curve. Given that the degree of $X$ on $\mathscr{E}$ is $2$ and the degree $Y$
on $\mathscr{E}$ is $3$ the polynomial $C^{E_{6}}$ is of degree $27$, i.e. the
equation $C^{E_{6}}=0$ defines $27$ points on the elliptic curve
$\mathscr{E}$.

The top degree coefficients of the polynomials $p_{i}(x)$ are determined
explicitly in terms of the coupling constants $\mathfrak{q}_{i}$. Indeed, the
$E_{6}$ characters, or more conveniently in the present case, the $E_{6}$
theta-functions, $c_{i}(\mathscr{Y}_{0},\mathbf{t},\mathfrak{q})$ as set in
(LABEL:eq:c-def), define a set of projective coordinates on
$\mathbb{W}\mathbb{P}^{1,1,1,2,2,2,3}$ (which differs slightly from the set
$({\widehat{\chi}}_{i})_{i=0}^{6}$):

$(c_{0}:c_{1}:c_{6}:c_{2}:c_{3}:c_{5}:c_{4})$ (7.172)

In these coordinates the solution of the theory has the canonical form (6.28)

$c_{i}(\mathscr{Y}_{0},\mathbf{t},\mathfrak{q})=\left(\prod_{j=0}^{r}\mathfrak{q}_{j}^{-\lambda_{i}(\lambda_{j}^{\vee})}\right)\,T_{i}(x)$
(7.173)

The ‘‘del Pezzo projective coordinates’’ $(p_{i})_{i=0}^{6}$ are related to
the theta-function coordinates $(c_{i})$ on
$\mathrm{Bun}_{E_{6}(\mathbb{C})}^{ss}(\mathscr{E})$ by a polynomial map of
the form

$p_{i}=\sum_{j_{1}\leq j_{2}\leq
j_{3}\dots}M_{i,\\{j_{1},j_{2},j_{3},\dots\\}}(\mathfrak{q})c_{j_{1}}c_{j_{2}}c_{j_{3}}\dots$
(7.174)

where $M_{i,\\{j_{1},j_{2},\dots,\\}}(\mathfrak{q})$ is certain modular
transformation matrix. This matrix can be explicitly computed by comparing the
spectral curve (7.168) and the $\widehat{A}_{26}$ spectral curve (7.95)
specialized to the the embedding $\widehat{E}_{6}\subset\widehat{A}_{26}$ by
fundamental representation. The coefficients
$M_{i,\\{j_{1},j_{2},j_{3},\dots\\}}$ are modular forms for modular group
$\Gamma(6)$ with a certain modular weights that can computed by observation
that the weights of variables $(X_{0},X_{1},X_{2},X_{3})$ under the modular
transformation $\tau\to-\tau^{-1}$ on $\mathscr{E}(\mathfrak{q})$ for
$\mathfrak{q}=\exp(2\pi\mathrm{i}\tau)$ are

$\begin{pmatrix}X_{0}&X_{1}&X_{2}&X_{3}\\\ 6&1&2&3\end{pmatrix}$ (7.175)

This implies that the modular weights of $p_{i}$ are

$\begin{pmatrix}&p_{0}&p_{1}&p_{6}&p_{2}&p_{3}&p_{5}&p_{4}\\\
&0&-2&-5&-6&-8&-9&-12\end{pmatrix}$ (7.176)

The $(c_{i})$ have modular weight $3$ because they are rank $6$ lattice theta-
functions. From this assignment of weights one finds the modular weights of
all coefficients $M_{i,\\{j_{1},\dots,j_{k}\\}}(\mathfrak{q})$; for example
$M_{4,\\{4\\}}$ has modular weight $15$. The space of modular forms for
$\Gamma(N)$ of a given weight $k$ is a finite dimensional vector space. (For
any integer $k>0$ the dimension is $k+1$ for $\Gamma(3)$ and $6k$ for
$\Gamma(6)$). Matching a finite number of the coefficients in the
$\mathfrak{q}$ expansion one finds explicitly the modular coefficients
$M_{i,\\{j_{1},\dots,\\}}(\mathfrak{q})$, see appendix LABEL:sec:E6-modular-
matrix.

####  7.6.2. Del Pezzo and $E_{7}$ bundles

The story for the $E_{7}$-bundles is similar. There is a family of del Pezzo
surfaces $\mathcal{S}_{7}\subset{\mathbb{W}}{{\mathbb{P}}}^{1,1,1,2}$,
described by the degree $4$ equation

${\Gamma}_{4}(X_{0},X_{1},X_{2},X_{3})=\sum_{i=0}^{4}X_{0}^{i}\,{\mathcal{G}}_{4-i}(X_{1},X_{2},X_{3})$
(7.177)

with

${\mathcal{G}}_{4}(X_{1},X_{2},X_{3})=-X_{3}^{2}+4X_{1}X_{2}^{3}-g_{2}X_{1}^{3}X_{2}-g_{3}X_{1}^{4}(X_{3}^{2}-4X_{1}X_{2}^{3}+g_{2}X_{2}X_{1}^{3}+g_{3}X_{1}^{4})$
(7.178)

and

$\displaystyle{\mathcal{G}}_{3}(X_{1},X_{2},X_{3})=p_{0}X_{1}^{3}+p_{7}X_{1}^{2}X_{2}(a_{1}X_{1}^{3}+a_{2}X_{2}X_{1}^{2})$
$\displaystyle{\mathcal{G}}_{2}(X_{1},X_{2},X_{3})=p_{1}X_{1}^{2}+p_{2}X_{1}X_{2}+p_{6}X_{2}^{2}(b_{1}X_{1}^{2}+b_{2}X_{1}X_{2}+b_{3}X_{2}^{2})$
$\displaystyle{\mathcal{G}}_{1}(X_{1},X_{2},X_{3})=p_{3}X_{1}+p_{5}X_{2}(c_{1}X_{1}+c_{2}X_{2})$
$\displaystyle{\mathcal{G}}_{0}=p_{4}$

Again, the divisor $X_{0}=0$ is the elliptic curve ${\mathscr{E}}$, which is
realized as a zero locus of the degree $4$ polynomial equation
${\mathcal{G}}_{4}(X_{1},X_{2},X_{3})=0$ in
${\mathbb{W}}{\mathbb{P}}^{1,1,2}$.

The isomorphism classes of $\mathcal{S}_{7}$, containing the fixed elliptic
curve ${\mathscr{E}}$, are in one-to-one correspondence with the points

$[p]=(p_{0}:p_{7}:p_{1}:p_{2}:p_{5}:p_{3}:p_{5}:p_{4})\in{\mathbb{W}\mathbb{P}}^{1,1,2,2,2,3,3,4}$
(7.179)

Again, we study the ‘‘(-1)-curves’’, which in a particular gauge look like:

${\mathbf{x}}({\zeta})=({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y})$
(7.180)

where $(X,Y;v_{X},v_{Y},w_{Y})$ obey a system of $5$ equations
${\Xi}_{i}(X,Y;v_{X},v_{Y},w_{Y})=0$, $i=0,\ldots,4$:

${\Gamma}_{4}({\zeta},1,X+{\zeta}v_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y})=\sum_{i=0}^{4}{\zeta}^{i}\,{\Xi}_{i}(X,Y;v_{X},v_{Y},w_{Y})$
(7.181)

where

$\displaystyle\Xi_{0}=-Y^{2}+4X^{3}-g_{2}Y-g_{3}$ (7.182)
$\displaystyle\Xi_{1}=p_{0}+Xp_{7}+12X^{2}v_{X}-g_{2}v_{X}-2Yv_{Y}$
$\displaystyle\Xi_{2}=p_{1}+Xp_{2}+X^{2}p_{6}+p_{7}v_{X}+12Xv_{X}^{2}-v_{Y}^{2}-Yw_{Y}$
$\displaystyle\Xi_{3}=p_{3}+Xp_{5}+p_{2}v_{X}+2Xp_{6}v_{X}+4v_{X}^{3}-v_{Y}w_{Y}$
$\displaystyle\Xi_{4}=p_{4}+p_{5}v_{X}+p_{6}v_{X}^{2}-\frac{w_{Y}^{2}}{4}$

We proceed similarly to the $E_{6}$ case: we solve for $v_{Y}$ and $w_{Y}$
from the equations $\Xi_{1}$ and $\Xi_{2}$, plug the solution into the
polynomial $\Xi_{3}$ and $\Xi_{4}$ and compute the resultant

$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{7})=\\\
-2^{10}Y^{18}\mathrm{res}_{v_{X}}(\Xi_{3}|_{v_{Y},w_{Y}:\Xi_{1,2}=0},\Xi_{4}|_{v_{Y},w_{Y}:\Xi_{1,2}=0})\mod\Xi_{0}$
(7.183)

which has the structure of the degree $28$ polynomial in $X$ with the
coefficients polynomial in $(p_{0},\dots,p_{7})$ of total degree 12:

$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0},\dots,p_{7})=(p_{0}^{12}+24g_{3}p_{0}^{10}p_{1}+\dots)+(24g_{2}p_{0}^{10}p_{1}+\dots)X+\\\
+\dots+(2^{16}p_{5}^{4}-2^{19}p_{4}p_{5}^{2}p_{6}+2^{20}p_{4}^{2}p_{6}^{2})X^{28}$
(7.184)

(A short `Mathematica` version of this formula is given in appendix
LABEL:sec:_E7-delPezzo-code.) The polynomial
$C^{E_{7}}(X,Y;g_{2},g_{3},p_{0}(x),\dots,p_{7}(x))$ together with the
Weierstraß cubic $\Xi_{0}(X,Y;g_{2},g_{3})$ defines the algebraic Seiberg-
Witten curve for $E_{7}$ quiver theory. Since on the elliptic curve
$\mathscr{E}$ the degree of $X$ is $2$, at each $x\in\mathbf{C}_{\left\langle
x\right\rangle}$ the spectral curve (7.184) defines $58$ points on
$\mathscr{E}$ encoding the vector bundle in the $\mathbf{58}$ representation
of $E_{7}$.

The relation between the del Pezzo parametrization $(p_{i})$ and the theta-
function parametrization $(c_{i})$ of
$\mathrm{Bun}_{E_{7}(\mathbb{C})}^{ss}(\mathscr{E})$ can be in principle
written in terms of a certain modular matrix $M(\mathfrak{q})$, as in the the
$E_{6}$ theory LABEL:sec:E6-modular-matrix. We do not record this
transformation in this work.

####  7.6.3. Del Pezzo and $E_{8}$ bundles

To get an effective description of $E_{8}$-bundles we study the family of del
Pezzo surfaces ${\mathcal{S}}_{8}\subset{\mathbb{W}}{\mathbb{P}}^{1,1,2,3}$:

$\displaystyle{\Gamma}_{6}(X_{0},X_{1},X_{2},X_{3})=$ $\displaystyle\ =\
-X_{3}^{2}+4X_{2}^{3}-X_{2}G_{2}(X_{0},X_{1})-G_{3}(X_{0},X_{1})=0$
$\displaystyle\qquad\qquad
G_{2}(X_{0},X_{1})=g_{2}X_{1}^{4}+\sum_{j=1}^{4}a_{j}X_{0}^{j}X_{1}^{4-j}$
$\displaystyle\qquad\qquad
G_{3}(X_{0},X_{1})=g_{3}X_{1}^{6}+\sum_{j=2}^{6}b_{j}X_{0}^{j}X_{1}^{6-j}$

The isomorphism classes of the del Pezzo surfaces ${\mathcal{S}}_{8}$
containing the fixed elliptic curve

$Y^{2}=4X^{3}-g_{2}X-g_{3}$

are parametrized by:

$[s]=(a_{1}:a_{2}:b_{2}:a_{3}:b_{3}:a_{4}:b_{4}:b_{5}:b_{6})\in{\mathbb{W}\mathbb{P}}^{1,2,2,3,3,4,4,5,6}$
(7.185)

The ‘‘-1’’-curves in ${\mathcal{S}}_{8}$ are described by the parametrizations
${\mathbf{x}}({\zeta})=({\zeta},1,X+{\zeta}v_{X}+\frac{1}{2}{\zeta}^{2}w_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y}+\frac{1}{6}{\zeta}^{3}u_{Y})$,
where $(X,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})$ to be found from the equations

${\Xi}_{i}(Y,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})=0,\qquad i=0,\ldots,6$ (7.186)

where

${\Gamma}_{6}({\zeta},1,X+{\zeta}v_{X}+\frac{1}{2}{\zeta}^{2}w_{X},Y+{\zeta}v_{Y}+\frac{1}{2}{\zeta}^{2}w_{Y}+\frac{1}{6}{\zeta}^{3}u_{Y})=$
$\qquad=\sum_{i=0}^{6}{\zeta}^{i}{\Xi}_{i}(X,Y,v_{X},v_{Y},w_{X},w_{Y},u_{Y})$

To find explicitly the equation of affine $E_{8}$ spectral curve, one shall
proceed in spirit similarly to the $E_{6},E_{7}$ cases considered above.
However the explicit computation becomes much more tedious as the minimal
representation of $E_{8}$ is $\bf{248}$, and the expected $x$-degree of the
curve is $60N$ (see below). We leave this task for future investigation.

##  Chapter 8 The integrable systems of monopoles and instantons

As we reviewed above, the ${\mathcal{N}}=2$ gauge theory compactified on a
circle ${\mathbb{S}}^{1}$ becomes, at low energy, the ${\mathcal{N}}=4$
supersymmetric sigma model with the hyperkäbler target space ${\mathfrak{P}}$.
The triplet of complex structures on ${\mathfrak{P}}$ is in correspondence
with the choices of a supercharge ${\mathcal{Q}}$, which is nilpotent up to an
infinitesimal translation along ${\mathbb{S}}^{1}$. The one supercharge which
is nilpotent even in the decompactified theory (it corresponds to the
topological supercharge of the Donaldson theory, for pure ${\mathcal{N}}=2$
super-Yang-Mills theory) corresponds to the complex structure $\bf I$. In this
complex structure ${{\mathfrak{P}}}$ has the structure of an algebraic
integrable system $({{\mathfrak{P}}},{\Omega},h)$:

$h:{{\mathfrak{P}}}\longrightarrow{{\mathfrak{M}}},\qquad{\Omega}|_{h^{-1}(u)}=0,\
u\in{{\mathfrak{M}}}$ (8.1)

The $\bf I$-holomorphic $(2,0)$ form ${\Omega}$ is the form which we
previously denoted by ${\Omega}_{\bf I}$.

We shall now describe these systems for the class I, II and II* theories we
studied so far. For some theories several presentations of the same integrable
system are possible.

In all cases we study the phase spaces ${\mathfrak{P}}$ have parameters
corresponding to the masses $m$ of the matter field in the gauge theory. The
cohomology class of $[{\Omega}]$ is linear in $m$. An explanation of this fact
in the symplectic geometry is the existence of a ‘‘larger’’ symplectic
manifold ${\mathfrak{P}}^{\mathrm{ext}}$ with the holomorphic Hamiltonian
torus ${\mathbb{T}}=({\mathbb{C}}^{\times})^{M}$ action, whose holomorphic
symplectic quotient of ${\mathfrak{P}}^{\mathrm{ext}}$ at some level $m$ of
the moment map produces ${\mathfrak{P}}$.

The explanation in gauge theory is the three dimensional mirror symmetry. Our
phase space ${\mathfrak{P}}$ is the Coulomb branch of the three dimensional
${\mathcal{N}}=4$ gauge theory, obtained by the ${\mathbb{S}}^{1}$
compactification of the four dimensional ${\mathcal{N}}=2$ theory. The masses
of the matter fields are the vacuum expectation values of the scalars in the
vector multiplet. Under the three dimensional mirror symmetry
[Intriligator:1996ex] these are exchanged with the Fayet-Illiopoulos terms,
which are the levels of the three moment maps in the hyperkähler quotient
construction [Hitchin:1986ea] of the Higgs branch of the mirror theory.

It is amusing to identify ${\mathfrak{P}}^{\mathrm{ext}}$ and the action of
the torus in the examples below. We shall treat the case of the class II
theories in some detail, leaving other examples to the interested reader.

###  8.1. Periodic Monopoles and the phase space of Class I theories

We shall now demonstrate that for the class I theories the phase space
${\mathfrak{P}}$ is the moduli space of the charge ${\mathbf{v}}$
$G$-monopoles on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with Dirac
singularities, whose location and the embedding of the Dirac $U(1)$-monopoles
into $G$ is parametrized by ${\mathbf{w}}$ and the masses
$m_{i,{\mathfrak{f}}}$.

Let us discuss the monopole moduli space in more detail.

The ordinary $G$-monopoles are the solutions of Bogomolny equation on
${\mathbb{R}}^{3}$:

$D_{A}{\phi}+\star F_{A}=0$ (8.2)

with finite $L^{2}$-energy:

${\mathcal{E}}(A,{\phi})=\int_{{\mathbb{R}}^{3}}\langle F_{A},\star
F_{A}\rangle+\langle D_{A}{\phi},\star D_{A}{\phi}\rangle$ (8.3)

One shows that as ${\vec{x}}\to\infty$, the conjugacy class of
$\phi({\vec{x}})$ approaches a fixed value. Equivalently,
${\phi}({\vec{x}})\longrightarrow
g^{-1}({\vec{x}})\,{\phi}_{\infty}\,g({\vec{x}})$, for some fixed
${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}$, the Cartan subalgebra of the
maximal compact subgroup $G$. Actually,
${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}/W({\mathfrak{g}})$, but, since
${\mathbb{S}}^{2}_{\infty}$ is simply connected, one can choose a uniform
representative ${\phi}_{\infty}\in\mathfrak{h}_{\mathbb{R}}$. This lift from
$\mathfrak{h}_{\mathbb{R}}/W({\mathfrak{g}})$ to $\mathfrak{h}_{\mathbb{R}}$
is going to be trickier in the case of periodic monopoles we shall study
below.

Suppose $\phi_{\infty}$ is generic, i.e. the only gauge transformations which
commute with it belong to the normalizer $N(T)$ of a maximal torus $T$. The
restriction of $\phi$ onto a two-sphere ${\mathbb{S}}^{2}_{\infty}$ of a very
large radius defines a map:

${\varphi}:{\mathbb{S}}^{2}_{\infty}\longrightarrow G/T$ (8.4)

and a $T$-subbundle $\mathbb{T}$ of the trivial $G$-bundle
$P=G\times{\mathbb{S}}^{2}_{\infty}$. The latter is characterized by its Chern
classes, which can be also identified with the class $[{\varphi}]$ of
${\varphi}$ in

${\pi}_{2}(G/T)\approx{\pi}_{1}(T)\approx{{\rm Q}}^{\vee}\,,$

also known as the magnetic charges of the monopole solution. The magnetic
charge can also be read off the solution of (8.2) by projecting the curvature
$F_{A}$ to the Cartan subalgebra $\mathfrak{h}$ defined by
$\phi\biggr{|}_{{\mathbb{S}}^{2}_{\infty}}$, and by taking the corresponding
integrals:

${\bf
m}_{\varphi}=\frac{1}{2\pi\mathrm{i}}\int_{{\mathbb{S}}^{2}_{\infty}}F_{A}^{\mathfrak{h}}$
(8.5)

Now let us compactify one of the spatial directions, i.e. replace
${\mathbb{R}}^{3}$ by $M^{3}={\mathbb{S}}^{1}\times{\mathbb{R}}^{2}$. Let
$\psi\in[0,2\pi)$ be the angular coordinate on $\mathbb{S}^{1}$ and let
$x=x_{1}+\mathrm{i}x_{2}$ be the complex coordinate on $\mathbb{R}^{2}$. Let
us normalize the metric on $M^{3}$ so that the circumference of
$\mathbb{S}^{1}$ is equal to one.

Consider the complex connection in the $\mathbb{S}^{1}$ direction (we use the
physical convention where $A$ is represented by Hermitian matrices, i.e. by a
real-valued one-form for the $U(1)$ gauge group):

${\nabla}={\partial}_{\psi}+\mathrm{i}A_{\psi}-{\phi}$ (8.6)

The Eq. (8.2) implies that the ${{\bar{x}}}$-variation of $\nabla$ is an
infinitesimal gauge transformation:

${\bar{\partial}}_{{\bar{x}}}{\nabla}+[A_{{\bar{x}}},{\nabla}]=0$ (8.7)

Therefore, the conjugacy class of the holonomy $g(x,{{\bar{x}}})$ of $\nabla$
around $\mathbb{S}^{1}$ varies holomorphically with $x$, and, in the gauge
where $g(x,{{\bar{x}}})\in\mathbf{T}$, it is locally holomorphic:

$[g(x)]=\left[P{\exp}\,\oint_{0}^{2\pi}\,\mathrm{i}d\psi\left(A_{\psi}({\psi},x,{{\bar{x}}})+\mathrm{i}{\phi}({\psi},x,{{\bar{x}}})\right)\right]\in{B({\mathfrak{g}})}$
(8.8)

As we shall clarify later, when $x\to\infty$,

$[g(x)]\to\left[\prod_{i=1}^{r}{\mathfrak{q}}_{i}^{-{\lambda}_{i}^{\vee}}\right]=b_{\infty}\in{B^{\text{ad}}({\mathfrak{g}})}\
.$

One is left with the quasimap $u:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{B^{\text{ad}}({\mathfrak{g}})}$. It is instructive to
calculate (8.8) for the Dirac monopole on $M^{3}$.

Recall that the Dirac monopole at ${\vec{p}}\in{\mathbb{R}}^{3}$ is the
connection in the $U(1)$ bundle over ${\mathbb{R}}^{3}\backslash{\vec{p}}$,
which is a pullback of the constant curvature connection on the Hopf bundle
over ${\mathbb{S}}^{2}$ via the projection map

${\pi}_{\vec{p}}:{\mathbb{R}}^{3}\backslash{\vec{p}}\longrightarrow{\mathbb{S}}^{2},\qquad{\pi}_{\vec{p}}({\vec{r}})=\frac{{\vec{r}}-{\vec{p}}}{|{\vec{r}}-{\vec{p}}|}$

The corresponding curvature two-form $F$ is given by:

$F=2\pi\mathrm{i}\,{\pi}_{\vec{p}}^{*}{\varpi}_{2}=\frac{\mathrm{i}}{2}\frac{({\vec{x}}-{\vec{p}})\cdot
d{\vec{x}}\times d{\vec{x}}}{|{\vec{x}}-{\vec{p}}|^{3}}$

where

$\int_{{\mathbb{S}}^{2}}{\varpi}_{2}=1$

The fact that up to the $|{\vec{r}}-{\vec{p}}|^{2}$ rescaling the two-form
${\varpi}_{2}$ coincides with the volume form on $\mathbb{S}^{2}$ obtained
from the flat metric on ${\mathbb{R}}^{3}$ implies that $F$ solves Maxwell
equations in ${\mathbb{R}}^{3}\backslash{\vec{p}}$ and moreover there is a
magnetic potential $\phi$, such that

$d{\phi}+\star_{3}F=0$

Moreover, if $\phi$ is normalized to approach zero at infinity, then

${\phi}=-\frac{1}{2|{\vec{r}}-{\vec{p}}|}$ (8.9)

The periodic Dirac monopole, i.e. the solution of Maxwell equations on
$M^{3}\backslash p=({\psi}_{0},x_{0},{{\bar{x}}}_{0})$ can be obtained from
the basic monopole in $\mathbb{R}^{3}$ by taking the superposition of the
fields of an infinite periodic array of monopoles, living on the universal
cover $\widetilde{M^{3}\backslash
p}={\mathbb{R}}^{3}\backslash({\psi}_{0}+2\pi{\mathbb{Z}},x_{0},{{\bar{x}}}_{0})$.
The magnetic potential is given by the regularized sum of potentials (8.9):

$\displaystyle{\phi}({\psi},x,{{\bar{x}}};p)={\phi}_{\infty}+\frac{{\rm
log}(\pi)-{\gamma}}{2\pi}$ (8.10)
$\displaystyle\qquad\qquad+\sum_{n\in{\mathbb{Z}}}\left(\varphi({\psi}-{\psi}_{0}-2\pi
n,x-x_{0})+\frac{1-{\delta}_{n,0}}{4\pi|n|}\right)$
$\displaystyle\qquad\qquad\qquad\qquad{\varphi}({\psi},x)=-\frac{1}{2\sqrt{{\psi}^{2}+x{{\bar{x}}}}}$

We calculate:

$\frac{1}{2\pi}\int_{0}^{2\pi}{\phi}({\psi},x,{{\bar{x}}};p)d{\psi}={\phi}_{\infty}+\frac{1}{2\pi}{\rm
log}\,|x-x_{0}|$ (8.11)

The calculation of

$\int_{0}^{2\pi}A_{\psi}({\psi},x,{{\bar{x}}};p)d{\psi}$

is a bit tricky. Fortunately its derivative is easy to compute:

$\displaystyle
d\int_{0}^{2\pi}A_{\psi}({\psi},x,{{\bar{x}}};p)d{\psi}=\oint_{{\mathbb{S}}^{1}}F=$
(8.12)
$\displaystyle\qquad\frac{\mathrm{i}}{4}\sum_{n\in{\mathbb{Z}}}\int_{0}^{2\pi}\,d\psi\wedge\,\frac{(x-x_{0})d{{\bar{x}}}-({{\bar{x}}}-{{\bar{x}}}_{0})dx}{\left(\,|x-x_{0}|^{2}+\left({\psi}-{\psi}_{0}+2\pi
n\right)^{2}\,\right)^{3/2}}=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\frac{\mathrm{i}}{2}d\,{\rm
log}\left(\frac{x-x_{0}}{{{\bar{x}}}-{{\bar{x}}}_{0}}\right)$

Thus

$d\oint(A_{\psi}+\mathrm{i}{\phi})d\psi=\mathrm{i}d\,{\rm log}(x-x_{0})$

and the monodromy is equal to

$g(x)=(x-x_{0})^{-1}$ (8.13)

up to some multiplicative constant. Now, if we have a superposition of several
Dirac monopoles, in the theory with the gauge group $T$, with the monopoles of
the type $i$, i.e. corresponding to the coweight
${\lambda}_{i}^{\vee}\in{\operatorname{Hom}}(U(1),T)$ located at the points
$({\psi}_{i,{\mathfrak{f}}},m_{i,\mathfrak{f}},{\bar{m}}_{i,\mathfrak{f}})$,
then the monodromy of the corresponding complexified connection
$A+\mathrm{i}{\phi}d\psi$ is equal to:

$g(x)\propto\prod_{i=1}^{r}\prod_{{\mathfrak{f}}=1}^{{\mathbf{w}}_{i}}(x-m_{i,\mathfrak{f}})^{-{\lambda}^{\vee}_{i}}$
(8.14)

Now let us consider the nonabelian Bogomolny equation on $M^{3}$. Instead of
solving the Eq. (8.2) modulo $G$-gauge transformations, let us solve two out
of three equations in (8.2), namely the equation

$[D_{{\bar{x}}},{\nabla}]=0$ (8.15)

modulo the action of the group ${\mathcal{G}}^{\mathbb{C}}$ of
$\mathbf{G}$-valued (complex) gauge transformations. In fact, (8.15) can be
viewed as the complex moment map for ${\mathcal{G}}^{\mathbb{C}}$, acting on
the space of $(A,{\Phi})$, endowed with the
${\mathcal{G}}^{\mathbb{C}}$-invariant holomorphic symplectic form:

${\Omega}=\frac{1}{2\pi}\int_{M^{3}}\langle{\delta}{\nabla}\wedge{\delta}A_{{\bar{x}}}\rangle\
d\psi dxd{{\bar{x}}}$ (8.16)

Let us now try to analyze the solutions to (8.15) in some domain
$D\times{\mathbb{S}}^{1}\subset M^{3}$ over $D\subset\mathbf{C}_{\left\langle
x\right\rangle}$. We fix the $\mathbf{G}$-gauge where
$A_{\psi}+\mathrm{i}\phi={\xi}(x,{{\bar{x}}})$, ${\partial}_{\psi}{\xi}=0$.
This gauge leaves some residual gauge freedom. Passing to

$g(x)={\exp}\,2\pi\mathrm{i}\xi(x)$

partially reduces the residual gauge invariance. The Eq. (8.15) implies that
${\bar{\partial}}_{{\bar{x}}}{\xi}=0$, and
$A_{{\bar{x}}}=a_{{\bar{x}}}\in\mathfrak{h}$. We then proceed with
constructing the cameral curve ${\mathcal{C}}_{u}$ which is the union of the
$W({\mathfrak{g}})$-orbits of $g(x)$ over all
$x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}={\mathbf{C}_{\left\langle x\right\rangle}}\cup\\{\infty\\}$.
The fiber ${\mathscr{A}}_{u}$ of the projection
$h:{{\mathfrak{P}}}\to{{\mathfrak{M}}}$ is the space of
$W({\mathfrak{g}})$-equivariant $\mathbf{T}$-bundles over $\mathcal{C}_{u}$ of
fixed multi-degree. We shall discuss in more detail the analogous situation
for the class II theories in the next section.

The asymptotics of the solution to (8.2) is characterized by a vector of
magnetic charges. Namely, over
${\mathbb{T}}^{2}_{\infty}={\mathbb{S}}^{1}\times{\mathbb{S}}^{1}_{\infty}$
where ${\mathbb{S}}^{1}_{\infty}$ is a large radius circle in
${\mathbb{R}}^{2}$, the gauge group $G$ is broken down to $T$. The gauge
bundle is therefore characterized by the vector of the first Chern classes:

${\bf m}\in H^{2}({\mathbb{T}}^{2}_{\infty},{\pi}_{1}({\mathbf{T}}))={{\rm
Q}}^{\vee}$ (8.17)

One can compute ${\bf m}$ by analyzing the behavior of the conjugacy class of
the holonomy $g(x)$ of the complexified connection ${\nabla}$. One can show
that the finite energy (8.3) solutions with non-trivial magnetic charge do not
exist. Indeed, macroscopically the system looks two dimensional, and
asymptotically it looks like a charged vortex, whose energy diverges
logarithmically at large distances.

However, infinite $3$-dimensional energy solutions may correspond to the
finite tension higher dimensional objects. For example, the noncommutative
$U(1)$ monopole describes a finite tension string, which is attached to the
worldvolume of the gauge theory [Gross:2000wc]. Similarly, the infinite energy
periodic monopole solutions have interpretation in the higher dimensional
theory, e.g. in the brane realization [Witten:1997sc] of the pure
${\mathcal{N}}=2$ $SU(N)$ gauge theory in four dimensions [Cherkis:2000cj].

One can actually make the energy finite in the infrared by allowing point-like
singularities in $M^{3}$. The idea is to screen the asymptotic magnetic charge
of the non-abelian solution by the opposite charge of the Dirac monopole
singularities.

Let us study the general situation. Suppose
${\gamma}\subset{\mathbf{C}_{\left\langle x\right\rangle}}$ is a closed
contour. For each point $x\in\gamma$ compute the holonomy $g(x,{{\bar{x}}})$
of the complexified connection $\nabla$, e.g. starting at the point ${\psi}=0$
on the fiber ${\mathbb{S}}^{1}$. It is a functional of the gauge field $A$ and
the Higgs field $\phi$: $(A,{\phi})\mapsto g(x,{{\bar{x}}})$. The gauge
transformed $(A,\phi)$ leads to the similarity-transformed function
$g(x,{{\bar{x}}})$: $(A^{h},{\phi}^{h})\mapsto
h^{-1}(0,x,{{\bar{x}}})g(x,{{\bar{x}}})h(0,x,{{\bar{x}}})$. We have a well-
defined map:

$[g_{\gamma}]:{\gamma}\longrightarrow B({\mathfrak{g}})$ (8.18)

which is a restriction on $\gamma$ of the holomorphic (cf. (8.15)) map
$U:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{B^{\text{ad}}({\mathfrak{g}})}$, $U:x\mapsto[g(x)]$.

Now let ${\Xi}({\mathfrak{g}})\subset{B^{\text{ad}}({\mathfrak{g}})}$ denote
the set of irregular orbits of $W({\mathfrak{g}})$ in ${\mathbf{T}}/Z$.
Generically the image $\Sigma_{u}$ of $[g(x)]$ crosses ${\Xi}({\mathfrak{g}})$
at some isolated points, which are the branch points of the cameral curve
$\mathcal{C}_{u}$:

${\Xi}_{x}=U^{-1}\left({\Sigma}_{u}\cap{\Xi}({\mathfrak{g}})\right)$

Let us consider the subvariety
$B({\mathfrak{g}})^{reg}={B^{\text{ad}}({\mathfrak{g}})}\backslash{\Xi}({\mathfrak{g}})$.
The fundamental group ${\pi}_{1}(B({\mathfrak{g}})^{reg})$ is related to Artin
braid group associated with the $\mathfrak{g}$ root system. Let us also define
${\mathbf{T}}^{reg}$ to be the subvariety in $\mathbf{T}$ consisting of the
regular elements, i.e. the elements of the maximal torus whose stabilizer in
$\mathbf{G}$ is the maximal torus $\mathbf{T}$. It is invariant under the
translations by $Z$. We have a map:
${\pi}_{*}:{\pi}_{1}({\mathbf{T}}^{reg})\to{\pi}_{1}(B({\mathfrak{g}})^{reg})$,
induced by the projection
${\pi}:{\mathbf{T}}^{reg}\to{\mathbf{T}}^{reg}/\left(Z\times
W({\mathfrak{g}})\right)$.

Now, let us go back to the loop $\gamma$. We would like to define a
$\mathbf{T}$-bundle over ${\mathbb{S}}^{1}\times{\gamma}$, by choosing a gauge
where $A_{\psi}+\mathrm{i}\phi$ is $\psi$-independent element of the Lie
algebra $\mathfrak{h}\subset\mathfrak{g}$. Then Bogomolny equations imply that
$A_{{\bar{x}}}$ also belongs to $\mathfrak{h}$, assuming that
$A_{\psi}+\mathrm{i}\phi$ is regular, i.e. its stabilizer in $\mathbf{G}$ is
the maximal torus $\mathbf{T}$.

There is an obstruction for such a gauge being possible throughout $\gamma$.
Namely the class of $\gamma$ in ${\pi}_{1}(B({\mathfrak{g}})^{reg})$ should
lie in the image of ${\pi}_{*}$. This is related to our solution of the four
dimensional ${\mathcal{N}}=2$ theory of the class I as follows.

For the asymptotically conformal theories, with the assignments of dimensions
${\mathbf{v}}$, ${\mathbf{w}}$, we have defined a ${\mathbf{G}}/Z$-valued
function on $\mathbf{C}_{\left\langle x\right\rangle}$ minus a finite number
of points:

$g(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\lambda}^{\vee}_{i}}{\mathscr{Y}}_{i}(x)^{{\alpha}_{i}^{\vee}}$
(8.19)

We identify $g(x)$ in (8.19) with the holonomy in (8.8). The factor

$g_{\infty}(x)=\prod_{i=1}^{r}{\mathscr{P}}_{i}(x)^{-{\lambda}^{\vee}_{i}}$

clearly corresponds to the Dirac monopoles sitting at some points
$({\psi}_{i,{\mathfrak{f}}},m_{i,{\mathfrak{f}}},{\bar{m}}_{i,\mathfrak{f}})$
with the charges ${\lambda}^{\vee}_{i}$. The remaining factor has to do with
the nonabelian monopoles.

Recall that the map $U:\mathbf{C}_{\left\langle
x\right\rangle}\backslash{\Xi}_{x}\to B({\mathfrak{g}})$ is determined by the
collection of gauge polynomials $T_{i}(x)$, $i=1,\ldots,r$. The singular locus
${\Xi}_{x}$ of $U$ are at the zeroes and poles of the discriminant

${\Delta}(x)=g(x)^{-2{\rho}}\prod_{{\alpha}\in
R_{+}}\left(g(x)^{\alpha}-1\right)^{2}$ (8.20)

where $R_{+}$ is the set of positive roots of $\mathfrak{g}$, and

${\rho}=\frac{1}{2}\sum_{{\alpha}\in R_{+}}{\alpha}\ .$

The discriminant ${\Delta}(x)$ is a rational function in $T_{i}(x)$’s and
${\mathscr{P}}_{i}(x)$’s. Now, given a loop ${\gamma}$ in
${\mathbf{C}_{\left\langle x\right\rangle}}\backslash{\Xi}_{x}$, when can we
lift $[g(x)]|_{\gamma}$ to the $\mathbf{T}$-valued loop?

For the simple root ${\alpha}_{i}$ let us denote by ${\Xi}_{x,i}$ the set of
solutions to the equation $g(x)^{{\alpha}_{i}}=1$ on the physical sheet of
$\mathcal{C}_{u}$.

${\Xi}_{x,i}=\left\\{\,x\,|\ g\left(x\right)^{{\alpha}_{i}}=1\,\right\\}$
(8.21)

so that

${\Xi}_{x}=\bigcup_{i=1}^{r}\ {\Xi}_{x,i}$ (8.22)

Actually the points of $\Xi_{x,i}$ are the endpoints of the cuts
$I_{i,{\mathbf{a}}}$.

Our claim is that for the loops ${\gamma}=A_{i{\mathbf{a}}}$ which encircle
the individual cuts $I_{i,\mathbf{a}}$ the class $[g(x)]\in
B({\mathfrak{g}})^{reg}$ lifts to ${\mathbf{T}}^{reg}$, and defines a
${\mathbf{T}}$-bundle over ${\mathbb{S}}^{1}\times{\gamma}$. Its
characteristic class is equal to ${\alpha}_{i}^{\vee}\in{{\rm Q}}^{\vee}$.

Thus the magnetic monopoles which correspond to the limit shape of the
${\mathcal{N}}=2$ theory have the Dirac monopole charges ${\bf q}_{\rm
Dir}=-\sum_{i=1}^{r}{\mathbf{w}}_{i}{\lambda}_{i}^{\vee}$ which are
distributed at the points
$({\psi}_{i,{\mathfrak{f}}},m_{i,\mathfrak{f}},{\bar{m}}_{i,{\mathfrak{f}}})$,
and the nonabelian monopole charges ${\bf
q}_{\rm{}^{\prime}tHP}=\sum_{i=1}^{r}{\mathbf{v}}_{i}{\alpha}_{i}^{\vee}$
which are located over the cuts $I_{i,{\mathbf{a}}}$. The net charge at
infinity is equal to

${\bf q}_{\rm Dir}+{\bf q}_{\rm{}^{\prime}tHP}=0$ (8.23)

for the asymptotically conformal theories.

For the asymptotically free theories the net charge at infinity is equal to
(cf. (3.1)):

${\bf q}_{\rm Dir}+{\bf
q}_{\rm{}^{\prime}tHP}=-\sum_{i=1}^{r}{\beta}_{i}{\lambda}_{i}{{}^{\vee}}$
(8.24)

The fact that ${\beta}_{i}\leq 0$ should follow from the positivity of energy
(as it does in the $A_{1}$ case) but we couldn’t find a simple proof for
general $\mathbf{G}$.

The relation of the monopole picture of ${\mathfrak{P}}$ to the Hitchin system
picture we had in the section 7.1.1 goes via the Nahm transform, or, since we
are ultimately working only in a particular complex structure of the moduli
space, via a version of Fourier-Mukai transform. The $U(k)$ monopoles on
$\mathbb{R}^{3}$ are mapped via Nahm’s transform to the solutions of a one-
dimensional system of Nahm’s equations. The $U(k)$ monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ are mapped via Nahm’s transform to
the solutions of a two-dimensional system of Hitchin’s equations, with
singularities. Indeed, our spectral curve in the form (7.123) captures the
solutions to two, ${\bar{\partial}}_{A}{\Phi}=0$, ${\partial}_{A}{\bar{\Phi}}$
out of three Hitchin’s equations. The remaining equation
$F_{A}+[{\Phi},{\bar{\Phi}}]=0$ away from the punctures would fix the
hyperkähler metric on ${\mathfrak{P}}$. Unfortunately we are not in the
position to discuss the metric on ${\mathfrak{P}}$ as long as we stay within
the realm of the four dimensional gauge theory. See [Nahm:1981nb,
Corrigan:1983sv, Hitchin:1983ay, Braam:1988qk].

We do need to discuss the holomorphic symplectic geometry of ${\mathfrak{P}}$.
The symplectic form on ${\mathfrak{P}}$ descends from the two-form (8.16) via
the Hamiltonian reduction with (8.15) being the moment map. The textbook
construction of the action-angle variables of the integrable system produces
the special coordinates
${\mathfrak{a}}^{\mathcal{I}},{\mathfrak{a}}_{\mathcal{I}}^{D}$ of the gauge
theory. We claim this construction is equivalent to the one using the periods
of the differentials $x\,d{\log}{\mathscr{Y}}_{i}$ on the cameral curve. The
essential points of the demonstration are identical for the class I and for
the class II theories.

We thus return to this question in the section 8.2.

Now let us study the $A_{1}$ case in some more detail. We wish to present yet
another perspective on the phase space ${\mathfrak{P}}$.

Consider the product of $N$ $A_{1}$ surfaces ${\mathcal{O}}_{a}$,
$a=1,\ldots,N$, the complex coadjoint orbits of $SL(2,{\mathbb{C}})$. Each
surface ${\mathcal{O}}_{i}$ is a quadric in ${\mathbb{C}}^{3}$, given by the
equation:

${\xi}^{2}_{a}+{\eta}_{a}^{2}+{\zeta}_{a}^{2}=s_{a}^{2}$ (8.25)

with some fixed constant $s_{a}\in\mathbb{C}$. The surface $\mathcal{O}_{a}$
has the holomorphic symplectic form:

${\varpi}_{a}=\frac{d{\xi}_{a}\wedge d{\eta}_{a}}{\zeta_{a}}$

which has the period $s_{a}$ along a non-contractible two-sphere in
$\mathcal{O}_{a}$. The moment map for the action of $SL(2,{\mathbb{C}})$ on
${\mathcal{O}}_{a}$ is a $2\times 2$ traceless matrix. Let us extend it into a
general $2\times 2$ matrix (which corresponds to the conformal extension of
the group):

$L_{a}(x)=\left(\begin{matrix}x+{\xi}_{a}&{\eta}_{a}+\mathrm{i}{\zeta}_{a}\\\
-{\eta}_{a}+\mathrm{i}{\zeta}_{a}&x-{\xi}_{a}\end{matrix}\right)$ (8.26)

and define the ‘‘monodromy matrix’’

$L(x)=\left(\begin{matrix}{\mathfrak{q}}&0\\\ 0&1\end{matrix}\right)\times
L_{N}(x-{\mu}_{N})L_{N-1}(x-{\mu}_{N-1})\ldots L_{1}(x-{\mu}_{1})$ (8.27)

It is actually better to work with the somewhat differently normalized ‘‘local
Lax operators’’:

$g_{a}(x)={\bf
1}_{2}+\frac{1}{x-s_{a}}\left(\begin{matrix}u_{a}^{+}&v_{a}^{+}\\\
v_{a}^{-}&u_{a}^{-}\end{matrix}\right)$ (8.28)

where

$u_{a}^{\pm}=s_{a}\pm{\xi}_{a},\qquad
v_{a}^{\pm}=\mathrm{i}{\zeta}_{a}\pm{\eta}_{a}$

obey

$u_{a}^{+}u_{a}^{-}-v_{a}^{+}v_{a}^{-}=0$ (8.29)

and define

$g(x)=\left(\begin{matrix}{\mathfrak{q}}&0\\\ 0&1\end{matrix}\right)\times
g_{N}(x-{\mu}_{N})g_{N-1}(x-{\mu}_{N-1})\ldots g_{1}(x-{\mu}_{1})$ (8.30)

Then, cf. [Gorsky:1996hs]:

${\rm
Det}(g(x))={\mathfrak{q}}\frac{{\mathscr{P}}^{+}(x)}{{\mathscr{P}}^{-}(x)}$
(8.31)

where we defined

${\mathscr{P}}^{\pm}(x)=\prod_{a=1}^{N}(x-{\mu}_{a}\pm s_{a})$

Now, define the phase space to be the complex symplectic quotient of the
product of the $A_{1}$ surfaces by the diagonal action of the
${\mathbb{C}}^{\times}$,

${{\mathfrak{P}}}=\times_{a=1}^{N}{\mathcal{O}}_{a}//{\mathbb{C}}^{\times}$
(8.32)

generated by

$H_{1}=\sum_{a=1}^{N}{\xi}_{a}$

The variety (8.32), defined by fixing the level ${\xi}$ of $H_{1}$ and
dividing the corresponding level set $H_{1}^{-1}({\xi})$ by
${\mathbb{C}}^{\times}$, carries the induced holomorphic symplectic structure.
The functions $h_{k}$ defined as:

${\operatorname{tr}}_{2}\,g(x)=(1+{\mathfrak{q}})\left(1+\sum_{k=1}^{\infty}x^{-k}h_{k}\right)$
(8.33)

Poisson-commute with respect to the induced Poisson structure. Moreover,

$h_{1}=H_{1}+\frac{{\mathfrak{q}}-1}{{\mathfrak{q}}+1}\sum_{a}s_{a}$

and the next $N-1$ $h_{k}$’s are independent. The rest of the expansion
coefficients can be expressed in terms of the first $N$.

We argue that the normalized Lax operator $g(x)$ is the complexified monodromy
field $g(x)$ in the corresponding periodic singular monopole problem, with the
monopole group $U(2)$ (the compact form of C$\mathbf{G}$) and ${\mu}_{a}\pm
s_{a}$ are the locations of $2N$ Dirac monopoles.

It is not difficult to convince oneself that the ‘‘local Lax operator’’ (8.28)
is indeed the complexified monodromy of a single $U(2)$ monopole screened by
two Dirac monopoles of the opposite $U(1)$ charges, located at $\pm s_{a}$.

What is amusing is that the Eq. (8.30) suggests that the complexified
monodromy of the charge $N$ $U(2)$ monopole screened by $2N$ Dirac monopoles
factorizes as the product of $N$ elementary monodromy matrices.

Note in passing that if we do not perform the reduction with respect to
${\mathbb{C}}^{\times}$ generated by $H_{1}$, i.e. work with the
$2N$-dimensional phase space $\tilde{\mathfrak{P}}$ (this is a first step
towards the extended phase space ${\mathfrak{P}}^{\mathrm{ext}}$), then
Sklyanin’s separation of variables [Sklyanin:1987ih, Sklyanin:1995bm]
identifies its open subset ${\tilde{\mathfrak{P}}}^{\circ}$ with the $N$’th
symmetric product of (which is most likely [Gorsky:1999rb] resolved into the
Hilbert scheme of $N$ points on) ${\mathbf{C}_{\left\langle
x\right\rangle}}\times{\mathbf{C}_{\left\langle t\right\rangle}}^{\times}$.
Incidentally [Donaldson:1985id], this manifold is symplectomorphic to the
moduli space of regular charge $N$ $SU(2)$ monopoles on ${\mathbb{R}}^{3}$.

In recent years the connection between the gauge theories and the spin chains,
and the inspired duality between the Gaudin-like integrable systems and the
Heisenberg spin chain was discussed in [Chen:2011sj, Dorey:2011pa,
Muneyuki:2011qu, Mironov:2012ba], building on the earlier work in
[Moore:1997dj, Gerasimov:2006zt, Gerasimov:2007ap, Nekrasov:2009zz,
Nekrasov:2009ui].

Before concluding this section, note that the masses of the bi-fundamental
matter hypermultiplets are encoded in the next-to-leading terms in the
asymptotics of the complexified connection $A_{\psi}+i{\phi}$ near
$x\to\infty$. We shall explain this in more detail in the similar context in
the section 8.2.

The moduli space of singular $\mathbf{G}$-monopoles with fixed conjugacy class
of the monodromy of $A_{\psi}+i{\phi}$ at $x\to\infty$, with unspecified
location of Dirac singularities of specified charges defines our extended
phase space ${\mathfrak{P}}^{\mathrm{ext}}$. It is acted upon by the torus
$T\times T_{M}$, where $T_{M}\subset{G_{M}}$ is the maximal torus of the
flavor group. The action of $T$ is via the constant gauge transformations
preserving the gauge field and the Higgs field at infinity, while $T_{M}$ acts
by changing the glueing data for the grafted Dirac monopoles. Fixing the level
of the corresponding moment maps and reducing with respect to the
complexification of the $T\times T_{M}$ action produces ${\mathfrak{P}}$.

Summarizing, we conclude with the conjecture: _the moduli space of singular
$\mathbf{G}$-monopoles on ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ is acted
upon by the Poisson-Lie group, which is a quasi-classical limit of the Yangian
$Y({\mathfrak{g_{\text{q}}}})$. The deformation quantization of
${\mathfrak{P}}^{\mathrm{ext}}$ (in the complex structure $I$) produces the
Yangian $Y({\mathfrak{g}})$. Fixing the asymptotics of the complexified gauge
field at infinity as well as the locations of the Dirac singularities would
specify._ It should be interesting to explore the holomorphic symplectic
geometry of ${\mathfrak{P}}$ in all of its complex structures and find the
analogue of the variety of opers (cf. [Nekrasov:2010ka, Nekrasov:2011bc]),
which in the $A_{r}$ case is the variety of local systems on the genus zero
curve coming from the $N$’th order differential operators with regular
singularities at $r+3$ punctures with the monodromies whose eigenvalues
coincide with our description of the residues of the Higgs field just above
the Eq. (7.23) (cf. [Frenkel:2003qx]).

Note that in [Gerasimov:2005qz] a connection between the representation theory
of Yangians and the moduli spaces of monopoles on ${\mathbb{R}}^{3}$ was
found. Also, in [Atiyah:1991gd, Atiyah:1996ij] the relations between the
monopole solutions and the solutions to the Yang-Baxter equations were
discussed. It remains to be seen, whether any of these connections carries
over to the ${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ case and whether it is
the one we need.

What happens if one tries to study periodic $\widehat{\mathbf{G}}$-monopoles
where $\widehat{\mathbf{G}}$ is Kac-Moody affine group? One naturally finds
double periodic instantons similar to relation between
$\widehat{\mathbf{G}}$-monopoles and periodic $\mathbf{G}$ instantons
[Garland_1988].

###  8.2. Double-periodic instantons and Class II theories

Let us now discuss the class II theories. Recall from the section 6.2.2 that
an elliptic curve
${\mathscr{E}}={\mathbb{C}}^{\times}/{\mathfrak{q}}^{\mathbb{Z}}$, with
$\mathfrak{q}$ given by 6.24, is associated with the class II gauge theory.
Also recall that the gauge group is the product of special unitary groups

${G_{\text{g}}}=\times_{i=0}^{r}\ SU(Na_{i})$

for some number $N$. Using (6.28) we have identified the extended moduli space
${\mathfrak{M}}^{\mathrm{ext}}_{N}$ of vacua of the theory (4.7),
$\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}_{N}=Nh^{\vee}$, with the
moduli space of degree $N$ framed quasimaps of
$({\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle},{\infty})$ to the
moduli space ${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$ of holomorphic
$\mathbf{G}$-bundles on $\mathscr{E}$, sending ${\infty}$ to a particular
bundle $[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]$. This space of quasimaps is a
natural base of the projection from the moduli space

${{\mathfrak{P}}^{\mathrm{ext}}}_{N}\approx\mathrm{Bun}^{ss}_{\mathbf{G};N}(S_{\mathscr{E}})^{{\mathcal{P}}_{\tilde{\mathfrak{q}}}}_{\infty}$
(8.34)

of framed semi-stable holomorphic principal $\mathbf{G}$-bundles
$\mathcal{P}$, $c_{2}({\mathcal{P}})=N$, on the surface
$S_{\mathscr{E}}\equiv{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}$, where the framing is the identification
of the restriction of $\mathcal{P}$ at the fiber at infinity:

${\mathcal{P}}\biggr{|}_{\\{\infty\\}\times{\mathscr{E}}}\approx{\mathcal{P}}_{\tilde{\mathfrak{q}}}$
(8.35)

In what follows we drop the subscript $N$.

The projection
${{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$ is defined
on the open dense subset of ${\mathfrak{P}}^{\mathrm{ext}}$ by restricting the
bundle ${\mathcal{P}}\in{\mathfrak{P}}^{\mathrm{ext}}$ to the fiber
${\mathscr{E}}_{x}$ and taking its equivalence class ${\bf
t}(x)=[{\mathcal{P}}|_{{\mathscr{E}}_{x}}]$ in
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$. This gives the desired quasimap
$U:x\mapsto{\bf t}(x)$. This quasimap is a map near $x=\infty$, approaching a
particular holomorphic bundle
$[{\mathcal{P}}_{\tilde{\mathfrak{q}}}]\in{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})$
over $\mathscr{E}_{\infty}=\\{{\infty}\\}\times{\mathscr{E}}$. The reason $U$
is not, in general, a map, has to do with the usual difference between the
stability condition in two complex dimensions and the fiberwise stability
condition, cf. [Losev:1999tu].

The semi-stable framed holomorphic principal bundles $\mathcal{P}$ on
$S_{\mathscr{E}}$, are in one-to-one correspondence with the $G$-instantons on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ with the flat metric (cf.
[Donaldson:1985zz]), i.e. connections on a principal $G$-bundle $P$ over
${\mathbb{S}}^{2}\times{\mathbb{T}}^{2}$, endowed with some metric, which is
conformally flat on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, which obey

$F_{A}=-\star F_{A}$

and have finite action,

$\int_{{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}}\left\langle F_{A}\wedge\star
F_{A}\right\rangle<\infty$

(as we explain later, these instantons correspond to the M2 branes and the
instanton action is the tension of the stack of M2 branes) and this forces the
curvature $F_{A}$ of the $G$-gauge field tend to zero as
$|{\vec{x}}|\to\infty$, ${\vec{x}}\in{\mathbb{R}}^{2}$. We fix the instanton
charge:

$N=-\frac{1}{8\pi^{2}h}\int_{{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}}\left\langle
F_{A}\wedge F_{A}\right\rangle$ (8.36)

(remember that $\left\langle,\right\rangle$ is the Killing form, which is the
trace in the adjoint representation). The real dimension of the moduli space
${\mathfrak{P}}^{\mathrm{ext}}$ of $G$-instantons of finite action on
$\mathbb{R}^{2}\times\mathbb{T}^{2}$ with fixed framing at infinity
$\mathbb{T}^{2}_{\infty}$ is equal to $4Nh$.

Actually, there is a subtlety here. The moduli space of charge $N$
$G$-instantons on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ may have several
components. This has to do with the fact that the moduli space of flat
$G$-connections on
${\mathbb{T}}^{3}={\mathbb{S}}^{1}_{\infty}\times{\mathbb{T}}^{2}$ may have
several components [Witten:1997bs]. We shall assume we are always in the
component of the trivial connection.

Note that ${\mathfrak{P}}^{\mathrm{ext}}$ is acted upon by the maximal torus
$T$ of $G$, the symmetry group of the flat connection at infinity. This action
lifts to the action of the algebraic torus ${\mathbf{T}}$ on the moduli space
of framed holomorphic bundles ${{\mathfrak{P}}^{\mathrm{ext}}}$. This is
entirely parallel to the action of the group $\mathbf{G}$ on the moduli space
${{\mathfrak{M}}}^{\mathrm{framed}}_{G}({\mathbb{R}}^{4})$ of framed
$G$-instantons on ${\mathbb{R}}^{4}$.

A holomorphic principal $\mathbf{G}$-bundle ${\mathcal{P}}$ on
$S_{\mathscr{E}}$ can be described using the transition functions
$g_{\alpha\beta}:U_{\alpha\beta}\to{\mathbf{G}}$ defined on the overlaps
$U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}$ of the open sets in the appropriate
open cover $(U_{\alpha})_{{\alpha}\in A}$ of $S_{\mathscr{E}}$. The transition
functions are holomorphic ${\bar{\partial}}g_{\alpha\beta}=0$, must obey the
cocycle condition $g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=1$ on
$U_{\alpha\beta\gamma}=U_{\alpha}\cap U_{\beta}\cap U_{\gamma}$, and the
cocycles differing by the holomorphic coboundaries define equivalent bundles:

$g_{\alpha\beta}\sim g_{\alpha}g_{\alpha\beta}g_{\beta}^{-1}$

where $g_{\alpha}:U_{\alpha}\to{\mathbf{G}}$ are holomorphic
$\mathbf{G}$-valued functions on the open sets $U_{\alpha}$ themselves.

Another way to describe the holomorphic bundle is to introduce a connection
$\nabla=d+A$ on a smooth bundle, such that its $(0,1)$-part is $(0,2)$-flat:

${\nabla}_{\bar{\partial}}^{2}=0$ (8.37)

The local holomorphic sections of
${\mathcal{P}}_{\alpha}={\mathcal{P}}|_{U_{\alpha}}$ are the solutions to
${\nabla}_{\bar{\partial}}s_{\alpha}=0$. Over an intersection
$U_{\alpha\beta}$ these solutions must differ by a holomorphic
$\mathbf{G}$-valued gauge transformation:

$s_{\alpha}({\zeta},{\bar{\zeta}})=g_{\alpha\beta}({\zeta})s_{\beta}({\zeta},{\bar{\zeta}})$

where $\zeta$ stand for local holomorphic coordinates. This is sometimes
expressed by saying that locally ${\bar{A}}$ is a pure $\mathbf{G}$-gauge

${\bar{A}}\biggr{|}_{U_{\alpha}}=-s_{\alpha}^{-1}{\bar{\partial}}s_{\alpha}$

In the case at hand ${\zeta}=(x,z)$, where $x\in\mathbf{C}_{\left\langle
x\right\rangle}\subset{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}$ and $z$ is the additive coordinate on $\mathscr{E}$. The Eq.
(8.37) reads:

${\bar{\partial}}_{{\bar{x}}}A_{{\bar{z}}}-{\bar{\partial}}_{{\bar{z}}}A_{{\bar{x}}}+[A_{{\bar{x}}},A_{{\bar{z}}}]=0$
(8.38)

This equation can be viewed as the complex moment map equation for the action
of the group $\mathcal{G}$ of the $\mathbf{G}$-gauge transformations on the
space $\mathcal{A}$ of connections on a given smooth principal
$\mathbf{G}$-bundle over $S_{\mathscr{E}}$, endowed with the holomorphic
symplectic form:

${\Omega}=\int_{S_{\mathscr{E}}}dx\wedge
dz\wedge\langle{\delta}{\bar{A}}\wedge{\delta}{\bar{A}}\rangle$ (8.39)

We have a little subtlety here. The two-form $dx\wedge dz$ was perfectly good
on $S_{\mathscr{E}}^{\circ}$ but on $S_{\mathscr{E}}$ is has a pole along the
divisor ${\mathscr{E}}_{\infty}=\\{{\infty}\\}\times{\mathscr{E}}$. We impose
the condition that $A_{{\bar{z}}}$ approaches a specific value as
$x\to\infty$. More specifically, recall (C.113) that generic $A_{{\bar{z}}}$
can be $\mathcal{G}$-transformed to the normal form
$A_{{\bar{z}}}\to\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi\in\mathfrak{h}$,
${\partial}_{z}{\xi}={\partial}_{{\bar{z}}}{\xi}=0$. We impose the boundary
conditions:

$A_{{\bar{z}}}(x,{{\bar{x}}})\to\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi_{\infty}+o(|x|^{-2}),\qquad|x|\to\infty$
(8.40)

for fixed $\xi_{\infty}$, which we relate to the gauge couplings
${\mathfrak{q}}_{i}$, $i=0,1,\ldots,r$ via:

${\xi}_{\infty}=-\frac{1}{2\pi\mathrm{i}}\sum_{i=1}^{r}{\log}{\mathfrak{q}}_{i}\,{\lambda}_{i}^{\vee}$
(8.41)

The limiting gauge field
${\bar{\partial}}_{{\bar{z}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}\xi_{\infty}$
corresponds to the holomorphic bundle ${\mathcal{P}}_{\tilde{\mathfrak{q}}}$
on $\mathscr{E}$. The decay rate (8.40) makes the integral (8.39) convergent.
One can impose weaker conditions, allowing even the $O(x^{-1})$ (but no
$x^{-1}{{\bar{x}}}^{-1}$ terms!) decay, which would make (8.39) convergent
with the principal value prescription. In fact, these subleading terms
correspond to the bi-fundamental masses of the $G_{\text{g}}$-theory. Let us
interpret them using the $\mathbf{T}$-action on
${{\mathfrak{P}}^{\mathrm{ext}}}$.

The constant $\mathbf{T}$-gauge transformation with the parameter
${\varepsilon}\in\mathfrak{h}$ acts on the $(0,1)$-gauge field ${\bar{A}}$ as
follows:

${\delta}{\bar{A}}=[{\varepsilon},{\bar{A}}]+{\nabla}_{\bar{\partial}}{\eta}_{\varepsilon}$
(8.42)

where ${\eta}_{\varepsilon}(x,{{\bar{x}}},z,{{\bar{z}}})$ is the compensating
gauge transformation which sufficiently fast decays at $x\to\infty$.
Contracting the vector field (8.42) with $\Omega$ in (8.39) gives us a closed
one-form ${\delta}{\bf m}_{\varepsilon}$ on the space of connections obeying
(8.38), where ${\bf m}_{\varepsilon}$ is linear in $\varepsilon$, ${\bf
m}_{\varepsilon}=\left\langle{\varepsilon},{\bf m}\right\rangle$ for some
${\bf m}\in\mathfrak{h}$:

$\displaystyle{\delta}{\bf m}_{\varepsilon}\,=$
$\displaystyle\,\int_{S_{\mathscr{E}}}\,dx\wedge
dz\wedge\left\langle\left([{\varepsilon},{\bar{A}}]+{\nabla}_{\bar{\partial}}{\eta}_{\varepsilon}\right){\delta}{\bar{A}}\right\rangle=\int_{S_{\mathscr{E}}}dx\wedge
dz\wedge{\bar{\partial}}\left\langle{\varepsilon},{\delta}{\bar{A}}\right\rangle$
(8.43) $\displaystyle\qquad\qquad{\delta}\oint_{x=\infty}dx\
\left\langle{\varepsilon},\int_{{\mathscr{E}}_{x}}d^{2}z\,A_{{\bar{z}}}\right\rangle$

Thus the moment map ${\bf m}$ for the $\mathbf{T}$ action is the residue at
$x=\infty$ of the zero mode of the $\mathfrak{h}$-projection of the
$A_{{\bar{z}}}$ gauge field. The analogous statement holds for the monopoles
of the section 8.1.

Now let us solve (8.38) locally over some domain $D$ in
$\mathbf{C}_{\left\langle x\right\rangle}$. We choose the gauge (C.113) over
each point $x\in D$:

$A_{{\bar{z}}}(z,{{\bar{z}}};x,{{\bar{x}}})\longrightarrow\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}{\xi}(x,{{\bar{x}}})\in\mathfrak{h}$
(8.44)

If $\infty\in D$, then, using (8.43), in the gauge (8.44):

${\xi}(x,{{\bar{x}}})={\xi}_{\infty}+\frac{{\bf m}}{x}+\ldots\,,\qquad
x\to\infty$ (8.45)

where ${\bf m}\in\mathfrak{h}$ is a linear function of the bi-fundamental
masses $({\bf m}_{e})_{e\in\mathrm{Edge}_{\gamma}}$.

Recall that $\mathbf{G}$ is a simple simply-connected Lie group. Therefore,
the restriction ${\mathcal{P}}|_{{\mathscr{E}}_{x}}$ is trivial as a smooth
$\mathbf{G}$-bundle. Therefore ${\bar{A}}$ is just a $\mathfrak{g}$-valued
$(0,1)$-form on $D\times\mathscr{E}$. Now let us decompose
$A_{{\bar{x}}}=a_{{\bar{x}}}+W_{{\bar{x}}}$, with
$a_{{\bar{x}}}\in\mathfrak{h}$ and
$W_{{\bar{x}}}\in{\mathfrak{h}}^{\perp}\subset\mathfrak{g}$, the orthogonal
decomposition being provided by the Killing form $\langle\cdot,\cdot\rangle$.
Then (8.38) implies: ${\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})=0$,
${\bar{\partial}}_{{\bar{z}}}a_{{\bar{x}}}=0$, $W_{{\bar{x}}}=0$, the latter
equation being valid for generic ${\xi}(x)$, corresponding to the irreducible
bundles on $\mathscr{E}$. Indeed, the $F^{0,2}=0$ equation (8.38) splits in
$\mathfrak{g}={\mathfrak{h}}\oplus{\mathfrak{h}}^{\perp}$ as follows:

$\displaystyle F^{0,2}|_{{\mathfrak{h}}}\ =$ $\displaystyle\
{\bar{\partial}}_{{\bar{x}}}{\xi}-{\bar{\partial}}_{{\bar{z}}}a_{{\bar{x}}}=0$
(8.46) $\displaystyle F^{0,2}|_{{\mathfrak{h}}^{\perp}}\ =$ $\displaystyle\
{\bar{\partial}}_{{\bar{z}}}W_{{\bar{x}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}[{\xi},W_{{\bar{x}}}]=0$

For generic $\xi\in\mathfrak{h}$ (irreducible ${\rm
ad}({\mathcal{P}})^{\perp}$) the $\mathfrak{h}^{\perp}$-equation (8.46) does
not have non-zero solutions, hence $W_{{\bar{x}}}=0$. As for the
$\mathfrak{h}$-equation (8.46) in our gauge its solution is:

$a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})=a_{{\bar{x}}}^{(0)}(x,{{\bar{x}}},z)+{{\bar{z}}}{\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})$

Since on $D\times{\mathscr{E}}$ the connection form
$A_{{\bar{z}}}d{{\bar{z}}}+a_{{\bar{x}}}d{{\bar{x}}}$ is simply an
$\mathfrak{h}$-valued $(0,1)$-form the component
$a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})$ must be $z$-periodic. This is
only possible if ${\bar{\partial}}_{{\bar{x}}}{\xi}(x,{{\bar{x}}})=0$, which
also implies $a_{{\bar{x}}}(x,{{\bar{x}}},z,{{\bar{z}}})$ is $z$-independent.

Of course, (8.44) is not the complete gauge fixing: there remain the
$z$-independent $\mathbf{T}$-valued gauge transformations and the
$W({\mathfrak{g}})$-transformations (C.114), which combine into the locally
$z$-constant $N({\mathbf{T}})$-gauge transformations. There are also the
shifts (C.115) by the lattice ${{\rm Q}}^{\vee}\oplus{\tau}{{\rm Q}}^{\vee}$,
generated by the $(z,{{\bar{z}}})$-dependent $\mathbf{T}$-valued gauge
transformations with discrete, as far as the $(x,{\bar{x}})$-dependence is
concerned, parameters ${\beta}_{1},{\beta}_{2}\in{{\rm Q}}^{\vee}$. Let us
expand:

${\xi}(x)=\sum_{i=1}^{r}{\xi}_{i}(x){\alpha}^{\vee}_{i}$ (8.47)

The residual shifts act on the components ${\xi}_{i}(x)$ by

${\xi}_{i}(x)\mapsto{\xi}_{i}(x)+{\beta}_{1,i}+{\beta}_{2,i}{\tau},\qquad
i=1,\ldots,r+1$

where ${\beta}_{1,i},{\beta}_{2,i}\in\mathbb{Z}$, for $i=1,\ldots,r$ are the
expansion coefficients:

${\beta}_{A}=\sum_{i=1}^{r}{\beta}_{A,i}{\alpha}_{i}^{\vee},\qquad A=1,2$
(8.48)

Let ${\bf
t}(x)=\left(t_{i}(x)=e^{2\pi\mathrm{i}\xi_{i}(x)}\right)_{i=1}^{r}\in({\mathbb{C}}^{\times})^{r}$
and $[{\bf t}(x)]\in{\mathscr{E}}^{r}$ be the equivalence classes for the
actions of the lattices ${{\rm Q}}^{\vee}$ and ${{\rm
Q}}^{\vee}\oplus{\tau}{{\rm Q}}^{\vee}$, respectively.

Thus, dividing by all but the locally $z$-independent $N({\mathbf{T}})$-gauge
transformations we arrive at the collection of $r$ points on $\mathscr{E}$,
or, in a more sophisticated fashion, a point $[{\bf t}(x)]$ in
${\mathscr{E}}\otimes{{\rm Q}}$, in addition to the $\mathfrak{h}$-valued
gauge field $a_{{\bar{x}}}(x,{{\bar{x}}})d{{\bar{x}}}$. This is all done over
the generic point $x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$. We haven’t completely fixed the gauge, though.

Let us forget for a moment about the gauge field

${\bar{a}}=a_{{\bar{x}}}(x,{{\bar{x}}})d{{\bar{x}}}$ (8.49)

Then we’d divide by the action $W({\mathfrak{g}})$, giving as back the point
$[{\mathcal{P}}|_{{\mathscr{E}}_{x}}]$ in the orbispace
${\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})={\mathscr{E}}\otimes{{\rm
Q}}/W({\mathfrak{g}})$, the holomorphic $\mathbf{G}$-bundle on $\mathscr{E}$
whose holomorphic structure is given by
${\bar{\partial}}_{{\bar{z}}}+\frac{2\pi\mathrm{i}}{{\tau}-{\bar{\tau}}}{\xi}(x)$.
The (quasi)map $U:x\mapsto U(x)=[{\mathcal{P}}_{x}]$ is point
$u\in{{\mathfrak{M}}^{\mathrm{ext}}}$ in the extended moduli space of the four
dimensional class II gauge theory.

The considerations similar to those in [Friedman:1997ih] show that the
instanton charge $c_{2}({\mathcal{P}})=N$ bundles on $S_{\mathscr{E}}$
correspond to the degree $N$ quasimaps
$u:{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\to{{\mathrm{Bun}}_{\mathbf{G}}({\mathscr{E}})}$. This is of
course in line with the original observations on the relations between the
sigma model and gauge instantons [Atiyah:1984tk].

By following the whole $W({\mathfrak{g}})$-orbit of $[{\bf t}(x)]$ in
${\mathscr{E}}^{r}$ as $x$ varies we obtain the curve ${\mathscr{C}}_{u}$,
which is a ramified $W({\mathfrak{g}})$-cover of
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. The fiber of the
projection ${\mathscr{C}}_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ over a point $x$ is the orbit

$W({\mathfrak{g}})\cdot[{\bf t}(x)]\subset{\mathscr{E}}\otimes{{\rm Q}}$

This leads us to the _obscured curve_ which we have encountered earlier in our
solution of the gauge theory using the limit shape equations.

Now, as in our prior discussion of Hitchin systems, let us recall the gauge
field ${\bar{a}}$ in (8.49). It looks like the $\mathfrak{h}$-valued one-form
on ${\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}$, since it is
locally $z$-independent, and therefore might be a $(0,1)$ part of a
$\mathbf{T}$-connection, defining a holomorpic $\mathbf{T}$-bundle over
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. However, we have
$|W({\mathfrak{g}})|$ worth of choices for ${\bar{\partial}}+{\bar{a}}$ at any
particular point $x$ since the residual $W({\mathfrak{g}})$-symmetry acts both
on $[{\bf t}(x)]$ and ${\bar{a}}$. This means that $\bar{a}$ becomes well-
defined when lifted to ${\mathscr{C}}_{u}$. The way it transforms under the
$W({\mathfrak{g}})$-action permuting the sheets of the cover
${\mathscr{C}}_{u}\to{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}$ makes it into the $W({\mathfrak{g}})$-equivariant gauge
field on $\mathscr{C}_{u}$, defining a $W({\mathfrak{g}})$-equivariant
holomorphic $\mathbf{T}$-bundle ${\mathscr{T}}$ over $\mathscr{C}_{u}$:

${\mathscr{T}}\in{\mathscr{A}}_{u}={\mathrm{Bun}}_{\mathbf{T}}({\mathscr{C}}_{u})^{W({\mathfrak{g}})}\approx{\operatorname{Hom}}_{W({\mathfrak{g}})}\left({\Lambda},{\rm
Pic}({\mathscr{C}}_{u})\right)\,,$ (8.50)

the idea of the last equality is that every weight
${\lambda}\in\Lambda=\operatorname{Hom}({\mathbf{T}},{\mathbb{C}}^{\times})$
defines a $\mathbb{C}^{\times}$-bundle ${\mathscr{T}}^{\lambda}$ on
$\mathscr{C}_{u}$, in the $W({\mathfrak{g}})$-compatible fashion.

Of course this discussion is not adequate at the branch points, yet hopefully
it can be extended similarly to other constructions of spectral covers
[Hitchin:1987mz], [Donagi:1995alg]. In particular, the analysis near the
branch points should demonstrate that the bundle ${\mathscr{T}}$ has a fixed
topological type, which we shall not attempt to determine in this paper.

We conjecture that (8.50) is the fiber of the projection
${\pi}:{{\mathfrak{P}}^{\mathrm{ext}}}\to{{\mathfrak{M}}^{\mathrm{ext}}}$,
sending the extended moduli space of vacua of the class II gauge theory on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ to that of the infinite volume four
dimensional gauge theory. Since the projection ${\mathcal{P}}\mapsto U$ is
roughly of the form $(A_{{\bar{x}}},A_{{\bar{z}}})\mapsto A_{{\bar{z}}}$, it
is Lagrangian with respect to (8.39). This is in agreement with
$\dim_{\mathbb{C}}{\mathfrak{P}}^{\mathrm{ext}}=2\dim_{\mathbb{C}}{\mathfrak{M}}^{\mathrm{ext}}=2Nh$.
From now on we fix the masses, divide by the global $\mathbf{T}$-action and
work with ${\mathfrak{P}}$ and ${\mathfrak{M}}$.

Thus the study of holomorphic bundles on $S_{\mathscr{E}}$ brought us the
following picture: over each point
$x\in{{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$ we hang the
$W({\mathfrak{g}})$-orbit of $[{\bf t}(x)]$ in ${\mathscr{E}}\otimes{{\rm
Q}}$. As $x$-varies, so does the orbit, spanning a curve ${\mathscr{C}}_{u}$
in ${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle
x\right\rangle}}\times{\mathscr{E}}\otimes{{\rm Q}}$, which projects down to
${{\mathbb{C}\mathbb{P}}^{1}_{\left\langle x\right\rangle}}$. However
${\mathscr{C}}_{u}$ is not yet the cameral curve $\mathcal{C}_{u}$. It is
however relatively straightforward to lift $\mathscr{C}_{u}$ to
$\mathcal{C}_{u}$ using the abelianized determinant line bundle over
${\mathscr{E}}^{r}$ we discussed in the section 6.4.

Now let us discuss the period map. The moduli space ${\mathfrak{P}}$ carries
the holomorphic symplectic structure $\Omega$ which descends from (8.39). From
the complexified textbooks on classical mechanics we learn, as we reviewed in
section 4.4, that choosing some basis $A_{\mathcal{I}},B^{\mathcal{I}}$, in
the integral homology $H_{1}({\mathscr{A}}_{u},{\mathbb{Z}})$ lattice, such
that the cup product of the basis vectors obeys (4.18) and then defining
${\mathfrak{a}}^{\mathcal{I}}$, ${\mathfrak{a}}_{\mathcal{I}}^{D}$ using
(4.16), (4.17), one verifies (4.19). It remains to compare this definition of
the special coordinates with the periods of the differentials
$x\,d{\log}{\mathscr{Y}}_{i}(x)$.

Let us work in the domain $|{\mathfrak{q}}_{i}|\ll 1$. The cycles
${\mathcal{A}}_{\mathcal{I}}$ and ${\mathcal{B}}^{\mathcal{I}}$ which we
defined (cf. Figs. 5.1, 5.3) on the cameral curve ${\mathcal{C}}_{u}$ define
the corresponding one-cycles on ${\mathscr{C}}_{u}={\mathcal{C}}_{u}/{{\rm
Q}}^{\vee}$ and consequently on ${\mathscr{A}}_{u}$.

Now let us compute the symplectic form $\Omega$ on the reduced phase space
${\mathfrak{P}}$. Using that in our gauge

$\int_{\mathscr{E}}d^{2}zA_{{\bar{z}}}(x,{{\bar{x}}})={\xi}(x)$

we obtain

${\Omega}=\int\,d^{2}x\left\langle{\delta}a_{{\bar{x}}}\wedge{\delta}{\xi}\right\rangle=\frac{1}{|W({\mathfrak{g}}|}\int_{{\mathscr{C}}_{u}}\left\langle{\delta}{\bar{a}}\wedge{\delta}{\varphi}\right\rangle$
(8.51)

where we denoted by $\bar{a}$ the $W({\mathfrak{g}}$-equivariant
$\mathbf{T}$-gauge field on $\mathscr{C}_{u}$ and by ${\delta}{\varphi}$ an
$\mathfrak{h}$-valued, $W({\mathfrak{g}})$-equivariant $(1,0)$-form on
$\mathscr{C}_{u}$, given by

${\delta}{\varphi}=x{\delta}{\xi}$

It is now easy to pin down the periods of $d{\mathbb{S}}$ among the periods of
${\delta}{\varphi}$. One uses the cycles in ${\mathscr{A}}_{u}$ where the
monodromy of the gauge field $\bar{a}$ along the cycle
${\mathcal{A}}_{\mathcal{I}}$ or ${\mathcal{B}}^{\mathcal{I}}$ changes by
$2\pi$.

To conclude this section, in parallel with the class I story we conjecture,
that the theories of class II lead to the representation theory of the
Yangians built on the Kac-Moody algebras, i.e. the toroidal algebras. It would
be nice to see whether the Kac-Moody symmetry of instanton moduli spaces
[Dolan:1982dc, Licata_2007] leads to the quasi-classical limit of the Yangian
action, at the level of the phase space ${\mathfrak{P}}^{\mathrm{ext}}$
symmetry.

###  8.3. Noncommutative instantons and Class II* theories

In this subsection we show that the data $t_{i}(x)$ with the cross-cut
transformations (5.16) correspond naturally to the charge $N$ instantons on
the noncommutative space ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$, with the
gauge group $U(r+1)$. We shall use the constructions analogous to the
constructions [Nekrasov:2000zz] of instantons on the noncommutative
${\mathbb{R}}^{4}$ adapted to the periodic case. The noncommutative solitons
on a cylinder were studied in [Demidov:2003xq].

The idea of the construction is the following. Consider first the commutative
situation. Let us denote the coordinates on ${\mathbb{R}}^{2}$ by
$x_{1},x_{2}$. Let
$(u_{1},u_{2})\in{\mathbb{T}}^{2}\subset{\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$,
$|u_{1}|=|u_{2}|=1$. Let us denote by $z,x$ the holomorphic local coordinates
on ${\mathbf{C}_{\left\langle x\right\rangle}}\times{\mathscr{E}}$,

$x=x_{1}+{\mathrm{i}}x_{2},\ z=\frac{1}{2\pi\mathrm{i}}\left({\rm
log}(u_{1})+{\tau}{\rm log}(u_{2})\right)$ (8.52)

Consider the $(0,1)$-component of $U(r+1)$ gauge field
${\bar{A}}=A_{{\bar{z}}}d{{\bar{z}}}+A_{{\bar{x}}}d{{\bar{x}}}$, where
$A_{{\bar{z}}}$, $A_{{\bar{x}}}$ are the $(r+1)\times(r+1)$ complex matrices.
They obey (8.38) and in the gauge (8.53) we have

$A_{{\bar{z}}}=\frac{2\pi i}{{\bar{\tau}}-{\tau}}{\rm
diag}\left({\xi}_{i}(x,{{\bar{x}}})\right)_{i=1}^{r+1}$ (8.53)

where, by the considerations analogous to those around (8.46) we conclude that
${\bar{\partial}}_{{\bar{x}}}{\xi}_{i}=0$. Again, there are the residual gauge
transformations

$A_{{\bar{z}}}\mapsto G^{-1}A_{{\bar{z}}}G+G^{-1}{\partial}_{{\bar{z}}}G$

which permute the eigenvalues ${\xi}_{i}(x)$:
$\\{{\xi}_{i}(x)\\}\mapsto\\{{\xi}_{{\sigma}(i)}(x)\\}$, for some
${\sigma}\in{\mathcal{S}}_{r+1}$, and shift them

${\xi}_{i}(x)\mapsto{\xi}_{i}(x)+l_{i}-{\tau}n_{i}$ (8.54)

The latter are generated by the diagonal gauge transformations:

$G={\rm diag}\left(u_{1}^{n_{i}}u_{2}^{l_{i}}\right)_{i=1}^{r+1}$ (8.55)

We can partially reduce the ambiguity (8.54) by passing to the exponential
variables:

$t_{j}(x)=e^{2\pi\mathrm{i}{\xi}_{j}(x)}\,\qquad j=1,\ldots,r+1$ (8.56)

The residual gauge transformations (8.54) become

$t_{j}(x)\mapsto{\mathfrak{q}}^{n_{j}}t_{j}(x),\qquad t_{j}(x)\mapsto
t_{{\sigma}(j)}(x)$ (8.57)

for some integers $n_{i}\in{\mathbb{Z}}$ and permutations
${\sigma}\in{\mathcal{S}}_{r+1}$.

Let us now consider the noncommutative case. Let us replace the algebra of
functions of $u_{1},u_{2},x_{1},x_{2}$ by the noncommutative algebra, with the
generators
${\widehat{u}}_{1},{\widehat{u}}_{2},{\widehat{x}}_{1},{\widehat{x}}_{2}$,
obeying:

$\displaystyle{\widehat{u}}_{1}^{-1}{\widehat{x}}_{i}{\widehat{u}}_{1}={\widehat{x}}_{i}+{\hbar}_{i}\cdot
1$ (8.58)
$\displaystyle{\widehat{u}}_{2}^{-1}{\widehat{x}}_{i}{\widehat{u}}_{2}={\widehat{x}}_{i},$
$\displaystyle{\widehat{u}}_{1}{\widehat{u}}_{2}={\widehat{u}}_{2}{\widehat{u}}_{1},$
$\displaystyle[{\widehat{x}}_{1},{\widehat{x}}_{2}]=0$

where

$\frac{\mathfrak{m}}{r+1}={\hbar}={\hbar}_{1}+{\mathrm{i}}{\hbar}_{2}$

Then the analogue of (8.54) for

$G={\rm
diag}\left({\widehat{u}}_{1}^{n_{i}}{\widehat{u}}_{2}^{l_{i}}\right)_{i=1}^{r+1}$
(8.59)

gives:

$t_{i}(x)\mapsto q^{n_{i}}t_{i}(x-\frac{n_{i}}{r+1}{\mathfrak{m}}),\qquad
t_{i}(x)\mapsto t_{{\sigma}(i)}(x)$ (8.60)

These are precisely the ${{}^{i}{\mathcal{W}}}$-transformations of the class
II* type ${\widehat{A}}_{r}$ gauge theory, with $t_{i}(x)$ given in (5.14),
(7.97).

Instead of giving more systematic discussion along the lines of
[Connes:1997cr, Nekrasov:1998ss, Astashkevich:1998uc, Nekrasov:2000zz,
Schwarz:2001ru, Douglas:2001ba] let us comment on the relation to the group
$\widehat{GL}_{\infty}$. We claim that the $U(r+1)$ instantons on the
noncommutative ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ with both
${\mathbb{R}}^{2}$ and ${\mathbb{T}}^{2}$ separately commutative, can be
interpreted as the commutative periodic monopoles on
${\mathbb{R}}^{2}\times{\mathbb{S}}^{1}$ with the gauge group
$\widehat{GL}_{\infty}$.

The idea is to interpret the noncommutative $U(r+1)$ gauge fields on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$ as the $\widehat{GL}_{\infty}$-gauge
fields on ${\mathbb{R}}^{2}_{x_{1},x_{2}}\times{\mathbb{S}}^{1}_{\psi}$, using
the relation of the group $\widehat{GL}_{\infty}$ to the quantization of the
volume-preserving diffeomorphisms of a cylinder
${\mathbb{R}}^{1}_{p}\times{\mathbb{S}}^{1}_{q}$, i.e. the pseudo-differential
operators ${\Psi}DO$ on ${\mathbb{S}}^{1}_{q}$. Here the generators
${\widehat{u}}_{1,2},{\widehat{x}}_{1,2}$ of the algebra (8.58) are related to
the commutative coordinates $(x_{1},x_{2},\psi)$ of the monopole theory and
the generators $(p,e^{\mathrm{i}q})$ of $\widehat{GL}_{\infty}$ in the via

$\displaystyle{\widehat{u}}_{2}=e^{\mathrm{i}\psi}$ (8.61)
$\displaystyle{\widehat{u}}_{1}=e^{\mathrm{i}q}$
$\displaystyle{\widehat{x}}_{1}=x_{1}-{\mathrm{i}}{\hbar}_{1}{\partial}_{q}$
$\displaystyle{\widehat{x}}_{2}=x_{2}-{\mathrm{i}}{\hbar}_{2}{\partial}_{q}$

##  Chapter 9 Higher dimensional theories

In this section we briefly go over the higher dimensional generalizations. We
shall consider the lifts of all our theories to five dimensions, and the lifts
of some of our theories to six dimensions. The latter restriction comes from
the possibility of encountering gauge and mixed anomalies in six dimensions,
which would prohibit the decoupling of the gauge sector from the supergravity
and ultimately string theory modes.

The classical ${\mathcal{N}}=2$ theories in four dimensions admit a canonical
lift to the ${\mathcal{N}}=1$ theories in six dimensions. Under this lift the
vector multiplets become the vector multiplets, and the hypermultiplets become
the hypermultiplets. The structure of the hypermultiplet does not change,
while the structure of the vector multiplet does change, for the complex
scalar $\Phi$ in the adjoint representation becomes the remaining component
$A_{5}+\mathrm{i}A_{6}$ of the gauge field.

We then compactify the theory on a two-torus ${\mathbb{T}}^{2}$. In addition
to the metric on the torus we shall also fix a background $B$-field, a
constant two-form. The combined metric and the two-form moduli are described
by a $2\times 2$ matrix $G$, with the positive definite symmetric part $g$. It
is convenient to parametrize $G$ by two complex numbers $T,U$, with ${\rm
Im}U,{\rm Im}T\geq 0$. The parameter $T$ encodes the complex structure of
$\mathbb{T}^{2}$, while the parameter $U$ is the complexified Kähler class:

$\displaystyle
U=\int_{\mathbb{T}^{2}}B+\mathrm{i}\,\int_{\mathbb{T}^{2}}\sqrt{{\rm
det}(g)},$ (9.1) $\displaystyle T=\frac{g_{12}+\mathrm{i}\sqrt{{\rm
det}(g)}}{g_{11}}$

Five (six) dimensional supersymmetric gauge theory compactified on a circle
(two-torus) would look four dimensional at low energy. The microscopic gauge
coupling ${\tau}$ in four dimensions is proportional to $U$ while $T$
determines the complex geometry of the $x$-plane $\mathbf{C}_{\left\langle
x\right\rangle}$. One can study the corresponding Seiberg-Witten geometry. Its
key feature compared to special geometry of more traditional four dimensional
models is the periodicity in the $x$-variable [Nekrasov:1996cz]. In the five
dimensional theory the $x$-plane becomes the cylinder
${\mathbf{C}_{\left\langle x\right\rangle}}^{\times}$, while in the six
dimensional theory the $x$-plane becomes the two-torus
${\mathbf{C}_{\left\langle
x\right\rangle}}/2{\pi}{\mathrm{i}}({\mathbb{Z}}\oplus T{\mathbb{Z}})$. It has
to do with the large gauge transformations. The result of these additional
symmetries is the relativistic nature of the corresponding integrable systems.
For example, the periodic Toda chain describing the pure ${\mathcal{N}}=2$
theory in four dimensions becomes the relativistic Toda chain. The
Hamiltonians of the relativistic systems have periodic dependence on momenta,
which are the rapidities of the particles. The resulting quantized
Hamiltonians are the difference operators.

Our discussion modifies in the case of five dimensional gauge theories in two
aspects. First, the notion of the amplitude function accommodates the large
gauge transformations:

${\mathscr{Y}}_{i}(x)={\exp}\,\langle\,{\operatorname{tr}}_{{\mathbf{v}}_{i}}\
{\log}\left(e^{{{\mathrm{i}\beta}}(x-{\Phi}_{i})}-1\right)\,\rangle_{u}$ (9.2)

where the dimension length $\beta$ characterizes the circumference of the
compactification circle. The limit shape integral equations (5.8) generalize
straightforwardly, with the kernel:

${\mathcal{K}}_{\beta}({\mathrm{x}})=\frac{{\mathrm{i}\beta}}{12}{\mathrm{x}}^{3}-\frac{{\rm
log}({{\mathrm{i}\beta}}{\Lambda}_{\rm
UV})}{2}{\mathrm{x}}^{2}-\frac{1}{{\mathrm{i}\beta}^{2}}\,{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}\right)$ (9.3)

Secondly, there are additional couplings in five dimensions: the levels
$k_{i}$ of the Chern-Simons interactions CS${}_{5}(A^{i})$,
$i\in\mathrm{Vert}_{\gamma}$. Effectively the Chern-Simons term changes the
gauge coupling ${\mathfrak{q}}_{i}$ to the $x$-dependent quantity:

${\mathfrak{q}}_{i}\longrightarrow{\mathfrak{q}}_{i}\,e^{{{\mathrm{i}\beta}}k_{i}x}$

In the six dimensional case the amplitude and the kernel (9.3) modify to:

${\mathscr{Y}}_{i}(x)={\exp}\,\langle\,{\operatorname{tr}}_{{\mathbf{v}}_{i}}\,{\log}\,{\theta}\left(e^{{{\mathrm{i}\beta}}(x-{\Phi}_{i})};Q\right)\,\rangle_{u}$
(9.4)
$\displaystyle{\mathcal{K}}_{\beta}({\mathrm{x}})=\frac{{\mathrm{i}\beta}}{12}{\mathrm{x}}^{3}-\frac{{\rm
log}({{\mathrm{i}\beta}}{\Lambda}_{\rm
UV})}{2}{\mathrm{x}}^{2}-\frac{1}{{\mathrm{i}\beta}^{2}}\,{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}\right)-$ (9.5)
$\displaystyle\qquad\qquad\qquad\sum_{n=1}^{\infty}\left({\rm
Li}_{3}\left(e^{{{\mathrm{i}\beta}}{\mathrm{x}}}Q^{n}\right)+{\rm
Li}_{3}\left(e^{-{{\mathrm{i}\beta}}{\mathrm{x}}}Q^{n}\right)\right)$

where $Q={\exp}\,2\pi\mathrm{i}T$, and now ${\beta}^{2}$ is the scale of the
area of the compactification torus ${\mathbb{T}}^{2}$. We will provide further
details on five and six dimensional theories in the companion paper [NP2012b].

##  Appendix A McKay correspondence, D-branes, and M-theory

###  A.1. From finite groups to Lie groups

McKay correspondence states [McKay:1980] that the affine ADE graphs $\gamma$
can be constructed from the representation theory of finite subgroups $\Gamma$
of $SU(2,{\mathbb{C}})$.

The vertices $i\in\mathrm{Vert}_{\gamma}$ in this case are the irreducible
representations of $\Gamma$, $i\mapsto{\mathcal{R}}_{i}$. One usually
enumerates them, $\mathrm{Vert}_{\gamma}=\\{0,1,\ldots,r\\}$, so that $0$
corresponds to the trivial representation ${\mathcal{R}}_{0}={\mathbb{C}}$. By
$r$ in this section we denote the number of nodes in the finite graph
$\gamma_{\text{fin}}$ obtained by discarding the node ‘‘0’’ from affine
$\gamma$. In all cases $r$ is the rank of the Cartan matrix associated to
Dynkin graph. Moreover, the Dynkin marks $a_{i}$ are the dimensions of
$V_{i}$, and the numbers of colors in affine quivers is

${\mathbf{v}}_{i}=Na_{i},\ \qquad$ (A.1)

where $a_{i}={\rm dim}{\mathcal{R}}_{i}$, and $N$ is some non-negative
integer. As we said above, $i=0$ corresponds to the trivial representation, so
$a_{0}=1$. In the table of McKay correspondence we write ‘‘$i(a_{i})$’’ by
each node to denote its label $i$ and Dynkin mark $a_{i}$. We label the nodes
in Bourbaki conventions.

The number of edges $I_{ij}$ in the McKay graph between the node $i$ and the
node $j$ is the multiplicity of representation $\mathcal{R}_{j}$ in the tensor
product of $\mathcal{R}_{i}$ with the defining representation $\mathbb{C}^{2}$

${\mathbb{C}}^{2}\otimes{\mathcal{R}}_{i}=\bigoplus_{j}{\mathbb{C}}^{I_{ij}}\otimes{\mathcal{R}}_{j}.$
(A.2)

The equation ${\beta}_{i}=0$ is verified by computing the dimensions of the
left and the right hand sides.

The order of $\Gamma$ agrees with the dimensions of the irreducible
representations / Dynkin marks of the McKay/Dynkin affine graph computed using
the standard relation from the orthogonality of characters

$|\Gamma|=\sum_{i}\dim\mathcal{R}_{i}^{2}=\sum_{i}a_{i}^{2}$ (A.3)

Polyhedron |  $\Gamma$ |  $T_{a,b,c}$ | Affine Dynkin graph $\gamma$ with labels $i(a_{i})$ |  Lie $\widehat{\mathfrak{g}}$
---|---|---|---|---
|  $\mathbb{Z}_{r+1}$ (r=5) |  $T_{r,1,1}$ |  |  $\widehat{A}_{r}$ (r=5)
|  $\mathbb{B}\mathbb{D}_{r-2}$ (r=7) |  $T_{r-2,2,2}$ |  |  $\widehat{D}_{r}$ (r=7)
|  $\mathbb{B}\mathbb{T}$ |  $T_{3,3,2}$ $(\widehat{T}_{3,3,3})$ |  |  $\widehat{E}_{6}$
|  $\mathbb{B}\mathbb{O}$ |  $T_{4,3,2}$ $(\widehat{T}_{4,4,2})$ |  |  $\widehat{E}_{7}$
|  $\mathbb{B}\mathbb{I}$ |  $T_{5,3,2}$ $(\widehat{T}_{6,3,2})$ |  |  $\widehat{E}_{8}$

McKay’s observation is that $\gamma$ is affine ADE Dynkin diagram, with the
trivial representation $\mathcal{R}_{0}$ associated with the affine node
‘‘0’’. The finite Dynkin graph $\gamma_{\text{fin}}$ is always a tri-star
graph $T_{a,b,c}$ with one trivalent vertex and three legs containing $a,b,c$
vertices (where in the counting we included the center trivalent vertex).
Hence the rank of the finite quiver $\gamma_{\mathrm{fin}}$ is $a+b+c-2$. In
fact $a,b,c$ have simple interpretation: the group $\Gamma$ is always a
Coxeter group $\mathrm{Cox}(a,b,c)$ defined on three generators $(x,y,z)$
subject to relation

$x^{a}=y^{b}=z^{c}=xyz$ (A.4)

The affine graph of $\widehat{E}_{r}$ series is also a trivalent graph denoted
by $(\widehat{T}_{a,b,c})$ in the table of McKay correspondence. Note that for
each of the three cases $\widehat{E}_{6},\widehat{E}_{7},\widehat{E}_{8}$ the
identity

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ (A.5)

holds, which makes contact with the unimodular parabolic singularities.

Now we shall list explicitly the discrete subgroups of $SU(2)$, the
classification going back to Plato and Klein [Klein:1884].

###  A.2. Platonic symmetries

####  A.2.1. Cyclic group

Let us present cyclic group $\mathbb{Z}_{r+1}$ in the $(a,b,c)$ notation:
consider group generated by $x,y,z$ subject to

$x^{r}=y=z=xyz$ (A.6)

Then $z=y$ and $xy=1$ and $y=x^{r}$ hence

$x^{r+1}=1$ (A.7)

hence $\mathbb{Z}_{r+1}=\mathrm{Cox}(r,1,1)=\mathrm{McKay}(A_{r})$. The order
of $\Gamma$ is

$|{\mathbb{Z}}_{r+1}|=r+1=\sum_{i\in\gamma}a_{i}^{2}=(r+1)\cdot 1$ (A.8)

####  A.2.2. Dihedral group

The dihedral group
$\mathbb{D}_{r-2}\simeq\mathbb{Z}_{r-2}\rtimes\mathbb{Z}_{2}$ of order
$2(r-2)$ is the symmetry group of regular $r-2$-gon considered as a subgroup
of $SO(3)$. The lift of dihedral group
$\mathbb{Z}_{2}\to\mathbb{B}\mathbb{D}_{r-2}\to\mathbb{D}_{r-2}$ to a subgroup
of $SU(2)$ is called bidihedral group $\mathbb{B}\mathbb{D}$, which has order
$4(r-2)$. The $\mathbb{B}\mathbb{D}_{r}$ is generated by $(x,y,z)$ subject to
relations

$x^{r-2}=y^{2}=z^{2}=xyz$ (A.9)

hence $\mathbb{B}\mathbb{D}_{r-2}=\mathrm{Cox}(r-2,2,2)=\mathrm{McKay}(D_{r})$
The order of $\Gamma$ is

$|\mathbb{B}\mathbb{D}_{r-2}|=4(r-2)=\sum_{i\in\gamma}a_{i}^{2}=(r-3)\cdot
4+4\cdot 1$ (A.10)

####  A.2.3. Tetrahedral group

The tetrahedral group $\mathbb{T}\simeq A_{4}$, of order 12, is the subgroup
of $SO(3)$ realizing the symmetries of tetrahedron. The lift of tetrahedral
group $\mathbb{Z}_{2}\to\mathbb{B}\mathbb{T}\to\mathbb{T}$ to a subgroup of
$SU(2)$ is called bitetrahedral group $\mathbb{B}\mathbb{T}$, which has order
$24$ and is generated by $(x,y,z)$ subject to the relations

$x^{3}=y^{3}=z^{2}=xyz$ (A.11)

Hence $\mathbb{B}\mathbb{T}=\mathrm{Cox}(3,3,2)=\mathrm{McKay}(E_{6})$. The
order of $\Gamma$ is

$|\mathbb{B}\mathbb{T}|=24=\sum_{i\in\gamma}a_{i}^{2}=3\cdot(1^{2}+2^{2})+3^{2}=24$
(A.12)

####  A.2.4. Octahedral group

The octahedral group $\mathbb{O}\simeq S_{4}$, of order $24$, is the subgroup
of $SO(3)$ realizing the symmetries of the cube/octahedron. The lift to
$SU(2)$ is called bioctahedral group $\mathbb{B}\mathbb{O}$, of order $48$;
the $\mathbb{B}\mathbb{O}$ can be generated by $(x,y,z)$ subject to relations

$x^{4}=y^{3}=z^{2}=xyz$ (A.13)

associated to the symmetry axes of the cube/octahedron of the order $4,3$ and
$2$. Hence $\mathbb{B}\mathbb{O}=\mathrm{Cox}(4,3,2)=\mathrm{McKay}(E_{7})$.
The order of $\Gamma$ is

$|\mathbb{B}\mathbb{O}|=48=\sum_{i\in\gamma}a_{i}^{2}=2\cdot(1^{2}+2^{2}+3^{2})+4^{2}+2^{2}$
(A.14)

####  A.2.5. Icosahedral group

The icosahedral group $\mathbb{I}\simeq A_{5}$, of order $60$, is the subgroup
of $SO(3)$ realizing the symmetries of the icosahedron/dodecahedron The lift
to $SU(2)$ is called biicosahedral group $\mathbb{B}\mathbb{O}$, of order
$120$; the biicosahedral group $\mathbb{B}\mathbb{I}$ can be generated by
$(x,y,z)$ subject to relations

$x^{5}=y^{3}=z^{2}=xyz$ (A.15)

associated to the symmetry axes of the icosahedron/dodecahedron of the order
$5,3$ and $2$. Hence
$\mathbb{B}\mathbb{O}=\mathrm{Cox}(5,3,2)=\mathrm{McKay}(E_{8})$. The order of
$\Gamma$ is

$|\mathbb{B}\mathbb{I}|=120=\sum_{i\in\gamma}a_{i}^{2}=(1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2})+2^{2}+4^{2}+3^{2}$
(A.16)

###  A.3. D-branes at singularities

The physical explanation of the relation between the
${\mathbb{C}}^{2}/{\Gamma}$-singularities [Kronheimer:1989] and the
superconformal theories with ${\mathcal{N}}=2$ supersymmetry is the following.
Consider the IIB string in the background
${\mathbb{C}}^{2}/{\Gamma}\times{\mathbb{C}}^{1}\times{\mathbb{R}}^{1,3}$,
where ${\Gamma}\subset SU(2)\subset SO(4)$ acts on
${\mathbb{C}}^{2}\approx{\mathbb{R}}^{4}$ by the hyperkähler rotations. This
background preserves half of the ten dimensional IIB supersymmetry. Now add a
stack of $N$ regular D3 branes. At the orbifold singularity the regular D3
brane splits as a collection of $r+1$ types of fractional branes, which
correspond to the irreducible representations of $\Gamma$ [Douglas:1996sw,
Johnson:1996py]. Moreover, the regular brane contains $a_{i}$ fractional
branes of the ${\mathcal{R}}_{i}$ type. The stack of $N$ regular brane splits,
therefore, as a collection of $Na_{i}$ fractional branes of $\mathcal{R}_{i}$
type, for all $i=0,\ldots,r$. Each cluster of fractional branes can be
classically moved anywhere in the two-plane, transverse to the singularity and
the D3 brane worldvolume. To summarize, the D3 branes are located at:

$0\,\times\,\\{{\mu}_{0}^{Na_{0}},{\mu}_{1}^{Na_{1}},\ldots,{\mu}_{r}^{Na_{r}}\\}\times{\mathbb{R}}^{1,3}$

Here $\\{{\mu}_{0}^{v_{0}},\ldots,{\mu}_{r}^{v_{r}}\\}$ represents the
positions of the $v_{i}$ copies of the $\mathcal{R}_{i}$ type of the
fractional branes in the two dimensional plane
${\mathbb{C}}^{1}\approx{\mathbb{R}}^{2}$ transverse to the singular ALE space
${\mathbb{C}}^{2}/{\Gamma}$ and the worldvolume ${\mathbb{R}}^{1,3}$ of the
branes. The positions ${\mu}_{i}$ are the ones which enter the Eq. (2.10). At
the low energy the worldvolume theory on the D3 branes is the quiver gauge
theory we are studying. If all ${\mu}_{i}$ parameters are scaled to zero (or
at least all coincide), the theory has no scale except for the string scale
which is absent in the low energy description. Therefore the theory has the
scaling invariance, which is promoted to the full ${\mathcal{N}}=2$
superconformal invariance. The couplings ${\tau}_{i}$, except for

$\tau=\sum_{i=0}^{r}a_{i}{\tau}_{i}$ (A.17)

which corresponds to the IIB dilaton-axion field, are not of geometric origin.
The gauge couplings $\tau_{i}$, $i=1,\ldots,r$, come from the twisted sector
fields:

$\tau_{i}=\int_{{\Sigma}_{i}}B_{RR}+{\tau}B_{NS}$ (A.18)

where $B_{NS}$, $B_{RR}$ are the Neveu-Schwarz(NSNS) and the Ramond-Ramond
two-form fields of the IIB supergravity, and $\Sigma_{i}$ stand for the non-
trivial two-cycles in the resolved $\widetilde{{\mathbb{C}}^{2}/{\Gamma}}$
geometry. These exceptional cycles are well-known to correspond to the simple
roots of $\mathfrak{g_{\text{q}}}$.

The 4d gauge theory on the stack of D3 branes in the above geometry is
$\mathcal{N}=2$ supersymmetric theory of the affine ADE quiver type. To find
the algebraic integrable system associated to the IR solution of such theory
we compactify the worldvolume of the D3 branes $(x_{0},x_{1},x_{2},x_{3})$ on
a circle $S^{1}$, say, along $x_{3}$. Hence, our setup is $N$ D3 branes along
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}$ inside the IIB string theory on
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma$. Here $\mathbf{C}_{\left\langle
x\right\rangle}$ is the real-two dimensional space associated with the scalars
of $\mathcal{N}=2$ vector multiplet in 4 dimensions. For 4d theories this is
affine complex line $\mathbf{C}_{\left\langle x\right\rangle}=\mathbb{C}$, for
5d theories on $S^{1}$ this is a cylinder $\mathbf{C}_{\left\langle
x\right\rangle}=\mathbb{C}^{\times}$ and for 6d theories on $T^{2}$ this is a
torus $\mathbf{C}_{\left\langle x\right\rangle}=\mathscr{E}_{\left\langle
x\right\rangle}$.

Given this type IIB string theory realization of the affine ADE quiver theory,
first we perform T-duality along $\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}$. The stack of $N$ D3-branes converts to the stack of $N$
D2-branes on $\mathbb{R}^{2,1}$ in type IIA string theory on
$\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma$. Next we lift the type IIA string
to M-theory on $\mathbb{R}^{1,2}\times\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbf{C}_{\left\langle
x\right\rangle}\times\mathbb{C}^{2}/\Gamma\times\mathbb{S}^{1}_{\left\langle
x_{10}\right\rangle}$ where $\mathbb{S}^{1}_{\left\langle
x_{10}\right\rangle}$ is the M-theory circle. The radius of M-theory circle is
determined by IIB coupling constant (A.17). We treat the product of two
circles along $x_{3}$ and along $x_{10}$ as the elliptic curve
$\mathscr{E}=\mathbb{S}^{1}_{\left\langle
x_{3}\right\rangle}\times\mathbb{S}^{1}_{\left\langle x_{10}\right\rangle}$
which has elliptic modulus $\mathfrak{q}=e^{2\pi\mathrm{i}\tau}$. The stack of
$N$ D2 branes converts to the stack of $N$ M2 branes along $\mathbb{R}^{1,2}$.
So finally we arrive to the M-theory picture.

That is, we consider the following configuration in M-theory: the space-time
background is

${\mathbb{R}}^{1,2}\times X_{4}\times{\mathbb{C}}^{2}/{\Gamma}$ (A.19)

to which we add a stack of $N$ M2 branes whose worldvolume is
${\mathbb{R}}^{1,2}$ in (A.19) localized at the orbifold singularity in
${\mathbb{C}}^{2}/{\Gamma}$, and anywhere in $X_{4}=\mathbb{C}_{\left\langle
x\right\rangle}\times\mathscr{E}_{\mathfrak{q}}$. The M-theory on
$\mathbb{C}^{2}/\Gamma$ is believed to contain the seven dimensional $G$-gauge
theory with sixteen supersymmetries, localized at the singular locus. The
maximal torus $T$ part of the gauge bosons and their superpartners descend
from the supergravity modes associated with the hyperkähler deformations of
the metric on ${\mathbb{R}}^{4}/{\Gamma}$. In particular, the gauge fields are
the modes of the three-form field $C_{3}$ integrated along the collapsed two-
cycles $\Sigma_{i}$, $i=1,\ldots,r$ in the deformed geometry
${\widetilde{{\mathbb{R}}^{4}/{\Gamma}}}$. The remaining $W$-bosons come from
the M2-branes wrapped on various two-cycles in
$H_{2}({\widetilde{{\mathbb{R}}^{4}/{\Gamma}}},{\mathbb{Z}})\approx{{\rm Q}}$.

In the IR, the stack of $N$ M2 branes stretched along $\mathbb{R}^{1,2}$ in
this 7d $G$-gauge theory on $\mathbb{R}^{1,2}\times X_{4}$ dissolve into the
charge $N$ $G$-instantons on $X_{4}=\mathbf{C}_{\left\langle
x\right\rangle}\times\mathscr{E}_{\mathfrak{q}}$ [Intriligator:1996ex,
deBoer:1996ck, Witten:1997kz, Blum:1997mm, Kapustin:1998fa, Kapustin:1998xn,
Katz:1997eq]. The moduli space of framed charge $N$ $G$-instantons on $X_{4}$
is the phase space of the algebraic integrable system (see chapter 4) we have
found from the microscopic four dimensional instanton counting in the quiver
gauge theory.

##  Appendix B Partitions and free fermions

Recall that a partition $\lambda$ is a non-increasing sequence of integers,
stabilizing at zero,

${\lambda}=({\lambda}_{1}\geq{\lambda}_{2}\geq\ldots\geq{\lambda}_{{\ell}({\lambda})}>0=0=0\ldots)$
(B.1)

The number ${\ell}({\lambda})\geq 0$ is called the _length_ of the partition
$\lambda$, while

$|{\lambda}|=\sum_{i=1}^{\ell({\lambda})}{\lambda}_{i}$ (B.2)

is called the _size_ of the partition $\lambda$.

Consider the theory of a single chiral complex fermion, which is a
$(1/2,0)$-differential, in two dimensions

${\mathcal{L}}=\int\,{\tilde{\psi}}{\bar{\partial}}{\psi}$ (B.3)

The theory (B.3) has a $U(1)$-symmetry

$({\psi},{\tilde{\psi}})\mapsto(e^{i{\alpha}}{\psi},e^{-i{\alpha}}{\tilde{\psi}})$
(B.4)

One can couple the fermion to the background abelian gauge field corresponding
to this symmetry:

${\bar{\partial}}\mapsto{\bar{\partial}}+{\bar{A}}$ (B.5)

so that the Lagrangian (B.3) deforms to

${\mathcal{L}}\to{\mathcal{L}}+\int{\bar{A}}J$

where $J={\tilde{\psi}}{\psi}$ is the $U(1)$ current, which we define as an
operator below.

In studying the space of states ${\mathcal{H}}$ corresponding to the
quantization of the theory (B.4) one can distinguish various sectors,
corresponding to twisted boundary conditions: Expand:

$\displaystyle{\psi}(z)=\left(\frac{dz}{z}\right)^{\frac{1}{2}}\sum_{r\in{\mathbb{Z}}+{\alpha}}{\psi}_{r}z^{-r}$
(B.6)
$\displaystyle{\tilde{\psi}}(z)=\left(\frac{dz}{z}\right)^{\frac{1}{2}}\sum_{r\in{\mathbb{Z}}-{\alpha}}{\tilde{\psi}}_{r}z^{-r}$

where ${\alpha}=\int_{{\mathbb{S}}^{1}}{\bar{A}}$. The fermion modes
${\psi}_{r},{\tilde{\psi}}_{s}$ form the Clifford algebra:

$\\{\,{\psi}_{r},\,{\tilde{\psi}}_{s}\,\\}={\delta}_{r+s,0}\,\qquad\\{\,{\psi}_{r},\,{\psi}_{s}\,\\}=\\{\,{\tilde{\psi}}_{r},\,{\tilde{\psi}}_{s}\,\\}=0$
(B.7)

The vacuum state $|{\emptyset};{\alpha}\rangle$ is annihilated by all the
${\alpha}$-positive harmonics

$\displaystyle{\psi}_{r}\,|{\emptyset};{\alpha}\rangle=0,\qquad r>{\alpha}$
(B.8) $\displaystyle{\tilde{\psi}}_{r}\,|{\emptyset};{\alpha}\rangle=0,\qquad
r>-{\alpha}$

The space of states is built by acting on $|{\emptyset};{\alpha}\rangle$ with
creation operators ${\psi}_{r}$, $r\leq{\alpha}$, and ${\tilde{\psi}}_{s}$,
$s\leq-{\alpha}$. The resulting Hilbert space ${\mathcal{H}}_{\alpha}$ has a
basis labeled by partitions:

$\displaystyle\qquad\qquad\qquad{\mathcal{H}}_{\alpha}=\bigoplus_{\lambda}{\mathbb{C}}|{\lambda};{\alpha}\rangle$
(B.9) $\displaystyle\qquad|{\lambda};{\alpha}\rangle=$
$\displaystyle{\psi}_{-{\lambda}_{1}+{\alpha}}{\psi}_{-{\lambda}_{2}+1+{\alpha}}\ldots{\psi}_{-{\lambda}_{{\ell}({\lambda})}+{\ell}({\lambda})-1+{\alpha}}\,{\tilde{\psi}}_{-{\alpha}}{\tilde{\psi}}_{-{\alpha}-1}\ldots{\tilde{\psi}}_{-{\alpha}+1-{\ell}({\lambda})}\,|{\emptyset};{\alpha}\rangle$

Define the normal ordering with respect to the ${\alpha}=0$ vacuum:

$:{\psi}_{i}{\tilde{\psi}}_{j}:\
=\left\\{\begin{matrix}{\psi}_{i}{\tilde{\psi}}_{j}\,,&j>0\\\ &\\\
-{\tilde{\psi}}_{j}{\psi}_{i}\,,&j\leq 0\end{matrix}\right.$ (B.10)

The $U(1)$ symmetry of the Lagrangian (B.3) is promoted to the
${\widehat{u}(1)}$ current algebra symmetry. It is generated by the operator

$\displaystyle J(z)=$ $\displaystyle\ :{\tilde{\psi}}(z){\psi}(z):\
=\sum_{n\in\mathbb{Z}}J_{n}z^{-n-1}dz$ (B.11) $\displaystyle
J_{n}=\sum_{r\in{\mathbb{Z}}+{\alpha}}:{\tilde{\psi}}_{r}{\psi}_{n-r}:$

The generating function

$\sum_{{\lambda}}{\mathfrak{q}}^{|{\lambda}|}t^{{\ell}({\lambda})}=\prod_{n=1}^{\infty}\frac{1}{1-t{\mathfrak{q}}^{n}}$
(B.12)

is a character of the fundamental $\widehat{u(1)}$ module,
${\mathcal{H}}_{0}$.

##  Appendix C Lie groups and Lie algebras

In this section we fix our notations for the notions from the Lie group and
Lie algebra theory we are using in the work. In this section we work over the
field $\mathbb{C}$ of complex numbers.

###  C.1. Finite dimensional Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simply-laced simple Lie algebra,
$\mathfrak{h}$ its Cartan subalgebra. Let $\mathbf{G}$ be the corresponding
_simply-connected_ simple Lie group, and $\mathbf{T}\subset\mathbf{G}$ the
corresponding to $\mathfrak{h}$ maximal torus. We have the exponential map
${\exp}_{\mathfrak{h}}:{\mathfrak{h}}\to\mathbf{T}$, which is a restriction on
$\mathfrak{h}$ of the exponential map
${\exp}_{\mathfrak{g}}:{\mathfrak{g}}\to\mathbf{G}$. We shall only use the
${\exp}_{\mathfrak{h}}$ map in this work and will omit the $\mathfrak{h}$
subscript in what follows. We also use the notation:

${\mathbf{e}}({\bf x})={\exp}\,(2\pi\mathrm{i}{\bf x})=e^{2\pi\mathrm{i}{\bf
x}}\in\mathbf{T},\qquad{\rm for}\qquad{\bf x}\in\mathfrak{h}$ (C.1)

####  C.1.1. The coroots

The kernel of the map (C.1), i.e. the set ${{\rm
Q}}^{\vee}\subset\mathfrak{h}$ which is mapped to the identity element in
$\mathbf{T}$, is called the coroot lattice (it is obviously an abelian group).
The coroot lattice ${{\rm Q}}^{\vee}$ has the rank $r={\rm rk}\mathbf{G}={\rm
dim}{\mathfrak{h}}$. Let us denote by ${\alpha}_{i}^{\vee}$, $i=1,\ldots,r$
its integral basis:

${{\rm
Q}}^{\vee}={\mathbb{Z}}{\alpha}_{1}^{\vee}\oplus{\mathbb{Z}}{\alpha}_{2}^{\vee}\oplus\ldots\oplus{\mathbb{Z}}{\alpha}_{r}^{\vee}$

The generators ${\alpha}_{i}^{\vee}$ are called the simple coroots.

Now, let $z\in{\mathbb{C}}^{\times}$, and ${\alpha}^{\vee}\in{{\rm
Q}}^{\vee}$, then

$z^{{\alpha}^{\vee}}\equiv{\mathbf{e}}\,\left(\frac{{\log}(z)}{2\pi\mathrm{i}}{\alpha}^{\vee}\right)\in\mathbf{T}$
(C.2)

is independent of the choice of the branch of the ${\log}(z)$. This identifies
the coroot lattice with the lattice of homomorphisms:

${{\rm Q}}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},\mathbf{T})$
(C.3)

Using Eq. (C.2) we can parametrize $\mathbf{T}$ by

${\mathbf{z}}=\left(z_{1},\ldots,z_{r}\right)\mapsto
t_{\mathbf{z}}=\prod_{i=1}^{r}z_{i}^{{\alpha}_{i}^{\vee}}$ (C.4)

where $z_{i}\in{\mathbb{C}}^{\times}$.

####  C.1.2. The weights

The dual lattice is called the lattice of weights:

${\Lambda}={\operatorname{Hom}}(\mathbf{T},{\mathbb{C}}^{\times})$ (C.5)

Let us represent element $t\in\mathbf{T}$ as

$t={\mathbf{e}}({\mathbf{x}})$ (C.6)

for some $\bf x\in\mathfrak{h}$, which is defined, as we recall, up to an a
shift by an element of the coroot lattice ${\rm Q}^{\vee}$. Then for any
$\lambda\in\Lambda$, the value of the homomorphism ${\lambda}$ on $t$, which
we denote by $t^{\lambda}\in{\mathbb{C}}^{\times}$, can be computed as:

$t^{\lambda}=e^{2\pi\mathrm{i}\lambda({\mathbf{x}})}$ (C.7)

where ${\lambda}({\bf x})$ is a linear function of $\bf x$. In this way we
view ${\lambda}$ as an element of $\mathfrak{h}^{*}$, so that
$\Lambda\subset\mathfrak{h}^{*}$. In order for (C.7) be independent of the
choice of $\bf x$ in (C.6), the value of $\lambda$ on any element of the
coroot lattice must be an integer:

${\lambda}({{\rm Q}}^{\vee})\subset{\mathbb{Z}}\ .$ (C.8)

which is another definition of the dual lattices. Let us fix the basis
$({\lambda}_{i})_{i=1}^{r}$ of $\Lambda$:

${\Lambda}={\mathbb{Z}}{\lambda}_{1}\oplus{\mathbb{Z}}{\lambda}_{2}\oplus\ldots\oplus{\mathbb{Z}}{\lambda}_{r}\,,\
$

dual to the basis $({\alpha}_{i}^{\vee})_{i=1}^{r}$ of simple coroots, so that

${\lambda}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$ (C.9)

####  C.1.3. The roots

The torus $\mathbf{T}$ acts on $\mathbf{G}$ via the adjoint action:

$Ad:\mathbf{T}\times\mathbf{G}\to\mathbf{G},\qquad(t,g)\mapsto t^{-1}gt$

Infinitesimally, it acts linearly on the Lie algebra

$Ad:\mathbf{T}\longrightarrow{\rm Aut}({\mathfrak{g}}),\qquad
Ad_{t}({\xi})=\frac{d}{ds}\biggr{|}_{s=0}t^{-1}{\exp}(s{\xi})t\in\mathfrak{g}$
(C.10)

for $\xi\in\mathfrak{g}$, and finally, this defines an action of
$\mathfrak{h}$ on $\mathfrak{g}$:

${\rm ad}:{\mathfrak{h}}\to{\rm End}({\mathfrak{g}}),\qquad{\rm ad}_{\bf
x}({\xi})=[{\xi},{\bf x}]\in\mathfrak{g}$ (C.11)

for ${\bf x}\in\mathfrak{h}$, ${\xi}\in\mathfrak{g}$.

This action gives us several structures: the root decomposition of
$\mathfrak{g}$:

${\mathfrak{g}}={\mathfrak{h}}\oplus\bigoplus_{{\alpha}\in
R}{\mathbb{C}}e_{\alpha}$ (C.12)

where $R\subset\Lambda$ is a set of non-vanishing weights of the adjoint
representation:

$ad_{t}(e_{\alpha})=t^{\alpha}e_{\alpha}$

These weights are called roots, and the lattice ${\rm Q}\subset\Lambda$ they
generate is a sublattice of $\Lambda$, called the _root lattice_. It has a
basis $({\alpha}_{i})_{i=1}^{r}$ of _positive simple roots_ :

${{\rm
Q}}={\mathbb{Z}}{\alpha}_{1}\oplus{\mathbb{Z}}{\alpha}_{2}\oplus\ldots\oplus{\mathbb{Z}}{\alpha}_{r}$
(C.13)

which allows us to define the _Cartan matrix_ $C^{\mathfrak{g}}$ of
$\mathfrak{g}$:

$C_{ij}^{\mathfrak{g}}={\alpha}_{i}({\alpha}_{j}^{\vee})$ (C.14)

which is non-degenerate. The additional requirement we impose on
$\mathfrak{g}$ is that it is _simply-laced_ , i.e. by an appropriate choice of
integral bases one can make $C^{\mathfrak{g}}$ symmetric:

$C_{ij}^{\mathfrak{g}}=C_{ji}^{\mathfrak{g}}$

####  C.1.4. The center

The quotient

$Z={\Lambda}/{{\rm Q}}$ (C.15)

is an abelian group, which is a subgroup both of $\mathbf{T}$ and
$\mathbf{G}$. In fact, it is the center of $\mathbf{G}$. Clearly, the center
does not act in the adjoint representation, so that in the Eq. (C.10) it is
the quotient $\mathbf{T}/Z$ which acts faithfully. Hence the root lattice

${{\rm Q}}\subset{\Lambda}\subset{\mathfrak{h}}^{*}$

can be also identified with the lattice of $\mathbf{T}/Z$ characters

${{\rm Q}}={\operatorname{Hom}}(\mathbf{T}/Z,{\mathbb{C}}^{\times})$ (C.16)

Finally, the coweight lattice ${\Lambda}^{\vee}\subset\mathfrak{h}$ is both
the integral dual to ${\rm Q}$, and the lattice of $\mathbf{T}/Z$
cocharacters:

${\Lambda}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},\mathbf{T}/Z)$
(C.17)

with the basis $({\lambda}_{i}^{\vee})_{i=1}^{r}$, dual to that of ${\rm Q}$:

${\alpha}_{i}({\lambda}_{j}^{\vee})={\delta}_{ij}$ (C.18)

The expression

$w^{{\lambda}^{\vee}}={\mathbf{e}}\,\left(\frac{{\log}(w)}{2\pi\mathrm{i}}{\lambda}^{\vee}\right)\in\mathbf{T}$

_does_ depend on the choice of the branch of the logarithm ${\log}(w)$,
however, the ambiguity is in the multiplicative $Z$-valued factor, since for
any ${\lambda}^{\vee}\in{{\rm Q}}^{\vee}$:

${\mathbf{e}}({\lambda}^{\vee})\in Z\subset\mathbf{T}$

Thus, for any $w\in{\mathbb{C}}^{\times}$, $w^{{\lambda}^{\vee}}$ is well-
defined as an element of $\mathbf{T}/Z$, as is claimed by (C.17).

The center $Z$, being a finite abelian group, is isomorphic to a product of
cyclic groups

$Z\approx\bigotimes_{{\xi}=1}^{z_{\mathfrak{g}}}\
{\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$ (C.19)

for some $z_{\mathfrak{g}}$, which is equal to $0$, $1$, or $2$.

$\bf G$ | $z_{\mathfrak{g}}$ |  ${\ell}_{\xi}$, ${\xi}=1,\ldots,a_{\mathfrak{g}}$
---|---|---
$A_{r}$ | $1$ | $r+1$
$D_{2s}$ | $2$ | $2,2$
$D_{2s+1}$ | $1$ | $4$
$E_{6}$ | $1$ | $3$
$E_{7}$ | $1$ | $2$
$E_{8}$ | $0$ |
Table C.1. The number of cyclic factors in $Z$ with their orders

The table (C.1) shows the values of $z_{\mathfrak{g}}$’s and ${\ell}_{\xi}$’s
for all simple simply-laced Lie groups.

Let ${\varpi}_{\xi}\in\mathbf{T}$ be the generator of the
${\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$ factor in $Z\subset\mathbf{T}$. In
other words

$\displaystyle{\varpi}_{\xi}^{k}\neq 1\ {\rm for}\
k=1,2,\ldots,{\ell}_{\xi}-1,$ (C.20) $\displaystyle\qquad{\rm and}\
{\varpi}_{\xi}^{{\ell}_{\xi}}=1\in T\ .$

Of course, the Eq. (C.20) does not characterize ${\varpi}_{\xi}$ uniquely.
Indeed, for any integer $s_{\xi}$, which is mutually prime with
${\ell}_{\xi}$, i.e. $(s_{\xi},{\ell}_{\xi})=1$, the $Z$ element
$\widetilde{{\varpi}_{\xi}}={\varpi}_{\xi}^{s_{\xi}}$ also generates
${\mathbb{Z}}/{\ell}_{\xi}{\mathbb{Z}}$. We write,

$\displaystyle{\mathbf{e}}({\lambda}_{i}^{\vee})=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\varpi}_{\xi}^{l_{i{\xi}}},\qquad
i=1,\ldots,r$ (C.21)
$\displaystyle{\varpi}_{\xi}=\prod_{j=1}^{r}{\mathbf{e}}\left(\frac{w_{{\xi}j}}{\ell_{\xi}}{\alpha}_{j}^{\vee}\right)$

for some integers $l_{i{\xi}},w_{{\xi}j}\in\mathbb{Z}$, which are normalized

$0\leq l_{i{\xi}}<{\ell}_{{\xi}},\qquad 0\leq w_{{\xi}j}<{\ell}_{\xi}$ (C.22)

and

$(w_{{\xi}j},{\ell}_{{\xi}})=1\qquad.$

Note that

${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{\mathfrak{g}})^{-1}_{ij}\in{\mathbb{Q}}$
(C.23)

By combining Eqs. (C.23) with (C.21), we derive:

$(C^{\mathfrak{g}})^{-1}_{ij}=\sum_{{\xi}=1}^{z_{\mathfrak{g}}}\frac{l_{i{\xi}}w_{{\xi}j}}{{\ell}_{{\xi}}}+{\mathcal{L}}^{\mathfrak{g}}_{ij}$
(C.24)

where ${\mathcal{L}}^{\mathfrak{g}}$ is some integral matrix. Note in passing
that if we were to study the group $G$ over the field ${\mathbb{F}}_{p^{n}}$
where $p$ divides ${\ell}_{y}$ then $C^{\mathfrak{g}}_{ij}$ would correspond
to an affine (or even double affine) root system.

####  C.1.5. Killing metric

Another structure we gain from the adjoint action of $T$ on $\mathfrak{g}$ is
the Killing metric on $\mathfrak{h}$,

$\langle
x,x\rangle=\frac{1}{h_{\mathfrak{g}}}\,{\operatorname{tr}}_{\mathfrak{g}}\,{\rm
ad}_{x}^{2}$ (C.25)

which identifies ${\mathfrak{h}}$ with ${\mathfrak{h}}^{*}$, ${{\rm
Q}}^{\vee}$ with ${{\rm Q}}$, and ${\Lambda}^{\vee}$ with $\Lambda$, and the
constant $h_{\mathfrak{g}}$ is chosen so that

$\langle{\lambda}_{i},{\alpha}_{j}\rangle={\delta}_{ij},\quad\langle{\alpha}_{i},{\alpha}_{j}\rangle=C_{ij}^{\mathfrak{g}},\quad\langle{\lambda}_{i},{\lambda}_{j}\rangle=(C^{\mathfrak{g}}_{ij})^{-1}$
(C.26)

####  C.1.6. The Weyl group

The torus $\mathbf{T}$ has some symmetries within $\mathbf{G}$. Namely, for
any $t\in\mathbf{T}$ there are transformations of the form:

$t\mapsto g^{-1}tg$ (C.27)

for some $g\in\mathbf{G}$, for which $g^{-1}tg\in\mathbf{T}$ also. Such
transformations form a group, which is called the _normalizer $N(\mathbf{T})$
of $\mathbf{T}$_. This group obviously contains $\mathbf{T}$, since the
transformation (C.27) with $g\in\mathbf{T}$ doesn’t even move $t$. It turns
out that there are additional nontrivial transformations. These additional
transformations form _the Weyl group_
$W({\mathfrak{g}})=N(\mathbf{T})/\mathbf{T}$.

For $t\in\mathbf{T}$ and $w\in W({\mathfrak{g}})$ let us denote by ${}^{w}t$
the result of the application of $g_{w}$ representing $w$ in
$N(\mathbf{T})\subset\mathbf{G}$:

${}^{w}t=g_{w}^{-1}tg_{w}$ (C.28)

By taking the limit $t\to 1$ we get the action of $W({\mathfrak{g}})$ on
$\mathfrak{h}$: for $\xi\in\mathfrak{h}$

${\xi}\mapsto\
^{w}{\xi}=\frac{d}{ds}\biggr{|}_{s=0}\,^{w}\left({\exp}\,s{\xi}\right)$ (C.29)

It is clear that $W({\mathfrak{g}})$ acts on $\mathfrak{h}$ by the orthogonal
transformations preserving the metric (C.25). The action of
$W({\mathfrak{g}})$ on $\mathfrak{h}^{*}$ ($\mathfrak{h}$) preserves both the
(co)root lattice and the (co)weight lattice.

The less trivial result is that the group $W({\mathfrak{g}})$ is generated by
reflections $r_{i}$ at the simple roots. The corresponding transformations on
$\mathfrak{h}$ and $\mathbf{T}$ are

${}^{r_{i}}x=x-{\alpha}_{i}(x){\alpha}_{i}^{\vee},\qquad^{r_{i}}t=t\left(t^{-{\alpha}_{i}}\right)^{{\alpha}_{i}^{\vee}}$
(C.30)

The $W({\mathfrak{g}})$-action on $\mathbf{T}$ can be also described in the
$z$-coordinates:

${}^{r_{i}}t_{\bf z}=t_{\bf z}(t_{\bf
z}^{-{\alpha}_{i}})^{{\alpha}_{i}^{\vee}}=t_{\tilde{\bf z}}\ ,$

where

$\displaystyle{\tilde{z}}_{j}=$ $\displaystyle z_{j},\qquad j\neq i$ (C.31)
$\displaystyle{\tilde{z}}_{i}=$ $\displaystyle
z_{i}\,\prod_{k=1}^{r}z_{k}^{-C^{\mathfrak{g}}_{ik}}$

####  C.1.7. Weyl group and the center

Since $W({\mathfrak{g}})$ acts on $\mathbf{T}$ by similarity transformations
in $\mathbf{G}$, the center $Z\subset\mathbf{T}\subset\mathbf{G}$ is fixed by
any $w\in W({\mathfrak{g}})$. In particular, $r_{i}$ preserves
${\varpi}_{\xi}$ for any $i$ and $\xi$, which is equivalent to

${\varpi}_{\xi}^{{\alpha}_{i}}=1$ (C.32)

Substituting into Eq. (C.32) the representation (C.22) and recalling the
definition (C.14) of $C^{\mathfrak{g}}$ we get:

$\frac{1}{{\ell}_{\xi}}\sum_{j=1}^{r}w_{{\xi}j}C_{ji}^{\mathfrak{g}}={\mathfrak{C}}_{{\xi}i}^{\mathfrak{g}}\in{\mathbb{Z}},\qquad{\xi}=1,\ldots,z_{\mathfrak{g}},\
i=1,\ldots,r\ .$ (C.33)

Combining this relation with Eq. (C.24) we obtain:

${\delta}_{ij}=\sum_{{\xi}=1}^{z_{\mathfrak{g}}}l_{i{\xi}}{\mathfrak{C}}_{{\xi}j}^{\mathfrak{g}}+\sum_{k=1}^{r}{\mathcal{L}}^{\mathfrak{g}}_{ik}C^{\mathfrak{g}}_{kj}$
(C.34)

On the other hand, using the Eq. (C.21), and the relations
$\sum_{j=1}^{r}C^{\mathfrak{g}}_{ij}{\lambda}_{j}^{\vee}={\alpha}_{i}^{\vee}$,
and ${\mathbf{e}}({\alpha}_{i}^{\vee})=1$ for any $i$, we derive:

$\prod_{j=1}^{r}\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\varpi}_{\xi}^{C_{ij}^{\mathfrak{g}}l_{j\xi}}=1$

which implies

$\frac{1}{{\ell}_{\xi}}\sum_{j=1}^{z_{\mathfrak{g}}}C_{ij}^{\mathfrak{g}}l_{j\xi}={\mathfrak{c}}_{i\xi}^{\mathfrak{g}}\in{\mathbb{Z}}$
(C.35)

####  C.1.8. Langlands dual, adjoint, and conformal groups

The simple Lie group $\mathbf{G}/Z$ has a trivial center, but it is not
simply-connected. This group is also denoted by $\mathbf{G}^{\text{ad}}$,
since it is represented faithfully in the adjoint representation
$\mathfrak{g}$. The maximal torus $\mathbf{T}^{\text{ad}}$ of
$\mathbf{G}^{\text{ad}}$, is the quotient $\mathbf{T}/Z={\mathfrak{h}}/{{\rm
Q}}$. Also, since the lattices of weights and roots of $\mathbf{G}$ and
$\mathbf{G}^{\text{ad}}$ are dual to each other, these groups are Langlands
duals, ${}^{L}\mathbf{G}=\mathbf{G}^{\text{ad}}$.

The group $\mathbf{G}^{\text{ad}}$ is not very convenient to work with. For
one thing, the center $Z$ looks differently for different groups. One defines
the _conformal extension_ $C\mathbf{G}$ of $\mathbf{G}$ as the group (see
[Levin:2010ve, Levin:2010mz, Morgan:2000math] for recent applications in a
related context),

${\mathrm{C}}\mathbf{G}=({\rm C}\times\,\mathbf{G})/Z$ (C.36)

where

${\rm C}=({\mathbb{C}}^{\times})^{z_{\mathfrak{g}}}$ (C.37)

where the center $Z$ acts on $\mathbf{G}$ in the usual way, and on $\rm C$ via
some character

${\chi}\in{\operatorname{Hom}}(Z,{\rm C})$ (C.38)

For example, we can choose

${\chi}({\varpi}_{x})=({\zeta}_{{\ell}_{y}}^{{\delta}_{x,y}})_{y=1}^{z_{\mathfrak{g}}}$

where ${\zeta}_{l}$ is the primitive $l$’th root of unity:

${\zeta}_{l}=e^{2\pi\frac{\mathrm{i}}{l}}$ (C.39)

The elements of the group $C\mathbf{G}$ are the classes of pairs
$[({\mathfrak{b}};g)]$, where ${\mathfrak{b}}\in\rm C$ is the
$z_{\mathfrak{g}}$-tuple of non-zero complex numbers,
${\mathfrak{b}}=({\mathfrak{b}}_{1},\ldots,{\mathfrak{b}}_{z_{\mathfrak{g}}})$,
and $g\in\mathbf{G}$, under the equivalence
$({\mathfrak{b}};g)\sim(\,^{x}{\mathfrak{b}};{\varpi}_{x}g\,)$, where

${}^{x}{\mathfrak{b}}=({\mathfrak{b}}_{y}{\zeta}_{{\ell}_{x}}^{-{\delta}_{x,y}})_{y=1}^{z_{\mathfrak{g}}}$

with the multiplication law

$[({\mathfrak{b}}_{1};g_{1})]\cdot[({\mathfrak{b}}_{2};g_{2})]=[({\mathfrak{b}}_{1}{\mathfrak{b}}_{2};g_{1}g_{2})]$

The maximal torus $C\mathbf{T}$ of $C\mathbf{G}$ has rank
$r+z_{\mathfrak{g}}$. The center $CZ$ of $C\mathbf{G}$ is the subgroup of
$C\mathbf{G}$ which consists of the equivalence classes containing
$({\mathfrak{b}};1)$, with $\mathfrak{b}\in{\rm C}$. Clearly, $CZ\approx{\rm
C}$. In this way, the centers of the conformal extensions look the same for
all groups of equal $z_{\mathfrak{g}}$.

The lattice

$C{{\rm Q}}^{\vee}={\operatorname{Hom}}({\mathbb{C}}^{\times},C\mathbf{T})$
(C.40)

is an extension of ${{\rm Q}}^{\vee}$ by the rank $z_{\mathfrak{g}}$ lattice
with the generators ${\beta}_{\xi}^{\vee}$, ${\xi}=1,\ldots,z_{\mathfrak{g}}$:

$C{{\rm
Q}}^{\vee}=\bigoplus_{i=1}^{r}{\mathbb{Z}}{\alpha}_{i}^{\vee}\oplus\bigoplus_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathbb{Z}}{\beta}_{\xi}^{\vee}$
(C.41)

The root subspace decomposition of ${\mathfrak{c}\mathfrak{g}}={\rm
Lie}(C\mathbf{G})={\mathfrak{g}}\oplus{\mathbb{C}}^{z_{\mathfrak{g}}}$ is easy
to compute. There are $r$ simple roots which we denote by ${\alpha}_{i}$. They
act on ${\mathfrak{c}}{\mathfrak{h}}={\rm Lie}(C\mathbf{T})$ as follows:

$\displaystyle{\alpha}_{i}({\alpha}_{j}^{\vee})=C_{ji}^{\mathfrak{g}}$ (C.42)
$\displaystyle{\alpha}_{i}({\beta}_{\xi}^{\vee})={\mathfrak{C}}_{{\xi}i}^{\mathfrak{g}}$

so that if we define

$K_{\xi}=-{\ell}_{\xi}{\beta}_{\xi}^{\vee}+\sum_{j=1}^{r}w_{{\xi}j}{\alpha}_{j}^{\vee}$
(C.43)

then

${\alpha}_{i}(K_{\xi})=0$ (C.44)

for any $i$, $\xi$. The meaning of (C.43) is the following. The lattice of
cocharacters of the maximal torus of $\mathrm{C}\times\mathbf{G}$ is the
direct sum of the lattice ${{\rm Q}}^{\vee}$ of coroots of $G$ and the lattice

$\displaystyle{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})=L\oplus{{\rm Q}}^{\vee}$ (C.45) $\displaystyle
L={\pi}_{1}({\rm
C})=\bigoplus_{{\eta}=1}^{z_{\mathfrak{g}}}\,{\mathbb{Z}}K_{\eta},$

so that a generic element of $\mathrm{C}\times\mathbf{T}$ can be represented
by:

${\widehat{t}}_{\mathfrak{b},{\bf
g}}=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{b}}_{\xi}^{K_{\xi}}\times\prod_{i=1}^{r}g_{i}^{{\alpha}_{i}^{\vee}}$
(C.46)

(again, in writing (C.46) we assume normalization ${\mathbf{e}}(K_{\xi})=1\in$
C.) The action of $Z$ on $\mathrm{C}\times\mathbf{G}$ translates to the action
on ${\mathfrak{b}},{\bf g}$:

${\varpi}_{\xi}:{\widehat{t}}_{{\mathfrak{b}},{\bf
g}}\mapsto{\widehat{t}}_{{\mathfrak{b}},{\bf
g}}\,{\zeta}_{{\ell}_{\xi}}^{-K_{\xi}+\sum_{i=1}^{r}w_{{\xi}i}{\alpha}_{i}^{\vee}}$
(C.47)

The quotient $C\mathbf{T}=({\rm C}\times\mathbf{T})/Z$ is coordinatized by

$\displaystyle[({\mathfrak{b}};t_{\mathbf{z}})]={\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}=\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{u}}_{\xi}^{{\beta}_{\xi}^{\vee}}\prod_{i=1}^{r}z_{i}^{{\check{\alpha}}_{i}^{\vee}}\in
CT,$ (C.48)
$\displaystyle{\mathfrak{u}}_{\xi}={\mathfrak{b}}_{\xi}^{-{\ell}_{\xi}},\qquad
z_{i}=g_{i}\prod_{{\xi}=1}^{z_{\mathfrak{g}}}{\mathfrak{b}}_{\xi}^{w_{{\xi}i}}$

in agreement with (C.40). Simply put,
${\beta}_{\xi}^{\vee}\in{\operatorname{Hom}}({\mathbb{C}}^{\times},C\mathbf{T})$,
${\ell}_{\xi}{\beta}_{\xi}^{\vee}\in{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})$, and
${\beta}^{\vee}_{\xi}\notin{\operatorname{Hom}}({\mathbb{C}}^{\times},{\rm
C}\times\mathbf{T})$.

Clearly, there are other choices of the $Z$-invariant coordinates, which
differ by multiplication of $z_{i}$ by any function of
${\mathfrak{u}}_{\xi}$’s.

Comparing the Eqs. (C.48) and (C.46) we arrive at (C.43). The Eq. (C.44)
simply reflects the fact that C acts trivially in the adjoint representation.
The lattice $C{{\rm Q}}$ generated by ${\alpha}_{i}$ is isomorphic to the root
lattice ${{\rm Q}}$.

The weight lattice $\widetilde{C\Lambda}$ of $\mathrm{C}\times\mathbf{G}$ is
the direct sum of the weight lattice of C and that of $\mathbf{G}$:

$\widetilde{C\Lambda}={\operatorname{Hom}}({\rm
C}\times\mathbf{T},{\mathbb{C}}^{\times})=L^{\vee}\oplus{\Lambda}$ (C.49)

The weight lattice $C\Lambda$ of $C\mathbf{G}$ is a sublattice of
$\widetilde{C\Lambda}$ which consists of the weights which are trivial on the
elements of $\mathrm{C}\times\mathbf{T}$ of the form

${\zeta}_{{\ell}_{\xi}}^{-K_{\xi}+\sum_{i=1}^{r}w_{{\xi}i}{\alpha}_{i}^{\vee}}$

for any $\xi$. In the $\widetilde{C\Lambda}$-basis
$({\mu}_{\xi};{\lambda}_{i})$, ${\xi}=1,\ldots,z_{\mathfrak{g}}$,
$i=1,\ldots,r$ such that

${\mu}_{\xi}(K_{\eta})={\delta}_{{\xi},{\eta}},\
{\mu}_{\xi}({\alpha}_{i}^{\vee})=0$ ${\lambda}_{i}(K_{\eta})=0,\
{\lambda}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$

the lattice $C\Lambda$ is spanned by

${\check{\mu}}_{\xi}=-{\ell}_{\xi}{\mu}_{\xi},\qquad{\check{\lambda}}_{i}={\lambda}_{i}+\sum_{\xi}w_{{\xi}i}{\mu}_{\xi}\
.$ (C.50)

An easy computation gives:

$\displaystyle{\check{\mu}}_{\xi}({\beta}_{\eta}^{\vee})={\delta}_{\xi\eta},\quad{\check{\lambda}}_{i}({\alpha}_{j}^{\vee})={\delta}_{ij}$
(C.51)
$\displaystyle{\check{\mu}}_{\xi}({\alpha}_{i}^{\vee})=0,\quad{\check{\lambda}}_{i}({\beta}_{\eta}^{\vee})=0$

Finally, the lattice $C\Lambda^{\vee}$ of coweights of $C\mathbf{G}$, which is
dual to the root lattice $C{{\rm Q}}$, is generated by the fundamental
coweights ${\check{\lambda}}_{i}^{\vee}$, which obey

${\alpha}_{i}({\check{\lambda}}_{j}^{\vee})={\delta}_{ij}$ (C.52)

The equations (C.52) define ${\check{\lambda}}_{i}^{\vee}$ up to the shifts by
the integer multiples of $K_{\xi}$. We choose the representative

${\check{\lambda}}_{i}^{\vee}=\sum_{j=1}^{r}{\mathcal{L}}_{ij}^{\mathfrak{g}}{\alpha}_{j}^{\vee}+\sum_{{\xi}=1}^{z_{\mathfrak{g}}}l_{i{\xi}}{\beta}_{\xi}^{\vee}$
(C.53)

Shifting ${\check{\lambda}}_{i}^{\vee}$ by $d_{\xi}K_{\xi}$ would change the
coefficient $l_{i{\xi}}\mapsto l_{i{\xi}}+{\ell}_{\xi}d_{\xi}$.

As we see, the coweights ${\check{\lambda}}_{i}^{\vee}$ are the integral
linear combinations of the coroots ${\alpha}^{\vee}$ and ${\beta}^{\vee}$, as
if $C\mathbf{G}$ were the simple simply-connected group with the trivial
center.

####  C.1.9. The D-homomorphism

By construction of the conformal group, there is a homomorphism

$\displaystyle D:C\mathbf{G}\longrightarrow{\rm C}/Z\approx{\rm C}$ (C.54)
$\displaystyle
D[({\mathfrak{b}};g)]=({\mathfrak{b}}_{\xi}^{{\ell}_{\xi}})_{{\xi}=1}^{z_{\mathfrak{g}}}$

In terms of the weights, the restriction of the homomorphism $D$ onto
$C\mathbf{T}$ is given by:

$D({\check{t}}_{{\mathfrak{u}},{\mathbf{z}}})=({\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}^{{\check{\mu}}_{\xi}})_{{\xi}=1}^{z_{\mathfrak{g}}}=({\mathfrak{u}}_{\xi})_{{\xi}=1}^{z_{\mathfrak{g}}}$
(C.55)

####  C.1.10. Twisted Weyl group action

Now we can define the twisted action of Weyl group $W({\mathfrak{g}})$. It is
nothing but the natural action of $W({\mathfrak{g}})$ on $C\mathbf{T}$.
However, we shall encounter a somewhat redundant parametrization of
$C\mathbf{T}$, by $\mathbf{T}\times\mathbf{T}$:

${\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}=\prod_{i=1}^{r}{\mathscr{P}}_{i}^{-\,{\check{\lambda}}_{i}^{\vee}}{\mathscr{Y}}_{i}^{{\check{\alpha}}_{i}^{\vee}}$
(C.56)

which is equal to ${\check{t}}_{{\mathfrak{u}},{\mathbf{z}}}$ with

${\mathfrak{u}}_{\xi}=\prod_{j}{\mathscr{P}}_{j}^{-l_{j\xi}},\qquad
z_{i}={\mathscr{Y}}_{i}\prod_{j}{\mathscr{P}}_{j}^{-{\mathcal{L}}_{ji}}$
(C.57)

Using (C.42) we compute the Weyl group $W({\mathfrak{g}})$ action on
$C\mathbf{T}$: under the reflection $r_{i}$ the group element
${\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}$ maps to
${}^{r_{i}}{\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}}={\mathbf{g}}_{{\mathscr{P}},{\tilde{\mathscr{Y}}}}$,
where

$\displaystyle{\tilde{\mathscr{Y}}}_{j}={\mathscr{Y}}_{j},\qquad j\neq i$
(C.58)
$\displaystyle{\tilde{\mathscr{Y}}}_{i}={\mathscr{P}}_{i}{\mathscr{Y}}_{i}\prod_{j}{\mathscr{Y}}_{j}^{-C_{ij}^{\mathfrak{g}}}$

The homomorphism $D$ is $W({\mathfrak{g}})$-invariant:

$D({\mathbf{g}}_{{\mathscr{P}},{\mathscr{Y}}})=D({\mathbf{g}}_{{\mathscr{P}},{\tilde{\mathscr{Y}}}})=(\prod_{j=1}^{r}{\mathscr{P}}_{j}^{-l_{j{\xi}}})_{{\xi}=1}^{z_{\mathfrak{g}}}$

###  C.2. Affine Lie algebras

The affine Lie algebras $\widehat{\mathfrak{g}}$ show up in the solution of
the theories of the class II. In preparing this section we consulted with
[Bardakci:1970nb], [Kac:1984]. Given a simple Lie group $\mathbf{G}$ with its
Lie algebra $\mathfrak{g}$ one defines the loop group $L\mathbf{G}$ and the
loop algebra $L\mathfrak{g}$ of (suitably defined, analytic, formal,
polynomial) maps of the neighborhood of a circle in $\mathbb{C}^{\times}$ into
$\mathbf{G}$ and $\mathfrak{g}$, respectively:

$L\mathbf{G}={\rm Maps}({\mathbb{C}}^{\times},\mathbf{G}),\quad
L{\mathfrak{g}}={\rm Maps}({\mathbb{C}}^{\times},{\mathfrak{g}}),$ (C.59)

with the point-wise product and Lie brackets, respectively. Then one defines
the central extension
$\widetilde{L{\mathfrak{g}}}=L{\mathfrak{g}}\oplus{\mathbb{C}}K$ by

$[f_{1}(t)\oplus a_{1}\,K,f_{2}(t)\oplus
a_{2}\,K]=[f_{1}(t),f_{2}(t)]\oplus\oint_{{\mathbb{S}}^{1}}\langle
df_{1}(t),f_{2}(t)\rangle\,K$

and the additional (non-central) extension $\widehat{\mathfrak{g}}$ of
$\widetilde{L{\mathfrak{g}}}$ by $\mathbb{C}$, which acts on $L\mathfrak{g}$
by the infinitesimal rotation of ${\mathbb{C}}^{\times}$ in (C.59). The
elements of $\widehat{\mathfrak{g}}$ can be represented as the
$\mathfrak{g}$-valued first order differential operators on $\mathbb{S}^{1}$
plus a constant:

${\tau}d+f(t)\oplus
aK\in\widehat{\mathfrak{g}},\qquad{\tau},a\in\mathbb{C},\,f(t)\in{\mathfrak{g}}$

with the commutation relations:

$\displaystyle[{\tau}_{1}d+f_{1}(t)\oplus a_{1}K,{\tau}_{2}d+f_{2}(t)\oplus
a_{2}K]=$ (C.60) $\displaystyle\quad 0\cdot
d+t\left({\tau}_{1}f_{2}^{\prime}(t)-{\tau}_{2}f_{1}^{\prime}(t)\right)+[f_{1}(t),f_{2}(t)]\,\oplus\oint_{{\mathbb{S}}^{1}}\langle
f_{1}(t),df_{2}(t)\rangle\,K$

Analogously, one defines the central extension $\widetilde{L\mathbf{G}}$ which
is a non-trivial ${\mathbb{C}}^{\times}$-bundle over $L\mathbf{G}$ (this is
analogous to the construction of the conformal group $C\mathbf{G}$ in the
previous section), and then the extension $\widehat{\mathbf{G}}$ of
$\widetilde{L\mathbf{G}}$ by ${\mathbb{C}}^{\times}$ which acts by rotation of
${\mathbb{C}}^{\times}$ in (C.59).

The algebra $\widehat{\mathfrak{g}}$ can also be defined by the general
construction of the generalized Kac-Moody algebras associated with the Cartan
matrix $C_{ij}^{\widehat{\mathfrak{g}}}$. The affine Cartan matrix has exactly
one eigenvector with the eigenvalue zero:

$\sum_{j=0}^{r}C_{ij}^{\widehat{\mathfrak{g}}}a_{j}=0$ (C.61)

The Cartan subalgebra $\widehat{\mathfrak{h}}$ of the corresponding affine Lie
algebra $\widehat{\mathfrak{g}}$ with the underlying finite dimensional
simply-laced Lie algebra $\mathfrak{g}$ of rank $r$ is the complex vector
space of dimension $r+2$. The dual space $\widehat{\mathfrak{h}}^{*}$ contains
the root lattice ${\widehat{\rm Q}}\subset\widehat{\mathfrak{h}}^{*}$, which
is generated by the simple roots ${\widehat{\alpha}}_{i}$, $i=0,\ldots,r$, of
which the simple roots ${\widehat{\alpha}}_{i}$ with $i>0$ generate the root
lattice of $\mathfrak{g}$. Likewise, the Cartan subalgebra
$\widehat{\mathfrak{h}}$ contains the coroot lattice ${\widehat{\rm
Q}}^{\vee}\subset\widehat{\mathfrak{h}}$, generated by the simple coroots
${\alpha}_{i}^{\vee}$, which obey:

${\widehat{\alpha}}_{i}({\widehat{\alpha}}_{j}^{\vee})=C_{ij}^{\widehat{\mathfrak{g}}}$
(C.62)

The following linear combination of the simple roots:

${\delta}=\sum_{i=0}^{r-1}a_{i}{\widehat{\alpha}}_{i}\in\widehat{\mathfrak{h}}^{*}$
(C.63)

is called the imaginary root. It annihilates the simple coroots, cf. (C.61):

${\delta}({\widehat{\alpha}}_{i}^{\vee})=0\,,\ i=0,\ldots,r-1$

The analogous linear combination of coroots

$K=\sum_{i=0}^{r-1}a_{i}{\widehat{\alpha}}_{i}^{\vee}\in\widehat{\mathfrak{h}}$
(C.64)

obeys

${\widehat{\alpha}}_{i}(K)=0\,,\ i=0,\ldots,r$ (C.65)

and generates the center of the affine Kac-Moody algebra since (C.65) implies
it commutes with everything in $\widehat{\mathfrak{g}}$. Then

${\widehat{\rm Q}}^{\vee}_{\mathbb{C}}={\mathfrak{h}}\oplus{\mathbb{C}}K\,,\
{\widehat{\rm Q}}_{\mathbb{C}}={\mathfrak{h}}^{*}\oplus{\mathbb{C}}{\delta}$
(C.66)

where $\mathfrak{h}$ and $\mathfrak{h}^{*}$ are the Cartan subalgebra and its
dual space of the corresponding finite dimensional simply-laced Lie algebra
$\mathfrak{g}$, respectively. In order to generate $\widehat{\mathfrak{h}}$
and $\widehat{\mathfrak{h}}^{*}$ as the vector spaces over $\mathbb{C}$ we
need to add one more generator in addition to the simple roots and the simple
coroots, respectively:

${\widehat{\mathfrak{h}}}^{*}={\widehat{\rm
Q}}_{\mathbb{C}}\oplus{\mathbb{C}}{\lambda}_{0}\,,\
{\widehat{\mathfrak{h}}}={\widehat{\rm
Q}}^{\vee}_{\mathbb{C}}\oplus{\mathbb{C}}{\delta}^{\vee}$ (C.67)

which obey:

${\widehat{\lambda}}_{0}({\widehat{\alpha}}_{i}^{\vee})={\widehat{\alpha}}_{i}({\delta}^{\vee})={\delta}_{i0}$
(C.68)

The generator ${\delta}^{\vee}$ is equal to the generator of the infinitesimal
loop rotation $d$ we used in (C.60).

The weight lattice ${\widehat{\Lambda}}$ of $\widehat{\mathfrak{g}}$ is
generated by the fundamental weights
${\lambda}_{i}\in\widehat{\mathfrak{h}}^{*}$, $i=0,1,\ldots,r$:

${\widehat{\Lambda}}={\Lambda}\oplus{\mathbb{Z}}{\lambda}_{0}\subset\widehat{\mathfrak{h}}^{*}$
(C.69)

which obey the following basic relations:

${\widehat{\lambda}}_{i}({\widehat{\alpha}}_{j}^{\vee})={\delta}_{ij},\
{\widehat{\lambda}}_{i}({\delta}^{\vee})=0,\
{\delta}({\delta}^{\vee})=a_{0}=1$ (C.70)

The fundamental coweights
${\widehat{\lambda}}_{i}^{\vee}\in\widehat{\mathfrak{h}}$ obey:

${\widehat{\alpha}}_{j}({\widehat{\lambda}}_{i}^{\vee})={\delta}_{ij},$

and form the coweight lattice

${\widehat{\Lambda}}^{\vee}=\bigoplus_{i=0}^{r}{\mathbb{Z}}{\lambda}_{i}^{\vee}\subset{\mathfrak{h}}$

The level of a weight ${\widehat{\lambda}}\in\widehat{\Lambda}$ is defined as
$k={\widehat{\lambda}}(K)$ so that the level of the $i$’th fundamental weight
is equal to

${\widehat{\lambda}}_{i}(K)=a_{i}={\delta}({\widehat{\lambda}}^{\vee}_{i})\ .$
(C.71)

Note that ${\widehat{\rm Q}}^{\vee}_{\mathbb{C}}$ is the Cartan subalgebra of
the central extension $\widehat{L{\mathfrak{g}}}$ of the loop algebra
$L{\mathfrak{g}}$. Adding ${\delta}^{\vee}$ makes up the Cartan of the affine
Kac-Moody algebra of $\mathfrak{g}$, which is an extension of
$\widehat{L{\mathfrak{g}}}$ by the operator of the infinitesimal loop
rotation, the zero mode of the Virasoro generator. In physics literature the
more common notation for ${\delta}^{\vee}$ is $L_{0}$.

Also we notice that

${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{\mathfrak{g}})^{-1}_{ij},\qquad
i,j=1,\ldots,r$ (C.72)

hence

${\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee})=\begin{pmatrix}0&0\\\
0&(C^{\mathfrak{g}})^{-1}\end{pmatrix}_{ij}.$ (C.73)

A useful relation obtained from identity operator
$|\Lambda_{i}\rangle\langle\alpha_{i}^{\vee}|+|\delta\rangle\langle\delta^{\vee}|$
is

$\sum_{i=0}^{r}{\widehat{\lambda}}_{i}({\widehat{\lambda}}_{j}^{\vee}){\widehat{\alpha}}_{k}({\widehat{\alpha}}_{i}^{\vee})={\widehat{\alpha}}_{k}({\widehat{\lambda}}_{j}^{\vee})-{\delta}({\widehat{\lambda}}_{j}^{\vee}){\alpha}_{k}({\delta}^{\vee})=\delta_{jk}-a_{j}\delta_{k0}$
(C.74)

Let $\lambda\in\mathfrak{h}^{*}$ for
$\widehat{\lambda}\in\widehat{\mathfrak{h}}^{*}$ denote the image of the
projection $\widehat{\mathfrak{h}}^{*}\to\mathfrak{h}^{*}$, i.e. forgetting
components spanned by $\delta$ and $\widehat{\lambda}_{0}$. Then

$\displaystyle\widehat{\lambda}_{i}^{\vee}=a_{i}\delta^{\vee}+\lambda_{i}^{\vee}$
(C.75)
$\displaystyle\widehat{\lambda}_{j}=a_{j}\widehat{\lambda}_{0}+\lambda_{j}$

###  C.3. ADE Cartan matrices, roots and weights

Our ADE conventions are summarized in the table A.

####  C.3.1. $A_{r}$ series

The Cartan matrix $C_{ij}^{\widehat{A}_{r}}$ with $i,j=0,\dots,r$

$C^{\widehat{A}_{r}}=\begin{pmatrix}2&-1&0&\dots&\dots&-1\\\
-1&2&-1&\dots&\dots&0\\\ 0&-1&2&-1&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&0&\dots&\dots&2&-1\\\
-1&0&\dots&\dots&-1&2\end{pmatrix}$ (C.76)

The affine simple roots are

${\widehat{\alpha}}_{0}=\delta-\theta,\widehat{\alpha}_{1}=e_{1}-e_{2},\widehat{\alpha}_{2}=e_{2}-e_{3},\dots,\widehat{\alpha}_{r}=e_{r}-e_{r+1}$

where $\theta=e_{1}-e_{r+1}$ is the highest root. The Dynkin marks are
$a_{i}=1,i=0,\dots,r$. The fundamental weights are

$\displaystyle\widehat{\lambda}_{0},$ (C.77)
$\displaystyle\widehat{\lambda}_{1}=\widehat{\lambda}_{0}+e_{1}-\frac{1}{r}{\bf
e}$
$\displaystyle\widehat{\lambda}_{2}=\widehat{\lambda}_{0}+e_{1}+e_{2}-\frac{2}{r}{\bf
e}$ $\displaystyle\dots$
$\displaystyle\widehat{\lambda}_{r}=\widehat{\lambda}_{0}+e_{1}+e_{2}+\dots+e_{r}-\frac{r}{r+1}{\bf
e}$

where

${\bf e}=e_{1}+\dots+e_{r}$

The inverse Cartan matrix of the block $i,j=1\dots r$, i.e. the inverse Cartan
matrix $(C^{A_{r}})^{-1}$ of ${\mathfrak{g}}=A_{r}$ is

${\lambda}_{i}({\lambda}_{j}^{\vee})=(C^{A_{r}})^{-1}_{ij}={\rm
max}(i,j)-\frac{ij}{r+1},\qquad i,j=1,\ldots,r$ (C.78)

####  C.3.2. $D_{r}$ series

In the standard basis $\\{e_{i}\,|\,i=1,\ldots,r\\}$ the simple roots of
$D_{r}$ are

$\displaystyle\alpha_{i}$ $\displaystyle=e_{i}-e_{i+1},\qquad i=1,\ldots,r-1$
(C.79) $\displaystyle\alpha_{r}$ $\displaystyle=e_{r-1}+e_{r}$

and the fundamental weights are

$\displaystyle\lambda_{i}$ $\displaystyle=e_{1}+e_{2}+\dots+e_{i},\quad
i=1\dots r-2$ (C.80) $\displaystyle\lambda_{r-1}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+\dots+e_{r-1}-e_{r})$
$\displaystyle\lambda_{r}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+\dots+e_{r-1}+e_{r})$

The highest root is $\theta=\sum_{i=1}^{r}\alpha_{i}a_{i}=e_{1}+e_{2}$.

In the basis $(e_{i})$ the root (coroot) lattice of $D_{r}$ is given by

$Q=Q^{\vee}=\\{\mathbf{n}\in\mathbb{Z}^{r}||{\bf n}|\in 2\mathbb{Z}\\}$ (C.81)

The weight (coweight) lattice of $D_{r}$ is

$\oplus_{i=1}^{r}\mathbb{Z}\lambda_{i}=\bigcup_{{\varepsilon}=0,\frac{1}{2}}\left({\mathbb{Z}}+{\varepsilon}\right)^{r}\
.$ (C.82)

The inverse Cartan matrix is

$\displaystyle(C^{D_{r}})^{-1}=$ (C.83)
$\displaystyle\qquad\qquad\|\lambda_{i}(\lambda_{j}^{\vee})\|_{i,j}^{r}=$
$\displaystyle\qquad\qquad\begin{pmatrix}1&1&1&1&\cdots&1&\frac{1}{2}\quad&\frac{1}{2}\quad\\\
1&2&2&2&\cdots&2&\quad\frac{2}{2}&\quad\frac{2}{2}\\\
1&2&3&3&\cdots&3&\frac{3}{2}\quad&\frac{3}{2}\quad\\\
1&2&3&4&\cdots&4&\quad\frac{4}{2}&\quad\frac{4}{2}\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&2&3&4&\cdots&r-2&\quad\frac{r-2}{2}&\quad\frac{r-2}{2}\\\
\frac{1}{2}&\frac{2}{2}&\frac{3}{2}&\frac{4}{2}&\cdots&\frac{r-2}{2}&\frac{r}{4}\quad&\frac{r-2}{4}\quad\\\
\quad\frac{1}{2}&\quad\frac{2}{2}&\quad\frac{3}{2}&\quad\frac{4}{2}&\cdots&\quad\frac{r-2}{2}&\quad\frac{r-2}{4}&\quad\frac{r}{4}\end{pmatrix}$

The basis of the affine Cartan dual is
$\\{\lambda_{0},\delta,e_{1},\dots,e_{r}\\}$. The simple roots are
$\widehat{\alpha}_{0}=\delta-\theta$ and the roots $\alpha_{i}$ of $D_{r}$.
The fundamental weights are

$\displaystyle\widehat{\lambda}_{0},$ (C.84)
$\displaystyle\widehat{\lambda}_{i}=a_{i}\widehat{\lambda}_{0}+{\lambda}_{i},\quad
i=1,\ldots,r$

where $\lambda_{i}$ denote the fundamental weights of $D_{r}$, and $a_{i}$ are
the Dynkin labels:

$(a_{0},\dots,a_{r})=(1,1,2,\dots,2,1,1)$ (C.85)

####  C.3.3. $E_{6}$

The Cartan matrix of $E_{6}$ is

$C^{E_{6}}=\begin{pmatrix}2&0&-1&0&0&0\\\ 0&2&0&-1&0&0\\\ -1&0&2&-1&0&0\\\
0&-1&-1&2&-1&0\\\ 0&0&0&-1&2&-1\\\ 0&0&0&0&-1&2\end{pmatrix}$ (C.86)

and the inverse is

$(C^{E_{6}})^{-1}=\begin{pmatrix}\frac{4}{3}&1&\frac{5}{3}&2&\frac{4}{3}&\frac{2}{3}\\\
1&2&2&3&2&1\\\ \frac{5}{3}&2&\frac{10}{3}&4&\frac{8}{3}&\frac{4}{3}\\\
2&3&4&6&4&2\\\ \frac{4}{3}&2&\frac{8}{3}&4&\frac{10}{3}&\frac{5}{3}\\\
\frac{2}{3}&1&\frac{4}{3}&2&\frac{5}{3}&\frac{4}{3}\end{pmatrix}$ (C.87)

In the affine Cartan matrix the affine node ‘‘0’’ of $\widehat{E}_{6}$
connects to the node ‘‘2’’ of $E_{6}$. The Dynkin marks

$(a_{0},\dots,a_{6})=(1,1,2,2,3,2,1).$ (C.88)

####  C.3.4. $E_{7}$

The Cartan matrix of $E_{7}$ is

$C^{E_{7}}=\begin{pmatrix}2&0&-1&0&0&0&0\\\ 0&2&0&-1&0&0&0\\\
-1&0&2&-1&0&0&0\\\ 0&-1&-1&2&-1&0&0\\\ 0&0&0&-1&2&-1&0\\\ 0&0&0&0&-1&2&-1\\\
0&0&0&0&0&-1&2\end{pmatrix}$ (C.89)

and the inverse is

$(C^{E_{7}})^{-1}=\begin{pmatrix}2&2&3&4&3&2&1\\\
2&\frac{7}{2}&4&6&\frac{9}{2}&3&\frac{3}{2}\\\ 3&4&6&8&6&4&2\\\
4&6&8&12&9&6&3\\\ 3&\frac{9}{2}&6&9&\frac{15}{2}&5&\frac{5}{2}\\\
2&3&4&6&5&4&2\\\ 1&\frac{3}{2}&2&3&\frac{5}{2}&2&\frac{3}{2}\end{pmatrix}$
(C.90)

In the affine Cartan matrix the affine node ‘‘0’’ of $\widehat{E}_{7}$
connects to the node ‘‘1’’ of $E_{7}$. The Dynkin marks

$(a_{0},\dots,a_{7})=(1,2,2,3,4,3,2,1).$ (C.91)

####  C.3.5. $E_{8}$

The Cartan matrix of $E_{8}$ is

$C^{E_{8}}=\begin{pmatrix}2&0&-1&0&0&0&0&0\\\ 0&2&0&-1&0&0&0&0\\\
-1&0&2&-1&0&0&0&0\\\ 0&-1&-1&2&-1&0&0&0\\\ 0&0&0&-1&2&-1&0&0\\\
0&0&0&0&-1&2&-1&0\\\ 0&0&0&0&0&-1&2&-1\\\ 0&0&0&0&0&0&-1&2\end{pmatrix}$
(C.92)

and the inverse is

$(C^{E_{8}})^{-1}=\begin{pmatrix}4&5&7&10&8&6&4&2\\\ 5&8&10&15&12&9&6&3\\\
7&10&14&20&16&12&8&4\\\ 10&15&20&30&24&18&12&6\\\ 8&12&16&24&20&15&10&5\\\
6&9&12&18&15&12&8&4\\\ 4&6&8&12&10&8&6&3\\\ 2&3&4&6&5&4&3&2\end{pmatrix}$
(C.93)

In the affine Cartan matrix the node ‘‘0’’ of $\widehat{E}_{8}$ connects to
the node ‘‘8’’ of $E_{8}$. The Dynkin marks

$(a_{0},\dots,a_{8})=(1,3,2,4,6,5,4,3,2).$ (C.94)

###  C.4. Affine Weyl group

For the Class II theories the relevant reflection group turns out to be the
affine Weyl group $W({\widehat{\mathfrak{g}}})$. As a group, it is a semi-
direct product of the finite Weyl group $W({\mathfrak{g}})$ and the root
lattice ${\rm Q}$ of $\mathfrak{g}$ (recall that for $\mathfrak{g}$ the root
and the coroot lattices are identified):

$W({\widehat{\mathfrak{g}}})=W({\mathfrak{g}})\ltimes{\rm Q}$ (C.95)

We can view $W({\widehat{\mathfrak{g}}})$ as the group acting in
$\widehat{\mathfrak{h}}$, preserving the non-degenerate scalar product:
$(\cdot,\cdot)$ which extends the Killing form on $\mathfrak{h}$ by the
pairing between $K$ and ${\delta}^{\vee}$, as follows. For

$x={\tau}{\delta}^{\vee}+{\sigma}K+{\xi},\qquad{\tau},{\sigma}\in{\mathbb{C}},\
{\xi}\in\mathfrak{h}$ (C.96) $(x,x)=\langle{\xi},{\xi}\rangle+2{\tau}{\sigma}$
(C.97)

The group $W({\widehat{\mathfrak{g}}})$ is generated by simple reflections
$r_{i}$, $i=0,1,\ldots,r$. The action of $r_{i}$ on $\widehat{\mathfrak{h}}$
is given by:

$r_{i}:x\mapsto x-{\alpha}_{i}(x){\alpha}_{i}^{\vee}\,,\qquad i=0,\ldots,r$
(C.98)

for $x\in{\widehat{\mathfrak{h}}}$. Similarly, the action of $r_{i}$ on
$\widehat{\mathfrak{h}}^{*}$ is given by

$r_{i}:p\mapsto p-p({\alpha}_{i}^{\vee}){\alpha}_{i}\,,\qquad i=0,\ldots,r$
(C.99)

Note that $K$ is invariant under the reflections (C.98), while ${\delta}$ is
invariant under the reflections (C.99), cf. (C.63).

On the hyperplane

$H_{\tau}=\\{\,x\,|\,{\delta}(x)={\tau}\,\\}\subset\widehat{\mathfrak{h}}\,$
(C.100)

the group $W({\widehat{\mathfrak{g}}})$ acts by orthogonal transformations,
generated by the ordinary reflections, and by translations. In the
decomposition (C.96) we have:

$r_{0}({\tau}{\delta}^{\vee}+{\sigma}K+{\xi})={\tau}{\delta}^{\vee}+{\sigma}K+{\xi}-({\tau}-{\theta}({\xi}))(K-{\theta}^{\vee})$
(C.101)

where we introduced the highest root $\theta\in\mathfrak{h}^{*}$, and the
highest coroot $\theta^{\vee}\in\mathfrak{h}$:

$n\theta={\delta}-{\alpha}_{0}=\sum_{i=1}^{r}a_{i}{\alpha}_{i},\qquad\theta^{\vee}=K-{\alpha}_{0}^{\vee}=\sum_{i=1}^{r}a_{i}{\alpha}_{i}^{\vee}$
(C.102)

which obey:
${\theta}({\theta}^{\vee})=\langle{\theta}^{\vee},{\theta}^{\vee}\rangle=2$,
and also ${\theta}({\xi})=\langle\xi,{\theta}^{\vee}\rangle$ for any
$\xi\in\mathfrak{h}$.

Now, to make the translational part of the $W({\widehat{\mathfrak{g}}})$
action explicit, let us perform an additional reflection at $\theta$:

$\displaystyle
r_{\theta}r_{0}(x)=r_{0}(x)-{\theta}({\xi}-({\tau}-{\theta}({\xi}))(-{\theta}^{\vee})){\theta}^{\vee}=$
(C.103)
$\displaystyle\qquad{\tau}{\delta}^{\vee}+({\sigma}-{\tau}+{\theta}({\xi}))K+{\xi}-{\tau}{\theta}^{\vee}$

Finally, the general element of $W({\widehat{\mathfrak{g}}})$ can be
represented as a pair $(w,{\beta})$, where $w\in W({\mathfrak{g}})$ and
$\beta\in{\rm Q}$, with the composition rule:

$(w_{1},{\beta}_{1})\cdot(w_{2},{\beta}_{2})=(w_{1}\cdot
w_{2},{\beta}_{1}+{\beta}_{2}^{w_{1}})$

The element ${\widehat{w}}=(w,{\beta})$ acts on $x\in H_{\tau}$ as follows:

$\displaystyle({\tau}{\delta}^{\vee}+{\sigma}K+{\xi})^{\widehat{w}}=$ (C.104)