^{+}(x){\mathscr{P}}^{-}(x)\right)&T(x)\end{matrix}\right)$
(7.29)

whose entries are polynomials. This is a particular case of a general
statement [Steinberg:1965], lifting a family of conjugacy classes in
${\mathbf{G}}_{\text{q}}$ to ${\mathbf{G}}_{\text{q}}$ itself (slightly
adapted for the conformal extension $C\mathbf{G}$). The lift (7.29) does not
depend on the split ${\mathscr{P}}(x)$ into the product of
${\mathscr{P}}^{\pm}$ factors.

There is yet another lift of $[g(x)]$ to $C\mathbf{G}$, which does depend on
the factorization, and makes closer contact with spin chains. We shall discuss
it in the section devoted to the study of the phase spaces of the integrable
systems corresponding to our gauge theories.

####  7.1.4. Duality

In the mapping to the Gaudin-Hitchin system we employed a particular lift
$g(x)$ of the conjugacy class $[g(x)]$ in
$SL(r+1,{\mathbb{Z}})/{\mathbb{Z}}_{r+1}$ to the conjugacy class in
$GL(r+1,{\mathbb{C}})$ by a judicious choice of the normalization factor
${\zeta}(x)$. More importantly, the spectral one-form describing the
eigenvalues of the Higgs field, is equal to $xdt/t$ where $x$ is the argument
of the amplitude function, and $t$ is the spectral variable describing the
eigenvalues of $g(x)$. For the group $GL(r+1,{\mathbb{C}})$ the eigenvalues of
$g(x)$ in some representation take values in ${\mathbf{C}_{\left\langle
t\right\rangle}}={\mathbb{C}}^{\times}$ which gets naturally compactified to
${\mathbb{C}\mathbb{P}}^{1}$ to allow the degenerations.

To summarize, the Lax operator of Gaudin-Hitchin system, the Higgs field
${\Phi}(t)dt/dt$ lives on the curve ${\mathbf{C}_{\left\langle
t\right\rangle}}$ of the eigenvalues of the ‘‘Lax operator’’ $g(x)$ of the
gauge theory. Vice versa, the ‘‘Lax operator’’ $g(x)$ of the gauge theory
lives on the curve ${\mathbf{C}_{\left\langle x\right\rangle}}$ of the
eigenvalues of the Higgs field of Hitchin system.

We shall encounter some versions of this ‘‘eigenvalue – spectral parameter’’
duality in other sections of this work.

###  7.2. Class I theories of $D$ type

These are the $SU({\mathbf{v}}_{1})\times\ldots\times SU({\mathbf{v}}_{r})$
theories whose quiver contains a trivalent vertex which connects two one-
vertex legs to a leg of the length $r-3$. The corresponding group
${\mathbf{G}}_{\text{q}}$ is $Spin(2r,{\mathbb{C}})$, its conformal version
$C\mathbf{G}$ is the extension of $\mathbf{G}$ by ${\mathbb{C}}^{\times}$ or
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$, depending on the parity of
$r$.

Passing from the $A$ type theories to the $D$ type theories we encounter new
phenomenon. In addition to the exterior powers $\wedge^{i}V$ of the vector
representation $V={\mathbb{C}}^{2r}$ of $Spin(2r)$ the fundamental
representations of the group ${\mathbf{G}}_{\text{q}}$ come also from spin
representations $S_{\pm}$. We should use the cameral curve ${\mathcal{C}}_{u}$
to get the special coordinates and the prepotential, however a lot of
information is contained in the spectral curve $C_{u}^{R}$ in some fundamental
representation $R$, which we shall take to be the vector $2r$ dimensional
representation $V=R_{\lambda_{1}}={\mathbb{C}}^{2r}$. In order to describe the
spectral curve we need to know the characters of the group element $g(x)$
(6.14) in the representations $\wedge^{i}V$, for $i=1,\ldots,2r$. When we deal
with $V$ and its exterior powers only, we do not see the full conformal
version of $\mathbf{G}$, only its one-dimensional extension (which we shall
denote simply by C$\mathbf{G}$) which consists of the matrices $g\in
GL(2r,{\mathbb{C}})$, such that $gg^{t}=D(g)\cdot{\bf 1}_{2r}$, with
$D(g)\in{\mathbb{C}}^{\times}$ a scalar.

The spectral curve $C_{u}=C^{V}_{u}$ in the vector representation can be
modified by the transformation similar to (7.10) to get the curve of minimal
degree in $x$. Let us label the vertices of the $D_{r}$ Dynkin diagram in such
a way, that the trivalent vertex is $r-2$, the tails are $r-1$, $r$, and the
end vertex of the ‘‘long leg’’ has the label $1$, see Appendix A. Then the
product of the matter polynomials ${\mathscr{P}}_{r-1}$ and
${\mathscr{P}}_{r}$ has degree

${\rm
deg}({\mathscr{P}}_{r-1}{\mathscr{P}}_{r})=2({\mathbf{v}}_{r-1}+{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2})$

Now we shall factorize ${\mathscr{P}}_{r-1}{\mathscr{P}}_{r}$ into a product
of two factors of equal degrees

${\mathscr{P}}_{r-1}{\mathscr{P}}_{r}={\mathscr{P}}^{+}{\mathscr{P}}^{-},\qquad{\rm
deg}{\mathscr{P}}^{+}={\rm
deg}{\mathscr{P}}^{-}={\mathbf{v}}_{r-1}+{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2}$
(7.30)

There are many possible factorizations. For example, if
${\mathbf{w}}_{r-1}\leq{\mathbf{w}}_{r}$, then we can take:
${\mathscr{P}}_{r}(x)={\mathscr{P}}^{+}(x)S(x)$,
${\mathscr{P}}^{-}(x)=S(x){\mathscr{P}}_{r-1}(x)$ for any degree
${\mathbf{v}}_{r}+{\mathbf{v}}_{r-1}-{\mathbf{v}}_{r-2}\leq{\mathbf{w}}_{r}=2{\mathbf{v}}_{r}-{\mathbf{v}}_{r-2}$
subfactor ${\mathscr{P}}^{+}(x)$ in ${\mathscr{P}}_{r}(x)$. We shall normalize
${\mathscr{P}}^{\pm}(x)$ so that the highest coefficient in both polynomials
equals

$\sqrt{{\mathfrak{q}}_{r-1}{\mathfrak{q}}_{r}}$

That there exist different decompositions (7.30) is a generalization of
$S$-duality of the ${\mathcal{S}}$-class ${\mathcal{N}}=2$ theories of the
$A_{r}$ type studied in [Gaiotto:2009we].

The spectral curve $C_{u}$ corresponding to the $2r$-dimensional vector
representation of $CSpin(2r,{\mathbb{C}})$ is mapped to the curve
$P_{D_{r}}^{C}(t,x)=0$ in the $(t,x)$-space, where

$P_{D_{r}}^{C}(t,x)=t^{-r}P_{\infty}(x){\det}_{R_{1}}(t\cdot 1_{2r}-g(x))$
(7.31)

with some polynomial $P_{\infty}(x)$ to be determined below. The group element
$g(x)$ in the vector representation ${\mathbb{C}}^{2r}$ of
$C{\mathbf{G}}_{\text{q}}$ is given by

$g(x)=E^{-1}\operatorname{diag}(g_{1}(x),\dots,g_{2r}(x))E$ (7.32)

with $E$ being any matrix such that

$(EE^{t})_{ij}={\delta}_{i,2r+1-j}$

represents the symmetric bilinear form on ${\mathbb{C}}^{2r}$ and

$\displaystyle g_{1}(x)=\zeta(x)\mathscr{Y}_{1}(x)$ (7.33) $\displaystyle
g_{i}(x)=\zeta(x)\mathscr{P}^{[i-1]}(x)\frac{\mathscr{Y}_{i}(x)}{\mathscr{Y}_{i-1}(x)},\quad
i=2,\dots,r-2$ $\displaystyle
g_{r-1}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\frac{\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x)}{\mathscr{Y}_{r-2}(x)}$
$\displaystyle
g_{r}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\mathscr{P}_{r-1}(x)\frac{\mathscr{Y}_{r}(x)}{\mathscr{Y}_{r-1}(x)}$
$\displaystyle
g_{r+1}(x)=\zeta(x)\mathscr{P}^{[r-2]}(x)\mathscr{P}_{r}(x)\mathscr{Y}_{r-1}(x)/\mathscr{Y}_{r}(x)$
$\displaystyle
g_{r+2}(x)=\zeta(x)\mathscr{P}^{[r]}(x)\mathscr{Y}_{r-2}(x)/(\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x))$
$\displaystyle
g_{2r+1-i}(x)=\zeta(x)\frac{\mathscr{P}^{[r]}(x)\mathscr{P}^{[r-2]}(x)}{{\mathscr{P}}^{[i-1]}(x)}\frac{\mathscr{Y}_{i-1}(x)}{\mathscr{Y}_{i}(x)},\quad
i=2,\dots,r-2$ $\displaystyle
g_{2r}=\zeta(x)\mathscr{P}^{[r]}(x)\mathscr{P}^{[r-2]}(x)\frac{1}{\mathscr{Y}_{1}(x)}$

The factor $\zeta(x)$ which likely gives the minimal degree curve is

$\zeta(x)^{-1}=\mathscr{P}^{+}(x)\mathscr{P}^{[r-2]}(x)$ (7.34)

Thus, the scalar $D(g(x))$ is equal to:

$D(g(x))=\frac{{\mathscr{P}}^{-}(x)}{{\mathscr{P}}^{+}(x)}$ (7.35)

and the prefactor in (7.31) is:

$P_{\infty}(x)=\mathscr{P}^{+}(x)^{r}\prod_{j=1}^{r-2}\mathscr{P}_{j}(x)^{j}.$
(7.36)

After some manipulations we find

$\displaystyle(-1)^{r}P_{D_{r}}^{C}(t,x)=T_{r}^{2}{\mathscr{P}}_{r-1}+T_{r-1}^{2}{\mathscr{P}}_{r}-{\eta}T_{r-1}T_{r}+$
(7.37)
$\displaystyle\qquad\sum_{l=1}^{\left[\frac{r}{2}\right]}T_{r-2l}\left(\prod_{j=r+1-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l}\right)\,{\xi}_{l}^{2}-$
$\displaystyle\qquad\qquad-\sum_{l=1}^{\left[\frac{r-1}{2}\right]}T_{r-2l-1}\left(\prod_{j=r-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l+1}\right)\,{\xi}_{l}{\xi}_{l+1}\,,$
$\displaystyle{\xi}_{l}=({\mathscr{P}}^{+}t)^{l}-({\mathscr{P}}^{-}t^{-1})^{l},\
{\eta}={\mathscr{P}}^{+}t+{\mathscr{P}}^{-}t^{-1}$

This equation has degree
$N=2({\mathbf{v}}_{r}+{\mathbf{v}}_{r-1})-{\mathbf{v}}_{r-2}$ in the $x$
variable. Note

${\mathbf{v}}_{r-2}\leq N\leq 2{\mathbf{v}}_{r-2}$ (7.38)

As in the $A_{r}$ case, the curve $C_{u}$ has branches going off to infinity
in the $x$-direction, over $2r$ points ${\check{t}}_{i},{\check{t}}_{i}^{-1}$,
$i=1,\ldots,r$ in the $t$-line ${\mathbb{C}\mathbb{P}}^{1}_{t}$ which
correspond to the weights of $R_{1}$

${\check{t}}_{i}=\frac{1}{\sqrt{{\mathfrak{q}}_{r-1}{\mathfrak{q}}_{r}}}\frac{{\mathfrak{q}}^{[i-1]}}{{\mathfrak{q}}^{[r-2]}}$
(7.39)

In addition, there are special points $t=0,\infty$. Over these points the
curve $C_{u}$ has $N$ branches, where $x$ approaches one of the roots of the
polynomial $P_{0}(x)$

$P_{0}(x)={\mathscr{P}}^{-}(x)^{r}\prod_{j=1}^{r-2}{\mathscr{P}}_{j}(x)^{j}\
.$ (7.40)

and $P_{\infty}(x)$, cf. (7.36), respectively.

The curve $C_{u}$ is invariant under the involution

$t\mapsto\frac{{\mathscr{P}}_{-}(x)}{{\mathscr{P}}_{+}(x)}t^{-1}$ (7.41)

The fixed points of (7.41) are the points of intersection of the curve $C_{u}$
and the curve

${\mathscr{P}}^{+}(x)t-{\mathscr{P}}^{-}(x)t^{-1}=0$ (7.42)

The equations ${\mathcal{R}}_{D_{r}}(t,x)=0$ (7.37) and (7.42) imply

$\displaystyle
T_{r}^{2}(x){\mathscr{P}}_{r-1}(x)+T_{r-1}^{2}(x){\mathscr{P}}_{r}(x)=$ (7.43)
$\displaystyle\qquad\qquad\qquad
T_{r-1}(x)T_{r}(x)({\mathscr{P}}^{+}(x)t+{\mathscr{P}}^{-}(x)t^{-1})$

and

$T_{r}^{2}(x){\mathscr{P}}_{r-1}(x)=T_{r-1}^{2}(x){\mathscr{P}}_{r}(x)$ (7.44)

Again, the curve $C_{u}$ is more economical then the full cameral curve
${\mathcal{C}}_{u}$. Again, the special coordinates
${\mathfrak{a}}_{i,{\mathbf{a}}}$ and the duals
${\mathfrak{a}}_{i,{\mathbf{a}}}^{D}$ are the linear combinations of the
periods of the differential $xdt/t$ and the masses.

Let us map the curve $C_{u}$ to the curve $\Sigma_{u}$ in the space $S$ which
is a ${\mathbb{Z}}_{2}$-quotient of the (blowup of the)
${\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbb{C}\mathbb{P}}^{1}_{t}$ space, parametrized by
$(x,s)$ where

$s=\frac{{\mathscr{P}}^{+}(x)}{{\mathscr{P}}^{-}(x)}t^{2}$

The curve $\Sigma_{u}$ is described by the equations $s+s^{-1}=2c$ and:

$P_{D_{r}}^{\Sigma}(x,c)\equiv{\bf
A}(x,c)^{2}-2{\mathscr{P}}_{r}(x){\mathscr{P}}_{r-1}(x)(c+1){\bf
B}(x,c)^{2}=0$ (7.45)

where ${\bf A},{\bf B}$ are the polynomials in $x$ and $c$ of bi-degrees
$(N,\left[\frac{r}{2}\right])$ and
$({\mathbf{v}}_{r-1}+{\mathbf{v}}_{r},\left[\frac{r-1}{2}\right])$,
respectively:

$\displaystyle{\bf
A}(x,c)=T_{r}^{2}{\mathscr{P}}_{r-1}+T_{r-1}^{2}{\mathscr{P}}_{r}+$ (7.46)
$\displaystyle\qquad\qquad+2\sum_{l=1}^{\left[\frac{r}{2}\right]}\,{\bf
C}_{l}(c)\,T_{r-2l}{\mathscr{P}}_{r-1}^{l}{\mathscr{P}}_{r}^{l}\prod_{j=r+1-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l},$
$\displaystyle{\bf
B}(x,c)=T_{r-1}T_{r}+2\sum_{l=1}^{\left[\frac{r-1}{2}\right]}\,{\bf
D}_{l}(c)\,T_{r-2l-1}{\mathscr{P}}_{r-1}^{l}{\mathscr{P}}_{r}^{l}\prod_{j=r-2l}^{r-2}{\mathscr{P}}_{j}^{j-r+2l+1},$

where the degree $l$ polynomials ${\bf C}_{l}(c)$, ${\bf D}_{l}(c)$ are
defined as follows:

$\displaystyle{\bf C}_{l}(c)=\frac{1}{2}(s^{l}+s^{-l})-1,\qquad s+s^{-1}=2c$
(7.47) $\displaystyle{\bf
D}_{l}(c)=\frac{(s^{l}-1)(s^{l+1}-1)}{2s^{l}(s+1)}=\sum_{j=0}^{l-1}(-1)^{j}{\bf
C}_{l-j}(c)$

Over the points $c=1$ and $c=-1$ the equation for $\Sigma_{u}$ becomes
reducible: at $c=1$:

$P_{D_{r}}^{\Sigma}(x,1)=({\mathscr{P}}_{r-1}T_{r}^{2}-{\mathscr{P}}_{r}T_{r-1}^{2})^{2}$
(7.48)

and at $c=-1$:

$P_{D_{r}}^{\Sigma}(x,-1)={\bf A}(x,-1)^{2}$ (7.49)

It is easy to see that the curve $\Sigma_{u}$ has double points at $(x,s)$
where either $s=1$ and $x$ being any of the $N$ roots of (7.48) or $s=-1$ and
$x$ is any of the $N$ roots of (7.49). The locations of these roots are not
fixed by the masses of the matter fields.

Let us normalize the equation of $\Sigma_{u}$ by dividing $P_{D_{r}}^{\Sigma}$
by the coefficient at $x^{2N}$:

${\mathcal{R}}_{D_{r}}(x,c)=\frac{P_{D_{r}}^{\Sigma}(x,c)}{\prod_{i=1}^{r}(s-{\check{t}}_{i}^{2})(1-s^{-1}{\check{t}}_{i}^{-2})}$
(7.50)

times a constant such that $\mathcal{R}_{D_{r}}(x,c)$ is monic in $x$ and a
rational function of $c$.

We thus arrive at the following interpretation of the curve $\Sigma_{u}$. It
is the spectral curve

${\mathcal{R}}_{D_{r}}\left(x,\frac{s^{2}+1}{2s}\right)\left(\frac{ds}{s}\right)^{2N}={\rm
Det}_{2N}\left(x\frac{ds}{s}-{\Phi}(s)\right)$ (7.51)

of the genus zero Higgs field

${\Phi}(s)=\sum_{s_{j}\in J}{\Phi}_{j}\frac{ds}{s-s_{j}}$ (7.52)

where $J\subset{\mathbb{C}\mathbb{P}}^{1}_{s}$ is the set of $2r+2$
singularities:

$J=\\{0,\infty\,\\}\cup\\{\,{\check{t}}_{i}^{2},\
{\check{t}}_{i}^{-2}\,|\,i=1,\ldots,r\\}$

Let ${\sigma}:{\mathbb{C}\mathbb{P}}^{1}_{s}\to{\mathbb{C}\mathbb{P}}^{1}_{s}$
be the involution ${\sigma}(s)=s^{-1}$. The Higgs field must obey:

${\sigma}^{*}{\Phi}={\Omega}{\Phi}^{t}{\Omega}^{-1}$ (7.53)

where ${\Omega}$ is a constant anti-symmetric matrix (cf. [Kapustin:1998fa]),
which defines the symplectic structure on $V={\mathbb{C}}^{2N}$. If we expand:

$\displaystyle{\Phi}(s)={\Phi}_{0}\frac{ds}{s}+\sum_{i=1}^{r}{\Phi}_{i}^{+}\frac{ds}{s-{\check{t}}_{i}^{2}}+\sum_{i=1}^{r}{\Phi}_{i}^{-}\frac{ds}{s-{\check{t}}_{i}^{-2}}\,,$
(7.54)
$\displaystyle\qquad\qquad{\Phi}_{\infty}=-{\Phi}_{0}-\sum_{i=1}^{r}({\Phi}_{i}^{+}+{\Phi}_{i}^{-})$

Then (7.53) implies:

$\displaystyle{\Phi}_{\infty}={\Omega}{\Phi}_{0}^{t}{\Omega}^{-1},\qquad{\Phi}_{i}^{+}={\Omega}({\Phi}_{i}^{-})^{t}{\Omega}^{-1},\qquad
i=1,\ldots,r$ (7.55)

Also, the matrices ${\Phi}^{+}_{i},{\Phi}^{-}_{i}$, $i=1,\ldots,r$, must have
rank one, while the matrices ${\Phi}_{0,\infty.\pm 1}$ have rank $2N$. We can
interpret

$\displaystyle{\mu}={\Phi}_{0}+{\Phi}_{\infty}+\sum_{i=1}^{r}({\Phi}_{i}^{+}+{\Phi}_{i}^{-})\,,$
(7.56)
$\displaystyle\qquad\qquad\qquad\qquad{\mu}^{t}={\Omega}^{-1}{\mu}{\Omega}$

as the moment map for the $Sp(2N)$ group action on the product of some orbits

${\mathcal{O}}_{0}\times{\mathcal{O}}_{-1}\times{\mathcal{O}}_{1}\times_{i=1}^{r}{\mathcal{O}}_{i}$

which generates the action ${\Phi}_{j}\mapsto g^{-1}{\Phi}_{j}g$ of $g\in
Sp(2N)$, such that:

$g{\Omega}g^{t}={\Omega}\ .$ (7.57)

It would be nice to develop further the theory of these orbifold Hitchin-
Gaudin systems. We shall encounter a genus one version of such theory in the
Class II $D_{r}$ section below.

The differential whose periods determine the special coordinates is equal to

$dS=x\frac{ds}{s}$ (7.58)

####  7.2.1. Freezing example

Here we will illustrate how the $D_{4}$ theory with
$v_{1}=v_{3}=v_{4}=v,v_{2}=2v$ and $w_{1}=w_{3}=w_{4}=0,w_{2}=v$ reduces to
$A_{3}$ with $v_{1}=v_{3}=v,v_{2}=2v$ and $w_{2}=2v$ when the node 4 freezes
under $\mathfrak{q}_{4}\to 0$. Keeping in mind unfreezing to the affine
$\widehat{D}_{4}$, let polynomial $Y_{0}$ of degree $v$ denote the fundamental
matter polynomial attached to the node ‘‘2’’.

The $D_{4}$ spectral curve for the node ‘‘1’’ from (7.37) in terms of variable
$\eta$

$\eta=t+\frac{\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}}{t}$
(7.59)

where $\mathscr{Y}_{2}=Y_{0}t$ is

$\mathcal{R}_{D_{4}}(\eta,x)=\eta^{4}Y_{0}^{2}-\eta^{3}T_{1}Y_{0}+\eta^{2}\left(\mathfrak{q}_{1}T_{2}-4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}Y_{0}^{2}\right)+\\\
\eta\left(-\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}T_{3}T_{4}+4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}T_{1}Y_{0}\right)-4\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}T_{2}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{4}T_{3}^{2}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}T_{4}^{2}$
(7.60)

Notice that the curve is polynomial of degree $4$ in $\eta$ with polynomial
coefficients in $x$ of degree $2v$. In the limit $x\to\infty$ we find the
limiting values of $\eta$ are

$1+\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}+\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}+\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}\mathfrak{q}_{4},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}+\mathfrak{q}_{1}\mathfrak{q}_{3}\mathfrak{q}_{4}$
(7.61)

Notice that the differential is

$\lambda=x\frac{dt}{t}=x\frac{d\eta}{(\eta^{2}-4\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4})^{\frac{1}{2}}}$
(7.62)

Also notice that at $\eta=\pm
2\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}}$
the curve factorizes as

$\mathcal{R}_{D_{4}}(\pm
2\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}},x)=\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}(\mathfrak{q}_{3}T_{4}(x)\mp\mathfrak{q}_{4}T_{3}(x))^{2}$
(7.63)

as well as it factorizes at $\eta=\infty$

$\mathcal{R}_{D_{4}}(\eta=\infty,x)=Y_{0}(x)^{2}$ (7.64)

We can interpret the multi-valued nature of $\lambda$ on the $\eta$-plane as
the deformation of the punctured sphere underlying the $A_{r}$-type theories
to the curve describing the $D_{r}$-type theories, by opening punctures into
cuts. Perhaps one can elevate this observation to the corresponding
deformation of the Liouville theory coupled to some conformal matter, along
the lines of [Knizhnik:1987xp, Gerasimov:1988gy].

We see that in the decoupling limit $\mathfrak{q}_{4}=0$ the above curve
reduces to

$\mathcal{R}_{A_{3}}(\eta,x)=\eta^{4}Y_{0}^{2}-\eta^{3}T_{1}Y_{0}+\eta^{2}\mathfrak{q}_{1}T_{2}-\eta\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}T_{3}Y_{4}+\mathfrak{q}_{1}^{3}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}Y_{4}^{2}$
(7.65)

where we just set that $\mathscr{Y}_{4}$ freezes and converts to a factor of
degree $v$ contributing to the fundamental matter polynomial for the node
‘‘2’’; we denote this factor by $Y_{4}\equiv\mathscr{Y}_{4}=T_{4}$. The curve
(7.65) is precisely the $A_{3}$ curve for the node ‘‘1’’ (7.12) in terms of
the variable $\mathscr{Y}_{1}=Y_{0}\eta$. This curve corresponds to the
$GL(2)$ Hitchin system with punctures at four punctures

$1,\quad\mathfrak{q}_{1},\quad\mathfrak{q}_{1}\mathfrak{q}_{2},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}$
(7.66)

Moreover, from the discussion after (7.21) (we have
$\mathbf{w}_{0}=0,\mathbf{w}_{2}=2,\mathbf{w}_{3}=0$ and $i_{*}=2$ and
$\mathbf{w}_{+}=\mathbf{w}_{-1}=1$) it is clear the the eigenvalues of the
Higgs field residues at $\eta=0$ and at $\eta=\infty$ are doubly degenerate
which effectively means that $SL(2,\mathbb{C})$ part of the Higgs field does
not have punctures at $\eta=0$ and $\eta=\infty$. We can continue the freezing
reduction and now we shall set $\mathfrak{q}_{3}=0$ declaring the function
$\mathscr{Y}_{3}$ as contributing to the fundamental matter at the node ‘‘2’’,
we denote $\mathscr{Y}_{3}=T_{3}=Y_{3}$. After factoring out $\eta$, the curve
(7.65) reduces to the $A_{2}$ curve

$\mathcal{R}_{A_{2}}(\eta,x)=\eta^{3}Y_{0}^{2}-\eta^{2}T_{1}Y_{0}+\eta\mathfrak{q}_{1}T_{2}-\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}Y_{3}Y_{4}$
(7.67)

The corresponding Gaudin system has punctures at $\eta=0$ and $\eta=\infty$
and at

$1,\quad\mathfrak{q}_{1},\quad\mathfrak{q}_{1}\mathfrak{q}_{2}$ (7.68)

Finally, we can freeze the node ‘‘1’’ by sending $\mathfrak{q}_{1}$ to zero
and rescaling $\eta=\tilde{\eta}\mathfrak{q}_{1}$ so that the former punctures
$\mathfrak{q}_{1},\mathfrak{q}_{1}\mathfrak{q}_{2}$ on the
$\tilde{\eta}$-plane in terms of $\tilde{\eta}$ become

$1,\quad\mathfrak{q}_{2}$ (7.69)

while the puncture $\eta=1$ is send away to $\tilde{\eta}=\infty$. We set
$Y_{1}\equiv\mathscr{Y}_{1}=T_{1}$ and find that (7.67) reduces to the
familiar $A_{1}$ curve with gauge polynomial $T_{2}$ of degree $2v$ and four
factors $(Y_{0},Y_{1},Y_{3},Y_{4})$ of degree $v$ which make fundamental
polynomial of degree $4v$

$\mathcal{R}_{A_{2}}(\eta,x)=-\tilde{\eta}^{2}Y_{1}Y_{0}+\tilde{\eta}T_{2}-\mathfrak{q}_{2}Y_{3}Y_{4}$
(7.70)

The punctures of the corresponding Gaudin model in $\tilde{\eta}$ plane are at
$(0,\mathfrak{q}_{2},1,\infty)$.

###  7.3. Class I theories of $E$ type

We are using Bourbaki conventions to label the nodes on the Dynkin graph of
$E_{r}$ series, see figures in the Appendix A. One can construct the analogues
of the spectral curves $C_{u}$ or $\Sigma_{u}$ using the minuscule
representations in the $E_{6}$ and $E_{7}$ cases. For $E_{8}$ one can
construct the spectral curve using the adjoint representation $\bf 248$.
However it seems more advantageous to use the degenerate version of del
Pezzo/$E$-bundle correspondence, which we review below in the discussion of
Class II theories of $E$ type. For the standard conformal $E_{r}$ quivers,
which are obtained by freezing of the node ‘‘0’’ in the affine $E_{r}$ quivers
with ranks ${\mathbf{v}}_{i}=Na_{i}$ where $a_{i}$ are Dynkin marks, we find
spectral curves of $(t,x)$-degree equal to $(27,6N)$ for $E_{6}$, $(56,12N)$
for $E_{7}$ and $(240,60N)$ for $E_{8}$. These degrees can be understood from
the degeneration of $\widehat{E}_{r}$ spectral curves computed in section 6.3.

####  7.3.1. The $E_{6}$ theory

The spectral curve in the fundamental representation $R_{6}=\mathbf{27}$
associated with the node ‘‘6’’, in which the group element of the conformal
extension of $E_{6}$ is $g(x)=(\mathscr{Y}_{6}(x),\dots)$ has the form

$\mathcal{R}_{E_{6}}(t,x)=0$ (7.71)

where the explicit expression is of the form111The explicit expression, which
we do not list here, is available upon a request; it is computed by the
straightforward expansion of the exterior powers $\bigwedge^{\bullet}R_{6}$ in
the representation ring $\mathrm{Rep}(E_{6})$ over the fundamental
representations $R_{1},\dots,R_{6}$.

$\mathcal{R}_{E_{6}}(t,x)={\det}_{R_{6}}(t\cdot
1_{27}-g(x))=t^{27}-t^{26}T_{6}+t^{25}\mathscr{P}_{6}T_{5}-t^{24}\mathscr{P}_{5}\mathscr{P}_{6}^{2}T_{4}+t^{23}\left(-\mathscr{P}_{2}^{2}\mathscr{P}_{3}^{2}\mathscr{P}_{4}^{4}\mathscr{P}_{5}^{4}\mathscr{P}_{6}^{4}T_{1}^{2}+\mathscr{P}_{1}\mathscr{P}_{2}^{2}\mathscr{P}_{3}^{2}\mathscr{P}_{4}^{4}\mathscr{P}_{5}^{4}\mathscr{P}_{6}^{4}T_{3}+\mathscr{P}_{4}\mathscr{P}_{5}^{2}\mathscr{P}_{6}^{3}T_{2}T_{3}-\mathscr{P}_{2}\mathscr{P}_{3}\mathscr{P}_{4}^{2}\mathscr{P}_{5}^{2}\mathscr{P}_{6}^{3}T_{1}T_{5}+\mathscr{P}_{1}^{2}\mathscr{P}_{2}^{3}\mathscr{P}_{3}^{4}\mathscr{P}_{4}^{6}\mathscr{P}_{5}^{5}\mathscr{P}_{6}^{4}T_{6}\right)+\dots-\mathscr{P}_{1}^{18}\mathscr{P}_{2}^{27}\mathscr{P}_{3}^{36}\mathscr{P}_{4}^{54}\mathscr{P}_{5}^{45}\mathscr{P}_{6}^{36}$
(7.72)

where we have omitted the explicit expressions for the terms from $t^{24}$ to
$t^{1}$, and we omitted the dependence on $x$ in the notations for the
polynomial coefficients so that $\mathcal{P}_{i}\equiv\mathcal{P}_{i}(x)$ and
$T_{i}\equiv T_{i}(x)$. The curve 7.72 has $x$-degree $27v_{6}$, and, of
course, is not the most economical. By rescaling $g(x)\to\zeta(x)g(x)$ with a
suitably chosen $\zeta(x)$ of degree $-v_{6}$ made of some powers of the
factors in fundamental polynomials we can reduce the degree of (7.72).

The most standard conformal $E_{6}$ quiver, which arises from the degenerate
limit $\mathfrak{q}_{0}\to 0$ in the node ‘‘0’’ of the affine
$\widehat{E}_{6}$ quiver, has matter polynomial
$\mathscr{P}_{2}=\mathfrak{q}_{2}Y_{0}$ of degree $N$ only at the node ‘‘2’’
to which the affine node ‘‘0’’ was attached, while the degrees of the gauge
polynomials are fixed by the Dynkin marks ${\mathbf{v}}_{i}=Na_{i}$, that is
$(\mathbf{v}_{1},\dots,\mathbf{v}_{6})=(N,2N,2N,3N,2N,N)$. For such conformal
$E_{6}$ quiver, the curve 7.72 has canonical reduced form under the choice
$\zeta^{-1}(x)=Y_{0}(x)$ and the degree of the reduced curve is
$6N=2\mathbf{v}_{*}$ where $\mathbf{v}_{*}\equiv\mathbf{v}_{4}=3N$ denotes the
rank in the trivalent node ‘‘4’’. The reduced curve of such special conformal
$E_{6}$ quiver is $\mathcal{R}_{E_{6}}(t,x)$, with
$\mathscr{P}_{i}=\mathfrak{q}_{i},i\neq
2;\mathscr{P}_{2}=\mathfrak{q}_{2}Y_{0}$ we find

$\mathcal{R}_{E_{6}}(t,x)=t^{27}Y_{0}^{6}-t^{26}Y_{0}^{5}T_{6}+t^{25}\mathfrak{q}_{6}Y_{0}^{4}T_{5}-t^{24}\mathfrak{q}_{5}\mathfrak{q}_{6}^{2}Y_{0}^{3}T_{4}+t^{23}\left(-\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{4}\mathfrak{q}_{5}^{4}\mathfrak{q}_{6}^{4}Y_{0}^{4}T_{1}^{2}+\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{4}\mathfrak{q}_{5}^{4}\mathfrak{q}_{6}^{4}Y_{0}^{4}T_{3}+\mathfrak{q}_{4}\mathfrak{q}_{5}^{2}\mathfrak{q}_{6}^{3}Y_{0}^{2}T_{2}T_{3}-\mathfrak{q}_{2}\mathfrak{q}_{3}\mathfrak{q}_{4}^{2}\mathfrak{q}_{5}^{2}\mathfrak{q}_{6}^{3}Y_{0}^{3}T_{1}T_{5}+\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{3}\mathfrak{q}_{3}^{4}\mathfrak{q}_{4}^{6}\mathfrak{q}_{5}^{5}\mathfrak{q}_{6}^{4}Y_{0}^{5}T_{6}\right)+\dots-t^{2}\mathfrak{q}_{1}^{15}\mathfrak{q}_{2}^{23}\mathfrak{q}_{3}^{30}\mathfrak{q}_{4}^{46}\mathfrak{q}_{5}^{39}\mathfrak{q}_{6}^{32}Y_{0}^{4}T_{3}+t\mathfrak{q}_{1}^{16}\mathfrak{q}_{2}^{25}\mathfrak{q}_{3}^{33}\mathfrak{q}_{4}^{50}\mathfrak{q}_{5}^{42}\mathfrak{q}_{6}^{34}Y_{0}^{5}T_{1}-\mathfrak{q}_{1}^{18}\mathfrak{q}_{2}^{27}\mathfrak{q}_{3}^{36}\mathfrak{q}_{4}^{54}\mathfrak{q}_{5}^{45}\mathfrak{q}_{6}^{36}Y_{0}^{6}$
(7.73)

where again we only indicated the middle terms but skipped the explicit
expressions. Indeed, one sees that the curve 7.73 of the $E_{6}$ quiver with
the standard rank assignments $\mathbf{v}_{i}=Na_{i}$ has degree $6N$. At the
limit $x\to\infty$ the 27 roots of $\mathcal{R}_{E_{6}}(t,x)$ in 7.73 approach
the set of points in the $t$-plane labeled by the weights $\lambda$ in the
$\mathbf{27}$ representation of $E_{6}$ and given explicitly by
$\prod_{i=1}^{6}\mathfrak{q}_{i}^{(\lambda_{i},\lambda-\lambda_{i})}$, or

$\left\\{\ \prod_{i=1}^{6}\mathfrak{q}_{i}^{n_{i}}\ |\
\sum_{i=1}^{6}n_{i}\alpha_{i}\
=\lambda_{6}-\lambda,\quad\lambda\in\mathrm{weights}(R_{6})\right\\}$ (7.74)

where $n_{i}$ are the coefficients of the expansion in the basis of simple
roots of the difference between a given weight in $\mathbf{27}$ and the
highest weight. One can associate a Higgs field to the spectral curve 7.73
with poles in the 27 punctures (7.74) with certain relations. In other words,
the curve 7.73 realizes a certain embedding of the standard conformal $E_{6}$
quiver theory with gauge group ranks $\mathbf{v}_{i}=(N,2N,2N,3N,2N,N)$ to
some specialization of the $A_{26}$ theory with ranks $(6N,6N,\dots,6N)$, and
this embedding can be lifted to the Higgs field spectral curve representation
of (7.73).

For non-standard assignments of $\mathbf{w}_{i}$ and $\mathbf{v}_{i}$ for the
conformal $E_{6}$ quiver we did not find a simple choice of $\zeta(x)$
reducing the curve 7.72 to the minimal degree. For small ranks
$\mathbf{v}_{i},\mathbf{w}_{i}$ we can find the reduced curve using the brute
search minimization problem on the total degree of the reduced curve under
$g(x)\to\zeta(x)g(x)$. We have found different chambers in the space of
parameters $\mathbf{w}_{i},\mathbf{v}_{i}$ with piece-wise linear dependence
of the reduced degree of $\mathbf{w}_{i}$ or $\mathbf{v}_{i}$’s but not a
simple expression. For example, in several examples we find

$(w_{i})$ | $(v_{i})$ | reduced curve $x$-degree
---|---|---
$(0,4,0,0,0,0)$ | $(4,8,8,12,8,4)$ | 24
$(3,0,0,0,0,3)$ | $(6,6,9,12,9,6)$ | 33
$(6,0,0,0,0,0)$ | $(8,6,10,12,8,4)$ | 40
$(4,0,0,0,0,1)$ | $(6,5,8,10,7,4)$ | 31
$(6,0,0,0,0,3)$ | $(10,9,14,18,13,8)$ | 53

where the first three lines list different conformal $E_{6}$ quivers sharing
the same $\mathbf{v}_{*}=12$, and one can see that the curve of the minimal
degree $2\mathbf{v}_{*}$ is obtained in the standard assignment
$\mathbf{w}_{i}=0,i\neq 2$ associated to the degenerate limit of the affine
$E_{6}$.

####  7.3.2. $E_{7}$ theory

We write the spectral curve in, for example, the $\mathbf{56}$ representation
of $E_{7}$ similar to the $E_{6}$ case. If
$(\mathbf{v}_{0},\dots,\mathbf{v}_{7})=Na_{i}$ where $a_{i}$ are Dynkin marks
of $E_{7}$ quiver, again, similar to $E_{6}$ quiver we find that the reduced
curve of the standard conformal $E_{7}$ quiver obtained from the degenerate
limit of the affine theory has $x$-degree $12N=3\mathbf{v}_{*}$ where
$\mathbf{v}_{*}=\mathbf{v}_{4}=4N$ is rank at the trivalent node. The standard
$E_{7}$ quiver spectral curve hence is realized as a specialization of the
spectral curve for $A_{55}$ quiver with ranks $(12N,12N,\dots,12N)$, or
Hitchin system with $56$ punctures on $t$-plane associated to the weights in
$\mathbf{56}$.

####  7.3.3. $E_{8}$ theory

For $E_{8}$ the minimal representation is adjoint $\mathbf{248}$. The reduced
curve in the adjoint representation for the standard conformal $E_{8}$ quiver
obtained from the degenerate limit of the affine theory has $x$-degree
$60N=10\mathbf{v}_{*}$ where $\mathbf{v}_{*}=6N$ is rank at the trivalent
node. Hence the standard conformal $E_{8}$ quiver spectral curve is realized
as a specialization of the spectral curve for $A_{247}$ quiver with ranks
$60(N,N,\dots,N)$, or Hitchin system with $240$ punctures on $t$-plane
associated to the non-zero adjoint weights in $\mathbf{248}$.

###  7.4. Class II theories of $A$ type and class II* theories

Let us start with the simplest nontrivial examples, and then pass onto a
general case.

####  7.4.1. Class II $\widehat{A}_{1}$ theory

For the class II theory we shift the arguments of ${\mathscr{Y}}_{i}(x)$ by
${\mu}_{i}$ to get rid of the bi-fundamental masses.

Let $g(x)\in\widehat{SL_{2}}$:

$g(x)={\mathfrak{q}}_{0}^{-{\widehat{\lambda}}_{0}^{\vee}}{\mathfrak{q}}_{1}^{-{\widehat{\lambda}}_{1}^{\vee}}{\mathscr{Y}}_{0}(x)^{{\widehat{\alpha}}_{0}^{\vee}}{\mathscr{Y}}_{1}(x)^{\widehat{\alpha}_{1}^{\vee}}$
(7.75)

We have: ${\mathfrak{q}}={\mathfrak{q}}_{0}{\mathfrak{q}}_{1}$,

$g(x)^{\alpha_{1}}=\frac{{\mathscr{Y}}_{1}^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}^{2}},\quad
g(x)^{-\delta}={\mathfrak{q}},\quad
g(x)^{\widehat{\lambda}_{0}}={\mathscr{Y}}_{0}(x)$ (7.76)

The normalized $\widehat{sl_{2}}$ characters (6.22) of the fundamental
representations ${\widehat{R}}_{0},{\widehat{R}}_{1}$ are equal to

$\displaystyle{{\mathscr{X}}}_{0}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=\,\frac{{\mathscr{Y}}_{0}(x)}{\phi({\mathfrak{q}})}\theta_{3}\left(\frac{{\mathscr{Y}}_{1}(x)^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}(x)^{2}};{\mathfrak{q}}^{2}\right)$
(7.77)
$\displaystyle{{\mathscr{X}}}_{1}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=\,\left(\frac{{\mathfrak{q}}_{1}}{\mathfrak{q}_{0}}\right)^{\frac{1}{4}}\frac{{\mathscr{Y}}_{0}(x)}{\phi({\mathfrak{q}})}\theta_{2}\left(\frac{{\mathscr{Y}}_{1}(x)^{2}}{{\mathfrak{q}}_{1}{\mathscr{Y}}_{0}(x)^{2}};{\mathfrak{q}}^{2}\right)$

(see the appendix for our conventions on elliptic functions). The characters
(7.77) are invariant under the Weyl transformations

$\displaystyle{\mathscr{Y}}_{0}\to{\mathfrak{q}}_{0}{\mathscr{Y}}_{0}^{-1}{\mathscr{Y}}_{1}^{2}$
(7.78)
$\displaystyle{\mathscr{Y}}_{1}\to{\mathfrak{q}}_{1}{\mathscr{Y}}_{1}^{-1}{\mathscr{Y}}_{0}^{2}$

and therefore we can equate them to the polynomials:

$\displaystyle{{\mathscr{X}}}_{0}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=T_{0}(x)$
(7.79) $\displaystyle\qquad\qquad
T_{0,0}=\frac{\theta_{3}\left({\mathfrak{q}}_{1}^{-1};{\mathfrak{q}}^{2}\right)}{\phi({\mathfrak{q}})}$
$\displaystyle{{\mathscr{X}}}_{1}({\mathscr{Y}}(x),{\bf\mathfrak{q}})=T_{1}(x)$
$\displaystyle\qquad\qquad
T_{1,0}=\left(\frac{{\mathfrak{q}}_{1}}{\mathfrak{q}_{0}}\right)^{\frac{1}{4}}\frac{\theta_{2}\left({\mathfrak{q}}_{1}^{-1};{\mathfrak{q}}^{2}\right)}{\phi({\mathfrak{q}})}$

The values of characters (7.77) and ${\mathfrak{q}}_{0},{\mathfrak{q}}_{1}$
define ${\mathscr{Y}}_{0}$ and ${\mathscr{Y}}_{1}$ up to an affine Weyl
transformation. To recover ${\mathscr{Y}}_{0}$ and ${\mathscr{Y}}_{1}$ we
invert the relations (7.77):

$\displaystyle{\mathscr{Y}}_{1}(x)={\mathfrak{q}}_{1}^{\frac{1}{2}}{\mathscr{Y}}_{0}(x)t$
(7.80)
$\displaystyle{\mathscr{Y}}_{0}(x)=\frac{\phi({\mathfrak{q}})}{\theta_{3}(t^{2};{\mathfrak{q}}^{2})}T_{0}(x)$

and express

$\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{{\theta}_{3}(t^{2};{\mathfrak{q}}^{2})}{{\theta}_{2}(t^{2};{\mathfrak{q}}^{2})}=\frac{T_{0}(x)}{T_{1}(x)}$
(7.81)

Actually, the ratio

${\xi}=\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{{\theta}_{3}(t^{2};{\mathfrak{q}}^{2})}{{\theta}_{2}(t^{2};{\mathfrak{q}}^{2})}$

is a meromorphic function on $\mathscr{E}$ with two first order poles at
$t=\pm\mathrm{i}$ and two simple zeroes at $t=\pm\mathrm{i}\mathfrak{q}$.
Therefore

$\xi={\xi}_{\infty}\frac{X(t,\mathfrak{q})-X_{0}}{X(t,\mathfrak{q})-X_{1}},\quad
X_{0}:=X(\mathrm{i}\mathfrak{q},\mathfrak{q}),\quad
X_{1}:=X(\mathrm{i},\mathfrak{q})$ (7.82)

with

${\xi}_{\infty}=\left(\frac{\mathfrak{q}_{0}}{{\mathfrak{q}}_{1}}\right)^{\frac{1}{4}}\frac{\theta_{3}(1,\mathfrak{q}^{2})}{\theta_{2}(1,\mathfrak{q}^{2})}$
(7.83)

and the explicit $\mathfrak{q}$-series for $X(t,\mathfrak{q})$ is given in
(LABEL:eq:weierx1),(LABEL:eq:weierx2). Hence, the algebraic Seiberg-Witten
curve $C_{u}$ describing the $\widehat{A}_{1}$ theory is a two-fold cover of
the rational curve ${\Sigma}_{u}$

$({\xi}_{\infty}T_{1}(x)-T_{0}(x))X-({\xi}_{\infty}T_{1}(x)X_{0}-T_{0}(x)X_{1})=0$
(7.84)

defined by the Weierstraß cubic (LABEL:eq:wxy). There are $4N$ branch points
of the $2:1$ cover $C_{u}\to{\Sigma}_{u}$:

$\displaystyle{\xi}_{\infty}T_{1}(x_{\infty,{\mathbf{a}}})-T_{0}(x_{\infty,{\mathbf{a}}})=0$
(7.85)
$\displaystyle({\xi}_{\infty}T_{1}(x_{{\alpha},{\mathbf{a}}})-T_{0}(x_{{\alpha},{\mathbf{a}}}))e_{\alpha}-({\xi}_{\infty}T_{1}(x_{{\alpha},{\mathbf{a}}})X_{0}-T_{0}(x_{{\alpha},{\mathbf{a}}})X_{1})=0$
$\displaystyle{\alpha}=1,2,3,\qquad{\alpha}=1,\ldots,N$

which can be split into $2$ groups of $N$ pairs, corresponding to the cycles
$A_{i{\mathbf{a}}}$ with $i=0,1$, e.g. $A_{0,{\mathbf{a}}}$ is a small circle
around the cut which connects $x_{1,{\mathbf{a}}}$ to $x_{2,{\mathbf{a}}}$,
while $A_{1,{\mathbf{a}}}$ is a small circle around the cut which connects
$x_{3,{\mathbf{a}}}$ to $x_{\infty,{\mathbf{a}}}$. The special coordinates are
computed by the periods of

$dS_{-}=x\,d{\log}\,(t)=x\,\frac{dX}{Y}$

The curve $C_{u}$ is the spectral curve. The cameral curve ${\mathcal{C}}_{u}$
is a $\mathbb{Z}$-cover of spectral curve $C_{u}$, which is given by the same
equations but now with $t\in{\mathbb{C}}^{\times}$ as opposed to
$t\in{\mathscr{E}}$. On cameral curve ${\mathcal{C}}_{u}$ we have the second
differential

$dS_{+}=x\,d{\log}\,{\theta}_{3}(t^{2};{\mathfrak{q}})$

which would be a multi-valued differential on spectral curve $C_{u}$ whose
periods are defined up to the periods of $dS_{-}$, similar to the
polylogarithm motives [Cartier:1987].

####  7.4.2. Class II* $\widehat{A}_{0}$ theory

This is a (noncommutative) $U(1)$ ${\mathcal{N}}=2^{*}$ theory. This theory
was solved in [Nekrasov:2003rj] by the similar method. There is only one
amplitude ${\mathscr{Y}}(x)={\mathscr{Y}}_{0}(x)$, with the single interval
$I$ as its branch cut, the single function

$t(x)\equiv
t_{0}(x)=\frac{{\mathscr{Y}}(x)}{{\mathscr{Y}}(x+{\mathfrak{m}})}.$

with two branch cuts $I$ and $I-{\mathfrak{m}}$. Crossing the $I$ cut maps
$t(x)\mapsto{\mathfrak{q}}t(x-{\mathfrak{m}})$. Crossing the cut
$I-{\mathfrak{m}}$ has the opposite effect:
$t(x)\mapsto{\mathfrak{q}}^{-1}t(x+{\mathfrak{m}})$. The extended functions

$t_{j}(x)={\mathfrak{q}}^{j}t(x-j{\mathfrak{m}})$

The analytically continued function $t(x)$ has cuts at
$I+{\mathfrak{m}}\mathbb{Z}$. The sheets of the Riemann surface of $t(x)$ are
labeled by $j\in\mathbb{Z}$, so that on the sheet $j$ the cuts are at
$I-j{\mathfrak{m}}$, and $I-(j+1){\mathfrak{m}}$. Upon crossing
$I+j{\mathfrak{m}}$ the $t_{j}(x)$ function transforms to $t_{j+1}(x)$
function. As $x\to\infty$ on this sheet the corresponding branch of $t(x)$
approaches ${\mathfrak{q}}^{j}$. These conditions uniquely fix the inverse
function to be the logarithmic derivative of $\theta_{1}$:

$x=a+{\mathfrak{m}}\,t\frac{d}{dt}{\rm log}\,{\theta}_{1}(t;{\mathfrak{q}})$
(7.86)

####  7.4.3. Class II $A_{r}$ theories

In order to solve the general rank $r$ theory, it is convenient to form a
linear combination of fundamental characters of ${\widehat{A}}_{r}$.
Ultimately we would like to define a regularized version of the characteristic
polynomial of $g(x)$, where, as in the general case, after the shift of the
arguments of ${\mathscr{Y}}_{i}(x)\to{\mathscr{Y}}_{i}(x+{\mu}_{i})$:

$g(x)=\prod_{i=0}^{r}{\mathfrak{q}}_{i}^{-{\widehat{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}(x)^{{\widehat{\alpha}}_{i}^{\vee}}$
(7.87)

Using $t_{i}(x)=g(x)^{e_{i}}$ (see the appendix), we compute:

$t_{i}(x)={\check{t}}_{i}\,\frac{{Y}_{i}(x)}{{Y}_{i-1}(x)},\quad
i=1,\dots,r+1$ (7.88)

where we extended the amplitude functions ${\mathscr{Y}}_{j}(x)$ defined for
$j=0,\ldots,r$ to be defined for all $j\in\mathbb{Z}$ by periodicity
${Y}_{j}(x)={\mathscr{Y}}_{j+(r+1)}(x)$ and where

${\bf t}(x)=(t_{1}(x),t_{2}(x),\ldots,t_{r+1}(x))$

represents an element of the maximal torus of $SL(r+1,{\mathbb{C}})$, i.e.

$\prod_{i=1}^{r+1}t_{i}(x)=1.$

The $\check{t}_{i}$ are the asymptotic values at $x\to\infty$ of $t_{i}(x)$
and are given by

$\check{t}_{i}=(\mathfrak{q}_{i}\dots\mathfrak{q}_{r})^{-1}(\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\dots\mathfrak{q}_{r}^{r})^{\frac{1}{r+1}},\quad
i=1\dots r+1$ (7.89)

and

$g(x)^{-\delta}={\mathfrak{q}},\quad
g(x)^{\widehat{\lambda}_{0}}={\mathscr{Y}}_{0}(x).$ (7.90)

Now we shall explore the relation between the conjugacy classes in Kac-Moody
group and the holomorphic bundles on elliptic curve $\mathscr{E}$. We will
consider a family of bundles on $\mathscr{E}$ parametrized by the
$\mathbf{C}_{\left\langle x\right\rangle}$-plane, as e.g. in
[Friedman:1997ih]. We start with individual bundles.

Let V be a rank $r+1$ polystable vector bundle of degree zero over the
elliptic curve ${\mathscr{E}}=\mathbb{C}^{\times}/\mathfrak{q}^{\mathbb{Z}}$,
with trivial determinant,

${\rm det}V\approx{\mathcal{O}}_{\mathscr{E}}$

Such bundle always splits as a direct sum of line bundles

$V=\bigoplus_{i=1}^{r+1}L_{i}\ .$

Each summand is a degree zero line bundle $L_{i}$ which can be represented as
$L_{i}={\mathcal{O}}(p_{0})^{-1}{\mathcal{O}}(t_{i})$ where ${\mathcal{O}}(p)$
is the degree one line bundle whose divisor is a single point $p\in E$ and
$p_{0}$ denotes the point $t=1$ corresponding to the identity in the abelian
group law on the elliptic curve $\mathscr{E}$. A meromorphic section $s_{i}$
of $L_{i}$ with a simple pole at $t=1$ and zero at $t=t_{i}$ can be written
explicitly using the theta-functions:

$s_{i}(t)=\frac{\theta(t/t_{i};\mathfrak{q})}{\theta(t;\mathfrak{q})}$ (7.91)

and is unique up to a multiplicative constant. To each degree zero vector
bundle $V$ with the divisor

$D_{V}=-(r+1)p_{0}+t_{1}+\dots+t_{r+1}$

of ${\rm det}V$ we associate a projectively unique section $s$ of its
determinant ${\det}V$ which has zeroes at $t_{1},\dots,t_{r+1}$ and a pole of
the order not greater than $r+1$ at $t=1$:

$s(t;{\bf
t})=\prod_{i=1}^{r+1}\frac{\theta(t/t_{i};\mathfrak{q})}{\theta(t;\mathfrak{q})}$
(7.92)

where we explicitly indicate the $\bf t$ dependence of the section $s$. Now
set $t_{i}=t_{i}(x)$ given by (7.88). The meromorphic sections
$s(t;{\mathbf{t}}(x);\mathfrak{q})$ can be expanded in terms of the theta-
functions $\Theta_{j}(\mathscr{Y}_{0}(x);{\mathbf{t}};\mathfrak{q})$ and
characters of $\widehat{A}_{r}$ (see (LABEL:eq:Archar)(LABEL:eq:Ar-theta)) as
follows

$\mathscr{Y}_{0}(x)\prod_{i=1}^{r+1}\frac{\theta(t/t_{i}(x);\mathfrak{q})}{\theta(t,{\mathfrak{q}})}=\\\
\sum_{i=0}^{r}\mathfrak{q}^{-\frac{i}{2}}\mathfrak{q}^{\frac{i^{2}}{2(r+1)}}\Theta_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})\phi_{i}(t;\mathfrak{q})=\\\
=\phi(\mathfrak{q})^{r}\sum_{i=0}^{r}\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})\phi_{i}(t;\mathfrak{q})$
(7.93)

where the functions $\phi_{i}(t;\mathfrak{q})$ are normalized meromorphic
elliptic functions defined in the appendix LABEL:subsubsec:phi. Hence we find
from (6.39) and (LABEL:eq:T-matrix) that the section $s(t,x)$ (7.92) obeys

${\mathscr{Y}}_{0}(x)s(t,x)=\phi(\mathfrak{q})^{r}\sum_{i=0}^{r}\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q})M_{i{\tilde{j}}}(\mathfrak{q})\tilde{\phi}_{{\tilde{j}}}(t;\mathfrak{q})$
(7.94)

where ${\tilde{\phi}}_{\tilde{j}}(t;{\mathfrak{q}})$ denotes the Weierstraß
monomials of Weierstraß elliptic functions $X(t,\mathfrak{q})$ and
$Y(t,\mathfrak{q})$; and $M_{i{\tilde{j}}}$ is a certain modular matrix as
defined in LABEL:subsubsec:phi. Recalling (6.5) that the characters
$(\chi_{i}(\mathscr{Y}_{0}(x);{\mathbf{t}(x)};\mathfrak{q}))$ evaluated on the
solutions $(\mathscr{Y}_{i}(x))$ are polynomials in $x$, from (6.22)(6.28) we
get

$\frac{{\mathscr{Y}}_{0}(x)s(t,x)}{\phi(\mathfrak{q})^{r}}=\sum_{i=0}^{r}\left(\prod_{j=0}^{r}\mathfrak{q}_{j}^{-\widehat{\lambda}_{i}(\widehat{\lambda}_{j}^{\vee})}\right)T_{i}(x)\sum_{{\tilde{j}}}M_{i\tilde{j}}(\mathfrak{q})\tilde{\phi}_{{\tilde{j}}}(t;\mathfrak{q})$
(7.95)

The section $s(t,x)$ vanishes at the $r+1$ points $t_{1}(x),\dots,t_{r+1}(x)$
for each $x\in\mathbf{C}_{\left\langle x\right\rangle}$, and hence defines the
$r+1$-folded spectral cover of $\mathbf{C}_{\left\langle x\right\rangle}$
plane by the equation

$R(t,x)=0$ (7.96)

where $R(t,x)$ is the right-hand side of (7.95). The curve (7.96) coincides
with the curve in [Witten:1997sc] constructed from by lifting to $M$-theory
the IIA brane arrangement realizing the elliptic model with $\mathfrak{m}=0$.

####  7.4.4. Class II* theory

Recall that in (6.37) we defined an infinite set of functions
$Y_{i}(x),i\in\mathbb{Z}$ . The analogue of the formula (1.2) is the matrix
$g(x)\in{\widehat{GL}_{\infty}}$, (cf. (6.39)):

$g(x)=Y_{0}(x)^{K}\times{\rm
diag}\,\left(\,t_{i}(x)\,\right)_{i\in{\mathbb{Z}}},\qquad
t_{i}(x)={\check{t}}_{i}\,\frac{Y_{i}(x)}{Y_{i-1}(x)}$ (7.97)

where ${\check{t}}_{i}$, $i\in\mathbb{Z}$ solve

${\check{t}}_{i+1}={\mathfrak{q}}_{i\,{\rm mod}\,(r+1)}{\check{t}}_{i},$

and are normalized as in (6.40)

$\prod_{j=1}^{r+1}{\check{t}}_{j}=1$

so that for $i=1,\ldots,r+1$ the ${\check{t}}_{i}$ coincide with those in
(6.40), and

${\check{t}}_{i+b(r+1)}={\check{t}}_{i}{\mathfrak{q}}^{b},\qquad{\check{t}}^{[i+b(r+1)]}={\check{t}}^{[i]}\left({\mathfrak{q}}^{r+1}\right)^{\frac{b(b-1)}{2}}$
(7.98)

The fundamental characters of $\widehat{GL}_{\infty}$ evaluated on $g(x)$,
${\chi}_{i}(g(x))$ are associated with representations ${\mathcal{R}}_{i}$ of
$\widehat{GL}_{\infty}$ with the highest weight taking value (cf.
(LABEL:eq:hwgli)):

$g(x)^{\tilde{\lambda}_{i}}=\ Y_{i}(x)\ {\check{t}}^{[i]}=\ Y_{0}(x)\
t(x)^{[i]}$ (7.99)

The characters are given by the infinite sums over all partitions
${\lambda}=({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots\geq{\lambda}_{{\ell}({\lambda})}>0)$
and so are the normalized invariants

$\displaystyle{\mathscr{X}}_{i}(\\{Y_{j}(x)\\},\mathfrak{q})=$
$\displaystyle\,\frac{1}{{\check{t}}^{[i]}}{\chi}_{i}(g(x))=$ (7.100)
$\displaystyle\sum_{{\lambda}}\prod_{j=1}^{{\ell}({\lambda})}\left({\mathfrak{q}}_{i-j+1}^{[{\lambda}_{j}]}\frac{Y_{i+{\lambda}_{j}-j+1}(x)}{Y_{i+\lambda_{j}-j}(x)}\right)Y_{i-{\ell}({\lambda})}(x)$
$\displaystyle\qquad=Y_{i}(x)+{\mathfrak{q}}_{i}\frac{Y_{i+1}(x)Y_{i-1}(x)}{Y_{i}(x)}+\ldots$

where we use the notation section 1.2.

The invariant ${\mathscr{X}}_{i}$ in (7.100) is a convergent series for
$|{\mathfrak{q}}_{i}|<1$ like the theta-series, if $t_{i}(x)$ is uniformly
bounded. In fact, for the periodic chain of arguments, i.e. for
$Y_{i}(x)=Y_{i+r+1}(x)$ the $\mathfrak{gl}_{\infty}$ character (7.100) reduces
to the usual affine character of $\widehat{\mathfrak{g}\mathfrak{l}}_{r}$. The
convergence of ${\mathscr{X}}_{i}$ in the class II* case is more subtle. We
shall comment on this below. For the moment let us view the invariants as the
formal power series in $\mathfrak{q}$ with coefficients in Laurent polynomials
in $Y_{i}(x)$.

For the class II* theory the extended amplitudes $Y_{i}(x)$ are quasi-periodic
in $i$, cf. (6.38), so

${{\mathscr{X}}}_{i+r+1}\left(\\{Y_{j}(x)\\},\mathfrak{q}\right)={{\mathscr{X}}}_{i}\left(\\{Y_{j}(x-(r+1){\mathfrak{m}})\\},\mathfrak{q}\right)$
(7.101)

The cameral curve ${\mathcal{C}}_{u}$ for the class II* $A_{r}$ theory is
defined by the system of $r+1$ functional equations

${\mathscr{X}}_{i}(\,\\{\,Y_{j}(x)\\},\mathfrak{q})=T_{i}(x),\quad
i=0,\dots,r$ (7.102)

with

$\displaystyle
T_{i}(x)=T_{i,0}x^{N}+T_{i,1}x^{N-1}+\sum_{{\mathbf{a}}=2}^{N}u_{i,{\mathbf{a}}}x^{N-\mathbf{a}}\,,$
(7.103) $\displaystyle\qquad\qquad
T_{i,0}=\sum_{\lambda}\prod_{j=1}^{{\ell}({\lambda})}{\mathfrak{q}}_{i-j+1}^{[\lambda_{j}]}$

Let us now describe the II* analogue of the spectral curve, and find its
realization in terms of some version of the Hitchin’s system. Along the way we
shall get an alternative derivation of (7.95) with the benefit of getting its
Hitchin’s form as well.

We form the generating function of ${\mathscr{X}}_{i}$’s and study its
automorphic properties. The idea is to regularize the infinite product

$\prod_{i\in\mathbb{Z}}(1-t_{i}(x)/t)/(1-{\check{t}}_{i}/t)$

while keeping the same set of zeroes and poles. Thus, we define

$R(t,x)=\frac{Y_{0}(x)}{D_{0}(t;{\bf\mathfrak{q}})}\prod_{k=1}^{\infty}(1-t_{k}(x)t^{-1})(1-t\,t_{1-k}(x)^{-1})$
(7.104)

where

$\displaystyle
D_{0}(t;{\bf\mathfrak{q}})=\prod_{k=1}^{\infty}(1-{\check{t}}_{k}t^{-1})(1-t\,{\check{t}}_{1-k}^{-1})$
(7.105)
$\displaystyle\qquad\qquad\qquad=\prod_{i=1}^{r+1}\frac{{\theta}(t/{\check{t}}_{i};{\mathfrak{q}})}{{\phi}({\mathfrak{q}})}$

First of all, given that at large $x$ the eigenvalues $t_{k}(x)$ approach
${\check{t}}_{k}$ which, in turn, behave as ${\mathfrak{q}}^{\frac{k}{r+1}}$,
we expect (7.104) to define the converging product, at least for large enough
$x$.

Secondly, let us check that (7.104) is ${}^{i}{\mathcal{W}}$-invariant. Let
$i=0,\ldots,r$, ${\mathbf{a}}=1,\ldots,N$. While crossing the
$I_{i,{\mathbf{a}}}$ cut the ‘‘eigen-value’’ $t_{i}(x)$ maps to $t_{i+1}(x)$,
which, in case $i\geq 1$ or $i<0$, leaves (7.104) manifestly invariant. For
$i=0$ several factors in $\Delta(t,x)$ transform, altogether conspiring to
make it invariant:

$\displaystyle
Y_{0}(x)\mapsto{\mathfrak{q}}_{0}Y_{-1}(x)Y_{1}(x)/Y_{0}(x)=t_{1}(x)/t_{0}(x),$
(7.106) $\displaystyle(1-t_{1}(x)t^{-1})(1-t\,t_{0}(x)^{-1})\mapsto$
$\displaystyle\qquad(1-t_{0}(x)t^{-1})(1-t\,t_{1}(x)^{-1})=\frac{t_{0}(x)}{t_{1}(x)}(1-t_{1}(x)t^{-1})(1-t\,t_{0}(x)^{-1})$

Thirdly, let us introduce the analogues of the spectral determinants for all
fundamental representations ${\mathcal{R}}_{i}$:

$\displaystyle{\Delta}_{i}(t,x)=\frac{Y_{i}(x)}{D_{i}(t;{\bf\mathfrak{q}})}\prod_{k=i+1}^{\infty}(1-t_{k}(x)t^{-1})(1-t\,t_{2i+1-k}(x)^{-1})$
(7.107) $\displaystyle\qquad\qquad
D_{i}(t;{\bf\mathfrak{q}})=\prod_{k=i+1}^{\infty}(1-{\check{t}}_{k}t^{-1})(1-t\,{\check{t}}_{2i+1-k}^{-1})$

Using
$D_{i+1}(t;{\mathfrak{q}})=-t{\check{t}}_{i+1}^{-1}D_{i}(t;{\mathfrak{q}})$,
$Y_{i+1}(x)=t_{i+1}(x){\check{t}}_{i+1}^{-1}Y_{i}(x)$ we derive:
${\Delta}_{i}(t,x)=R(t,x)$ for all $i\in\mathbb{Z}$.

Then, the quasi-periodicity (7.101), (7.98) implies

$R({\mathfrak{q}}t,x+{\mathfrak{m}})={\Delta}_{r+1}(t,x)=R(t,x)$ (7.108)

Given the large $x$ asymptotics of $Y_{0}(x)$ and $t_{i}(x)$, we conclude:

$R(t,x)=x^{N}+\sum_{k=1}^{N}{\delta}_{k}(t)x^{N-k}$ (7.109)

where ${\delta}_{k}(t)$ are the quasi-elliptic functions, which have the first
order poles at $t={\check{t}}_{i}$, $i=0,\ldots r$ on the elliptic curve
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$. Indeed, the
poles come from the $D_{0}(t;{\mathfrak{q}})$ denominator, while the quasi-
ellipticity of $\delta_{k}(t)$ follows from (7.108):

${\delta}_{i}({\mathfrak{q}}t)-{\delta}_{i}(t)={\mathfrak{m}}^{i}+{\rm
polynomial\ in}\ {\mathfrak{m}}\ {\rm linear\ in}\
{\delta}_{k}({\mathfrak{q}}t),\qquad k<i$ (7.110)

Now use (LABEL:eq:fermch), (LABEL:eq:chari) to rewrite $R(t,x)$ as:

$\displaystyle R(t,x)\,=$
$\displaystyle\frac{\sum_{i\in\mathbb{Z}}(-t)^{i}{\check{t}}^{[i]}{{\mathscr{X}}}_{i}\left(\\{Y_{j}(x)\\},\mathfrak{q}\right)}{D_{0}(t;{\bf\mathfrak{q}})}$
(7.111)
$\displaystyle\qquad\qquad=\frac{1}{D_{0}(t;{\bf\mathfrak{q}})}\sum_{i\in\mathbb{Z}}(-t)^{i}{\check{t}}^{[i]}T_{i}(x)$

where we extended the definition of gauge polynomials $T_{i}(x)$ to
$i\in\mathbb{Z}$ by quasi-periodicity implied by (7.101):

$T_{i+r+1}(x)=T_{i}(x-{\mathfrak{m}})$ (7.112)

Armed with (7.112), (7.98) we reduce (7.111) to a finite sum: let

$r(t,x)=\sum_{i=0}^{r}(-t)^{i}{\check{t}}^{[i]}T_{i}(x)\,,$

then (cf. (LABEL:eq:phi_p))

$\displaystyle
R(t,x)=\frac{1}{D_{0}(t;{\mathfrak{q}})}\sum_{b\in{\mathbb{Z}}}r(t,x-b{\mathfrak{m}})\left((-t)^{b}{\mathfrak{q}}^{\frac{b(b-1)}{2}}\right)^{r+1}$
(7.113)
$\displaystyle\qquad\qquad=\frac{1}{D_{0}(t;{\mathfrak{q}})}\left({\theta}\left(-(-t)^{r+1};{\mathfrak{q}}^{r+1}\right)\ast_{{\mathfrak{m}}}r(t,x)\right)$

where the $\ast_{\hbar}$-product is defined by the usual Moyal formula:

$\left(f\ast_{\hbar}g\right)(t,x)=e^{\hbar\frac{\partial^{2}}{{\partial}{\xi}_{1}{\partial}{\eta}_{2}}-\hbar\frac{\partial^{2}}{{\partial}{\xi}_{2}{\partial}{\eta}_{1}}}|_{{\xi}={\eta}=0}\,f(t+{\eta}_{1},x+{\xi}_{2})g(t+{\eta}_{2},x+{\xi}_{2})$
(7.114)

The appearance of the $\ast$-product is the first hint that the class II*
theory has something to do with the noncommutative geometry. We shall indeed
soon see that a natural interpretation of the solution to the limit shape
equations of the class II* theory involves instantons on the noncommutative
four-manifold ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, where the
noncommutativity is ‘‘between’’ the $\mathbb{R}^{2}$ and the $\mathbb{T}^{2}$
components.

####  7.4.5.  Hitchin system on $T^{2}$

The above solution can be represented by the affine $GL(N)$ Hitchin system on
$\mathscr{E}$:

${\Phi}({\mathfrak{q}}t)={\Phi}(t)+N{\mathfrak{m}}\cdot{\bf 1}_{N}$ (7.115)

with $r+1$ rank $1$ punctures ${\check{t}}_{j}$:

$\displaystyle{\Phi}(t)\sim{\Phi}_{j}\frac{dt}{t-{\check{t}}_{j}},\qquad
j=1,\ldots,r+1$ (7.116)
$\displaystyle\qquad\qquad{\Phi}_{j}={\mathbf{u}}_{j}\otimes{\mathbf{v}}_{j}^{t},\qquad{\mathbf{u}}_{j},{\mathbf{v}}_{j}\in{\mathbb{C}}^{N}$

whose eigenvalues are fixed in terms of masses:

${\mathbf{v}}_{j}^{t}{\mathbf{u}}_{j}={\operatorname{tr}}{\Phi}_{j}=Nm_{j}\ .$
(7.117)

Actually, the vectors and covectors ${\mathbf{v}}_{j},{\mathbf{u}}_{j}$ are
defined up to the ${\mathbb{C}}^{\times}$-action

$({\mathbf{v}}_{j},{\mathbf{u}}_{j})\mapsto(z_{j}{\mathbf{v}}_{j},z_{j}^{-1}{\mathbf{u}}_{j}),\qquad
z_{j}\in{\mathbb{C}}^{\times}$ (7.118)

and (7.117) is the corresponding moment map equation, defining the coadjoint
orbit ${\mathcal{O}}_{j}$ of $SL(N,{\mathbb{C}})$. We can shift ${\Phi}(t)$ by
the meromorphic scalar matrix

${\bf\Phi}(t)={\Phi}(t)-\sum_{j=1}^{r+1}m_{j}{\xi}(t/{\check{t}}_{j})\frac{dt}{t}\,{\bf
1}_{N}\ ,$

which gives the following traceless meromorphic Higgs field (see
[Nekrasov:1995nq]):

${\bf\Phi}({t})=\left\|p_{a}{\delta}_{a}^{b}+\sum_{j=0}^{r}u_{j}^{b}v^{j}_{a}(1-{\delta}_{a}^{b})\frac{{\theta}_{1}({t}/{t}_{j}w_{b}/w_{a}){\theta}_{1}^{\prime}(1)}{{\theta}_{1}({t}/{t}_{j}){\theta}_{1}(w_{b}/w_{a})}\right\|_{a,b=1}^{N}$
(7.119)

which depends, in addition to the $SL(N,{\mathbb{C}})$-orbits
${\mathcal{O}}_{1},\ldots,{\mathcal{O}}_{r+1}$ on the choice
$(w_{1},\ldots,w_{N})$ of a holomorphic $SL(N,{\mathbb{C}})$ bundle on
$\mathscr{E}$, and the dual variables $(p_{1},\ldots,p_{N})$, subject to

$\sum_{a=1}^{N}p_{a}=0,\qquad\prod_{a=1}^{N}w_{a}=1$

There are additional constraints:

$\sum_{j=1}^{r+1}u_{j}^{a}v_{a}^{j}={\mathfrak{m}}$ (7.120)

which generate the action of the residual gauge transformations in the maximal
torus ${\bf T}=({\mathbb{C}}^{\times})^{N-1}$ of $SL(N,{\mathbb{C}})$. The
dimension of the corresponding phase space ${\mathfrak{P}}$, whose open subset
${{\mathfrak{P}}}^{\circ}$ is isomorphic to

${{\mathfrak{P}}}^{\circ}\approx\left(T^{*}\mathrm{Bun}_{SL(N,{\mathbb{C}})}({\mathscr{E}})\times\times_{j=1}^{r+1}{\mathcal{O}}_{j}\right)//{\bf
T}$ (7.121)

is equal to

${\rm dim}{{\mathfrak{P}}}=2(N-1)+(r+1)(2(N-1))-2(N-1)=2(r+1)(N-1)=2{\bf r}$
(7.122)

which is twice the dimension of the moduli space ${\mathfrak{M}}$ of vacua of
the class II* $A_{r}$ theory with the gauge group
${G_{\text{g}}}=SU(N)^{r+1}$. The remaining $r+1$ mass parameters are encoded
in the symplectic moduli of the coadjoint orbits ${\mathcal{O}}_{j}$, as
expected.

The relation to our solution is in the equality of two spectral determinants:

$R(t,x)={\rm
Det}_{N}\,\left[\left(x-\sum_{j}m_{j}{\xi}(t/t_{j})\right)\cdot{\bf
1}_{N}-{\bf\Phi}({t})\right]=0$ (7.123)

which is established by comparing the modular properties and the residues of
the left and the right hand sides.

Note the duality of the twisted periodicities of the gauge theory and
Hitchin’s system Lax operators:

$\displaystyle{\Phi}({\mathfrak{q}}t)=w^{-1}{\Phi}(t)w+{\mathfrak{m}}\cdot{\bf
1}_{N}\in{\mathfrak{sl}}(N,{\mathbb{C}})$ (7.124)
$\displaystyle{\mathfrak{q}}\cdot
g(x-{\mathfrak{m}})=S^{-1}g(x)S\in\widehat{GL}_{\infty}$

where $S$ is the shift operator $S=\sum_{i\in\mathbb{Z}}E_{i,i+r+1}$, and
$w={\rm diag}(w_{1},\ldots,w_{N})$. The Eq. (7.123) can be suggestively
written as:

${\rm Det}_{N}(x-{\Phi}(t))\approx{\rm Det}_{H}(t-g(x))$ (7.125)

where $H$ is the single-particle Hilbert space of a free fermion $\psi$.

####  7.4.6. Relation to many-body systems and spin chains

The parameters of the spectral curve (7.123) are holomorphic functions on
${{\mathfrak{P}}}^{\circ}$, which Poisson-commute, and define the integrable
system. One way of enumerating the Hamiltonians of the integrable system is to
mimic the construction of Hamiltonians (4.22) of the higher genus Hitchin
system. For example, the quadratic Casimir is a meromorphic $2$-differential
on $\mathscr{E}$ with the fixed second order poles at $t={\check{t}}_{j}$

${\operatorname{tr}}{\bf\Phi}(t)^{2}=\sum_{j=1}^{r+1}N^{2}m_{j}^{2}{\wp}(t/{\check{t}}_{j})dt^{2}+\sum_{j=1}^{r+1}U_{2,1,j}{\xi}(t/{\check{t}}_{j})+U_{2,0}$

The Hamiltonians $U_{2,0}$, $U_{2,1,j}$ are computed explicitly in
[Nekrasov:1995nq]. They describe the motion of $N$ particles on $\mathscr{E}$
with the coordinates $w_{1},\ldots,w_{N}$, which have additional
$GL(r+1,{\mathbb{C}})$-spin degrees of freedom. However, in view of our gauge
theory analysis, it seems more natural to view this system as the
$\widehat{GL}_{\infty}$-spin chain. We conjecture that the deformation
quantization of the properly compactified phase space ${\mathfrak{P}}$ will
contain the subalgebra ${\mathcal{A}}_{\mathfrak{m}}$ of the Yangian
$Y(\widehat{GL}_{\infty})$ algebra, which is a deformation of the Yangian of
the affine ${\widehat{A}}_{r}$.

The relation of many-body systems and spin chains based on finite dimensional
symmetry groups was discussed in the context of Hecke symplectic
correspondences in [Levin:2001nm, Olshanetsky:2008uu]. One can also interpret
the results of [Felder:1995iv] as the quantum version of this correspondence.

###  7.5. Class II theories of $D$ type

In this section $\mathfrak{g_{\text{q}}}=\widehat{D}_{r}$.

The fundamental weights of $\widehat{D}_{r}$ are
$\lambda_{0},\widehat{\lambda}_{i}=a_{i}^{\vee}\lambda_{0}+\lambda_{i},i=1,\dots,r$
where $\lambda_{i}$ are fundamental weights of $D_{r}$, and Dynkin labels are
$(a_{0},\dots,a_{r})=(1,1,2,\dots,2,1,1)$ (see Appendix C.3.2).
Correspondingly,

$\displaystyle t_{1}(x)=$
$\displaystyle\,\check{t}_{1}\mathscr{Y}_{1}(x)/\mathscr{Y}_{0}(x),$ (7.126)
$\displaystyle t_{2}(x)=$
$\displaystyle\,\check{t}_{2}\mathscr{Y}_{2}(x)/(\mathscr{Y}_{1}(x)\mathscr{Y}_{0}(x)),$
$\displaystyle t_{i}(x)=$
$\displaystyle\,\check{t}_{i}\mathscr{Y}_{i}(x)/\mathscr{Y}_{i-1}(x),\quad
i=3,\ldots,r-2$ $\displaystyle t_{r-1}(x)=$
$\displaystyle\,\check{t}_{r-1}\mathscr{Y}_{r-1}(x)\mathscr{Y}_{r}(x)/\mathscr{Y}_{r-2}(x),$
$\displaystyle t_{r}(x)=$
$\displaystyle\,\check{t}_{r}\mathscr{Y}_{r}(x)/\mathscr{Y}_{r-1}(x)$

with

$\displaystyle\check{t}_{i}$
$\displaystyle=\left(\mathfrak{q}_{i}\mathfrak{q}_{i+1}\dots\mathfrak{q}_{r-2}\right)^{-1}\left(\mathfrak{q}_{r-1}\mathfrak{q}_{r}\right)^{-\frac{1}{2}}$
(7.127) $\displaystyle\qquad i=1,\ldots,r-2$
$\displaystyle\check{t}_{r-1}=(\mathfrak{q}_{r-1}\mathfrak{q}_{r})^{-\frac{1}{2}}\,,\quad\check{t}_{r}=(\mathfrak{q}_{r-1}/\mathfrak{q}_{r})^{\frac{1}{2}}$

There are $4$ irreducible $\widehat{D}_{r}$ highest weight modules
$\widehat{R}_{0},\widehat{R}_{1},\widehat{R}_{r-1},\widehat{R}_{r}$ at level
$1$, and $r-3$ irreducible $\widehat{D}_{r}$ highest weight modules
$\widehat{R}_{2},\dots,\widehat{R}_{r-2}$ at level $2$. In this section, to
shorten formulae, we are using not the characters of $\widehat{R}_{i}$
themselves but the closely related affine Weyl invariant functions
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ at level $2$ and
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ at level $1$ expressed terms
of theta-functions explicitly as given in the appendix (LABEL:eq:Dr-inv). Such
functions ${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ and
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}$ differ from the actual
characters by a simple power of Euler function $\phi(\mathfrak{q})$ and some
$\mathfrak{q}$-dependent constant, also
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{0},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{1}$
appear as a linear combination of $\widehat{R}_{0}$ and $\widehat{R}_{1}$
characters, while
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r-1}$
appear as linear combination of $\widehat{R}_{r-1}$ and $\widehat{R}_{r}$
characters (see appendix (LABEL:eq:Dr-inv)).

$\begin{cases}{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}(\mathscr{Y}_{0}(x),{\mathbf{t}(x)};\mathfrak{q})=T_{j}(x)\\\
{}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{j}(\mathscr{Y}_{0}(x),{\mathbf{t}(x)};\mathfrak{q})=T_{j}(x)\\\
\end{cases}$ (7.128)

where polynomials $T_{j}(x)$ are of degree $N$ for $j=0,1,r-1,r$ and of degree
$2N$ for $j=2,\dots,r-2$. so that ${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}$
is of degree $1$ in $\mathscr{Y}_{0}$ for $j=0,1,r-1,r$ and
${}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}$ is of degree $2$ in
$\mathscr{Y}_{0}$ for $j=2,\dots,r-2$. Also, in this section the highest
coefficient of the polynomial $T_{j}(x)$ is normalized differently then in
6.22; one can find it as theta-series evaluating (LABEL:eq:Dr-inv) on
$\check{t}_{i}$.

Using the standard embedding $so(2r)\subset sl(2r)$ we construct the algebraic
equation of the spectral curve of the $\widehat{D}_{r}$ theory as the
specialization of the spectral curve for $\widehat{A}_{2r-1}$ theory. Indeed,
a vector bundle V associated to the vector representation of $SO(2r)$ splits
as the sum of $r$ pairs of line bundles

$L_{t_{i}}\oplus L_{t_{i}^{-1}}$

with the degree zero line bundle $L_{t}$ being

$L_{t}=\mathcal{O}(p_{0})^{-1}\mathcal{O}(t)$ (7.129)

and $p_{0}\in{\mathscr{E}}$ is our friend $t=1$. Then we proceed as in
(LABEL:eq:phi_p)(LABEL:eq:s-ThetaD)(7.92) by considering a meromorphic section
of the determinant bundle ${\rm det}$V$\approx{\mathcal{O}}_{\mathscr{E}}$

$s(t,x)=\prod_{i=1}^{r}\frac{\theta(t/t_{i}(x);\mathfrak{q})}{\theta(t;\mathfrak{q})}\frac{\theta(t/t_{i}(x)^{-1};\mathfrak{q})}{\theta(t;\mathfrak{q})}$
(7.130)

From LABEL:se:phiD we find

$\mathscr{Y}_{0}^{2}s(t,x)=\sum_{i=0}^{r}\Xi_{i}(\mathscr{Y}_{0};\mathbf{t}(x);\mathfrak{q})M_{ij}(\mathfrak{q})X^{j}(t;\mathfrak{q})$
(7.131)

where $X^{j}(t,\mathfrak{q})$ are powers of Weierstraß monomials forming a
basis in the space $H^{0}_{\text{even}}({\mathscr{E}},\mathcal{O}(2rp_{0}))$
of meromorphic functions on elliptic curve symmetric under the reflection
$t\to t^{-1}$ and with a pole of order no greater then $2r$ at the origin, and
$M_{ij}(\mathfrak{q})$ is a certain modular matrix.

The linear relations (LABEL:eq:D-theta-relations) allow to express $\Xi_{i}$
in terms of

$\displaystyle\tilde{\Xi}_{i}={}_{2}\tilde{{\mathscr{X}}}^{\widehat{D}}_{i}\quad
i=2,\dots,r-2$ (7.132)
$\displaystyle\tilde{\Xi}_{i}=({}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{i})^{2}\quad
i=0,1,r-1,r$

as

$\Xi_{i}=\sum_{{\tilde{i}}=0}^{r}{\tilde{\Xi}}_{\tilde{i}}{\tilde{M}}_{{\tilde{i}}i}(\mathfrak{q})$
(7.133)

where $\tilde{M}_{i\tilde{i}}(\mathfrak{q})$ is a certain modular
transformation matrix. Using the character equations (7.128) the spectral
curve (7.131) turns into

$\mathscr{Y}_{0}^{2}s(t,x)=\sum_{\tilde{i},j}\tilde{T}_{\tilde{i}}(x)\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})X^{j}(t,\mathfrak{q})$
(7.134)

where
$\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})=\tilde{M}_{\tilde{i}i}(\mathfrak{q})M_{ij}(\mathfrak{q})$
and

$\displaystyle\tilde{T}_{\tilde{i}}(x)=T_{\tilde{i}}(x)\quad\tilde{i}=2,\dots,r-1$
(7.135)
$\displaystyle\tilde{T}_{\tilde{i}}(x)=(T_{\tilde{i}}(x))^{2}\quad\tilde{i}=0,1,r-1,r$

The spectral curve of the $\widehat{D}_{r}$ theory is the algebraic equation
$R(t,x)=0$ where $R(t,x)$ is the right hand side of 7.134 combined with the
Weierstraß cubic equation (LABEL:eq:wxy). The $\widehat{D}_{r}$ curve is the
specialization of the $\widehat{A}_{2r-1}$ curve in two ways. First, there are
no odd in $Y$ monomials in (7.134), and, second, the polynomial coefficients
$\tilde{T}_{\tilde{i}}(x)$ of degree $2N$ in $x$ satisfy factorization
condition: they are full squares for $\tilde{i}=0,1,r-1,r$.

To interpret the curve in Hitchin-Gaudin formalism we will rewrite it in a
slightly different form. First, notice that222Indeed, the LHS and RHS is the
meromorphic elliptic function with $2r$ zeroes at points $X_{i},Y$ and
$X_{i},-Y$ and the pole of order $2r$ at $t=1$, or $X=\infty$. Such function
is unique up to a normalization which is fixed by the asymptotics at $t=1$.

$\prod_{i=1}^{r}\frac{\theta_{1}(t/\check{t}_{i};\mathfrak{q})}{\theta_{1}(t;\mathfrak{q})}\frac{\theta_{1}(t/\check{t}_{i}^{-1};\mathfrak{q})}{\theta_{1}(t;\mathfrak{q})}=\prod_{i=1}^{r}\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})(X-\check{X}_{i})$
(7.136)

We used here the notations (LABEL:eq:weierx2), (LABEL:eq:eal) for the
Weierstraß functions and

$\displaystyle\check{X}_{i}=X(\check{t}_{i};{\mathfrak{q}}),\quad\check{Y}_{i}^{2}=4\prod_{{\alpha}=1}^{3}(\check{X}_{i}-e_{\alpha})$

Then, if we divide (7.130) by (7.136) we find333And use
$\theta_{1}(t,\mathfrak{q})$ in lieu of $\theta(t,\mathfrak{q})$ as the basic
function, so that strictly speaking there is slightly different transformation
matrix $\tilde{\tilde{M}}_{\tilde{i}j}$ compared to (7.130)(LABEL:eq:s-ThetaD)

$\mathscr{Y}_{0}^{2}(x)\prod_{i=1}^{r}\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})\times\\\
\prod_{i=1}^{r}\frac{{\theta}_{1}(t/t_{i}(x);\mathfrak{q})}{{\theta}_{1}(t/\check{t}_{i};\mathfrak{q})}\frac{{\theta}_{1}(t/t_{i}^{-1}(x);\mathfrak{q})}{{\theta}_{1}(t/\check{t}_{i}^{-1};\mathfrak{q})}=R(x,X(t,{\mathfrak{q}}))\\\
R(x,X):=\frac{\sum_{\tilde{i},j=0}^{r}\tilde{T}_{\tilde{i}}(x)\tilde{\tilde{M}}_{{\tilde{i}}j}(\mathfrak{q})X^{j}}{\prod_{i=1}^{r}(X-\check{X}_{i})}$
(7.137)

Now, at the order two points on $\mathscr{E}$444e.g. the points
$(1,-1,-\mathfrak{q}^{-1/2},\mathfrak{q}^{1/2})$ in the $t$-parametrization,
where vanish the respective theta functions
$\theta_{1}(t;\mathfrak{q}),\theta_{2}(t;\mathfrak{q}),\theta_{3}(t;\mathfrak{q}),\theta_{4}(t;\mathfrak{q})$,
or, equivalently, at the four branch points in the $X$ plane:
$(\infty,e_{1},e_{2},e_{3})$, the value of the section $R(x,X)$ can be
expressed in terms of the weight $1$ invariants
${}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r-1},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{r},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{0},{}_{1}\tilde{{\mathscr{X}}}^{\widehat{D}}_{1}$
(compare with (LABEL:eq:Dr-inv) and (LABEL:eq:D1-theta)(LABEL:eq:D1-theta-
jacobi)), and it factorizes as

$\displaystyle R(x,X)|_{X\to\infty}=(T_{r-1}(x))^{2}$ (7.138) $\displaystyle
R(x,X)|_{X\to e_{1}}=c_{2}({\tilde{\mathfrak{q}}})\,(T_{r}(x))^{2}$
$\displaystyle R(x,X)|_{X\to
e_{2}}=c_{3}({\tilde{\mathfrak{q}}})\,(T_{0}(x))^{2}$ $\displaystyle
R(x,X)|_{X\to e_{3}}=c_{4}({\tilde{\mathfrak{q}}})\,(T_{1}(x))^{2}$

where

$c_{k}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{r}\frac{\theta_{1}(\check{t}_{i};\mathfrak{q})\theta_{1}(\check{t}_{i}^{-1};\mathfrak{q})}{\theta_{k}(\check{t}_{i};\mathfrak{q})\theta_{k}(\check{t}_{i}^{-1};\mathfrak{q})},\quad
k=2,3,4$ (7.139)

The Seiberg-Witten differential is given by:

${\lambda}=x\frac{dX}{Y}$ (7.140)

It is defined on the two fold cover $C_{u}$ of the curve $R(x,X)=0$, which is
a curve in the product
${\mathbb{C}\mathbb{P}}^{2}_{(X:Y:Z)}\times\mathbf{C}_{\left\langle
x\right\rangle}$, given by the equations:

$\displaystyle Y^{2}Z=4(X-e_{1}Z)(X-e_{2}Z)(X-e_{3}Z)$ (7.141) $\displaystyle
F(x,Z,X)=Z^{r}R(x,X/Z)=0$

The curve $C_{u}$ can be interpreted at the spectral curve of $GL(2N)$
Hitchin-Gaudin system on the orbifold ${\mathscr{E}}/\mathbb{Z}_{2}$, such
that at the fixed point $X=\infty,e_{1},e_{2},e_{3}$ the $GL(2N)$ system
reduces to the $Sp(2N)$ system. For more details on the Hitchin system, Nahm
transform and the brane construction of the the spectral curve for the
${\widehat{D}}_{r}$ quiver see [Kapustin:1998fa, Kapustin:1998xn]. Our main
result is the rigorous derivation of the spectral curve and its periods from
the gauge theory considerations.

####  7.5.1. Deforming the $N_{f}=4$ SU(2) theory

The $\widehat{D}_{4}$ theory can be interpreted as the theory obtained from
gauging the flavor group of the $D_{4}$ theory with
$({\mathbf{v}}_{1},{\mathbf{v}}_{2},{\mathbf{v}}_{3},{\mathbf{v}}_{4})=(N,2N,N,N)$
theory, and with
$({\mathbf{w}}_{1},{\mathbf{w}}_{2},{\mathbf{w}}_{3},{\mathbf{w}}_{4})=(0,N,0,0)$
matter multiplets. In the limit $\mathfrak{q}_{0}\to 0$ the elliptic curve
$\mathscr{E}$ degenerates to the cylinder $\mathbb{C}_{\left\langle
t\right\rangle}^{\times}$, while Seiberg-Witten curve (7.141) degenerates to
the Seiberg-Witten curve of the $D_{4}$ theory (7.50).

Let us consider the case $N=1$. Let us parametrize the polynomials
$T_{0},T_{1},T_{3},T_{4}$ as:

$T_{i}(x)=T_{i,0}({\tilde{\mathfrak{q}}})(x-m_{i}),\qquad i=0,1,3,4$ (7.142)

and

$T_{2}(x)=T_{2,0}({\tilde{\mathfrak{q}}})(x^{2}-m_{2}x+u)$

where parameters $q_{i},m_{i}$ and $u$ are related to the microscopic
couplings ${\mathfrak{q}}_{i}$ and the $U(1)^{4}\times SU(2)$ Coulomb moduli

$\displaystyle
T_{3,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{1}(\check{t}_{i}),\quad$
$\displaystyle
T_{4,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{2}(\check{t}_{i}),$
(7.143) $\displaystyle
T_{0,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{3}(\check{t}_{i}),\quad$
$\displaystyle
T_{1,0}({\tilde{\mathfrak{q}}})=\prod_{i=1}^{4}\theta_{4}(\check{t}_{i}),$
$\displaystyle\qquad\qquad\qquad\qquad
T_{2,0}({\tilde{\mathfrak{q}}})=\Xi_{2}(1,\mathbf{\check{t}},\mathfrak{q})$

where $\check{t}_{i}$ are defined in (7.127). Then the spectral curve of the
${\widehat{D}}_{4}$ theory (7.137)(LABEL:eq:consD) has the generic form:

$R(x,X)=T_{3}^{2}(x)+\sum_{i=1}^{4}\frac{b_{i}(x)}{X-\check{X}_{i}}$ (7.144)

where $b_{i}(x)$ are some polynomials of degree $2$ that we want to relate to
the coupling constants and Coulomb parameters. The first thing to notice is
that $R(x,X)$ in (7.144) obtained from (7.137) does not have poles at
$X=\check{X}_{i}$ at $x\to\infty$ in the leading order $x^{2}$. Therefore, the
polynomials $b_{i}(x)$ are actually degree 1 polynomials containing $8$
coefficients. There are 6 linear equations on these coefficients coming from 3
factorization equations (LABEL:eq:consD) viewed as coefficients at $x^{1}$ and
$x^{0}$ (and notice that the equations at $x^{2}$ are identically satisfied
because of (7.143) and (7.139))

$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{1}-\check{X}_{i}}=c_{2}T_{4}^{2}(x)-T_{3}^{2}(x)$
(7.145)
$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{2}-\check{X}_{i}}=c_{3}T_{0}^{2}(x)-T_{3}^{2}(x)$
$\displaystyle\sum_{i=1}^{4}\frac{b_{i}(x)}{e_{3}-\check{X}_{i}}=c_{4}T_{1}^{2}(x)-T_{3}^{2}(x)$

The above three equations determine four linear functions $b_{i}(x)$ up to a
single linear function, which depends on two parameters
${\tilde{m}}_{2},{\tilde{u}}$:

$\tilde{b}_{j}(x)=(-1)^{j}(-\tilde{m}_{2}x+\tilde{u})\,{\rm
Det}\left\|\begin{matrix}\frac{1}{e_{a}-\check{X}_{b}}\end{matrix}\right\|_{a=1,\ldots
3}^{b=1,\ldots 4,\,b\neq j}$ (7.146)

From (7.137) it is clear that $\tilde{m}_{2},\tilde{u}$ are proportional to
$m_{2},u_{2}$. To summarize, we can describe the spectral curve (7.144) of
$\widehat{D}_{4}$ theory by the coupling constants
$\mathfrak{q}_{i},i=0,\dots,4$, which define the elliptic curve
$\mathscr{E}(\mathfrak{q})$ with modulus
$\mathfrak{q}=\mathfrak{q}_{0}\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}\mathfrak{q}_{4}$
and positions of 4 punctures $\check{X}_{i}$ in the $\mathbb{C}_{\left\langle
X\right\rangle}$ plane for Weierstraß cubic using (7.127), the 4 parameters
$m_{i},i=0,1,3,4$ entering into relations (7.145) through (7.142) and 2
parameters $\tilde{m}_{2},\tilde{u}$ in (7.146).

Now consider the limit $\mathfrak{q}_{0}\to 0$ which turns the
$\widehat{D}_{4}$ class II quiver theory to the $D_{4}$ class I quiver theory.
In this limit the Weierstraß cubic degenerates: $e_{1}=-2e_{3},\quad
e_{2}=e_{3}=1/12$,

$Y^{2}=4\left(X-e_{3}\right)^{2}\left(X+2e_{3}\right)^{2}$ (7.147)

with

$X=\frac{t}{(1-t)^{2}}+\frac{1}{12},\quad Y=\frac{t(1+t)}{(1-t)^{3}}$ (7.148)

The Seiberg-Witten differential $x\frac{dX}{Y}$ becomes $x\frac{dt}{t}$. The
elliptic curve $\mathscr{E}$ degenerates to the rational curve which is the
double cover $t\mapsto X$ of the complex projective line
$\mathbb{C}\mathbb{P}^{1}_{X}$. To make contact with the Seiberg-Witten curve
of the the $D_{4}$ quiver theory it is convenient to work in the coordinate
which is related to the coordinate $X$ by rational transformation

$\eta=2+\frac{1}{X-e_{3}}=t+t^{-1}$ (7.149)

The function $\eta(X)$ is degree two meromorphic function on $\mathscr{E}$
with values at the four $\mathbb{Z}_{2}$ invariant points given by

$\eta(e_{2})=\eta(e_{3})=\infty,\quad\eta(e_{1})=-2,\quad\eta(\infty)=2$
(7.150)

Rewriting (7.137) in terms of $\eta(x)$ we find the equation of spectral curve
$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)=0$ for

$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)=\sum_{i=0}^{4}\eta^{i}p_{i}(x)$ (7.151)

where $p_{i}(x)$ are some polynomials of degree $2$ in $x$. Moreover, the
factorization conditions (LABEL:eq:consD) translates to the statement that
$\tilde{\mathcal{R}}^{D_{4}}(\eta,x)$ is full square at $\eta=\infty$ and at
$\eta=\pm 2$ in the polynomial ring of $x$. Notice that this is precisely the
factorization conditions (7.63)(7.64) of the curve (7.60) for the $D_{4}$
quiver. (The variables $t$ and $\eta$ in the equations (7.60), (7.59)
correspond to $t$ and $\eta$ of this section multiplied by a factor
$\mathfrak{q}_{1}\mathfrak{q}_{2}\mathfrak{q}_{3}^{\frac{1}{2}}\mathfrak{q}_{4}^{\frac{1}{2}}$.)

Given the above discussion and the section 7.2.1 let us summarize the freezing
hierarchy $\widehat{D}_{4}\to D_{4}\to A_{3}\to A_{2}\to A_{1}$. For
$\widehat{D}_{4}$ theory we start with elliptic curve
$\mathscr{E}(\mathfrak{q})$ with $\mathbb{Z}_{2}$ reflection symmetry $t\to
t^{-1}$ (or $Y\to-Y$) and $8$ $\mathbb{Z}_{2}$-symmetrically located punctures
in 4 pairs $(\check{t}_{i},\check{t}_{i}^{-1})$. As we freeze
$\mathfrak{q}_{0}\to 0$, the elliptic curve $\mathscr{E}(\mathfrak{q})$
degenerates to a $\mathbb{Z}_{2}$-symmetrical cylinder
$\mathbb{C}^{\times}_{t}$ with 4 old pairs
$(\check{t}_{i},\check{t}_{i}^{-1})$ of punctures. The cylinder
$\mathbb{C}^{\times}_{t}$ double covers its $\mathbb{Z}_{2}$-quotient
$\mathbb{C}\mathbb{P}^{\times}_{\eta}$. This is the situation of $D_{4}$
quiver theory (7.60). As we freeze $\mathfrak{q}_{4}\to 0$ the second sheet of
the double cover $\mathbb{C}^{\times}_{t}\to\mathbb{C}^{\times}_{\eta}$ is
removed to infinity and we are left with $4$ punctures of $A_{3}$ quiver at
$(\mathfrak{q}_{1}^{-1},1,\mathfrak{q}_{2},\mathfrak{q}_{2}\mathfrak{q}_{3})$.555Keeping
in mind the ultimate configuration of the $A_{1}$ quiver dynamical at node “2”
we have rescaled the position of punctures by a factor of $\mathfrak{q}_{1}$.
Notice, that as discussed after (7.65) the $SL(2,\mathbb{C})$ residues of the
Higgs field vanish at the punctures in $0$ and $\infty$. As we freeze
$\mathfrak{q}_{3}\to 0$ the puncture at $\mathfrak{q}_{2}\mathfrak{q}_{3}$
(with non-trivial $SL(2,\mathbb{C})$ residue of Higgs field) merges with the
puncture $0$ and we are in the situation of the $A_{2}$ quiver with
$SL(2,\mathbb{C})$ punctures at $(\mathfrak{q}_{1}^{-1},1,\mathfrak{q}_{2},0)$
and $GL(1,\mathbb{C})$ puncture at $\infty$. Finally as we freeze
$\mathfrak{q}_{1}\to 0$ the puncture at $\mathfrak{q}_{1}^{-1}$ (with non-
trivial residue of the $SL(2,\mathbb{C})$ Higgs field) is merged with the
puncture at $\infty$ and we are left with $\mathbb{C}\mathbb{P}^{1}$ with
$SL(2,\mathbb{C})$ punctures at $(\infty,1,\mathfrak{q}_{2},0)$ for the
$A_{1}$ quiver theory defined at the dynamical node ‘‘2’’. See figure 7.3.

Figure 7.3. The freezing $\widehat{D}_{4}\to D_{4}\to A_{3}\to A_{2}\to
A_{1}$. The live nodes are denoted by red, the frozen nodes are denoted by
blue. The nodes are labeled as $i_{\mathbf{v}_{i}}$

###  7.6. Class II theories of $E$ type

The main technical tool is the natural isomorphism between the moduli space of
the $E_{k}$-bundles on elliptic curve $\mathscr{E}$ and the moduli space of
del Pezzo surfaces $\mathcal{S}_{k}$, which are obtained by blowing up $k$
points in ${\mathbb{C}}{\mathbb{P}}^{2}$, and have the fixed elliptic curve
$\mathscr{E}$ as the anticanonical divisor. The spectral curve is found using
the ‘‘cylinder map’’ [Kanev], and see [Donagi:2008kj, Donagi:2008ca,
Curio:1998bva] for applications.

Another way of encoding the geometry of the moduli space the $E_{k}$-bundles
is in the unfolding of the parabolic unimodular singularities [Arnold:1985]
${\widehat{T}}_{a,b,c}$ with

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

which are:

$\displaystyle{\tilde{E}}_{6}=P_{8}={\widehat{T}}_{3,3,3}\,:$
$\displaystyle\,x^{3}+y^{3}+z^{3}+mxyz,\qquad\qquad m^{3}+27\neq 0,$ (7.152)
$\displaystyle{\tilde{E}}_{7}=X_{9}={\widehat{T}}_{2,4,4}\,:$
$\displaystyle\,x^{4}+y^{4}+z^{2}+mxyz,\qquad\qquad m^{4}-64\neq 0,$
$\displaystyle{\tilde{E}}_{8}=J_{10}={\widehat{T}}_{2,3,6}\,:$
$\displaystyle\,x^{6}+y^{3}+z^{2}+mxyz,\qquad\qquad 4m^{6}-432\neq 0$

We shall not pursue this direction in this work.

######  Remark.

Another important question left for future work is the connection between our
description of the special geometry via the periods of $d{\mathbb{S}}$ and the
periods of non-compact Calabi-Yau threefolds of [Katz:1997eq].

####  7.6.1. Del Pezzo and $E_{6}$ bundles

The Del Pezzo surface
$\mathcal{S}_{6}\subset\mathbb{W}\mathbb{P}^{1,1,1,1}={\mathbb{C}\mathbb{P}}^{3}$
is a zero locus of a homogeneous degree $3$ polynomial:

${\Gamma}_{3}(X_{0},X_{1},X_{2},X_{3})=\sum_{i=0}^{3}X_{0}^{3-i}\,{\mathcal{G}}_{i}(X_{1},X_{2},X_{3})$
(7.153)

where ${\mathcal{G}}_{i}$ is the degree $i$ homogeneous polynomial in
$X_{1},X_{2},X_{3}$. In particular,

${\mathcal{G}}_{3}(X_{1},X_{2},X_{3})=-X_{1}X_{3}^{2}+4X_{2}^{3}-g_{2}X_{1}^{2}X_{2}-g_{3}X_{1}^{3}$
(7.154)

defines the elliptic curve ${\mathscr{E}}$, which determines the gauge
coupling $\mathfrak{q}={\exp}\,2\pi\mathrm{i}\tau$, cf. (LABEL:eq:wxy):

${\tau}=\frac{\oint_{B}dX/Y}{\oint_{A}dX/Y}\,,\qquad
X=X_{2}/X_{1},\,Y=X_{3}/X_{1}$ (7.155)

The rest of the coefficient functions ${\mathcal{G}}_{0,1,2}$ is parametrized
as follows:

$\displaystyle{\mathcal{G}}_{2}(X_{1},X_{2},X_{3})=p_{0}X_{1}^{2}+p_{1}X_{1}X_{2}+p_{6}X_{2}X_{3}$
(7.156)
$\displaystyle{\mathcal{G}}_{1}(X_{1},X_{2},X_{3})=p_{2}X_{1}+p_{3}X_{2}+p_{5}X_{3}$
(7.157) $\displaystyle{\mathcal{G}}_{0}(X_{1},X_{2},X_{3})=p_{4}$ (7.158)

The isomorphism classes of $\mathcal{S}_{6}$ surfaces containing the fixed
elliptic curve ${\mathscr{E}}$ are in one-to-one correspondence with the
points

$[p]=(p_{0}:p_{1}:p_{6}:p_{2}:p_{3}:p_{5}:p_{4})\in{\mathcal{M}}$ (7.159)

in the weighted projective space
${\mathcal{M}}={\mathbb{W}\mathbb{P}}^{1,1,1,2,2,2,3}$, which is also
isomorphic, by E.Loojienga’s theorem [Loojienga:1976], to the moduli space
$\mathrm{Bun}_{E_{6}({\mathbb{C}})}^{ss}(\mathscr{E})$ of holomorphic semi-
stable principal $E_{6}$-bundles on $\mathscr{E}$. We label the projective
coordinates $p_{i}$ in such a way that the projective weight of $p_{i}$ equals
Dynkin mark $a_{i}$ in our conventions Appendix A. The correspondence between
the $E_{6}$-bundles on $\mathscr{E}$ and the del Pezzo surfaces
$\mathcal{S}_{6}$ is geometric: there are precisely $27$ degree $1$ rational
curves (‘‘the $(-1)$-lines’’) $C_{a}$ on $\mathcal{S}_{6}$, $a=1,\ldots,27$,
which are the divisors of the line bundles ${\mathscr{L}}_{a}$ on
$\mathcal{S}_{6}$. The direct sum

${\mathcal{U}}=\bigoplus_{a=1}^{27}{\mathscr{L}}_{a}\,,$ (7.160)

has no infinitesimal deformations, as a bundle on $\mathcal{S}_{6}$. The
mapping class group of $\mathcal{S}_{6}$ acts on the $(-1)$-lines by the
$E_{6}$ Weyl transformations. As a result, the bundle ${\mathcal{U}}$ is a
vector bundle associated to a canonical principal $E_{6}({\mathbb{C}})$-bundle
${\mathcal{P}}_{\mathcal{S}_{6}}$ over $\mathcal{S}_{6}$ with the help of a
${\bf 27}$ representation:

${\mathcal{U}}={\mathcal{P}}_{\mathcal{S}_{6}}\times_{E_{6}({\mathbb{C}})}{\bf
27}$ (7.161)

The restriction of ${\mathcal{P}}_{\mathcal{S}_{6}}|_{E}$ is the holomorphic
principal $E_{6}({\mathbb{C}})$ bundle over $E$ which corresponds to the point
$[s]$ in Loojienga’s theorem. Again, the associated rank $27$ vector bundle
${\mathcal{U}}|_{E}$ splits

${\mathcal{U}}|_{\mathscr{E}}=\bigoplus_{a=1}^{27}{\mathscr{L}}_{a}$ (7.162)

The line subbundles ${\mathscr{L}}_{a}$ can be expressed as:

${\mathscr{L}}_{a}=\bigotimes_{i=1}^{6}\,{\mathbb{L}}_{i}^{w_{a,i}}$ (7.163)

where $w_{a,i}$, $i=1,\ldots,6$, $a=1,\ldots,27$ are the components of the
weight vector. The line bundles ${\mathbb{L}}_{i}$, $i=1,\ldots,6$ are defined
up to the action of the $E_{6}$ Weyl group. Let us now compute the
${\mathscr{L}}_{a}$’s. The rational curve of degree one in $\mathcal{S}_{6}$
is a rational curve of degree one in ${\mathbb{C}\mathbb{P}}^{3}$ which is
contained in $\mathcal{S}_{6}$. A parametrized rational curve of degree one in
${\mathbb{C}\mathbb{P}}^{3}$ is a collection of $4$ linear functions:
${\zeta}\mapsto{\mathbf{X}}({\zeta})$,

${\mathbf{X}}({\zeta})=\left(X_{0}+{\zeta}v_{0},X_{1}+{\zeta}v_{1},X_{2}+{\zeta}v_{2},X_{3}+{\zeta}v_{3}\right)$
(7.164)

The two quadruples

${\mathbf{X}}({\zeta})\qquad{\rm
and}\qquad(c{\zeta}+d)\,{\mathbf{X}}\left(\frac{a{\zeta}+b}{c{\zeta}+d}\right)$

for

$\left(\begin{matrix}a&b\\\ c&d\\\ \end{matrix}\right)\in{\rm
GL}_{2}({\mathbb{C}})$ (7.165)

define identical curves in ${\mathbb{C}\mathbb{P}}^{3}$. We can fix the
GL${}_{2}({\mathbb{C}})$ gauge by choosing the parameter $\zeta$ so that:

${\mathbf{X}}({\zeta})=\left({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}\right)$
(7.166)

The requirement that the curve lands in
$\mathcal{S}_{6}\subset{\mathbb{C}\mathbb{P}}^{3}$ reads as

${\Gamma}_{3}\left({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}\right)=\sum_{i=0}^{3}{\zeta}^{i}{\Xi}_{i}(X,Y;v_{X},v_{Y})\equiv
0$ (7.167)

which is a system of $4$ equations

${\Xi}_{i}(X,Y;v_{X},v_{Y})=0,\qquad i=0,\ldots,3$

on $4$ unknowns $X,Y,v_{X},v_{Y}$:

$\displaystyle\Xi_{0}$ $\displaystyle=-Y^{2}+4X^{3}-g_{2}X-g_{3}$
$\displaystyle\Xi_{1}$
$\displaystyle=-g_{2}v_{X}+p_{6}XY+p_{1}X+p_{0}+12X^{2}v_{X}-2Yv_{Y}$
$\displaystyle\Xi_{2}$
$\displaystyle=p_{6}Yv_{X}+p_{1}v_{X}+p_{6}Xv_{Y}+p_{3}X+p_{5}Y+p_{2}+12Xv_{X}^{2}-v_{Y}^{2}$
$\displaystyle\Xi_{3}$
$\displaystyle=p_{6}v_{X}v_{Y}+p_{3}v_{X}+p_{5}v_{Y}+p_{4}+4v_{X}^{3}$

The equation $\Xi_{0}=0$ in the above system is the equation of the elliptic
curve $\mathscr{E}$. To find the equation of the spectral cover associated
with the vector bundle $\mathcal{U}|_{\mathscr{E}}$ in the $\mathbf{27}$
representation we can express $v_{Y}$ from the equation $\Xi_{1}=0$, then plug
it into the remaining equations $\Xi_{2}=0$ and $\Xi_{3}=0$, compute the
resultant of these two polynomials with respect to the variable $v_{X}$,
reduce modulo the equation $\Xi_{0}=0$ defining the elliptic curve
$\mathscr{E}$, arriving at:

$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})=-4Y^{4}\mathrm{res}_{v_{X}}(\Xi_{2}|_{v_{Y}:\Xi_{1}=0},\Xi_{3}|_{v_{Y}:\Xi_{1}=0})\mod\Xi_{0}$
(7.168)

The resultant $C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})$ is a polynomial
in $X,Y$ with polynomial coefficients in $g_{2},g_{3},p_{0},\dots,p_{6}$ of
the form

$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{6})=\\\
(p_{0}^{6}+\dots)+(6p_{0}^{5}p_{1}+\dots)X+\dots+(-256p_{3}^{3}+\dots)X^{12}\\\
+(12g_{3}p_{0}^{4}p_{5}+\dots)Y+(32g_{3}p_{0}^{4}p_{5}+\dots)XY+\dots+(-256p_{5}^{3}+\dots)X^{12}Y$
(7.169)

(A short `Mathematica` version of this formula is given in appendix
LABEL:sec:_E6-delPezzo-code.) Now let us imagine having a family
${\mathcal{U}}$ of the $E_{6}$-bundles on $E$.

In our solution the vacuum $u$ of the gauge theory is identified with the
degree $N$ quasimap:

$\displaystyle p:{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}\to\mathrm{Bun}_{E_{6}(\mathbb{C})}(\mathscr{E})\simeq\mathbb{W}\mathbb{P}^{1,1,1,2,2,2,3}$

given by the polynomials $p_{i}(x)$ of degree $Na_{i}$

$p_{i}=p_{i}(x),\quad i=0,\dots,6$ (7.170)

Together with the equation of the Weierstraß cubic
$\Xi_{0}(X,Y,g_{2},g_{3})=0$, the equation

$C^{E_{6}}(X,Y;g_{2},g_{3},p_{0}(x),\dots,p_{6}(x))=0$ (7.171)

defines the Seiberg-Witten curve of the affine $E_{6}$ theory as an algebraic