ITEP-TH-14/20

IITP-TH-10/20

Generalizations of parabolic Higgs bundles, real structures

and integrability
A. Levin${}^{\,\natural\,\,\flat}$ M. Olshanetsky${}^{\,\flat\,\S}$ A.
Zotov${}^{\,\diamondsuit\,\flat\,\natural}$
♭ \- ITEP of NRC ”Kurchatov Institute”,
B. Cheremushkinskaya, 25, Moscow, 117259, Russia
§ \- Institute for Information Transmission Problems RAS (Kharkevich
Institute),
Bolshoy Karetny per. 19, Moscow, 127994, Russia
♮ \- National Research University Higher School of Economics, Russian
Federation,
NRU HSE, Usacheva str. 6, Moscow, 119048, Russia
♢ \- Steklov Mathematical Institute of Russian Academy of Sciences,
Gubkina str. 8, Moscow, 119991, Russia

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves
with marked points. They are endowed with additional structures, which replace
the parabolic structures at marked points in the parabolic Higgs bundles. The
latter means that the coadjoint orbits are attached to the marked points. The
moduli spaces of parabolic Higgs bundles are the phase spaces of complex
completely integrable systems. In our case the coadjoint orbits are replaced
by the cotangent bundles over some special symmetric spaces in such a way that
the moduli space of the modified Higgs bundles are still phase spaces of
complex completely integrable systems. We show that the moduli space of the
parabolic Higgs bundles is the symplectic quotient of the moduli space of the
quasi-antisymmetric Higgs bundle with respect to the action of product of
Cartan subgroups. Also, by changing the symmetric spaces we introduce quasi-
compact and quasi-normal Higgs bundles. Then the fixed point sets of real
involutions acting on their moduli spaces are the phase spaces of real
completely integrable systems. Several examples are given including integrable
extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic Calogero-
Moser system and the rational ${\rm SL}(2)$ Gaudin system together with its
real reductions.

###### Contents

  1. 1 Introduction and summary
  2. 2 General construction
    1. 2.1 Vector bundles
    2. 2.2 Higgs bundles
      1. 2.2.1  $\alpha$-model
      2. 2.2.2 $\beta$-model
      3. 2.2.3 Hierarchy of moduli spaces
    3. 2.3 Integrals of motion
  3. 3 Real structures
    1. 3.1 From Type I to Type IV
    2. 3.2 From Type II to Type III
    3. 3.3 Mixed systems
  4. 4 Double coset construction
    1. 4.1 Affine flags and affine symmetric spaces
    2. 4.2 Moduli space ${\rm Bun}(\Sigma_{g,n},G^{\mathbb{C}})$
    3. 4.3 Moduli space of Higgs bundles $\mathcal{M}(\Sigma_{g,n},G^{\mathbb{C}})$
    4. 4.4 Real structures
  5. 5 Examples
    1. 5.1 Elliptic case
      1. 5.1.1 Extension of the Euler-Arnold top
      2. 5.1.2 Real reductions
    2. 5.2 Extension of Calogero-Moser system
    3. 5.3 Rational case
  6. 6 Conclusion
  7. A Two equivalent constructions of symplectic quotients
  8. B Simple Lie groups: notations and decompositions
  9. C Cotangent bundles and coadjoint orbits

##  1 Introduction and summary

Let $G^{\mathbb{C}}$ be a simple complex Lie group, $\Sigma_{g}$ is a Riemann
curve of genus $g$ with a canonical class $\kappa$ and $E$ is a vector
$G^{\mathbb{C}}$-bundle over $\Sigma_{g}$. The Higgs bundle is pair
$(d_{\bar{A}},\Phi)$, where $d_{\bar{A}}$ is a holomorphic structure on the
bundle $E$, and $\Phi$ is the Higgs field – a section of the bundle
$End(E)\otimes\kappa$. The Higgs bundles are a symplectic spaces. The
symplectic quotient with respect to the action of gauge group
$\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})=\\{C^{\infty}(\Sigma_{g})\to
G^{\mathbb{C}}\\}$ is the moduli space of Higgs bundles
$\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})$.

If the curve is smooth then the moduli space of the Higgs bundles are the
phase spaces of classical complex integrable systems called the Hitchin
systems [16]. If at some points $(x_{1},\ldots,x_{n})$ of $\Sigma_{g,n}$ the
gauge group is reduced to the Borel subgroup, or more generally to parabolic
subgroups [22, 32] then the corresponding Higgs bundle is called parabolic.
This construction is equivalent to attachment to the marked points the
coadjoint orbits 111In Appendix we prove this equivalence in general setting.
In this paper we use both constructions.. The moduli space
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ of parabolic Higgs bundles is
also a phase spaces of an integrable systems [3, 26]. This approach to
integrable systems allows one to construct the Lax operators, integrals of
motion, action-angle variables, classical r-matrices and so on. Many
integrable systems including the Calogero-Moser systems, the Euler-Arnold
elliptic tops and the Gaudin type systems can be described in this way.

The moduli space of the Higgs bundles over smooth curves is a symplectic
quotient of the moduli space of parabolic bundles with respect to the action
of some finite-dimensional group $\tilde{G}_{0}$

$\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})=\tilde{G}_{0}\setminus\setminus\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})\,,\qquad\tilde{G}_{0}=\prod_{a}G^{\mathbb{C}}\,,$
(1.1)

see (2.31).

In this paper we modify the parabolic Higgs bundles in such a way that their
moduli spaces also become the phase spaces of integrable systems (real or
complex). Namely, we replace the Borel subgroups $B^{\mathbb{C}}$ with some
special reductive subgroups. If these subgroups are some special complex group
$U^{\mathbb{C}}\subset G^{\mathbb{C}}$ (for example, for $G^{\mathbb{C}}={\rm
SL}(N,{\mathbb{C}})$, $U^{\mathbb{C}}=$SO$(N,{\mathbb{C}})$) then we come to
the complex integrable systems as before.

Another possibility is to choose this subgroup to be the maximal compact
subgroup or the normal subgroup of $G^{\mathbb{C}}$. The resulting moduli
spaces are real varieties. To come to integrable system we use involutions
defined on the moduli space of Higgs bundles proposed in [1]. The fixed point
set of the involutions are the phase spaces of classical real completely
integrable systems.

First examples of such system were proposed in [9]. It is the Calogero-
Sutherland system with the two types of spin variables. In our previous work
[21] we defined these systems using this new type of the Higgs bundles. In
this paper we develop a general approach to this Higgs bundles and consider
new examples.

Outline of the paper

Our goal is to investigate the moduli spaces of these modifications of the
parabolic Higgs bundles and to construct completely integrable systems. For
this purpose we replace the co-adjoint orbits attached to the marked points in
the parabolic Higgs bundles by symplectic manifolds in such a way that the
corresponding moduli spaces are still the phase spaces of integrable systems.
These symplectic spaces are cotangent bundles to some special symmetric spaces
defined as follows.

Consider a maximal compact subgroup $C$ of $G^{\mathbb{C}}$ (and its normal
form $G^{\mathbb{R}}$ [15]). For example, for $G^{\mathbb{C}}={\rm
SL}(N,{\mathbb{C}})$ the maximal compact subgroup is SU$(N)$ and
$G^{\mathbb{R}}$ is SL$(N,{\mathbb{R}})$. The subgroups $C$ and
$G^{\mathbb{R}}$ are fixed point sets of the commuting anti-holomorphic
involutive automorphisms $\rho$ and $\sigma$ of $G^{\mathbb{C}}$

$\rho(C)=C\,,~{}~{}\sigma(G^{\mathbb{R}})=G^{\mathbb{R}}\,.$

Let $U$ be a maximal compact subgroup of $G^{\mathbb{R}}$:

$U=\\{g\in G^{\mathbb{R}}\,|\,\rho(g)=g\\}\,.$

Or, equivalently,

$U=\\{g\in C\,|\,\sigma(g)=g\\}\,.$

For the group SL$(N,{\mathbb{C}})$ the subgroup $U$ is SO$(N,{\mathbb{R}})$.
Let $U^{\mathbb{C}}$ be the complexification of the real group $U$. It is the
fixed point set in $G^{\mathbb{C}}$ of the involutive automorphism
$\theta=\sigma\circ\rho$

$U^{\mathbb{C}}=\\{g\in G^{\mathbb{C}}\,|\,\theta(g)=g\\}\,.$

For $G^{\mathbb{C}}={\rm SL}(N,{\mathbb{C}})$ the corresponding subgroup is
$U^{\mathbb{C}}=$SO$(N,{\mathbb{C}})$.

Define five types of the coset spaces:

$\begin{array}[]{lll}{\bf I.}&~{}~{}\mathcal{X}_{I}=C\backslash
G^{\mathbb{C}}&~{}~{}\rho\,(C)=C\\\ {\bf
II.}&~{}~{}\mathcal{X}_{II}=G^{\mathbb{R}}\backslash
G^{\mathbb{C}}&~{}~{}\sigma\,(G^{\mathbb{R}})=G^{\mathbb{R}}\\\ {\bf
III.}&\mathcal{X}_{III}=U\backslash C&~{}~{}\sigma\,(U)=U\\\ {\bf
IV.}&~{}~{}\mathcal{X}_{IV}=U\backslash G^{\mathbb{R}}&~{}~{}\rho\,(U)=U\\\
{\bf V.}&~{}~{}\mathcal{X}_{V}=U^{\mathbb{C}}\backslash
G^{\mathbb{C}}&~{}~{}\theta\,(U^{\mathbb{C}})=U^{\mathbb{C}}\end{array}$

Table 1

They are symmetric spaces [4, 15]. It means that their stationary subgroups
are the fixed point sets of the corresponding involutive automorphisms. The
symmetric spaces of types I, III and IV are Riemannian manifolds, while the
spaces of type II and V are pseudo-riemannian manifolds. The action of
involutions leads to the following interrelations between symmetric spaces:

$\displaystyle\mathcal{X}_{IV}=U\backslash
G^{\mathbb{R}}\stackrel{{\scriptstyle\sigma}}{{\hookrightarrow}}\mathcal{X}_{I}=C\backslash
G^{\mathbb{C}}\,,$ (1.2a) $\displaystyle\mathcal{X}_{IV}=U\backslash
G^{\mathbb{R}}\stackrel{{\scriptstyle\sigma}}{{\hookrightarrow}}\mathcal{X}_{V}=U^{C}\backslash
G^{\mathbb{C}}\,,$ (1.2b)

where the arrows mean the embedding as the fixed points set. Similarly,

$\displaystyle\mathcal{X}_{III}=U\backslash
C\stackrel{{\scriptstyle\rho}}{{\hookrightarrow}}\mathcal{X}_{II}=G^{{\mathbb{R}}}\backslash
G^{\mathbb{C}}\,,$ (1.3a) $\displaystyle\mathcal{X}_{III}=U\backslash
C\stackrel{{\scriptstyle\rho}}{{\hookrightarrow}}\mathcal{X}_{V}=U^{C}\backslash
G^{\mathbb{C}}\,.$ (1.3b)

Let $\mathcal{D}=(x_{1},\ldots,x_{n})$ be the set of the marked points on the
curve $\Sigma_{g,n}$. Recall that _the parabolic structure_ of the Higgs
bundles over $\Sigma_{g,n}=\Sigma_{g}\setminus\mathcal{D}$ means that we
attach to the marked points coadjoint $G^{\mathbb{C}}$-orbits
$\mathcal{O}_{a}$ $(a=1,\ldots,n)$. As we mentioned above the moduli space of
the parabolic Higgs bundles $\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$
are phase spaces of completely integrable system. The Poisson commuting
independent integrals of motion are constructed by means of the generalized
Beltrami differentials on $\Sigma_{g,n}$. The number of integrals is
independent of structures attached to the marked points. It turns out that it
is equal to
$\frac{1}{2}\dim\,(\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$. This
provides the Liouville integrability of the systems related to the parabolic
Higgs bundles.

There is another way to define the parabolic Higgs bundles. Instead of
attaching the coadjoint orbits to the marked points (the symplectic charges)
one can reduce the gauge group (in an appropriate way). For the parabolic
bundles the gauge group takes values in Borel subgroups at the marked points.
We call this construction the $\beta$-model, while the first construction is
called the $\alpha$-model 222 These two descriptions of charges are known in
gauge theories. It is ”electric”, or ”magnetic” points of view. Unfortunately,
we did not find the proof of their equivalence in the literature. For this
reason we give it in Appendix A..

We define the following modifications of the parabolic Higgs bundles that
eventually leads to the integrability.

The quasi-antisymmetric structure of the Higgs bundles means that we attach to
the marked points the cotangent bundles
$T^{*}(\mathcal{X}_{V,a})=T^{*}(U^{{\mathbb{C}}}_{a}\backslash
G^{\mathbb{C}})$. We prove that the moduli spaces
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ of the quasi-antisymmetric
Higgs bundles are the phase spaces of complex completely integrable systems.
The moduli space of the parabolic Higgs bundles
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ is a symplectic quotient of
the moduli space the quasi-antisymmetric Higgs bundles
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ with respect to the action of a
finite-dimensional group (see $\mathcal{H}$ in (2.53)):

$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{H}\setminus\\!\setminus\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(1.4)

The latter means that the systems of type V are integrable extensions of the
parabolic Hitchin systems. The coordinates on
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ play the role of collective
coordinates on the phase space $\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$.

The real integrable systems arise in the following way. Assume that
$\Sigma_{g,n}$ admits an anti-holomorphic involution
$\jmath\,:\,\Sigma_{g,n}\to\Sigma_{g,n}$. In a neighborhood of fixed points of
this action one can find a local coordinate $z$ such that the involution
$\jmath$ is given by the complex conjugation $\jmath(z)=\bar{z}$ [11]. The
fixed point set can be seen as a union $\Sigma^{0}=\cup S^{1}_{a}$ of copies
of the unit circle $S^{1}$ embedded in $\Sigma_{g,n}$. Assume that the set
$\mathcal{D}$ of the marked points is invariant with respect to the involution
$\jmath(\mathcal{D})=\mathcal{D}$. Following [1] define two involutions on the
moduli space of Higgs bundles $\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})$ 333In
[1] they are denoted as $\iota_{2}$ and $\iota_{3}$.

$\iota^{\rho}(d_{\bar{A}}),\Phi)=(\jmath^{*}\rho(d_{\bar{A}}),-\jmath^{*}\rho(\Phi))\,,~{}~{}\,\,\iota^{\sigma}(d_{\bar{A}}),\Phi)=(\jmath^{*}\sigma(d_{\bar{A}}),\jmath^{*}\sigma(\Phi))\,.$
(1.5)

The fixed point set of $\iota^{\sigma}$ is the Higgs $G^{\mathbb{R}}$-bundle
over $\Sigma^{0}$ and the fixed point set of $\iota^{\rho}$ is the Higgs
$C$-bundle over $\Sigma^{0}$. As an intermediate step we construct two types
of Higgs bundles using the cotangent bundles to symmetric spaces of type I or
type II.

##### Type I.

Attach to the marked points the cotangent bundles
$T^{*}(\mathcal{X}_{I,a})=T^{*}(C_{a}\setminus G^{\mathbb{C}})$. We call these
Higgs bundles quasi-compact, because at the marked points the gauge group of
the bundle is reduced to the maximal compact subgroups $C_{a}$. The moduli
space of the quasi-compact Higgs bundles
$\mathcal{M}_{I}(\Sigma_{g,n},G^{\mathbb{C}})$ is a real space. Its dimension
exceeds the number of integrals of motion. To come to integrable systems we
use the involution $\iota^{\sigma}$ (1.5) acting on
$\mathcal{M}_{I}(\Sigma_{g,n},G^{\mathbb{C}})$. It follows from (1.2a) that
the fixed set of involution $\iota^{\sigma}$ is the moduli space
$\mathcal{M}_{IV}(\Sigma^{0}_{n},G^{\mathbb{R}})$ of the $G^{\mathbb{R}}$
Higgs bundles over $\Sigma^{0}$, where the cotangent bundles attached to the
marked points are $T^{*}(\mathcal{X}_{IV,a})=T^{*}(U_{a}\setminus
G^{\mathbb{R}})$. Acting by the same involutions on the moduli space of the
antisymmetric Higgs bundles $\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ and
using (1.2b) we come to the same fixed point set
$\mathcal{M}_{IV}(\Sigma_{g,n},G^{\mathbb{C}})$ in
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$. Thus we have the embeddings

$\mathcal{M}_{IV}(\Sigma^{0}_{n},G^{\mathbb{R}})\stackrel{{\scriptstyle\iota^{\sigma}}}{{\hookrightarrow}}\mathcal{M}_{I}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}\mathcal{M}_{IV}(\Sigma^{0}_{n},G^{\mathbb{R}})\stackrel{{\scriptstyle\iota^{\sigma}}}{{\hookrightarrow}}\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(1.6)

We prove that moduli space $\mathcal{M}_{IV}$ is a phase space of a real
completely integrable system.

#####  Type II.

Replace in the above construction the cotangent bundles
$T^{*}(\mathcal{X}_{I,a})$ with the cotangent bundles
$T^{*}(\mathcal{X}_{II,a})=T^{*}(G^{\mathbb{R}}_{a}\setminus G^{\mathbb{C}})$.
We call these Higgs bundles _quasi-normal_ since $G^{\mathbb{R}}_{a}$ is the
normal subgroup of $G^{\mathbb{C}}$. The moduli space of the quasi-normal
Higgs bundles $\mathcal{M}_{II}(\Sigma_{g,n},G^{\mathbb{C}})$ is too large to
be a phase space of completely integrable system. The integrable systems arise
on the fixed point set of the involution $\iota^{\rho}$ (1.5) acting on
$\mathcal{M}_{II}(\Sigma_{g,n},G^{\mathbb{C}})$ and on
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$. Using (1.3a) and (1.3b) we
come to the moduli space $\mathcal{M}_{III}(\Sigma^{0}_{n},C)$ of the Higgs
$C$-bundles over $\Sigma^{0}$ embedded in the moduli spaces $\mathcal{M}_{II}$
and $\mathcal{M}_{V}$

$\mathcal{M}_{III}(\Sigma^{0}_{n},C)\stackrel{{\scriptstyle\iota^{\rho}}}{{\hookrightarrow}}\mathcal{M}_{II}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}\mathcal{M}_{III}(\Sigma^{0}_{n},C)\stackrel{{\scriptstyle\iota^{\rho}}}{{\hookrightarrow}}\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(1.7)

$\mathcal{M}_{III}$ is also a phase space of a real completely integrable
system.

In the absence of the marked points both involutions and their fixed point
sets as the phase space of real integrable systems were considered in [1, 2]

Schematically, we have the following interrelations between the moduli spaces:

---
$\textstyle{\mathcal{M}_{II}}$$\textstyle{\mathcal{M}_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{H}\setminus\\!\setminus}$$\textstyle{\mathcal{M}_{I}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{III}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\rho}}$$\scriptstyle{\iota^{\rho}}$$\textstyle{\mathcal{M}_{par}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{IV}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\sigma}}$$\scriptstyle{\iota^{\sigma}}$
(1.8)

Here the passage from $\mathcal{M}_{V}$ to $\mathcal{M}_{par}$ means the
symplectic reduction (1.4), while the other arrows denote the embeddings as
the fixed point sets. While the moduli spaces $\mathcal{M}_{I}$ and
$\mathcal{M}_{II}$ play a supporting role, the remaining moduli spaces are the
phase spaces of integrable systems.

There is a universal generalization of the parabolic Higgs bundles, which we
call type 0 Higgs bundles. For these bundles we attach to the marked points
the cotangent bundles $T^{*}G^{\mathbb{C}}$. The moduli space
$\mathcal{M}_{0}(\Sigma_{g,n},G^{\mathbb{C}})$ of the type 0 Higgs bundles
allows one to define the moduli spaces of the types I,II and V Higgs bundles
by symplectic reductions with respect to action of the groups $G_{I}$,
$\,G_{II}$ and $G_{V}$ (2.29):

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{II}\setminus\\!\setminus}$$\scriptstyle{G_{V}\setminus\\!\setminus}$$\scriptstyle{G_{I}\setminus\\!\setminus}$$\textstyle{\mathcal{M}_{II}}$$\textstyle{\mathcal{M}_{V}}$$\textstyle{\mathcal{M}_{I}}$

Similarly to the parabolic bundles the Higgs bundles with the cotangent
bundles attached to the marked points after symplectic reductions become the
Higgs bundles over the smooth curves (see (1.1))

$\mathcal{M}_{J}(\Sigma_{g,n},G^{\mathbb{C}})\stackrel{{\scriptstyle\tilde{G}_{0}\setminus\\!\setminus}}{{\longrightarrow}}\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})\,,~{}~{}(J=0,I,II,V)\,,$
(1.9)

where $\tilde{G}_{0}$ is from (1.1). It is possible to attach to the marked
points the cotangent bundles to the Euclidean symmetric spaces of zero
curvature. The corresponding systems are analogues of rational integrable
systems, while the systems related to $\mathcal{M}_{III}$ and
$\mathcal{M}_{IV}$ are analogues of the hyperbolic and trigonometric systems.
We don’t consider the rational case here.

##### Organization of the paper.

In Section 2 we construct the Higgs bundles over curves with marked points of
types $J=0,I,II,V$ based on the $\alpha$ and $\beta$ models. For the bundles
of type V we construct the integrals of motion and prove the complete
integrability of the systems.

In Section 3 we consider the real involutions for the systems of type V. The
fixed point sets of the involutions are the real integrable systems of types
III and IV. We consider the mixing of these systems with the systems of the
parabolic type singularities.

In Section 4 we describe the Higgs bundles using the double coset construction
of the moduli space of holomorphic bundles. This construction allows one to
define topologically non-trivial Higgs bundles described by their
characteristic classes. The construction of the parabolic bundles is related
to the affine flag varieties [29]. For the other types of the Higgs bundles
the similar role is played by the affine symmetric spaces.

In Section 5 we consider three examples of quasi-antisymmetric and real Higgs
bundles. First we consider a system on $\Sigma_{0,n}$ $\,({\mathbb{C}}P^{1}$
with $n$ marked points). It is an extension of the rational Gaudin system
corresponding to the symplectic reduction (1.4). A special configuration of
the marked points has two involutions corresponding to the real integrable
systems of type III and IV. Next example is the system of type V on
$\Sigma_{1,1}$ (an elliptic curve with one marked point) and group
$G^{\mathbb{C}}=$SL$(2,{\mathbb{C}})$. The trivial bundles lead to the
generalization of the two-body elliptic Calogero-Moser system. For the quasi-
antisymmetric Higgs bundles the coupling constant acquires internal degree of
freedom with explicit hamiltonian dynamics. For non-trivial bundles the moduli
space of the quasi-antisymmetric Higgs bundles $\mathcal{M}_{V}$ is isomorphic
to the cotangent bundle $T^{*}($SL$(2,{\mathbb{C}})/{\mathbb{C}}^{*})$,
($\dim\,(\mathcal{M}_{V})=4)$. We construct the Lax operator and two commuting
integrals of motion and establish the connection of this model to the ${\rm
SL}(2,{\mathbb{C}})$ Euler-Arnold top coming from the diagram (1.4).

In the concluding Section we discuss interpretation of the presented results
in terms of the twisted $N=4$, $d=4$ SUSY Yang-Mills theory [20, 12, 13]. In
this context the quasi-antisymmetric Higgs bundles are related to the surface
operators that apparently break $N=4$ supersymmetry to $N=2$.

In the Appendix we prove the equivalence of $\alpha$ and $\beta$ models
descriptions of the Higgs bundles over multipoint curves. Then we provide the
necessary information regarding symmetric spaces and flag varieties. Finally,
we describe the symplectic structures on the coadjoint orbits and on the
cotangent bundles to symmetric spaces by the symplectic reduction from the
cotangent bundle $T^{*}G^{\mathbb{C}}$ and define the Darboux coordinates on
cotangent bundles to symmetric spaces.

##### Acknowledgements

The work of M. Olshanetsky was funded by the Russian Science Foundation (Grant
No. 16-12-10344). The work of A. Levin was partially supported partially
supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag.
14.641.31.0001. The research of A. Zotov was supported by the HSE University
Basic Research Program and by the Young Russian Mathematics award.

##  2 General construction

###  2.1 Vector bundles

Let $G^{\mathbb{C}}$ be a complex simple Lie group, $\Sigma_{g}$ is a Riemann
curve of genus $g$ and $\mathcal{P}(G^{\mathbb{C}})$ is the principle
$G^{\mathbb{C}}$ \- bundle over $\Sigma_{g}$. Define associated vector bundle
$E(G^{\mathbb{C}})=\mathcal{P}\times_{G^{\mathbb{C}}}V$, where $V$ is a
$G^{\mathbb{C}}$ module. Let $d_{\bar{A}}$ be a holomorphic structure on the
bundle $E(G^{\mathbb{C}})$. The moduli space of the holomorphic bundles
$Bun(\Sigma_{g},G^{\mathbb{C}})$ is the quotient

$Bun(\Sigma_{g},G^{\mathbb{C}})=\\{d_{\bar{A}}\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,,$
(2.1)

where

$\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})=\\{C^{\infty}(\Sigma_{g})\to
G^{\mathbb{C}}\\}$ (2.2)

is the gauge group. It acts on $d_{\bar{A}}$ by the affine transformations

$d_{\bar{A}}\to fd_{\bar{A}}f^{-1}\,.$ (2.3)

Here we consider the topologically trivial bundles. The non-trivial bundles
will be introduced in Section 5.

The dimension of $Bun(\Sigma_{g},G^{\mathbb{C}})$ can be extracted from the
Riemann-Roch theorem. For stable bundles we have

$\dim\,Bun(\Sigma_{g},G^{\mathbb{C}})=(g-1)\dim\,G^{\mathbb{C}}\,.$ (2.4)

Consider a multipoint curve $\Sigma_{g,n}$ with $n$ marked points
$\mathcal{D}=(x_{1},\ldots,x_{n})$. In the $\alpha$-model we attach to them
different data, which are not necessarily holomorphic. The known example is
the holomorphic bundle with the _parabolic structure_. In this case we attach
the flag varieties $Fl^{\mathbb{C}}_{a}=B_{a}^{\mathbb{C}}\setminus
G^{\mathbb{C}}$, where $B^{\mathbb{C}}_{a}$ are Borel subgroups of
$G^{\mathbb{C}}$. The moduli space of holomorphic bundles with parabolic
structure is defined as the quotient space

$Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\\}\,,\,\prod
Fl^{\mathbb{C}}_{a})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$

From (2.4) and (B.27) we have

$\dim\,Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1)\dim\,G^{\mathbb{C}}+n\dim\,Fl^{\mathbb{C}}=(g-1)\dim\,G^{\mathbb{C}}+n\sum_{j=1}^{l}(d_{j}-1)\,,$
(2.5)

where $d_{j}$ are the orders of the invariants of the algebra
${\mathfrak{g}}^{\mathbb{C}}$ (B.17) and $l={\rm
rank}({\mathfrak{g}}^{\mathbb{C}})$.

It follows from Proposition A.1 that $Bun_{par}$ can be equivalently defined
in the $\beta$-model by reducing the gauge group to the subgroup

$\mathcal{G}_{par}(\Sigma_{g,n},G^{\mathbb{C}}):=(\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})|_{z=x_{a}}=B^{\mathbb{C}}_{a})\,.$
(2.6)

Then

$Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})=\\{d_{\bar{A}}\\}/\mathcal{G}_{par}(\Sigma_{g,n},G^{\mathbb{C}})\,,$
(2.7)

where (in the notations of the Proposition A.1) we identify
$X=\\{d_{\bar{A}}\\}$, $G=\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})$ and
$H=\mathcal{G}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ so that $H\setminus
G=\prod_{a}{\rm Fl}_{a}^{\mathbb{C}}$. Let

$G_{0}=\prod_{a}G^{\mathbb{C}}\,.$ (2.8)

Since the group $G^{\mathbb{C}}$ acts transitively on the flag variety we find
that

$Bun(\Sigma_{g},G^{\mathbb{C}})=G_{0}\setminus
Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})\,,$ (2.9)

where the leave of the of the projection
$\pi\,:\,Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})\to
Bun(\Sigma_{g},G^{\mathbb{C}})$ is $\prod_{a}Fl_{a}$.

We are going to modify this construction in the following ways. Replace the
flag varieties with four different types of quotient spaces.

##### Type 0.

Consider two isomorphic representations $V$ and $V_{a}$ of $G^{\mathbb{C}}$,
where $V_{a}$ is the fiber over the marked point $x_{a}$ and $V$ is fixed.
Denote by $Isom(V_{a},V)$ the space of $G^{\mathbb{C}}$-isomorphisms between
$V_{a}$ and $V$. This space is a principal homogeneous space over
$G^{\mathbb{C}}$.

For curves with marked points we replace the moduli space
$Bun(\Sigma_{g},G^{\mathbb{C}})$ by the moduli space of holomorphic bundles
with the trivializations $g_{a}:V_{a}\to V$ of fibers at the marked points

$Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})=\Big{\\{}d_{\bar{A}}\,,\,\prod_{a=1}^{n}Isom(V_{a},V)\Big{\\}}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$
(2.10)

Then

$\dim\,Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1+n)\dim\,G^{\mathbb{C}}\,.$
(2.11)

In the $\beta$-model the gauge group is

$\mathcal{G}_{0}(\Sigma_{g},G^{\mathbb{C}})=\\{f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,|\,f(x_{a})=Id\,,~{}(a=1,\ldots,n)\\}\,,$
(2.12)

and $Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})$ is defined as

$Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})=\\{d_{\bar{A}}\\}/\mathcal{G}_{0}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(2.13)

As above we have the projection
$\pi\,:\,Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\to
Bun(\Sigma_{g},G^{\mathbb{C}})$ and

$Bun(\Sigma_{g},G^{\mathbb{C}})=G_{0}\setminus
Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}G_{0}=\prod_{a}G^{\mathbb{C}}\,.$
(2.14)

##### Type I.

Let $C$ be the maximal compact subgroup of $G^{\mathbb{C}}$, defined by the
involutive automorphism $\rho$ (B.9). In the $\alpha$-model we attach to the
marked points the symmetric spaces $\mathcal{X}_{I,a}=C_{a}\setminus
G^{\mathbb{C}}$. By definition, the moduli space of bundles with _quasi-
compact structure_ is the quotient space

$Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\\}\,,\,\prod\mathcal{X}_{I,a})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$
(2.15)

Equivalently, in the $\beta$-model

$Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\\}/\mathcal{G}_{I}(\Sigma_{g,n},G^{\mathbb{C}})\,,$

where

$\mathcal{G}_{I}(\Sigma_{g,n},G^{\mathbb{C}})=\\{f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,|\,f(x_{a})\in
C_{a})\,,~{}(a=1,\ldots,n)\\}\,.$ (2.16)

Note that the holomorphic structure is broken at the marked points, and
$Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})$ is a real space. From (B.21) we find

$\dim_{\mathbb{R}}\,Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+n\dim_{\mathbb{R}}\,X_{I}=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+n\sum_{j=1}^{l}(2d_{j}-1)\,.$
(2.17)

Since $\mathcal{X}_{I}=C\setminus G^{\mathbb{C}}$ and the left action commutes
with the right action of $\mathcal{G}_{I}(\Sigma_{g,n},G^{\mathbb{C}})$ we
have

$Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})=G_{I}\setminus
Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}G_{I}=\prod_{a}C_{a}\,.$ (2.18)

The group $G^{\mathbb{C}}$ acts transitively on the symmetric space
$\mathcal{X}_{I}$. Thereby, similarly to (2.14)

$Bun(\Sigma_{g},G^{\mathbb{C}})=G_{0}\setminus
Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})\,,$ (2.19)

where $G_{0}$ is (2.8).

##### Type II.

Let $G^{\mathbb{R}}$ be the normal subgroup of $G^{\mathbb{C}}$ defined by the
involutive automorphism $\sigma$ (B.10). Attach to the marked points the
symmetric spaces $\mathcal{X}_{II,a}=G^{\mathbb{R}}_{a}\setminus
G^{\mathbb{C}}$. The moduli space of bundles with _the quasi-normal structure_
is the quotient space

$Bun_{II}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\\}\,,\,\prod\mathcal{X}_{IIa})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,,$
(2.20)

or equivalently,

$Bun_{II}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\\}/\mathcal{G}_{II}(\Sigma_{g,n},G^{\mathbb{C}})\\}\,,$

where

$\mathcal{G}_{II}(\Sigma_{g,n},G^{\mathbb{C}})=\\{f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,|\,f(x_{a})\in
G^{\mathbb{R}}_{a}\\}\,.$ (2.21)

Again, $Bun_{II}(\Sigma_{g,n},G^{\mathbb{C}})$ is a real space and its
dimension coincides with $\dim\,Bun_{I}$:

$\dim_{\mathbb{R}}\,Bun_{II}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+n\dim_{\mathbb{R}}\,X_{II}=\dim_{\mathbb{R}}\,Bun_{I}(\Sigma_{g,n},G^{\mathbb{C}})\,,$
(2.22)

(see (B.20)). In addition, as in (2.18) we have

$Bun_{II}(\Sigma_{g,n}G^{\mathbb{C}})=G_{II}\setminus
Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}G_{II}=\prod_{a}G^{\mathbb{R}}_{a}\,,$
(2.23)

and

$Bun(\Sigma_{g},G^{\mathbb{C}})=G_{0}\setminus
Bun_{II}(\Sigma_{g,n},G^{\mathbb{C}})\,.$ (2.24)

##### Type V.

Let $U^{\mathbb{C}}$ be the complex subgroup of $G^{\mathbb{C}}$ defined by
the involutive automorphism $\theta=\sigma\rho$. Attach to the marked points
the symmetric spaces $\mathcal{X}_{V,a}=U^{\mathbb{C}}_{a}\setminus
G^{\mathbb{C}}$. The moduli space of bundles with _quasi-antisymmetric
structure_ is the quotient space

$Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})=\\{(d_{\bar{A}}\,,\,\prod\mathcal{X}_{Va})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,,$

or equivalently,

$Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})=\\{d_{\bar{A}}\\}/\mathcal{G}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,,$

where

$\mathcal{G}_{V}(\Sigma_{g,n},G^{\mathbb{C}})=\\{f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,|\,f(x_{a})\in
U^{\mathbb{C}}_{a}\\}\,.$ (2.25)

The space $Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ is complex. Its dimension is
equal to

$\dim_{\mathbb{C}}\,Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1)\dim_{\mathbb{C}}\,G^{\mathbb{C}}+n\dim_{\mathbb{C}}\,\mathcal{X}_{V}=$
(2.26) $=(g-1)\dim_{\mathbb{C}}\,G^{\mathbb{C}}+n\sum_{j=1}^{l}d_{j}\,,$

(see (B.25)). Finally,

$Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})=G_{V}\setminus
Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\,,~{}~{}G_{V}=\prod_{a}U^{\mathbb{C}}_{a}$
(2.27)

and

$Bun(\Sigma_{g},G^{\mathbb{C}})=G_{0}\setminus
Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,,$ (2.28)

where $G_{0}$ is (2.8).

We summarize the interrelations between the moduli spaces of the bundles. Let
$\mathcal{G}_{0}$, $\mathcal{G}_{J}$ are the gauge groups defined above and

$G_{J}=\prod_{a}K_{a,J}\,,~{}~{}K_{a,J}=\left\\{\begin{array}[]{cc}Id&J=0\,,\\\
C_{a}&J=I\,,\\\ G_{a}^{\mathbb{R}}&J=II\,,\\\
U_{a}^{\mathbb{C}}&J=V\,.\end{array}\right.$ (2.29)

Then

$\mathcal{G}_{J}(\Sigma_{g,n},G^{\mathbb{C}})=\\{f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,|\,f(x_{a})\in
G_{J}\\}\,.$ (2.30)

Define the finite-dimensional group

$\tilde{G}_{0}=\prod_{a}G^{\mathbb{C}}\,.$ (2.31)

There is the the commutative diagram

$\textstyle{\\{d_{\bar{A}}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{G}_{J}}$$\scriptstyle{\mathcal{G}_{0}}$$\textstyle{\quad
Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{J}}$$\scriptstyle{\tilde{G}_{0}}$$\textstyle{Bun_{J}(\Sigma_{g,n},G^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{G}_{0}}$$\textstyle{\quad
Bun(\Sigma_{g},G^{\mathbb{C}})}$ (2.32)

Here arrows mean the passage to the quotient spaces. The action of
$\tilde{G}_{0}$ is the action on the symmetric spaces $\mathcal{X}_{a,J}$. We
have a natural ”forgetting” projection
$Bun_{J}(\Sigma_{g,n},G^{\mathbb{C}})\to Bun(\Sigma_{g},G^{\mathbb{C}})$. The
fiber of this projection is the product of the symmetric spaces
$\mathcal{X}_{a,J}$.

This construction is valid for the parabolic bundles, if the gauge group
$\mathcal{G}_{J}$ is replaced by the group $\mathcal{G}_{par}$ (2.6) and
$G_{J}$ on $G_{par}=\prod_{a}B_{a}^{\mathbb{C}}$. In particular,

$Bun_{0}(\Sigma_{g,n},G^{\mathbb{C}})\stackrel{{\scriptstyle
G_{par}}}{{\longrightarrow}}Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})\stackrel{{\scriptstyle\tilde{G}_{0}}}{{\longrightarrow}}Bun(\Sigma_{g},G^{\mathbb{C}})\,,$
(2.33)

where again the arrows mean the passage to the quotient spaces. On can
consider arbitrary combinations of parabolic and quasi-antisymmetric
structures. Eventually, they lead to algebraically integrable systems. The
quasi-compact (type I) and the quasi-normal (type II) systems are related to
real integrable systems. We come to this point later.

###  2.2 Higgs bundles

Let $\kappa$ be a canonical class on the curve $\Sigma_{g}$. The Higgs field
$\Phi$ is a ${\rm End}(E(G^{\mathbb{C}}))\otimes\kappa(\Sigma_{g})$-valued
section of the bundle of distributions on $\Sigma_{g}$. The pair
$(\Phi\,,d_{\bar{A}})$ forms the Higgs bundle over the smooth curve
$\Sigma_{g}$ [16]. The Higgs bundle is a symplectic space with the form

$\Omega=\int_{\Sigma_{g}}(D\Phi,D\bar{A})\,.$ (2.34)

It is invariant under the transformations (2.3) and

$\Phi\to f\Phi f^{-1}\,,~{}~{}f\in\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$
(2.35)

The moduli spaces of the Higgs bundles are symplectic quotients with respect
to this action.

Now consider the Higgs bundles over punctured curves. As for the moduli spaces
of holomorphic bundles we have $\alpha$ and $\beta$ realizations of these
objects.

####  2.2.1  $\alpha$-model

Consider first the parabolic Higgs bundles [22, 32]. To this end extend the
upstairs symplectic space. The $\alpha$-model for the parabolic Higgs bundles
is defined by the data $(\Phi\,,d_{\bar{A}},\cup\mathcal{O}_{a})$, where
$\mathcal{O}_{a}=\\{{\bf S}(\nu_{a},g_{a})\\}$ are coadjoint
$G^{\mathbb{C}}$-orbits (C.17) attached to the marked points. The symplectic
form on this data is

$\Omega_{par}=\Omega+\sum_{a}\omega^{KK}_{a}\delta(z-x_{a},\bar{z}-\bar{x}_{a})\,,$
(2.36)

where $\omega^{KK}_{a}=\omega^{KK}(\nu_{a})$ is the Kirillov-Kostant form
(C.18). The corresponding Poisson brackets are the canonical brackets on the
affine space $(D_{\bar{A}},\Phi)$ and the Lie-Poisson brackets for the orbit
variables.

The gauge group $\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})$ (2.2) is generated by
symplectomorphisms of the parabolic Higgs bundles. It acts on the variables as
(2.3), (2.35) and $g_{a}\to g_{a}f^{-1}(x_{a})$ in the representation (C.18)
of the Kirillov-Kostant form $\omega^{KK}$. These transformations lead to the
moment map equation

$d_{\bar{A}}\Phi+\sum_{a=1}^{n}\delta(z-x_{a},\bar{z}-\bar{x}_{a}){\bf
S}(\nu_{a},g_{a})=0\,.$ (2.37)

It means that in the neighborhood of the marked point $x_{a}$ the Higgs field
has a simple pole

$\Phi(z)\sim\frac{{\bf S}_{J,a}}{z-x_{a}}+{\bf S}^{0}_{J,a}+(z-x_{a}){\bf
S}^{1}_{J,a}+\cdots\,.$ (2.38)

Imposing the moment map constraints (2.37) and fixing the gauge with respect
to the action (2.35) we come to the moduli space of parabolic Higgs bundles:

$\mathcal{M}^{\alpha}_{par}(\Sigma_{g,n},G^{\mathbb{C}})=\left((d_{\bar{A}}\,,\,\Phi)\,,\,\prod_{a=1}^{n}\mathcal{O}_{a}\right)//\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})=\\{(\ref{mom1})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$
(2.39)

Similarly, the Higgs bundles of types O , I , II and V are defined by the data

$\begin{array}[]{c}(\Phi\,,d_{\bar{A}});\,\prod_{a=1}^{n}T^{*}G^{\mathbb{C}}_{a}\,,~{}~{}J=0\,,\\\
(\Phi\,,d_{\bar{A}});\,\prod_{a=1}^{n}T^{*}\mathcal{X}_{J,a}\,,~{}~{}J=I\,,\,II\,,\,V\,,\end{array}$
(2.40)

where $T^{*}\mathcal{X}_{J,a}$ are the cotangent bundles to the symmetric
spaces $\mathcal{X}_{J,a}$. The sections of the cotangent bundles
$T^{*}\mathcal{X}_{J,a}$ are described in terms of the variables $g$ and ${\bf
X}$ (C.28), or in terms of the Darboux variables $\mathcal{P}$ and
$\mathcal{Q}$ (C.30). These Higgs bundles are equipped with the symplectic
forms

$\Omega^{J}=\Omega+\sum_{a=0}^{n}\omega_{a}^{J}\delta(z-x_{a},\bar{z}-\bar{x}_{a})\,,$
(2.41)

where $\omega_{a}^{0}$ is the symplectic form on the cotangent bundle
$T^{*}G^{\mathbb{C}}_{a}$ (C.2), and $\omega_{a}^{J}=\omega_{a}^{X_{J}}$ is
the symplectic form on the cotangent bundles $T^{*}\mathcal{X}_{J,a}$ (C.29),
(C.34).

The symplectomorphisms corresponding to $\Omega^{J}$ act as $g\rightarrow
gf^{-1}$, ${\bf X}\rightarrow f{\bf X}f^{-1}$, see (C.36). Using (C.38) we
come to the moment map equation:

$d_{\bar{A}}\Phi+\sum_{a=1}^{n}{\bf
X}_{J}(g_{a},\zeta_{a})\delta(z-x_{a},\bar{z}-\bar{x}_{a})=0\,.$ (2.42)

The latter equation means that the Higgs field $\Phi$ is a meromorphic section
of the bundle ${\rm End}(E(G^{\mathbb{C}}))\otimes\kappa(\Sigma)$ with simple
poles at $x_{a}$ and the residues ${\rm Res}\,\Phi_{z=x_{a}}={\bf X}_{J,a}$:

$\Phi(z)\sim\frac{{\bf X}_{J,a}}{z-x_{a}}+{\bf X}^{0}_{J,a}+(z-x_{a}){\bf
X}^{1}_{J,a}+\cdots\,.$ (2.43)

Imposing the moment constraints (2.42) and fixing the gauge we come to the
moduli space

$\mathcal{M}^{\alpha}_{J}(\Sigma_{g,n},G^{\mathbb{C}})=\left((d_{\bar{A}}\,,\,\Phi)\,,\,\prod_{a=1}^{n}T^{*}\mathcal{X}_{J,a}\right)//\mathcal{G}(\Sigma_{g},G^{\mathbb{C}}))=\\{(\ref{mome})\\}/\mathcal{G}(\Sigma_{g},G^{\mathbb{C}})\,.$
(2.44)

####  2.2.2 $\beta$-model

In the ${\bf\beta}$-model the Higgs bundle is defined by the pair
$(\Phi\,,d_{\bar{A}})$, the symplectic form (2.34) and the gauge groups
$\mathcal{G}_{par}$ (2.6), $\mathcal{G}_{0}$ (2.8), $\mathcal{G}_{I}$ (2.16),
$\mathcal{G}_{II}$ (2.21), $\mathcal{G}_{V}$ (2.25). Then

$\mathcal{M}^{\beta}_{par}(\Sigma_{g,n},G^{\mathbb{C}})=(d_{\bar{A}}\,,\,\Phi)//_{\nu}\mathcal{G}_{par}(\Sigma_{g},G^{\mathbb{C}})\,,$
(2.45)
$\mathcal{M}^{\beta}_{J}(\Sigma_{g,n},G^{\mathbb{C}})=(d_{\bar{A}}\,,\,\Phi)//\mathcal{G}_{J}(\Sigma_{g},G^{\mathbb{C}})\,,~{}~{}J=0,I,II,V\,.$
(2.46)

Proposition A.2 claims that the $\alpha$-model is equivalent to the
$\beta$-model. Let us describe this isomorphism explicitly. Consider the one-
point case. Let $D_{\varepsilon}$ be a small disk with the local coordinate
$z$ and $z=0$ is the marked point. Then the form $\Omega$ (2.34) can be
rewritten as

$\Omega=D\vartheta\,,~{}~{}\vartheta=\int_{\Sigma_{g}\setminus
D_{\varepsilon}}(\Phi,D\bar{A})+\int_{D_{\varepsilon}}(\Phi,D\bar{A})\,.$
(2.47)

Consider the map $h=C^{\infty}(D_{\varepsilon})\to G^{\mathbb{C}}$. On the
disk $D_{\varepsilon}$ the connection form $\bar{A}$ can be represented as

$\bar{A}=h^{-1}\bar{\partial}h\,.$ (2.48)

Then

$\int_{D_{\varepsilon}}(\Phi,D\bar{A})=\int_{D_{\varepsilon}}(\Phi,D(h^{-1}\bar{\partial}h))\,.$
(2.49)

Since

$D(h^{-1}\bar{\partial}h)=\bar{\partial}(h^{-1}Dh)-[h^{-1}Dh,h^{-1}\bar{\partial}h]\,,$

(2.49) takes the form

$\int_{D_{\varepsilon}}(\Phi,D\bar{A})=-\int_{D_{\varepsilon}}(d_{\bar{A}}\Phi),h^{-1}Dh)+\oint_{\Gamma_{\varepsilon}}(h^{-1}Dh,\Phi)\,,$

where the contour $\Gamma_{\varepsilon}$ is the boundary of $D_{\varepsilon}$.
Define the gauge group

$\mathcal{G}_{J,\varepsilon}=\\{\,f\in C^{\infty}(D_{\varepsilon})\to
G^{\mathbb{C}}\,|\,f(0)\in K_{J}\\}\,,$

where $K_{J}$ is (2.29). The group $\mathcal{G}_{J,\varepsilon}$ is the
restriction of the gauge group $\mathcal{G}_{J}$ (2.30) on $D_{\varepsilon}$.
It acts on $\bar{A}$ (2.48) and, therefore, on $h$ as $h\to hf^{-1}$. Let
$\epsilon_{J}\in$Lie$(\mathcal{G}_{J,\varepsilon}$. This action is generated
by the Hamiltonian

$\int_{\Sigma_{g}}\left(\epsilon_{J},\bar{\partial}\Phi+[\bar{A},\Phi]\right)+\oint_{\Gamma_{\varepsilon}}(\epsilon_{J},\Phi)\,.$

Let $Res_{z=0}\,\Phi(z)=\zeta_{J}$. Since
$\epsilon_{J}(0)\in$Lie$(K_{J})={\mathfrak{k}}_{J}$ the vanishing of the
moment map leads to the equation

$d_{\bar{A}}\Phi+\delta(z,\bar{z})\zeta_{J}=0\,,~{}~{}Pr|_{{\mathfrak{k}}^{*}_{J}}\zeta_{J}=0\,.$
(2.50)

This equation coincides with the moment map constraint equation (2.42) for
$n=1$. Let in (2.48) $g=h(0)$. Then the form (2.47)

$\Omega=\int_{\Sigma_{g}}(D\Phi,D\bar{A})+D(\zeta_{J},g^{-1}Dg)$

coincides with (2.41).

####  2.2.3 Hierarchy of moduli spaces

The moduli space $\mathcal{M}_{J}(\Sigma_{g,n},G^{\mathbb{C}})$ is a complex
symplectic manifold for $J=0$ and $V$. Its holomorphic coordinates are
$(\Phi,\bar{A})$, the holomorphic coordinates on $T^{*}G^{\mathbb{C}}$ are
$(\zeta_{0},g)$, and $(\zeta_{V},g)$ are holomorphic coordinates on
$T^{*}X_{V}$. Here $g\in G^{\mathbb{C}}$,
$\zeta_{0}\in{\mathfrak{g}}^{\mathbb{C}}$ and
$Pr\,\zeta_{V}|_{{\mathfrak{u}}^{\mathbb{C}}}=0$.

From (2.4) and (B.25) we find

$dim_{\mathbb{C}}\,\mathcal{M}_{0}(\Sigma_{g,n},G^{\mathbb{C}})=2(g-1+n)\dim\,G^{\mathbb{C}}\,.$
$dim_{\mathbb{C}}\,\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,=2(g-1)\dim\,G^{\mathbb{C}}+2n\sum_{j=1}^{l}d_{j}\,.$

There is the symplectic analog of diagrams (2.32), (2.33) for the moduli
spaces $\mathcal{M}_{J}$. In this diagram the moduli spaces $Bun_{J}$ are
replaced by the moduli spaces $\mathcal{M}_{J}$ and the actions of the group
are replaced by the symplectic actions

$\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})=\tilde{G}_{0}\setminus\setminus\mathcal{M}_{J}(\Sigma_{g,n},G^{\mathbb{C}})~{}~{}(J=0,I,II,V)\,,$
(2.51)
$\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})=\tilde{G}_{0}\setminus\setminus\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(2.52)

There exists a relation between the moduli spaces
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ and
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ based on the construction of
the coadjoint orbits (C.41). Define an analog $\mathcal{H}$ of the group
$G_{J}$ (2.29)

$\mathcal{H}=\prod_{a=1}^{n}H_{a}^{\mathbb{C}}\,,$ (2.53)

where $H_{a}^{\mathbb{C}}$ are Cartan subgroups of $G^{\mathbb{C}}$. Notice
that they are simultaneously the Cartan subgroups of the symmetric spaces
$H_{a}^{\mathbb{C}}\subset\mathcal{X}_{V,a}$. We claim that

$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{H}\setminus\setminus_{\vec{\nu}}\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})\,.$
(2.54)

The action of $H_{a}^{\mathbb{C}}$ on $T^{*}\mathcal{X}_{V,a}$ is defined in
(C.39). The value of the moments $\vec{\nu}=(\nu_{1},\ldots,\nu_{n})$
satisfies the conditions

$Pr\,|_{{\mathfrak{h}}_{a}^{\mathbb{C}}}\,\zeta_{a,V}=\nu_{a}\,.$ (2.55)

(see (C.40)). To prove (2.54) consider the coadjoint orbits
$\mathcal{O}_{\nu}$ (C.41) obtained by the symplectic reduction
$\mathcal{O}_{\nu}=H^{\mathbb{C}}\setminus\setminus_{\nu}T^{*}\mathcal{X}_{V}$.
Using the representation (2.44) for
$\mathcal{M}_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ and (2.39) for
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ we derive from (C.41) the
statement (2.54).

The moduli space $\mathcal{M}_{0}$ is a progenitor of the module spaces
$\mathcal{M}_{J}$. From the diagram (2.32) and (2.33) we find

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\quad\mathcal{M}_{0}(\Sigma_{g,n},G^{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{G}_{J}\setminus\setminus}$$\scriptstyle{\mathcal{G}_{par}\setminus\setminus}$$\scriptstyle{G_{0}\setminus\setminus}$$\textstyle{\mathcal{M}_{J}(\Sigma_{g,n},G^{\mathbb{C}})}$$\textstyle{\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})}$$\textstyle{\quad\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})}$

where the arrows mean the symplectic quotients. As will be shown below,
$\mathcal{M}_{I,II}$ eventually lead to real integrable systems.

The insertion of cotangent bundles $T^{*}X_{J,a}$ for $J=I\,,\,II$ at the
points $x_{1},\dots,x_{q}$ breaks the complex structures of the Higgs bundles.
The moduli spaces $\mathcal{M}_{I,II}(\Sigma_{g,n},G^{\mathbb{C}})$ are real
symplectic spaces. Their dimensions (see (2.17), (2.22)) are

$\dim_{\mathbb{R}}\,\mathcal{M}_{I}(\Sigma_{g,n},G^{\mathbb{C}})=\dim_{\mathbb{R}}\,\mathcal{M}_{II}(\Sigma_{g,n},G^{\mathbb{C}})=$
(2.56)
$=2(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+2n\dim_{\mathbb{R}}\,\mathcal{X}_{I,II}=2(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+2n\sum_{j=1}^{l}(2d_{j}-1)\,.$

###  2.3 Integrals of motion

Let ${\cal V}$ be the algebra of vector fields on $\Sigma_{g,n}$ vanishing at
the marked points. Consider the generalized Beltrami differentials as elements
of the cohomology group

$\mu_{d_{j}}\in H^{1}(\Sigma_{g,n},{\cal
V}^{\otimes(d_{j}-1)}\otimes\bar{\kappa})\,,$ (2.57)

where $\bar{\kappa}$ is the anti-canonical class on $\Sigma_{g,n}$ and $d_{j}$
are the orders of invariants of the algebra ${\mathfrak{g}}^{\mathbb{C}}$. In
particular, $\mu_{2}$ is the standard Beltrami differential. Let
$n_{j}=\dim\,H^{1}(\Sigma_{g,n},{\cal
V}^{\otimes(d_{j}-1)}\otimes\bar{\kappa})$ and $\mu^{k}_{d_{j}}$,
$k=0,\ldots,d_{j}-1$ be a basis in $H^{1}(\Sigma_{g},{\cal
V}^{\otimes(d_{j}-1)}\otimes\bar{\kappa})$. From the Riemann-Roch theorem

$n_{j}=(2d_{j}-1)(g-1)+nd_{j}\,,$ (2.58)

where the last term is the contribution of the marked points. By means of the
generalized Beltrami differentials we construct the gauge invariant quantities

$I_{jk}=\int_{\Sigma_{g}}(\Phi^{d_{j}})\mu^{k}_{d_{j}}\,.$ (2.59)

They are independent and Poisson commute with respect to brackets coming from
the symplectic form (2.41). In accordance with the Liouville theorem for
complete integrability we need $\dim\,(Bun)$ integrals. From (2.58) the total
number of integrals is

$\mathcal{N}_{G}=\sum_{j=1}^{l}n_{j}=(g-1)\sum_{j=1}^{l}(2d_{j}-1)+n\sum_{j=1}^{l}d_{j}$

or (see (B.18), (B.27))

$\mathcal{N}_{G}=(g-1)\dim_{\mathbb{C}}\,G^{\mathbb{C}}+n\sum_{j=1}^{l}d_{j}=(g-1)\dim_{\mathbb{C}}\,G^{\mathbb{C}}+n(\dim_{\mathbb{C}}\,(Fl(G^{\mathbb{C}}))+l)\,.$
(2.60)

It seems that for the parabolic bundles we have $nl$ extra integrals of motion
(see (2.5)). This is actually not the case. To demonstrate it consider the
dependence of differentials $\mu$ on positions of the marked points. Let
${\mathcal{U}}^{\prime}_{a}$ be neighborhoods of the marked points $x_{a}\,$,
$(a=1,\ldots,n)$ such that
${\mathcal{U}}^{\prime}_{a}\cap{\mathcal{U}}^{\prime}_{b}=\emptyset$ for
$a\neq b$. Define a smooth function $\chi_{a}(z,\bar{z})$

$\chi_{a}(z,\bar{z})=\left\\{\begin{array}[]{cl}1,&\mbox{$z\in{\mathcal{U}}_{a}$
},~{}{\mathcal{U}}^{\prime}_{a}\supset{\mathcal{U}}_{a}\\\
0,&\mbox{$z\in\Sigma_{g,n}\setminus{\mathcal{U}}^{\prime}_{a}$}\,.\end{array}\right.$
(2.61)

In a neighborhood of the marked point $x_{a}$ the differentials behave as

$\mu_{d_{j},a}=\sum_{k=0}^{d_{j}-1}\mu^{d_{j}}_{k,a}\,,~{}~{}\,\mu^{d_{j}}_{k,a}=t^{d_{j}}_{a,k}(z-x_{a})^{k}\bar{\partial}\chi_{a}(z,\bar{z})\,.$
(2.62)

Here $t^{d_{j}}_{a,k}$ can be considered as coordinates on the space
$H^{1}(\Sigma_{g},{\cal V}^{\otimes(d_{j}-1)}\otimes\bar{\kappa})$. The
corresponding integrals (2.59) assume the form:

$I_{j,k,a}\sim\int_{\Sigma_{g}}(\Phi^{d_{j}})\mu^{d_{j}}_{k,a}\sim\int_{\Sigma_{g}}(\Phi^{d_{j}})(z-x_{a})^{k}\bar{\partial}\chi_{a}(z,\bar{z})\,.$
(2.63)

Then from (2.43) we conclude

$I_{j,d_{j}-1,a}\sim({\bf
T}_{a})^{d_{j}}\,,~{}~{}I_{j,d_{j}-2,a}\sim\sum_{b\neq a}\frac{({\bf
T}_{a}^{d_{j}-1}{\bf T}_{b})}{x_{a}-x_{b}}+\sum_{b}({\bf T}_{a}^{d_{j}-1}{\bf
T}^{0}_{b})+\cdots\,,$ (2.64)

where ${\bf T}_{a}={\bf S}_{a}$ for the parabolic bundles and ${\bf
T}_{a}={\bf X}_{J,a}$ for the bundles of type $J$.

For the parabolic bundles the integrals $I_{j,d_{j}-1,a}\sim({\bf
S}_{a}^{d_{j}})$ $(j=1,\ldots,l)$, $(a=1,\ldots,n)$ are the Casimir functions
defining the orbits and thereby they are irrelevant. In this way we come to
the Liouville integrability in the parabolic case.

For the bundles of type V there are the needed number of integrals of motion,
that is $\mathcal{N}_{G}=\dim\,Bun_{V}(\Sigma_{g,n},G^{\mathbb{C}})$ (see
(2.26)). For these bundles $I_{j,d_{j}-1,a}\sim({\bf X}_{V,a}^{d_{j}})$ are no
longer the Casimir functions. Thus, the moduli space of Higgs quasi-
antisymmetric bundles is a phase space of integrable systems.

It follows from this construction that there exist the mixed complex
integrable systems with insertion of coadjoint orbits $\mathcal{O}_{a}$ at
some marked points and the cotangent bundles $T^{*}\mathcal{X}_{V,b}$ at other
marked points.

As we noticed, the moduli space of the Higgs bundles of type I and II are real
spaces. The integrals related to the marked points (2.63) are real quantities,
because the quantities $({\bf T}_{a})^{d_{j}}$, $({\bf T}_{a}^{d_{j}-1}{\bf
T}_{b}),\ldots$ are real. We should compare

$\mathcal{N}^{\mathbb{R}}_{G}=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+n\sum_{j=1}^{l}d_{j}$
(2.65)

and (2.17), (2.22)

$\dim_{\mathbb{R}}\,Bun_{I,II}(\Sigma_{g,n},G^{\mathbb{C}})=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{C}}+n\sum_{j=1}^{l}(2d_{j}-1)\,.$

Then we have the deficiency $\delta_{G}$ of integrals for the system to be
completely integrable:

$\delta_{G}=\dim_{\mathbb{R}}\,Bun_{I,II}(\Sigma_{g,n},G^{\mathbb{C}})-\mathcal{N}^{\mathbb{R}}_{G}=n\sum_{j=1}^{l}(d_{j}-1)\,.$
(2.66)

In the next Section we come to integrable systems by passing to the real forms
of types III and IV.

##  3 Real structures

Assume that the Riemann surface $\Sigma_{g}$ is equipped with an anti-
holomorphic involution

$\jmath\,:\,\Sigma_{g}\to\Sigma_{g}.$

In a neighborhood of a fixed point one can find a local coordinate $z$ such
that the involution $\iota$ is given by the complex conjugation
$\jmath(z)=\bar{z}$ [11]. The fixed point set can be seen as a union
$\Sigma^{0}_{g}=\cup S^{1}_{j}$ of copies of the unit circle $S^{1}$ embedded
in $\Sigma_{g}$. We assume that the set of marked points
$\mathcal{D}=(x_{1},\ldots,x_{n})$ is invariant with respect to the involution

$\jmath(\mathcal{D})=\mathcal{D}\,.$ (3.1)

There are two possibilities:

$1.\,\jmath(x_{a})=x_{a}\,,~{}(a=1,\ldots,n)\,,\qquad
2.\,\jmath(x_{a})=x_{a+1}\,,~{}x_{a}\,,\,x_{a+1}\in\mathcal{D}\,.$ (3.2)

Either $\mathcal{D}\subset\Sigma^{0}_{g}$ or some elements of $\mathcal{D}$
are conjugated under the involution $\iota$.

###  3.1 From Type I to Type IV

Let $d\sigma$ be the antiholomorphic involutive automorphism (B.10) acting on
the complex Lie algebra ${\mathfrak{g}}^{\mathbb{C}}$. The fixed point set of
${\mathfrak{g}}^{\mathbb{C}}$ is the normal real form of
${\mathfrak{g}}^{\mathbb{R}}$. Attach to the points from $\mathcal{D}$ the
cotangent bundles $T^{*}\mathcal{X}_{I,a}$. Following the general scheme [1]
we combine the anti-holomorphic involution of the curve $\jmath$ with an
involutive automorphism $\sigma$ of the group $G^{\mathbb{C}}$ into the
operator $i_{\Sigma}$ acting on the Higgs data of the moduli space
$\mathcal{M}_{I}(\Sigma^{0}_{g,n},G^{\mathbb{C}})$ 444In [1] it is denoted as
$i_{3}$.:

$\imath^{\sigma}(\Phi\,,d_{\bar{A}}(z)\,,\,\prod
T^{*}\mathcal{X}_{I,a})=(\jmath^{*}\sigma(\Phi)\,,\jmath^{*}\sigma(d_{\bar{A}})\,,\,\prod
T^{*}\sigma(\mathcal{X}_{I,a}))\,.$ (3.3)

First, consider the case 1 in (3.2) ($\mathcal{D}\subset\Sigma^{0}_{g}$).
Since the invariant subgroups of $G^{\mathbb{C}}$ and $C_{a}$ with respect to
$\sigma$ are $G^{\mathbb{R}}$ and $U_{a}$, the fixed point set of the
$\sigma$-action at the marked points is the product of cotangent bundles to
the symmetric spaces $\prod(T^{*}\mathcal{X}_{IV,a}=T^{*}(U_{a}\setminus
G^{\mathbb{R}})$. In this way we come to the moduli space
$\mathcal{M}_{IV}(\Sigma^{0}_{g,n},G^{\mathbb{R}})$ of $G^{\mathbb{R}}$-Higgs
bundles over $\Sigma^{0}_{g,n}$ with the cotangent bundles
$T^{*}\mathcal{X}_{IV,a}$ attached to the marked points. Its real dimension
equals

$\dim_{\mathbb{R}}\,\mathcal{M}_{IV}(\Sigma^{0}_{g,n},G^{\mathbb{R}})=2(g-1)\dim\,G^{\mathbb{R}}+2n\dim\,\mathcal{X}_{IV}\,.$
(3.4)

Let $\Phi^{0}$ be the invariant Higgs field
$\Phi^{0}=\iota^{*}(\sigma(\Phi^{0}))$. The involution $\imath^{\sigma}$ acts
on the generalized Beltrami differentials (2.57) and, in particular, on the
differentials related to the marked points (2.62)

$\mu\to\jmath^{*}(\mu)\,.$ (3.5)

Let $\mu(0)^{k}_{d_{j}}$ be the differentials, which are invariant with
respect to this involution. Then the quantities (2.59)

$I^{0}_{jk}=\int_{\Sigma_{g}}((\Phi^{0})^{d_{j}})\mu(0)^{k}_{d_{j}}$ (3.6)

are independent involutive real integrals of motion. Since
$\dim_{\mathbb{R}}\,G^{\mathbb{R}}=\dim_{\mathbb{C}}\,G^{\mathbb{C}}$ for the
number of integrals we have

$\mathcal{N}^{\mathbb{R}}_{G}=(g-1)\dim_{\mathbb{R}}\,G^{\mathbb{R}}+n\sum_{j=1}^{l}d_{j}\,.$
(3.7)

From (B.26) and (3.4) we find that

$\frac{1}{2}\dim_{\mathbb{R}}\,\mathcal{M}_{IV}(\Sigma^{0}_{g,n},G^{\mathbb{R}})=\mathcal{N}_{G}^{\mathbb{R}}\,.$
(3.8)

It means that the moduli space $\mathcal{M}_{IV}$ is the phase space of a real
integrable system with the hierarchy of commuting integrals (3.6).

Now consider the case 2. in (3.2). For simplicity take $n=2$, Locally the
fixed point set is $\Im m\,z=0$ and $x_{2}=\bar{x}_{1}=\bar{x}$. We have two
cotangent bundles $T^{*}\mathcal{X}_{I,1}$ and $T^{*}\mathcal{X}_{I,2}$
attached to these points, where $\mathcal{X}_{I,2}=\sigma(\mathcal{X}_{I,1})$.
In total we have $\sum_{j}(2d_{j}-1)$ degrees of freedom (B.21). Using
representation (2.43) we write the Higgs field as

$\Phi(z)\sim\frac{{\bf X}^{(-1)}_{I}}{z-x}+{\bf X}^{(0)}_{I}+(z-x){\bf
X}^{(1)}_{I}+\frac{\sigma({\bf X}^{(-1)}_{I})}{z-\bar{x}}+\sigma({\bf
X}^{(0)}_{I})+(z-\bar{x})\sigma({\bf X}^{(1)}_{I})+\cdots\,.$ (3.9)

As in (2.64) we have two families of integrals related to the points $x=x_{1}$
and $x_{2}=\bar{x}$

$I_{j,d_{j}-1,1}\sim({\bf
X}^{(-1)}_{I})^{d_{j}}\,,~{}~{}I_{j,d_{j}-2,1}\sim\frac{(({\bf
X}^{(-1)}_{I})^{d_{j}-1}\sigma({\bf X}_{I}))}{x-\bar{x}}+(({\bf
X}^{(-1)}_{I})^{d_{j}-1}(\sigma({\bf X}^{(0)}_{I})+{\bf
X}^{(0)}_{I}))+\cdots\,,$ $I_{j,d_{j}-1,2}\sim(\sigma{\bf
X}^{(-1)}_{I})^{d_{j}}\,,~{}~{}I_{j,d_{j}-2,2}\sim\frac{((\sigma({\bf
X}^{(-1)}_{I}))^{d_{j}-1}{\bf X}_{I})}{\bar{x}-x}+((\sigma({\bf
X}^{(-1)}_{I}))^{d_{j}-1}({\bf X}^{(0)}_{I})+\sigma({\bf
X}^{(0)}_{I}))\cdots\,.$

The integrals $I_{j,d_{j}-1,1}=({\bf X}^{(-1)}_{I})^{d_{j}}$ and
$I_{j,d_{j}-1,2}=(\sigma{\bf X}^{(-1)}_{I})^{d_{j}}$ coincide. To justify this
fact we use representation (C.28) ${\bf X}_{J}=g\zeta_{I}g^{-1}$. There is a
basis in the Lie algebra ${\mathfrak{g}}^{\mathbb{C}}$ such that $\zeta_{I}$
are traceless Hermitian matrices and $\sigma$ acts as the transposition. In
this way these integrals are equal. Thereby, we have $\sum_{j}(2d_{j}-1)$
integrals. They provide the complete integrability, because this number is
equal to the dimension of $\mathcal{X}_{I}$ (B.21).

###  3.2 From Type II to Type III

Consider the involutive automorphism $d\rho$ of the algebra
${\mathfrak{g}}^{\mathbb{C}}$, which was the fixed set subalgebra. It is the
compact form ${\mathfrak{c}}$ (B.9). Similarly to (3.3) define the operator
$\imath^{\rho}$ acting on the moduli space
$\mathcal{M}_{II}(\Sigma^{0}_{g,n},G^{\mathbb{C}})$:

$\imath^{\rho}(\Phi\,,d_{\bar{A}}(z)\,,\,\prod
T^{*}\mathcal{X}_{II,a})=(\jmath^{*}\rho(\Phi)\,,\jmath^{*}\rho(d_{\bar{A}})\,,\,\prod
T^{*}\rho(\mathcal{X}_{II,a}))\,.$ (3.10)

In the case 1. (3.2) the fixed points set of the $\imath^{\rho}$ action is the
moduli space $\mathcal{M}_{III}(\Sigma^{0}_{g,n},C)$ of the Higgs $C$-bundles
over $\Sigma^{0}_{g,n}$ with the cotangent bundles $T^{*}\mathcal{X}_{III,a}$
($\mathcal{X}_{III,a}=U_{a}\setminus C$) attached to the marked points. The
symmetric spaces $\mathcal{X}_{III,a}=U_{a}\setminus C$ and
$\mathcal{X}_{IV,a}=U_{a}\setminus G^{\mathbb{R}}$ are Cartan dual and have
the same dimensions. Since the $\imath^{\rho}$ action on the generalized
Beltrami differentials still have the form (3.5) we have (as above) the same
number of invariant integrals. Therefore, as in (3.8) we have

$\frac{1}{2}\dim_{\mathbb{R}}\,\mathcal{M}_{III}(\Sigma^{0}_{g,n},C)=\mathcal{N}_{G}^{\mathbb{R}}\
$

and we come to the real completely integrable systems of type III.

In the case 2 (3.2) we have the same form of the Higgs field as in (3.9),
where ${\bf X}_{I}$ is replaced by ${\bf X}_{II}$ and the involutive
automorphism $\sigma$ – by the involutive automorphism $\rho$. To find the
correct number of integrals of motion one should prove that $({\bf
X}^{(-1)}_{II})^{d_{j}}=(\rho({\bf X}^{(-1)}_{II}))^{d_{j}}$. Again we use
representation ${\bf X}_{II}=g\zeta_{II}g^{-1}$, where $\zeta_{II}$ can be
represented as traceless imaginary matrices and $\rho$ acts as a
transposition. In this way we come to the needed equality, and thereby, to the
complete integrability for this structure of poles of the Higgs field.

The complexification of the moduli spaces of types III and IV is the moduli
space $\mathcal{M}_{V}$ (2.44). Acting in the opposite direction one can start
with the moduli space $\mathcal{M}_{V}$, and then by applying the involutions
$\imath^{\rho}$ and $\imath^{\sigma}$ obtain (as the fixed point sets) real
moduli spaces $\mathcal{M}_{III}$ and $\mathcal{M}_{IV}$. Taking into account
the diagram (B.16) we describe the real reductions by the diagram:

---
$\textstyle{\mathcal{M}_{II}}$$\textstyle{\mathcal{M}_{V}}$$\textstyle{\mathcal{M}_{I}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{III}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\imath^{\rho}}$$\scriptstyle{\imath^{\rho}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{M}_{IV}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\imath^{\sigma}}$$\scriptstyle{\imath^{\sigma}}$
(3.11)

where the arrows mean the embeddings as fixed points sets of the involutions.

###  3.3 Mixed systems

In the complex case we have constructed the mixed integrable systems of
parabolic and quasi-antisymmetric types. Here we consider the mixed systems of
types III or IV with the systems with parabolic singularities. Assume that
corresponding marked point are located on the invariant subset
$\Sigma^{0}_{g,n}$.

Consider first the action of the involution $\imath^{\sigma}$ on the parabolic
bundles. The invariant subgroup of this action is the normal form
$G^{\mathbb{R}}$, and we come to the $G^{\mathbb{R}}$-bundles over
$\Sigma^{0}_{g,n}$. It means that at the marked points we have
$B^{\mathbb{R}}$ flag varieties ${\rm
Fl}_{a}^{\mathbb{R}}=B^{\mathbb{R}}\setminus G^{\mathbb{R}}$, where
$B^{\mathbb{R}}=B^{\mathbb{C}}\cap G^{\mathbb{R}}$. For the Higgs bundles they
correspond to the $G^{\mathbb{R}}$ coadjoint orbits
$\mathcal{O}^{\mathbb{R}}_{a}$. Thus we come to the fixed point set
$\mathcal{M}_{par}(\Sigma^{0}_{g,n},G^{\mathbb{R}})$. It is a phase space of
real integrable systems. It means that there exist mixed systems
$\mathcal{M}_{par,IV}(\Sigma^{0}_{g,n},G^{\mathbb{R}})$ with insertion of real
orbits $\mathcal{O}^{\mathbb{R}}_{a}$ at some marked points and cotangent
bundles $T^{*}\mathcal{X}_{IV,b}$ at the remaining points.

The action of the involution $\imath^{\rho}$ leads to the $C$-bundles over
$\Sigma^{0}_{g,n}$. Then the intersection of the complex orbits with the
compact subalgebra $\mathcal{O}_{a}\cap{\mathfrak{c}}$ is isomorphic to the
orbits of compact subgroup $\mathcal{O}^{C}_{a}=T_{a}\setminus C$, where
$T_{a}=B^{\mathbb{C}}\cap C$ is the Cartan torus of the compact group $C$. Due
to the Iwasawa decomposition $G^{\mathbb{C}}=CB^{\mathbb{C}}$ the latter is
isomorphic to a complex flag variety
$Fl_{a}^{\mathbb{C}}=B^{\mathbb{C}}\setminus G^{\mathbb{C}}$. Therefore, the
fixed point set is the moduli space $\mathcal{M}_{par}(\Sigma^{0}_{g,n},C)$ of
the Higgs $C$-bundles over $\Sigma^{0}_{g,n}$ with the flag varieties
$Fl_{a}^{\mathbb{C}}$ attached to some marked points and cotangent bundles
$T^{*}\mathcal{X}_{III,b}$ at the remaining points.

##  4 Double coset construction

Let us give an alternative descriptions of the moduli spaces
$Bun_{J}(\Sigma_{g,n},G^{\mathbb{C}})$ and
$\mathcal{M}_{J}(\Sigma_{g,n},G^{\mathbb{C}})$, $J=0,\ldots V$. It is based on
the double coset construction [5, 7] of
$Bun_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ and
$\mathcal{M}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$. For simplicity we assume
that $G^{\mathbb{C}}$ is simply-connected, although in fact everything works
in the general case.

As above, there are two equivalent ways to define these moduli spaces. We
describe both of them (models $\alpha$ and $\beta$). Before coming to the
moduli spaces we present two constructions of the affine flag varieties and
affine symmetric spaces.

###  4.1 Affine flags and affine symmetric spaces

First consider the $\beta$-model description of the affine spaces (Appendix
A). Let $K$ be a subgroup of $G^{\mathbb{C}}$. We will consider two cases.
First $K=B^{\mathbb{C}}$ for the flag variety
$Fl^{\mathbb{C}}=B^{\mathbb{C}}\setminus G^{\mathbb{C}}$. and the second
$K=K_{J}$

$K_{0}=Id\,,\,K_{I}=C\,,\,K_{II}=G^{\mathbb{R}}\,,\,K_{V}=U^{\mathbb{C}}$
(4.1)

(see Table 1). The quotients $\mathcal{X}_{J}=K_{J}\setminus G^{\mathbb{C}}$
are the symmetric spaces from Table 1. In fact, the following construction
does not depend on the choice of $K_{J}$. Define the loop group

$L(G^{\mathbb{C}})=G^{\mathbb{C}}\otimes{\mathbb{C}}[t^{-1},t]]\,,$

its subgroups

$L^{+}(G^{\mathbb{C}})=G^{\mathbb{C}}\otimes{\mathbb{C}}[[t]]\,,$
$L^{+}(G^{\mathbb{C}},K)=\\{K(Id+tL^{+}(G^{\mathbb{C}}))\\}$ (4.2)

and the quotient space
$\mathcal{K}^{aff}(K)=L(G^{\mathbb{C}})/L^{+}(G^{\mathbb{C}},K)$. In
particular,

$\displaystyle{\rm Affine~{}flag~{}variety}~{}{\rm
Fl}^{aff}=L(G^{\mathbb{C}})/L^{+}(G^{\mathbb{C}},B^{\mathbb{C}})\,,$ (4.3a)
$\displaystyle{\rm
Affine~{}symmetric~{}space}~{}\mathcal{X}_{J}^{aff}(K_{J})=L(G^{\mathbb{C}})/L^{+}(G^{\mathbb{C}},K_{J})\,,~{}~{}J=I,II,V\,.$
(4.3b)

###### Remark 4.1

The affine flag variety is a particular case of the affine Grassmannian [29].
The definition of the affine symmetric space $\mathcal{K}^{aff}(K_{J})$
differs from the conventional one [14].

Let $P^{\vee}\subset{\mathfrak{h}}^{\mathbb{C}}$ be the coweight lattice (see
Appendix B). Consider the affine Weyl group

$W^{aff}=\\{\hat{w}=wt^{\gamma}\,|\,w\in W\,,\,\gamma\in P^{\vee}\\}$

and define the subgroup $L^{-}(G^{\mathbb{C}})\subset L(G^{\mathbb{C}})$

$L^{-}(G^{\mathbb{C}})=\\{n_{-}(Id+t^{-1}G^{\mathbb{C}}\otimes[t^{-1}])\,,~{}n_{-}\in
N^{-}\\}\,.$

The affine Bruhat decomposition [29]

$L(G^{\mathbb{C}})=\left(L^{-}(G^{\mathbb{C}})\bigcup_{\hat{w}\in
W^{aff}}\hat{w}\right)L^{+}(G^{\mathbb{C}},B^{\mathbb{C}})$

is the generalization of the Bruhat decomposition (B.5). Then we have explicit
description of the Affine flag variety (4.3a)

${\rm Fl}^{aff}=\left(L^{-}(G^{\mathbb{C}})\bigcup_{\hat{w}\in
W^{aff}}\hat{w}\right)\,.$

This representation defines the stratification of ${\rm Fl}^{aff}$ on the
Schubert cells corresponding to the elements of the affine Weyl group.

An alternative definition of these quotient spaces is based on the
$\alpha$-model. We identify

$X=L(G^{\mathbb{C}})\,,~{}~{}H=L^{+}(G^{\mathbb{C}},K)\,,~{}~{}G=L^{+}(G^{\mathbb{C}})\,.$

Then the $\alpha$-model is the quotient
$(L(G^{\mathbb{C}})\times(L^{+}(G^{\mathbb{C}},K)\setminus
L^{+}(G^{\mathbb{C}})))/L^{+}(G^{\mathbb{C}})$. Its equivalence to the
$\beta$-model is provided by the Proposition A.1. In these terms the
Proposition claims that

$\mathcal{K}^{aff}(K)=\left(L(G^{\mathbb{C}})\times\left(L^{+}(G^{\mathbb{C}},K)\setminus
L^{+}(G^{\mathbb{C}})\right)\right)/L^{+}(G^{\mathbb{C}})\,.$

Notice that

$L^{+}(G^{\mathbb{C}},K)\setminus L^{+}(G^{\mathbb{C}})\sim K\setminus
G^{\mathbb{C}}=\mathcal{X}~{}{\rm or}~{}{\rm Fl}^{\mathbb{C}}\,.$

Thereby (compare with (4.3))

$\displaystyle{\rm Fl}^{aff}=\left(L(G^{\mathbb{C}})\times{\rm
Fl}^{\mathbb{C}}\right)/L^{+}(G^{\mathbb{C}})\,,$ (4.4a)
$\displaystyle\mathcal{X}_{J}^{aff}=\left(L(G^{\mathbb{C}})\times\mathcal{X}_{J}\right)/L^{+}(G^{\mathbb{C}})\,.$
(4.4b)

###  4.2 Moduli space ${\rm Bun}(\Sigma_{g,n},G^{\mathbb{C}})$

Let $\mathcal{D}=(x_{1},\ldots,x_{n})$ be the set of marked points. Using the
$\beta$-model description define the moduli space ${\rm
Bun}(\Sigma_{g,n},G^{\mathbb{C}})$ in the following way.

The $E(G^{\mathbb{C}})$-bundle can be trivialized over small disjoint disks
$D=\cup_{a=1}^{n}D_{a}$ around the marked points and over the complement to
the set of the marked points $\Sigma_{g,n}\setminus\mathcal{D}$. Then
$E(G^{\mathbb{C}})$ bundle is defined by the transition holomorphic functions
on

$D^{\times}=\cup_{a=1}^{n}(D_{a}^{\times})\,,~{}~{}D_{a}^{\times}=D_{a}\setminus
x_{a}\,.$

Let $G^{\mathbb{C}}(X)$ be the holomorphic maps from $X\subset\Sigma_{g,n}$ to
$G^{\mathbb{C}}$. Describe these maps in terms of local coordinates $t_{a}$ on
the disks $D_{a}$. The group $G^{\mathbb{C}}(D)$ has the form of the
$G^{\mathbb{C}}$-valued polynomials

$G^{\mathbb{C}}(D)=\prod_{a=1}^{n}G^{\mathbb{C}}(D_{a})=\prod_{a=1}^{n}L_{a}^{+}(G^{\mathbb{C}})\,,~{}~{}L_{a}^{+}(G^{\mathbb{C}})=G^{\mathbb{C}}\otimes{\mathbb{C}}[[t_{a}]]\,.$
(4.5)

To simplify notations in the definition of $G^{\mathbb{C}}(D^{\times})$ we
consider first the one point case. Let $\gamma$ be an element of the coweight
lattice $P^{\vee}$ . Define the group of maps

$G_{\gamma}^{\mathbb{C}}(D^{\times})=\\{k_{\gamma}(t)\in
t^{\gamma}L(G^{\mathbb{C}})\,|\,L(G^{\mathbb{C}})=G^{\mathbb{C}}\otimes{\mathbb{C}}[t^{-1},t]]\\}\,.$
(4.6)

If $\gamma\in\mathcal{Q}^{\vee}$ then $k(t)$ has a trivial monodromy
$k(t\exp\,2\pi\imath)=k(t)$. Otherwise, the monodromy is nontrivial
$k(t\exp\,2\pi\imath)=\exp\,(\frac{2\pi\imath}{m})k(t)$, where
$\exp\,(\frac{2\pi\imath}{m})$ represents an element of the center
$\mathcal{Z}(G^{\mathbb{C}})\sim\mathcal{P}^{\vee}/\mathcal{Q}^{\vee}$. It
means that $k(t)$ is the map of $(D^{\times})$ to the adjoint group $G^{ad}$
(B.4), but not to $G^{\mathbb{C}}$. In this case there is an obstruction
described by the second cohomology group
$H^{2}(\Sigma,\mathcal{Z}(G^{\mathbb{C}}))\sim\mathcal{Z}(G^{\mathbb{C}})$ to
lift the $G^{ad}$-bundle to $G^{\mathbb{C}}$-bundle. The group
$H^{2}(\Sigma,\mathcal{Z}(G^{\mathbb{C}}))$ defines _the characteristic
classes_ of the holomorphic bundles [30]. The construction of the
corresponding Higgs bundles and related Hitchin systems were discussed in
[25]. In the general case

$G_{\vec{\gamma}}^{\mathbb{C}}(D^{\times})=\prod_{a=1}^{n}t_{a}^{\gamma_{a}}L_{a}(G^{\mathbb{C}})=\prod_{a=1}^{n}\\{k_{\gamma_{a}}(t_{a})\\}\,,~{}~{}\vec{\gamma}=(\gamma_{1},\ldots,\gamma_{n})\,.$
(4.7)

The characteristic class of the bundle is defined by the sum
$\sum_{a}\gamma_{a}$.

Define two gauge groups

$\mathcal{G}_{out}=G^{\mathbb{C}}(\Sigma_{g}\setminus\mathcal{D})$ (4.8)

and

$\mathcal{G}_{int}=G^{\mathbb{C}}(D)=\prod_{a=1}^{n}L^{+}_{a}(G^{\mathbb{C}})\,,~{}~{}L^{+}_{a}(G^{\mathbb{C}})=G^{\mathbb{C}}\otimes{\mathbb{C}}[[t_{a}]]\,,~{}t_{a}\in
D_{a}\,.$ (4.9)

Modify the last group as

$\widetilde{\mathcal{G}}^{J}_{int}=\prod_{a=1}^{n}L^{+}_{a}(G^{\mathbb{C}},K_{a})\,,~{}~{}(\widetilde{\mathcal{G}}^{J}_{int}\subset\mathcal{G}_{int})\,,$
(4.10)

where $L^{+}_{a}(G^{\mathbb{C}},K_{a})$ is defined through (4.2), and $K_{a}$
is either equal to $B_{a}^{\mathbb{C}}$, or has the form (4.1)
$K_{a}=K_{J,a}$.

The group $\mathcal{G}_{out}$ is holomorphic on
$\Sigma_{g}\setminus\mathcal{D}$ and acts from left on
$G^{\mathbb{C}}(D^{\times})$. The group $\mathcal{G}^{J}_{int}$ acts from the
right. Define the moduli spaces in the $\beta$-model description

${\rm
Bun}_{J,\gamma}(\Sigma_{g}\setminus\mathcal{D},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\left(\prod_{a=1}^{n}t_{a}^{\gamma_{a}}L_{a}(G^{\mathbb{C}})/L^{+}_{a}(G^{\mathbb{C}},K_{a})\right)\,.$
(4.11)

For $\gamma_{a}\in Q^{\vee}$ this definition is equivalent to the given above
for $Bun_{par}$ (2.7) and ${\rm Bun}_{J}$ (2.13), (2.15), (2.20), (2.26). In
this case the connection $d_{\bar{A}}$ can be reconstructed from $k(t)$.

Due to (4.3) and (4.9) the moduli spaces for the one marked point case can be
rewritten as in the $\alpha$-model:

$\begin{array}[]{l}{\rm
Bun}_{par,\gamma}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\left(t^{\gamma}({\rm
Fl}^{\mathbb{C}})^{aff}\right)\equiv\mathcal{G}_{out}\setminus\left((t^{\gamma}L(G^{\mathbb{C}})\times{\rm
Fl}^{\mathbb{C}})\right)/\mathcal{G}_{int}\,,\\\ {\rm
Bun}_{J}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\left(t^{\gamma}\mathcal{X}_{J}^{aff}\right)\equiv\mathcal{G}_{out}\setminus
t^{\gamma}\left(L(G^{\mathbb{C}})\times\mathcal{X}_{J}\right)/\mathcal{G}_{int}\,.\end{array}$
(4.12)

###  4.3 Moduli space of Higgs bundles
$\mathcal{M}(\Sigma_{g,n},G^{\mathbb{C}})$

To define the moduli space of the Higgs bundles
$\mathcal{M}_{par,\gamma}(\Sigma_{g,n},G^{\mathbb{C}})$ and
$\mathcal{M}_{J,\gamma}(\Sigma_{g,n},G^{\mathbb{C}})$ in the double coset
construction we introduce analogues of the Higgs fields

$\eta_{a}(t_{a})\in(L_{a}({\mathfrak{g}}^{\mathbb{C}})\otimes
dt_{a})\,,~{}~{}a=1,\ldots,n\,.$ (4.13)

and replace the symmetric spaces and the flag varieties with the corresponding
cotangent bundles and the coadjoint orbits respectively. In the $\alpha$-model
the Higgs data corresponding to the description of(4.12) are of the form:

$\bigcup_{a}\,\left(\eta_{a},t^{\gamma_{a}}k_{a}(t_{a}))\,,\,\left(\begin{array}[]{ll}T^{*}\mathcal{X}_{J,a}&J=0,I,II,V\,,\\\
\mathcal{O}_{a}&\end{array}\right)\right)\,,~{}k_{a}(t_{a})\in
L_{a}(G^{\mathbb{C}})\,.$

These data are endowed with the symplectic form

$\Omega_{J}=\sum_{a}\left(\oint_{\Gamma_{a}}D(\eta_{a}k_{a}^{-1}Dk_{a})+\omega_{a}\right)\,.$
(4.14)

Here $\Gamma_{a}$ is a contour around $t_{a}=0$ in $D^{\times}_{a}$,
$\omega_{a}=\omega^{KK}$ (C.18) for the parabolic bundles, and
$\omega_{a}=\tilde{\omega}(\mathcal{X}_{J,a})$ (C.2) or (C.29) for the
cotangent bundles.

The action of $\mathcal{G}_{int}=\prod_{a}G^{\mathbb{C}}(D_{a})$ (4.9) and
$\mathcal{G}_{out}=G^{\mathbb{C}}(\Sigma_{g}\setminus\mathcal{D})$
$G^{\mathbb{C}}(D^{\times})$ is lifted to the Higgs fields (4.13) as

$\mathcal{G}_{a,int}\,:\,\eta_{a}\to
f_{a,int}\eta_{a}f^{-1}_{a,int}\,,~{}~{}k_{a}\to k_{a}f^{-1}_{a,int}\,,$
(4.15) ${\bf X}_{J,a}\to f_{a,int}(0){\bf
X}_{J,a}f_{a,int}^{-1}(0)\,,~{}~{}{\bf S}_{a}\to f_{a,int}(0){\bf
S}_{a}f^{-1}_{a,int}(0)\,,~{}~{}g_{a}\to f_{a,int}(0)g_{a}\,,$
$\mathcal{G}_{out}\,:\,\eta_{a}\to\eta_{a}\,,~{}k_{a}\to f_{a,out}k_{a}\,.$
(4.16)

The $\mathcal{G}_{out}$-action is generated by the Hamiltonian

$F_{out}=\sum_{a}\oint_{\gamma_{a}}(\epsilon_{out},{\rm
Ad}_{k^{-1}_{a}}\eta_{a})\,,$ (4.17)

where $\epsilon_{out}\in Lie(G(\Sigma_{g}\setminus\mathcal{D})$. The
corresponding moment map is

$\mu_{out}=Pr_{Lie^{*}(G(\Sigma_{g}\setminus\mathcal{D})})\sum_{a}{\rm
Ad}_{k^{-1}_{a}}\eta_{a}(t)\,.$ (4.18)

The moment map constraint $\mu_{out}=0$ means that

${\rm Ad}_{k^{-1}_{a}}\eta_{a}~{}{\rm
has~{}holomorphic~{}continuation~{}on~{}}\Sigma_{g}\setminus\mathcal{D}\,.$
(4.19)

The action of $\mathcal{G}_{a,int}$ on $\omega_{a}$ leads to the moments
$\mu^{L}$ (C.14) and (C.38). Then we come to the Hamiltonians

$F_{a,int}=\left\\{\begin{array}[]{ll}\oint_{\Gamma_{a}}(\epsilon_{a,int},\eta_{a}-{\bf
X}_{J,a})\,,&{\rm for~{}}\omega_{a}=\tilde{\omega}_{a}\,,\\\
\oint_{\Gamma_{a}}(\epsilon_{a,int},\eta_{a}-{\bf S}_{a})\,,&{\rm
for~{}}\omega_{a}=\omega_{a}^{KK}\,,\end{array}\right.$ (4.20)

where $\epsilon_{a,int}\in Lie(L_{a}^{+}(G{\mathbb{C}}))$.

Similarly the condition $\mu_{int}=0$ implies that $\eta_{a}$ can be continued
on $D_{a}$ with simple poles at the marked points $t_{a}=0$ and

$Res\,\eta_{a}|_{t_{a}=0}=\left\\{\begin{array}[]{ll}{\bf
X}_{J,a}&J=0,I,II,V\,,\\\ {\bf S}_{a}&\,.\end{array}\right.$ (4.21)

The conditions (4.19), (4.21) define the moduli space as the symplectic
quotient. For a single marked point case we have

$\mathcal{M}_{J,\gamma}(\Sigma_{g}\setminus\mathcal{D},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\setminus(\eta,t^{\gamma}T^{*}{\bf
X}^{aff}_{J}\,,\,{\bf S})//\mathcal{G}_{int}\,,~{}~{}k(t)\in
t^{\gamma}L(G^{\mathbb{C}})\,.$

Similarly to (4.12) we have

$\begin{array}[]{l}\mathcal{M}_{par,\vec{\gamma}}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\left(\prod_{a}(t_{a}^{\gamma_{a}},\mathcal{O}^{\mathbb{C}}_{a})^{aff})\right)\,,\\\
\mathcal{M}_{J,\vec{\gamma}}(\Sigma_{g,n},G^{\mathbb{C}})=\mathcal{G}_{out}\setminus\left(\prod_{a}(t_{a}^{\gamma_{a}},T^{*}\mathcal{X}_{J,a}^{aff})\right)\,.\end{array}$
(4.22)

It was proved in [24] that the data which were used to define the moduli
spaces $\mathcal{M}^{\alpha}_{par}(\Sigma_{g,n},G^{\mathbb{C}})$ in the
previous description (2.39) can be reconstructed from
$\mathcal{M}_{par,\vec{\gamma}}(\Sigma_{g,n},G^{\mathbb{C}})$.

Consider the $\beta$-model construction. We replace the form $\Omega_{j}$
(4.14) on

$\tilde{\Omega}_{J}=\sum_{a}\oint_{\Gamma_{a}}D(\eta_{a}k_{a}^{-1}Dk_{a})\,,$

the gauge group $\mathcal{G}^{J}_{int}$ (4.9) on
$\widetilde{\mathcal{G}}^{J}_{int}$ (4.10). Similarly to (4.20) the
Hamiltonian generating the gauge transformation is of the form:

$\tilde{F}_{a,int}=\oint_{\Gamma_{a}}(\tilde{\epsilon}_{a,int},\eta_{a})\,,$
(4.23)

where

$\tilde{\epsilon}_{a,int}\in
Lie(L_{a}^{+}(G^{\mathbb{C}}))\,,~{}~{}\tilde{\epsilon}_{a,int}=x_{a,0}+t_{a}x_{a,1}+\ldots\,.$
(4.24) $x_{a,0}\in\,Lie\,(B^{\mathbb{C}})={\mathfrak{b}}\,,~{}{\rm
or~{}}Lie\,(K_{J,a})={\mathfrak{k}}_{J}\,.$

The moment map constraints $\tilde{F}_{a,int}=0$ assume the form

$1.\,Pr_{{\mathfrak{k}}^{*}_{J}}\eta_{a}=0\,,~{}{\rm
or}~{}2.\,Pr_{{\mathfrak{b}}^{*}_{J}}\eta_{a}=\nu\in{\mathfrak{h}}^{\mathbb{C}}\,.$
(4.25)

In the case 1. the symplectic quotient is described by the pairs
$(\eta_{a},k_{a})$, where $\eta_{a}$ satisfies this condition, and due to
(4.15) $k_{a}$ is invariant under the action $k_{a}\to k_{a}f^{-1}_{a}$,
$f_{a}\in K_{J,a}$. This pair defines the cotangent $T^{*}\mathcal{X}_{J,a}$
as in (C.25), (C.26), where the right action of the gauge group is replaced on
the left one.

In the case 2. we come to the coadjoint orbit (C.10) passing through the
Cartan element $\nu\in{\mathfrak{h}}^{\mathbb{C}}$.

###  4.4 Real structures

Here we investigate the passages from the moduli spaces $\mathcal{M}_{I,II}$
to $\mathcal{M}_{III,IV}$ in terms of the double coset construction. In
section 3 it was performed for trivial bundles. We investigate the
compatibility of the involutions $\imath^{\sigma}$ (3.10) and $\imath^{\rho}$
(3.3) with non-trivial characteristic classes of underlying bundles.

Assume that the set $\mathcal{D}$ of marked points belongs to the invariant
set $\Sigma_{g}^{0}$ of the anti-holomorphic involution $\jmath$ of
$\Sigma_{g,n}$. Consider the one point case $\mathcal{D}=x$ and the local
coordinate $t$ on the disc $D=\\{t\,|\,|t|\leq 1\\}$ is $t=z-x$. The
involution $\jmath$ acts as $t\to\bar{t}$. The representation (4.6) defines
the gluing function

$t^{\gamma}k(t)=t^{\gamma}(k_{0}+k_{1}t+\ldots)\,.$

The involutions $\imath^{\rho}$ (3.3) and $\imath^{\sigma}$ (3.10) act on
$k(t)$ as $k(t)\to\jmath^{*}\rho(k(t)$ and $k(t)\to\jmath^{*}\sigma(k(t)$ and
on the multiplier as $t^{\gamma}\to t^{-\gamma}$. It means that the fixed
point sets $\mathcal{M}_{III}$ and $\mathcal{M}_{IV}$ of the moduli spaces
$\mathcal{M}_{I}$ and $\mathcal{M}_{II}$ exist in the case $2\gamma\in
Q^{\vee}$ only (see Proposition 6 in [2]). It follows from Table 4 in the
Appendix B that it happens for the ${\rm SL}(N,{\mathbb{C}})$-bundles ($N$ is
even), for the bundles with the remaining classical gauge groups and for the
E7-bundles.

In the general case, for $\mathcal{D}=(x_{1},\ldots,x_{n})$ there are $n$
gluing functions $t^{\gamma_{a}}(k_{0,a}+k_{1,a}t+\ldots)$ defined on the
contours $\Gamma_{a}$. We have $n$ conditions $2\gamma_{a}\in Q^{\vee}$,
$(a=1,\ldots,n)$. Type III or IV bundles are nontrivial if the sum
$\left(\sum_{a}\gamma_{a}\right)\notin Q^{\vee}$.

##  5 Examples

###  5.1 Elliptic case

Consider the elliptic curve
$\Sigma_{\tau}={\mathbb{C}}/({\mathbb{Z}}+\tau{\mathbb{Z}})$ and let $E$ be
the ${\rm SL}(2,{\mathbb{C}})$ vector bundle over $\Sigma_{\tau}$. We consider
one point case. The marked point is $z=0$. Since $\mathcal{Z}({\rm
SL}(2,{\mathbb{C}}))\sim{\mathbb{Z}}_{2}$ there are two types of bundles
corresponding to the trivial and non-trivial characteristic classes. In the
parabolic situation the first case corresponds to the two-particle elliptic
Calogero-Moser system and the second case is the ${\rm SL}(2,{\mathbb{C}})$
Euler-Arnold top. Since the dimensions of both phase spaces equal two, the
systems are integrable. Here we consider their quasi-antisymmetric extensions
and their real reductions.

####  5.1.1 Extension of the Euler-Arnold top

Consider first non-trivial bundle of type V. The sections $s\in\Gamma(E)$ of
the rank two bundle have the quasi-periodicities $s(z+1)=\sigma_{3}s(z)$,
$s(z+\tau)=\sigma_{1}s(z)$, where $\sigma$-matrices are defined below.
Therefore, the Higgs field takes value in the Lie algebra ${\rm
sl}(2,{\mathbb{C}})$ and satisfy the following conditions

$\Phi(z+1)=\sigma_{3}\Phi(z)\sigma_{3}\,,~{}~{}\Phi(z+\tau)=\sigma_{1}\Phi(z)\sigma_{1},.$
(5.1)

In addition

$Res\,\Phi(z)|_{z=0}={\bf X}\,,$ (5.2)

where ${\bf X}=g^{-1}\zeta g\in{\rm sl}(2,{\mathbb{C}})$ and $\zeta^{T}=\zeta$
(see also (C.25) and (C.28)). It means that we attach to the point $z=0$ the
cotangent bundle $T^{*}\mathcal{X}_{V}$ to the space of complex symmetric
matrices $\mathcal{X}_{V}=$SO$(2,{\mathbb{C}})\setminus{\rm
SL}(2,{\mathbb{C}})$. Instead of this bundle we consider the cotangent bundle
$T^{*}({\rm GL}(2,{\mathbb{C}})/$O$(N,{\mathbb{C}}))\sim\\{(\zeta,g)\\}$,
where $g\in{\rm GL}(2,{\mathbb{C}})$ and $\zeta=\zeta^{T}$.

Before passing to $T^{*}\mathcal{X}_{V}$ note that the form $\omega$ (C.2)
defined on $T^{*}{\rm GL}(2,{\mathbb{C}})$ is invariant under the
multiplication $g\to\lambda g$, $\lambda\in{\mathbb{C}}^{*}$. The quotient
${\rm GL}(2,{\mathbb{C}})/{\mathbb{C}}^{*}$ is the group P${\rm
GL}(2,{\mathbb{C}})$. The corresponding moment constraint ${\rm tr}\,\zeta=0$,
generating this action, is ${\rm tr}\,\zeta=0$. The group P${\rm
GL}(2,{\mathbb{C}})$ is covered by ${\rm SL}(2,{\mathbb{C}})=\\{g\in{\rm
GL}(2,{\mathbb{C}})\,|\,\det\,g=1\\}$ (P${\rm GL}(2,{\mathbb{C}})={\rm
SL}(2,{\mathbb{C}})/{\mathbb{Z}}_{2}$). Fixing the ${\mathbb{C}}^{*}$ action
as $\det\,g=1$ we come to the symplectic quotient (C.1):

$T^{*}{\rm GL}(2,{\mathbb{C}})//{\mathbb{C}}^{*}=T^{*}{\rm
SL}(2,{\mathbb{C}})=\\{\zeta,g)\,|\,{\rm tr}\,\zeta=0\,,~{}\det\,g=1\\}$ (5.3)

with the form $\omega$ (C.2) Construct the moduli space of the quasi-
antisymmetric bundles $\mathcal{M}_{V}(\Sigma_{\tau}\setminus 0,{\rm
SL}(2,{\mathbb{C}}))$. Consider first the group GL$(2,{\mathbb{C}})$. We use
the basis of the Pauli matrices $\sigma_{a}$ $(a=0,\dots,3)$.

$\sigma_{0}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\,,~{}\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\,,~{}\sigma_{2}=\left(\begin{array}[]{cc}0&-\imath\\\
\imath&0\\\ \end{array}\right)\,,~{}\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\\\ \end{array}\right)\,.$

Now pass to the cotangent bundle $T^{*}\mathcal{X}_{V}=T^{*}({\rm
SO}(2,{\mathbb{C}})\setminus{\rm SL}(2,{\mathbb{C}}))$ (5.3). In this case
$g^{*}=g^{T}$. Due to (C.30)

$\mathcal{P}=g^{-1}\zeta(g^{T})^{-1}\,,~{}~{}\mathcal{Q}=g^{T}g$ (5.4)

are complex symmetric matrices. Their decomposition in the basis of the Pauli
matrices assumes the form

$\begin{array}[]{c}\mathcal{P}(p_{0},p_{1},p_{3})=p_{0}\sigma_{0}+p_{1}\sigma_{1}+p_{3}\sigma_{3}\,,~{}~{}p_{j}\in{\mathbb{C}}\,,\\\
\mathcal{Q}(q_{0},q_{1},q_{3})=q_{0}\sigma_{0}+q_{1}\sigma_{1}+q_{3}\sigma_{3}\,,~{}~{}q_{j}\in{\mathbb{C}}\,.\end{array}$
(5.5)

The condition $\det\,\mathcal{Q}\neq 0$ implies that
$q_{0}^{2}-q_{1}^{2}-q_{3}^{2}\neq 0$. It follows from (C.34) that
$(p_{0},p_{1},p_{3})$ and $(q_{0},q_{1},q_{3})$ are the Darboux coordinates on
the cotangent bundle
$T^{*}($SO$(2,{\mathbb{C}})\setminus$GL$(2,{\mathbb{C}}))$.

The passage to ${\rm SL}(2,{\mathbb{C}})$ means that $\det\,g=1$. Then

$\det\,\mathcal{Q}=q_{0}^{2}-q_{1}^{2}-q_{3}^{2}=1\,.$

Let ${\bf X}(p,q)=\mathcal{P}\mathcal{Q}=g^{-1}\zeta g$.

${\bf X}=X_{0}\sigma_{0}+X_{1}\sigma_{1}+\imath
X_{2}\sigma_{2}+X_{3}\sigma_{3}\,.$ (5.6)

Then

$\begin{array}[]{l}X_{0}=q_{0}p_{0}+q_{1}p_{1}+q_{3}p_{3}\\\
X_{1}=q_{0}p_{1}+q_{1}p_{0}\,,\\\ X_{2}=q_{3}p_{1}-q_{1}p_{3}\,,\\\
X_{3}=q_{0}p_{3}+q_{3}p_{0}\,.\end{array}$ (5.7)

Since ${\bf X}\in$sl$(2)$, $\,{\rm tr}\,{\bf X}={\rm tr}\,\zeta=0$. Therefore,
$X_{0}=q_{0}p_{0}+q_{1}p_{1}+q_{3}p_{3}=0$. Thus, we impose two constraints

$1.\,c_{1}=q_{0}^{2}-q_{1}^{2}-q_{3}^{2}-1=0\,,~{}~{}2.\,c_{2}=q_{0}p_{0}+q_{1}p_{1}+q_{3}p_{3}=0\,.$
(5.8)

These constrains are the second class constraints (recall that the upstairs
brackets are canonical $\\{p_{i},q_{j}\\}=\delta_{ij}$ ):

$\\{c_{1},c_{2}\\}|_{\varphi_{1}=0}=-2\,,$

i.e. the matrix of the Poisson brackets between the constraints
$\\{c_{i},c_{j}\\}$ takes the form

$\left(\begin{array}[]{cc}{0}&{-2}\\\ {2}&{0}\end{array}\right)$

on-shell (5.8). Taking into account the brackets

$\begin{array}[]{ccc}\\{p_{0},c_{1}\\}=2q_{0}\,,&\\{p_{1,2},c_{1}\\}=-2q_{1,2}\,,&\\{q_{j},c_{1}\\}=0\,,\\\
\\{p_{j},c_{2}\\}=p_{j}\,,&\\{q_{j},c_{2}\\}=-q_{j}&\end{array}$

we come to the following reduced Dirac brackets on-shell the constraints
(5.8):

$\begin{array}[]{c}\displaystyle{\\{p_{0},q_{0}\\}=1-q_{0}^{2}\,,\quad\\{p_{0},q_{1}\\}=-q_{0}q_{1}\,,\quad\\{p_{0},q_{3}\\}=-q_{0}q_{3}\,,}\\\
\\\
\displaystyle{\\{p_{1},q_{0}\\}=q_{0}q_{1}\,,\quad\\{p_{1},q_{1}\\}=1+q_{1}^{2}\,,\quad\\{p_{1},q_{3}\\}=q_{1}q_{3}\,,}\\\
\\\
\displaystyle{\\{p_{3},q_{0}\\}=q_{0}q_{3}\,,\quad\\{p_{3},q_{1}\\}=q_{1}q_{3}\,,\quad\\{p_{3},q_{3}\\}=1+q_{3}^{2}\,.}\end{array}$
(5.9)

These brackets along with the constraints (5.8) defines the Poisson structure
on the cotangent bundle $T^{*}\mathcal{X}_{V}=T^{*}({\rm
SO}(2,{\mathbb{C}})\setminus{\rm SL}(2,{\mathbb{C}}))$.

It can be checked directly from (5.7) that the coefficients $X_{\alpha}$
satisfies the Poisson-Lie sl$(2)$ algebra

$\\{X_{\alpha},X_{\beta}\\}=c_{\alpha\beta\gamma}X_{\gamma}\,,~{}~{}(\alpha\,,\beta\,,\gamma=1,2,3)\,,$
(5.10)

or in details

$\\{X_{1},X_{2}\\}=-X_{3}\,,~{}~{}\\{X_{2},X_{3}\\}=-X_{1}\,,~{}~{}\\{X_{3},X_{1}\\}=X_{2}\,.$

The Casimir function of this algebra is

$\mathcal{C}={\rm tr}\,{\bf X}^{2}=X_{1}^{2}-X_{2}^{2}+X_{3}^{2}\,.$ (5.11)

It should be emphasized that $\mathcal{C}$ is not the Casimir function of the
Poisson algebra on
$T^{*}($GL$(2,{\mathbb{C}}/$SO$(2,{\mathbb{C}}))$, generated by
$(p_{j},q_{k})$ ($j,k=0,1,3)$.

For the ${\rm SL}(2,{\mathbb{C}})$ Higgs bundles over the elliptic curve
$\Sigma_{\tau}$ with the quasi-periodicities (5.1) one can take $\bar{A}=0$ as
the gauge fixing condition. After the gauge transformation the Higgs field
becomes the Lax operator $L_{V}$. It satisfies the holomorphicity condition
(2.42) and has a first order pole at $z=0$. The moduli space is defined by the
residue $res\,L(z)|_{z=0}={\bf X}$

$\mathcal{M}_{V}(\Sigma_{\tau}\setminus\\{0\\},{\rm
SL}(2,{\mathbb{C}}))\sim(T^{*}\mathcal{X}_{V}=T^{*}({\rm
SL}(2,{\mathbb{C}})/{\rm SO}(2,{\mathbb{C}}))=$ (5.12)
$\\{((p_{0},q_{0}),(p_{1},q_{1})(p_{3},q_{3}))\,|\,(c_{1}=0\,,\,c_{2}=0)\\}\,.$

It has complex dimension 4.

The Lax operator can be defined in terms of the Kronecker function. The
Kronecker function $\phi(u,z)$ is related to the elliptic curve
$\Sigma_{\tau}$ and takes the form

$\phi(u,z)=\frac{\vartheta(u+z)\vartheta^{\prime}(0)}{\vartheta(u)\vartheta(z)}\,,$
(5.13)

where $\vartheta(z)$ is the theta-function

$\vartheta(z|\tau)=q^{\frac{1}{8}}\sum_{n\in{\bf Z}}(-1)^{n}e^{\pi
i(n(n+1)\tau+2nz)}\,.$ (5.14)

The Kronecker function has the following quasi-periodicities:

$\phi(u,z+1)=\phi(u,z)\,,~{}~{}~{}\phi(u,z+\tau)=e^{-2\pi\imath
u}\phi(u,z)\,,$ (5.15)

and has the first order pole at $z=0$

$\phi(u,z)\sim\frac{1}{z}\ldots\,.$ (5.16)

It is related to the Weierstrass function $\wp$ as follows:

$\phi(u,z)\phi(-u,z)=\wp(z)-\wp(u)\,.$ (5.17)

Let

$\varphi_{1}(z)=\phi(\frac{1}{2},z)\,,~{}~{}\varphi_{2}(z)=\exp(\pi\imath
z)\phi(\frac{1+\tau}{2},z)\,,~{}~{}\varphi_{3}(z)=\exp(\pi\imath
z)\phi(\frac{\tau}{2},z)\,.$

Then from (5.1), (5.2), (5.15) and (5.16) we find the Lax operator

$L_{V}=X_{1}\varphi_{1}(z)\sigma_{1}+X_{2}\varphi_{2}(z))\sigma_{2}+X_{3}\varphi_{3}(z))\sigma_{3}\,,$
(5.18)

where $X_{\alpha}$ are given by (5.7). Since ${\rm tr}\,L^{2}$ is a double
periodic function it can be expressed in terms of the $\wp$-function. By means
of (5.17) define the expansion ${\rm
tr}\,L_{V}^{2}=2H^{V}_{2}\wp(z)+2H^{V}_{0}$, where the coefficients are two
Hamiltonians

$H^{V}_{2}=\frac{1}{2}(X^{2}_{1}-X_{2}^{2}+X^{2}_{3})=$ (5.19)
$=\frac{1}{2}\Big{(}(q_{0}p_{1}+q_{1}p_{0})^{2}-(q_{1}p_{3}-q_{3}p_{1})^{2}+(q_{0}p_{3}+q_{3}p_{0})^{2}\Big{)}\,,$
$H^{V}_{0}=\frac{1}{2}(X^{2}_{1}\wp(\frac{1}{2})-X^{2}_{2}\wp(\frac{1+\tau}{2})+X^{2}_{3}\wp(\frac{\tau}{2})=$
(5.20)
$=\frac{1}{2}(q_{0}p_{1}+q_{1}p_{0})^{2}\wp(\frac{1}{2})-\frac{1}{2}(-q_{1}p_{3}+q_{3}p_{1})^{2}\wp(\frac{\tau+1}{2})+\frac{1}{2}(q_{0}p_{3}+q_{3}p_{0})^{2}\wp(\frac{\tau}{2})\,.$

Since $H^{V}_{2}$ (5.19) is proportional to the Casimir function (5.11), the
Hamiltonians $H^{V}_{2}$ and $H^{V}_{0}$ Poisson commute.

We replace the Hamiltonian $H^{V}_{0}$ by the Hamiltonian $\tilde{H}^{V}_{0}$,
where the values of the $\wp$-functions are replaced by an arbitrary complex
coefficients

$\tilde{H}^{V}_{0}(J_{1},J_{2},J_{3})=\frac{1}{2}(X^{2}_{1}J_{1}-X^{2}_{2}J_{2}+X^{2}_{3}J_{3})=$
(5.21)
$=\frac{1}{2}(q_{0}p_{1}+q_{1}p_{0})^{2}J_{1}-\frac{1}{2}(-q_{1}p_{3}+q_{3}p_{1})^{2}J_{2}+\frac{1}{2}(q_{0}p_{3}+q_{3}p_{0})^{2}J_{3}\,.$

It commutes with $H^{V}_{2}=\tilde{H}^{V}_{0}(1,1,1)$ as before.

These Hamiltonians being restricted on the phase space $T^{*}({\rm
SL}(2,{\mathbb{C}})/$SO$(2,{\mathbb{C}}))$ also Poisson commute, because the
symplectic reduction (5.3) does not break the involutivity of the
Hamiltonians. For $J_{k}\neq 1$ they are independent. Therefore, we come to
the complex completely integrable system on $T^{*}({\rm
SL}(2,{\mathbb{C}})/$SO$(2,{\mathbb{C}}))$, because $\dim\,({\rm
SL}(2,{\mathbb{C}})/$SO$(2,{\mathbb{C}}))=2$. The system is described by the
two commuting Hamiltonians $\tilde{H}^{V}_{0}$, $H^{V}_{2}$, the brackets
(5.9) and constraints (5.8).

Consider the Hamiltonian $\tilde{H}^{V}_{0}$ and the corresponding equations
of motion before the passage to the group ${\rm SL}(2,{\mathbb{C}})$. Using
(5.9) we obtain

$\begin{array}[]{ll}\dot{q}_{0}=J_{1}X_{1}q_{1}+J_{3}X_{3}q_{3}\,,&\dot{p}_{0}=-J_{1}X_{1}p_{1}-J_{3}X_{3}p_{3}\,,\\\
\dot{q}_{1}=J_{1}X_{1}q_{0}-J_{2}X_{2}q_{3}\,,&\dot{p}_{1}=-J_{1}X_{1}p_{0}-J_{2}X_{2}p_{3}\,,\\\
\dot{q}_{3}=J_{2}X_{2}q_{1}+J_{3}X_{3}q_{0}\,,&\dot{p}_{3}=-J_{3}X_{3}p_{0}+J_{2}X_{2}p_{1}\,.\end{array}$
(5.22)

On the other hand since the variables $X_{\alpha}$ (5.7) satisfy the sl$(2)$
Poisson algebra (5.10), $\tilde{H}^{V}_{0}$ (5.21) is the Hamiltonian of the
Euler-Arnold top on the GL$(2,{\mathbb{C}})$ orbit corresponding to the value
of $H^{V}_{2}$ (5.19). The equations of motion are of the form:

$\dot{X}_{\alpha}=-c_{\alpha\beta\gamma}J_{\beta}X_{\beta}X_{\gamma}\,.$

Comparing these equations with (5.22) we conclude that
$X_{\alpha}(p_{0},\ldots,q_{3})$ play the role of collective coordinates on
the phase space $T^{*}({\rm SO}(2)\setminus$GL$(2,{\mathbb{C}}))$. As in the
general case the coadjoint orbit arises here as the symplectic quotient of the
cotangent bundle $T^{*}({\rm SO}(2)\setminus{\rm SL}(2,{\mathbb{C}}))$ with
respect to the action of the diagonal subgroup of ${\rm SL}(2,{\mathbb{C}})$.

####  5.1.2 Real reductions

Consider the real symmetric spaces

$\mathcal{X}_{III}=S^{2}={\rm SO}(2)\setminus{\rm
SU}(2)\,,~{}~{}\mathcal{X}_{IV}={\rm SO}(2)\setminus{\rm
SL}(2,{\mathbb{R}})\,.$

Here $\mathcal{X}_{IV}$ is the Lobachevsky plane.

In the first case in (5.7) $(q_{0},p_{0})\in{\mathbb{R}}$,
$(q_{1},p_{1}),(q_{3},p_{3})\in\imath{\mathbb{R}}$. The coefficients
$X_{\alpha}\in\imath{\mathbb{R}}$ and satisfy the su$(2)$ Poisson-Lie algebra

$\\{X_{\alpha},X_{\beta}\\}=-\imath\epsilon_{\alpha\beta\gamma}X_{\gamma}\,.$

The sum

$H^{III}_{2}=-\frac{1}{2}(X^{2}_{1}+X_{2}^{2}+X^{2}_{3})$

is positive. It is the Casimir function of this algebra.

In the second case all coordinates and momenta $(p_{j},q_{j})$ are real. Then
the coefficients $X_{1}$, $X_{3}$ are real and $X_{2}$ is imaginary. The
Poisson brackets coincide with (5.10). Consequently, the Casimir function,
which will play the role of the Hamiltonian assumes the form

$H^{IV}_{2}=\frac{1}{2}(X^{2}_{1}-X_{2}^{2}+X^{2}_{3})\,.$

This Casimir function is real.

Our goal is to construct two Lax operators as a result of involutions (1.6)
and (1.7). To define the involution of the ${\rm SL}(2,{\mathbb{C}})$ Higgs
bundles we need elliptic curve with anti-holomorphic involutions $\jmath$. If
$\tau=\imath t$, $t\in{\mathbb{R}}^{+}$ then the involution is conjugation
$\jmath\,(z)=\bar{z}$. The fixed point set $\Sigma^{0}_{\imath t}$ is $\Re
e\,z=x$, or $z=\frac{1}{2}\imath t+x$, where $0\leq x<1$. Another choice of
the fixed point set is $z=a+\imath x$, $a=0$, or $a=\frac{1}{2}$. The
coefficients $\wp(\frac{1}{2})$, $\wp(\frac{1+\imath t}{2})$, $\wp(\imath
t/2)$ of the Hamiltonian $H^{V}_{0}$ (5.20) belong to the fixed points set.
They are real.

Another possibility is to define the elliptic curve as the quotient
$\Sigma_{\omega_{1}\omega_{2}}={\mathbb{C}}/(2\omega_{1}{\mathbb{Z}}+2\omega_{2}{\mathbb{Z}})$,
$(\Re e\,\omega_{j}>0)$ and $\omega_{2}=\bar{\omega}_{1}$. We will not analyze
here this case. Real involutions for many marked points on elliptic curves and
integrability were considered in [10].

The real Higgs SU$(2)$-bundles (for the system of type III) and
SL$(2,{\mathbb{R}})$-bundles (for the system of type IV) are results of
reductions (1.8) of $\mathcal{M}_{V}$ (5.12). We have two types of the moduli
spaces

$\mathcal{M}_{III}(\Sigma^{0}_{\imath t}\setminus 0,{\rm
SU}(2))\sim(T^{*}\mathcal{X}_{III}=T^{*}({\rm SO}(2)\setminus{\rm SU}(2))=$
(5.23)
$\\{((p_{0},q_{0})\in{\mathbb{R}},(p_{1},q_{1})(p_{3},q_{3})\in\imath{\mathbb{R}})\,|\,(c_{1}=0\,,\,c_{2}=0)\\}\,.$
$\mathcal{M}_{IV}(\Sigma^{0}_{\imath t}\setminus 0,{\rm
SL}(2,{\mathbb{R}}))\sim(T^{*}\mathcal{X}_{IV}=T^{*}({\rm SO}(2)\setminus{\rm
SL}(2,{\mathbb{R}}))=$ (5.24)
$\\{((p_{0},q_{0})(p_{1},q_{1})(p_{3},q_{3})\in{\mathbb{R}})\,|\,(c_{1}=0\,,\,c_{2}=0)\\}\,.$

The Lax operators (the Higgs fields) assume the forms

$L^{III,IV}=X_{1}\varphi_{1}(x)\sigma_{1}+X_{2}\varphi_{2}(x))\sigma_{2}+X_{3}\varphi_{3}(x))\sigma_{3}\,,~{}~{}x\in\Sigma^{0}_{\imath
t}\,,$ (5.25)

where $X_{\alpha}$ are imaginary for the systems of type III and $X_{1},X_{3}$
are real and $X_{2}$ are imaginary for the systems of type IV.

As before we come to the Hamiltonians

$\tilde{H}^{III}_{0}=\frac{1}{2}\sum_{\alpha}X_{\alpha}^{2}J_{\alpha}\,,~{}~{}\tilde{H}^{IV}_{0}=\frac{1}{2}(X_{1}^{2}J_{1}-X_{2}^{2}J_{2}+X_{3}^{2}J_{3})$

commuting with $H^{III}_{2}$ and $H^{IV}_{2}$.

The system of type III corresponds to a motion of a particle on the two-
dimensional sphere $S^{2}$ and of type IV to a motion on the Lobachevsky
plane. The moduli spaces $\mathcal{M}_{III}(\Sigma_{\imath t}^{0}\setminus
0,{\rm SU}(2))=T^{*}S^{2}$ (see 5.12)) is the Eguchi-Hanson gravitational
instanton [8]. It has the hyper-Kahler structure [18]. In the general case we
did not find the hyper-Kahler structure on the space
$\mathcal{M}_{V}(\Sigma_{\tau}\setminus\\{0\\},{\rm SL}(2,{\mathbb{C}}))$.

Consider the corresponding quantum system. The Hamiltonian $\tilde{H}_{0}^{V}$
(5.21) is quantized by the canonical quantization rules and subsequent
imposing of (the quantization) constraints (5.8). The quantization in terms of
the generators $\hat{X}_{\alpha}$ of the algebra su$(2)$ is straightforward
[33]. The generators act on the space of functions on $S^{2}$ that are
quadratically integrable with respect to the invariant measure. The
Hamiltonian $\hat{H}^{III}_{2}$ is the Laplace-Beltrami operator defined by
the su$(2)$ invariant metric on $S^{2}$. In the spherical coordinates on
$S^{2}$ the operator $\hat{H}^{III}_{0}$ assume the form

$\hat{H}^{III}_{0}=(\cos\psi\partial_{\theta}-\tan^{-1}\theta\sin\partial_{\psi})^{2}J_{1}+(-\sin\psi\partial_{\theta}-\tan^{-1}\theta\cos\psi\partial_{\psi})^{2}J_{2}+\partial^{2}_{\psi}J_{3}\,.$
(5.26)

The eigen-subspaces $V_{l}$ of $H^{III}_{2}$ are irreducible unitary
representations of su$(2)$ of spin $l$, ($l=0,\frac{1}{2},1,\ldots$) and
$\dim\,V_{l}=2l+1$. The basis of $V_{l}$ is generated by the functions

$\Psi_{lm}=P_{l}^{m}(\cos\theta)\exp\,(-\imath m\psi)\,,\quad-l\leq m\leq l,$

where $P_{l}^{m}(\cos\theta)$ are the associated Legendre polynomials. Since
$\hat{H}^{III}_{2}$ and $\hat{H}^{III}_{0}$ commute we can restrict the latter
operator on the space $V_{l}$. Therefore, the eigenfunctions of
$\hat{H}^{III}_{0}$ are the linear combination of the functions $\Psi_{lm}$.

###  5.2 Extension of Calogero-Moser system

Consider the trivial type V SL$(2,{\mathbb{C}})$ Higgs bundle over
$\Sigma_{\tau}$. The section of the bundle have the quasi-periodicities
(compare with (5.1))

$\Phi(z+1)=\Phi(z)\,,~{}~{}\Phi(z+\tau)=Q(u)\Phi(z)Q^{-1}(u)\,,~{}~{}Q(u)={\rm
diag}({\bf e}(u),{\bf e}(-u))\,,~{}({\bf e}(u)=\exp\,2\pi\imath u)\,.$

By the gauge transforms the connection can be trivialised $\bar{A}=0$. The
system has residual symmetry acting on the variables as in (2.3), (2.35) by
constant diagonal matrices. This action produces the moment constraint
$\bar{\partial}\Phi|_{{\rm diag}}=0$. Therefore the diagonal part of the
residue ${\bf X}$ (5.2) vanishes. In this way we come to the constraint (see
(5.7))

$X_{3}=0\,,~{}~{}(q_{0}p_{3}+q_{3}p_{0}=0)\,.$ (5.27)

Let $\sigma_{\pm}=\frac{1}{2}(\sigma_{1}\pm\imath\sigma_{3})$ and
$X_{\pm}=(X_{1}\pm X_{2})$. The Lax operator assumes the form

$L^{V}(z)=v\sigma_{3}+X_{\pm}{\bf e}(\pm 2u)\phi(\pm 2u,z)\sigma_{\pm}\,.$
(5.28)

Expanding as above ${\rm tr}\,(L^{2}(z))$ we find two Hamiltonians

$2H^{V}_{2}=X_{+}X_{-}=X_{1}^{2}-X_{2}^{2}\,,$ (5.29)
$H^{V}_{0}=\frac{1}{2}v^{2}+X_{+}X_{-}\wp(2u)\,,$ (5.30)

where $(v,u)$ are the canonical variables in the Calogero-Moser system. Fix
the value of $H_{2}=\mu^{2}$. Due to (5.27) the Hamiltonian $H_{2}$ is
proportional to the Casimir function (5.11). It plays the role of the coupling
constant in the Hamiltonian $H_{0}$. The constant has the internal degrees of
freedom

$X_{+}X_{-}=(q_{0}p_{1}+q_{1}p_{0})^{2}-(-q_{1}p_{3}+q_{3}p_{1})^{2}\,.$

Thereby, the variables $(p_{j},q_{j})$ have a nontrivial dynamic with respect
to the Hamiltonian $H_{0}$. This dynamic depends on the Calogero-Moser
coordinate $u$, while the dynamic of the Calogero-Moser system depends on the
value of Hamiltonian $H_{2}$ only.

The reductions to the type III and IV bundles lead to the corresponding
generalizations of the Sutherland type systems

$H^{III}_{0}=\frac{1}{2}v^{2}+\frac{X_{+}X_{-}}{\sinh^{2}(2u)}\,,~{}~{}\,\,H^{IV}_{0}=\frac{1}{2}v^{2}+\frac{X_{+}X_{-}}{\sin^{2}(2u)}\,.$

###  5.3 Rational case

Consider ${\rm SL}(2,{\mathbb{C}})$ quasi-antisymmetric Higgs bundles over
${\mathbb{C}}P^{1}$ with $n$ marked points $\mathcal{D}=(x_{1},\ldots,x_{n})$.
The moment equation (2.42) takes the form

$\bar{\partial}L(z,\bar{z})=\sum_{a=1}^{n}\delta(z-x_{a},\bar{z}-\bar{x}_{a}){\bf
X}_{a}\,,$ (5.31)

where (see (5.6))

${\bf X}_{a}=X^{a}_{1}\sigma_{1}+X^{a}_{2}\sigma_{2}+X^{a}_{3}\sigma_{3}\,.$

The moduli space is a result of the symplectic reduction

$\mathcal{M}_{V}({\mathbb{C}}P^{1}\setminus\mathcal{D},{\rm
SL}(2,{\mathbb{C}}))={\rm
SL}(2,{\mathbb{C}})\setminus\setminus(\prod_{a}T^{*}\mathcal{X}{V,a})\,.$

Thereby the dimension of the moduli space is
$\dim\,\mathcal{M}_{V}({\mathbb{C}}P^{1}\setminus\mathcal{D},{\rm
SL}(2,{\mathbb{C}}))=2(2n-3)$.

As above we consider first the GL$(2,{\mathbb{C}})$ bundles. Let ${\bf
X}_{a}=\mathcal{Q}^{a}\mathcal{P}^{a}$ where (see (5.5))

$\begin{array}[]{c}\mathcal{P}^{a}(p^{a}_{0},p^{a}_{1},p^{a}_{3})=p^{a}_{0}\sigma_{0}+p^{a}_{1}\sigma_{1}+p^{a}_{3}\sigma_{3}\,,~{}~{}p^{a}_{j}\in{\mathbb{C}}\,,\\\
\mathcal{Q}^{a}(q^{a}_{0},q^{a}_{1},q^{a}_{3})=q^{a}_{0}\sigma_{0}+q^{a}_{1}\sigma_{1}+q^{a}_{3}\sigma_{3}\,,~{}~{}q^{a}_{j}\in{\mathbb{C}}\,.\end{array}$
(5.32)

Then

$\begin{array}[]{l}X^{a}_{1}=q^{a}_{0}p^{a}_{1}+q^{a}_{1}p^{a}_{0}\,,\\\
X^{a}_{2}=-q^{a}_{1}p^{a}_{3}+q^{a}_{3}p^{a}_{1}\,,\\\
X^{a}_{3}=q^{a}_{0}p^{a}_{3}+q^{a}_{3}p^{a}_{0}\,.\end{array}$ (5.33)

As above $(p^{a}_{j},q^{b}_{k})$ ($j,k=0,1,3)$ are the Darboux coordinates
$\\{p^{a}_{j},q^{b}_{k}\\}=\delta^{ab}\delta_{jk}$. As a consequence for the
collective coordinates we have (see (5.10))

$\\{X^{a}_{\alpha},X^{b}_{\beta}\\}=\delta^{ab}c_{\alpha\beta\gamma}X_{\gamma}\,.$
(5.34)

The Lax operator

$L(z,\bar{z})=\sum_{a=1}^{n}\frac{{\bf X}_{a}}{z-x_{a}}$

is a solution of the equation (5.31). Since

${\rm tr}\,(L^{2})=\sum\frac{({\bf
X}^{2}_{a})}{(z-x_{a})^{2}}+\left(\sum_{b\neq a}\frac{({\bf X}_{a}{\bf
X}_{b})}{x_{b}-x_{a}}\right)\frac{1}{z-x_{a}}$

we come to Gaudin type Hamiltonians:

$H_{2}^{a}=({\bf X}^{2}_{a})\,,~{}~{}H_{1}^{a}=\sum_{b\neq a}\frac{({\bf
X}_{a}{\bf X}_{b})}{x_{b}-x_{a}}\,.$

The Hamiltonians $H_{2}^{a}$ are the Casimir functions with respect to the
brackets (5.34), and thereby they commute with the Hamiltonians $H_{1}^{b}$.
But they are not the Casimir functions in the Poisson algebra defined on the
phase space
$\prod_{a}\,T^{*}($GL$(2,{\mathbb{C}}/$SO${}_{a}(2,{\mathbb{C}}))$. In terms
of ${\bf X}_{a}$ ($a=1,\dots,n$) $L(z)$ is the standard Lax operator for the
Gaudin system. The latter is contained in the phase space
$\mathcal{M}_{V}({\mathbb{C}}P^{1}\setminus\mathcal{D},{\rm
SL}(2,{\mathbb{C}}))$. The number of independent integrals is $2n-3$, while
for the Gaudin system it is equal to $n-3$.

Let $\partial_{a,2}=\\{H_{2}^{a},~{}\\}$ and
$\partial_{a,1}=\\{H_{1}^{a},~{}\\}$. In terms of the Darboux matrices
$\mathcal{P}_{a}$, $\mathcal{Q}^{a}$ (5.32) the equations of motion take the
form

$\begin{array}[]{l}\partial_{a,2}\mathcal{Q}^{b}=\delta^{ab}(\mathcal{Q}^{a}\mathcal{P}^{a}\mathcal{Q}^{a}+(\mathcal{Q}^{a})^{2}\mathcal{P}^{a})\,,\\\
\partial_{a,2}\mathcal{P}^{b}=-\delta^{ab}(\mathcal{Q}^{a}(\mathcal{P}^{a})^{2}+\mathcal{P}^{a}\mathcal{Q}^{a}\mathcal{P}^{a})\,,\\\
\partial_{a,1}\mathcal{Q}^{b}=\frac{1}{x_{b}-x_{a}}\mathcal{Q}^{a}\mathcal{P}^{a}\mathcal{Q}^{b}\,,\\\
\partial_{a,1}\mathcal{Q}^{a}=\sum_{b\neq
a}\frac{1}{x_{b}-x_{a}}\mathcal{Q}^{a}\mathcal{Q}^{b}\mathcal{P}^{b}\,,\\\
\partial_{a,1}\mathcal{P}^{b}=\frac{1}{x_{a}-x_{b}}\mathcal{Q}^{a}\mathcal{P}^{a}\mathcal{P}^{b}\,,\\\
\partial_{a,1}\mathcal{P}^{a}=\sum_{b\neq
a}\frac{1}{x_{a}-x_{b}}\mathcal{P}^{a}\mathcal{Q}^{b}\mathcal{P}^{b}\end{array}$

Consider the anti-involution $\jmath(z)=\bar{z}$ of ${\mathbb{C}}P^{1}$. The
marked points of the invariant set

$(x_{1},\ldots,x_{n})\in\mathcal{D}$

are either real $x_{a}\in{\mathbb{R}}$, or combined in the conjugated pairs
$x_{a}=\bar{x}_{a+1}$.

For the systems of type III the residues satisfy conditions

$\begin{array}[]{l}{\bf X}_{a}=-\bar{\bf X}_{a}~{}{\rm
for~{}}x_{a}\in{\mathbb{R}}\,,\\\ {\bf X}_{a}=-\bar{\bf X}_{a+1}~{}{\rm
for~{}}x_{a}=\bar{x}_{a+1}\,.\end{array}$

The case when all $x_{a}\in{\mathbb{R}}$ corresponds to the real reduction of
the ${\rm SL}(2,{\mathbb{C}})$ Gaudin system.

For the systems of type IV the residues satisfy conditions

$\begin{array}[]{l}{\bf X}_{a}=\bar{\bf X}_{a}~{}{\rm
for~{}}x_{a}\in{\mathbb{R}}\,,\\\ {\bf X}_{a}=\bar{\bf X}_{a+1}~{}{\rm
for~{}}x_{a}=\bar{x}_{a+1}\,.\end{array}$

In both cases the moduli spaces has the same structures as in (5.23) and
(5.24), where the coordinates and momenta acquire the additional index $"a"$
related to the position of the marked point.

##  6 Conclusion

The structures described in this paper are related to the $\mathcal{N}=4$ SUSY
Yang-Mills theory on a four-dimensional space [20, 12, 13].

First, the Higgs bundles are related to the Hitchin equations that can be
defined in the following way. Let $(\Phi,\bar{A})$ be the Higgs data and
$\rho$ is the involutive automorphism $\rho$ of $G^{\mathbb{C}}$ with the
fixed point set to be the maximal compact subgroup $C$ of $G^{\mathbb{C}}$
(B.9). Using the connection $d_{\bar{A}}$ define the unitary connection
$\nabla_{A}=d_{\bar{A}}+d\rho(d_{\bar{A}})$,
$(d\rho(d_{\bar{A}})=\partial_{z}+d\rho(\bar{A}))$

$d_{\bar{A}}=(\partial_{\bar{z}}+A_{\bar{z}})d\bar{z}\,,~{}d_{A}=(\partial_{z}+A_{z})dz\,,~{}~{}A_{\bar{z}}=\bar{A}\,,~{}A_{z}=d\rho(\bar{A})\,,$
(6.1)

where $d\rho$ is the corresponding involutive automorphism acting on the Lie
algebra ${\mathfrak{g}}^{\mathbb{C}}$.

Similarly, for the Higgs field $\Phi$ define the adjoint field
$\Phi^{*}=-d\rho(\Phi)$ and

$\phi=\phi_{z}dz+\phi_{\bar{z}}d\bar{z}\,,~{}~{}\phi_{z}=\Phi\,,~{}\phi_{\bar{z}}=\Phi^{*}\,.$
(6.2)

In these terms the Hitchin equations assume the form [16]

$\displaystyle F(A)-\phi\wedge\phi=0\,,$ (6.3a) $\displaystyle
d_{A}\phi_{\bar{z}}+d_{\bar{A}}\phi_{z}=0\,,$ (6.3b)
$\displaystyle\imath(d_{A}\phi_{\bar{z}}-d_{\bar{A}}\phi_{z})=0\,.$ (6.3c)

The gauge group $\mathcal{G}(\Sigma_{g},C)=\,Map^{C^{\infty}}(\Sigma_{g}\to
C)$ acts on the space of solutions of the Hitchin equations. Let

${\mathfrak{M}}(\Sigma_{g},C)=\mathcal{G}^{C}\setminus\\!\\{\rm
solutions~{}of~{}(\ref{heq})\\}\,.$

be the moduli space. This construction endows ${\mathfrak{M}}(\Sigma_{g},C)$
with the hyper-Kahler structure [19]. Thereby, ${\mathfrak{M}}(\Sigma_{g},C)$
has a $CP^{1}$ family of complex structures.

The last two equations in (6.3) can be combined into the complex form
$d_{A_{\bar{z}}}\phi_{z}=0$. This equation coincides with the moment equation
for the Higgs bundles (2.42) with $n=0$. Recall that the moduli space of the
Higgs bundles is the quotient with respect to the action of the complex gauge
group

$\mathcal{M}(\Sigma_{g},G^{\mathbb{C}})=\\{d_{A_{\bar{z}}}\phi_{z}=0\\}/\mathcal{G}(\Sigma_{g},C)\,.$

In one of these complex structures $I$ the moduli space
${\mathfrak{M}}(\Sigma_{g},C)$ is isomorphic to a smooth part of the Higgs
moduli space $\mathcal{M}^{0}(\Sigma_{g},G^{\mathbb{C}})$ [16, 31].

On the other hand, the Hitchin equations (6.3) arise in the topological
version of the twisted $\mathcal{N}=4$ SUSY Yang-Mills theory on the four-
dimensional space-time $M$ [20]. Let $M={\mathbb{R}}^{2}\times\Sigma_{g,n}$.
There is a reduction of this four-dimensional gauge theory to the two-
dimensional sigma-model defined by the map of ${\mathbb{R}}^{2}$ to the moduli
space ${\mathfrak{M}}(\Sigma_{g},C)$.

The moduli space ${\mathfrak{M}}(\Sigma_{g,n},C)$ of the Hitchin equations
defined on multipoint curves is a result of including in the four-dimensional
gauge theory two-dimensional surface operators [12]. Let $\tilde{D}$ be a two-
dimensional surface $\tilde{D}\subset M$ that intersects transversally with
the curve $\Sigma_{g,n}$ along the set of the marked points
$\tilde{D}\cap\Sigma_{g,n}=\mathcal{D}(x_{1},\ldots,x_{n})$. As it is
explained in [13] the parabolic Higgs bundles over multipoint curves are
described by the interactions of the $\mathcal{N}=4$ Yang-Mills theory with a
sigma-model supported on the surface $\tilde{D}$. The sigma-model on
$\tilde{D}$ is called the surface operator. For the parabolic bundles the
target spaces of the sigma-model are the co-adjoint $G^{\mathbb{C}}$-orbits.
These spaces are hyper-Kahler and the total space
${\mathfrak{M}}_{par}(\Sigma_{g,n},C)$ is also hyper-Kahler [12]. It was
established in [23] that the cotangent bundles $T^{*}G^{\mathbb{C}}$ is a
hyper-Kahler variety. Apparently, the whole theory describing the interaction
of the $\mathcal{N}=4$ Yang-Mills theory with a sigma-model with the target
spaces $T^{*}G^{\mathbb{C}}$ is not hyper-Kahler, but only Kahler. This
phenomenon is most likely to occur in the case of $T^{*}\mathcal{X}_{V}$. It
means that including the surface operators of these types breaks the
$\mathcal{N}=4$ SUSY to $\mathcal{N}=2$.

It was found in [1] that the fixed point sets of the real involutions (1.5)
are special branes $\mathcal{B}$ in the space
$\mathcal{M}^{0}(\Sigma_{g},G^{\mathbb{C}})$ corresponding to different
complex structures in ${\mathfrak{M}}(\Sigma_{g},G^{\mathbb{C}})$. We can
apply the same involutions to the parabolic bundles and come to integrable
systems. Let $H_{a}$ be the Cartan subgroup of $G^{\mathbb{R}}$ and $T_{a}$ is
the Cartan torus of the compact group $C$. The spin variables belong to the
following types of orbits. The real orbits $H_{a}^{\mathbb{R}}\setminus
G^{\mathbb{R}}$ are attached to the marked points of the fixed point set of
the involution $\iota^{\sigma}$ and the flag varieties $T_{a}\setminus C$ are
attached to the marked of the fixed point set of the involution
$\iota^{\rho}$.

APPENDIX

##  Appendix A Two equivalent constructions of symplectic quotients

Let $H$ be a subgroup if a group $G$ and $X$ be a $G$-set. We consider two
quotient spaces: the $\alpha$-model $(X\times H\backslash G)/G$ and the
$\beta$-model $X/H$ Our aim is to prove they are isomorphic.

###### Proposition A.1

The $\alpha$ and the $\beta$ models are isomorphic ($X/H\sim(X\times
H\backslash G)/G$).

_Proof_
First we construct a map

$\zeta:X/H\to(X\times H\backslash G)/G\,.$

Denote by $[g]$ the coset $Hg$. The map

$\tilde{\eta}\,:\,X\to X\times H\backslash G\,,~{}x\to(x,[1_{G}])$

is $H$-invariant. Hence we have the induced map

$\eta:X/H=(X\times H\backslash G)/H\,.$

The required map $\alpha$ is composition of $\eta$ with the natural projection
$\zeta=\pi\circ\eta$

$\pi\,:\,(X\times G/H)/H\to(X\times H\backslash G)/G\,.$

Second we prove that $\zeta$ is injective. Let $\zeta(x)=\zeta(y)$. So
$(y,[1_{G}])$ belongs to $G$ orbit of $(x,[1_{G}])$:
$(y,[1_{G}])=(g(x)x,g([1_{G}]))$ for some $g\in G$. But $g([1_{G}])=[g]$ so
$[g]=[1_{G}]$ and $g\in H$. We get $y=g(x)\in Hx$.

Third we prove surjectivity. Any element $(X,[g])$ in $X\times G/H$ belongs to
$G$ orbit of the element $g^{-1}(X,[g])=(g^{-1}(X),[1_{G}])\in\eta(X)$. $\Box$

Let $G$ and $H$ be Lie groups and $M$ is a symplectic manifold with
Hamiltonian action of $G$.

###### Proposition A.2

$M//_{\nu}H=(M\times H\backslash\backslash_{\nu}T^{*}G)//G$

If $\mu\in$Lie${}^{*}(H)$ is the moment map defining the symplectic reductions
then in the notations $\backslash\backslash_{\nu}$, $//_{\nu}$ $\nu$ means the
value of $\mu$.

_Proof._ We start from construction of a map. Denote by $\mu_{G}$ ($\mu_{H}$)
the momentum map to Lie coalgebra ${\mathfrak{g}}^{*}$ (${\mathfrak{h}}^{*}$).
Since ${\mathfrak{h}}\subset{\mathfrak{g}}$ one has dual projection
$\pi:{\mathfrak{g}}^{*}\to{\mathfrak{h}}^{*}$. As the action of $H$ is the
restriction of the action of $G$, $\mu_{H}=\pi\circ\mu_{G}$

The total space of the cotangent bundle $T^{*}G$ equals to
$G\times{\mathfrak{g}}^{*}$ and the right action of the group on itself
induces coadjoint action ${\rm Ad}^{*}$ on the coalgebra. The momentum of this
action is just the opposite to identical map on the second factor. Hence an
element of $H\backslash\backslash_{\nu}T^{*}G$ can be described as a pair
$([g],\theta)$, $[g]$ be a coset as above and $\theta\in{\mathfrak{g}}^{*}$
such that $\pi(\theta)=-\nu$.

The Hamiltonian quotient $M//_{\nu}H$ is the quotient by action of $H$ of the
momentum level $M^{\nu}=\mu_{H}^{-1}(\nu)$.

Define the map

$\tilde{\eta}_{H}:M\to M\times
H\backslash\backslash_{\nu}T^{*}G\,:X\to(X,([1_{G}],-\mu_{G}(X)))\,.$

Since $\pi(-\mu_{G}(X))=-\mu_{H}(X)=-\nu$ the second component
$([1_{G}],-\mu_{G}(X))$ lies in $H\backslash\backslash_{\nu}T^{*}GH$.

The $G$-momentum of an element $(X,([g],\theta))$ in the product $M\times
H\backslash\backslash_{\nu}T^{*}G$ is equal to $\mu_{G}(X)+\theta$. So the
image of the map $\tilde{\eta}_{H}$ lies in the zero $G$-momentum level
$(M\times H\backslash\backslash_{\nu}T^{*}G)^{\mu_{G}=0}$.

Evidently, the map $\tilde{\eta}_{H}$ is $H$-invariant. Hence we have induced
map

$\eta_{H}:HM^{\nu}/H=(M\times
H\backslash\backslash_{\nu}T^{*}G)^{\mu_{G}=0}/H\,.$

Its composition with natural projection

$H/(M\times H\backslash\backslash_{\nu}T^{*}G)^{\mu_{G}=0}\to G/(M\times
H\backslash\backslash_{\nu}T^{*}G)^{\mu_{G}=0}$

is the required map $\zeta_{H}$.

Injectivity of the map $\zeta_{H}$ follows from the fact that the stabilizer
of the trivial coset $[1_{G}]$ is exactly $H$, so the proof is very close to
the proof of injectivity in Prop 1.

For surjectivity we (again like in Prop 1) we choose in $G$\- orbit a
representative $(X,([1_{G}],\theta))$ with trivial coset as component. As the
$G$-momentum $\mu_{G}(X)+\theta$ should vanishes, $\theta=-\mu_{G}(X)$, hence
this element is $\tilde{\eta}_{H}(X)$. $\Box$

###### Proposition A.3

The isomorphism $\zeta_{H}$ from the previous proposition is symplectic.

_Proof_
The symplectic form on the symplectic quotient is uniquely determined by the
following property: its pull-back to the momentum level coincides with
restriction of the initial symplectic form.

Since the restriction of the symplectic form on $T^{*}G$ to the fiber
${\mathfrak{g}}^{*}$ vanishes, the map $\tilde{\eta}_{H}$ is symplectic. Since
$\zeta$ is injective, a $G$-orbit in $M\times
H\backslash\backslash_{\nu}T^{*}G$ cut on the image of $M$ the image under
$\tilde{\eta}_{H}$ of some $H$-orbit in $M$ on the image of $M$. Hence the
pullback of the restriction of the form to $(M\times
H\backslash\backslash_{\nu}T^{*}G)^{\mu_{G}=0}$ coincides with the restriction
to $M^{\nu}$ and the corresponding projections to Hamiltonian quotients are
equal. These forms are pullback under projections of the symplectic forms on
the Hamiltonian quotients, hence this symplectic forms coincides. $\Box$

##  Appendix B Simple Lie groups: notations and decompositions

Let $G^{\mathbb{C}}$ be a simple complex Lie group,
${\mathfrak{g}}^{\mathbb{C}}$ its Lie algebra of rank $l$ and
${\mathfrak{h}}^{\mathbb{C}}$ is a Cartan subalgebra.

##### Chevalley basis in ${\mathfrak{g}}$.

Let $\\{\alpha\in R\\}$ be the root system, and $R^{\pm}$ are subsystems of
positive (negative) roots with respect to some ordering.

The algebra ${\mathfrak{g}}^{\mathbb{C}}$ has the root basis

$\\{E_{\alpha}\,,\,\alpha\in R\\}\,,\,\\{e_{1},\ldots,e_{l}\\},,$ (B.1)

where $(e_{j})$ is a basis in ${\mathfrak{h}}^{\mathbb{C}}$.

There are two sets of generators we use here:

$\\{(E_{\alpha}-E_{-\alpha})\,,\,\imath(E_{\alpha}+E_{-\alpha})\,,~{}\alpha\in
R^{+}\\}\,,\,\\{\imath e_{1},\ldots,\imath e_{l}\\}\,.$ (B.2)
$\\{(E_{\alpha}-E_{-\alpha})\,,\,(E_{\alpha}+E_{-\alpha})\,,~{}\alpha\in
R^{+}\\}\,,\,\\{e_{1},\ldots,e_{l}\\}\,.$ (B.3)

_Centers of simple groups_.
Consider the dual root system
$R^{\vee}=\\{\alpha^{\vee}=\frac{2\alpha}{(\alpha,\alpha)}\,|\,\alpha\in R$,
and let $Q^{\vee}$ be the coroot lattice in ${\mathfrak{h}}^{\mathbb{C}}$. The
coweight lattice $P^{\vee}\subset{\mathfrak{h}}^{\mathbb{C}}$ is the dual
lattice to the root lattice $Q$ generating by $R$. Let $G^{\mathbb{C}}$ be an
universal covering group. The group $G^{\mathbb{C}}$ is simply-connected and
in all cases apart $G_{2}$, $F_{4}$ and $E_{8}$ has a non-trivial center
$\mathcal{Z}(G^{\mathbb{C}})$. The center is isomorphic to the quotient group
$P^{\vee}/Q^{\vee}$.

$G^{\mathbb{C}}$ | $\mathcal{Z}(\bar{G})$
---|---
SL$(n,{\mathbb{C}})$ | $\mu_{n}$
Spin${}_{2n+1}({\mathbb{C}})$ | $\mu_{2}$
Sp${}_{n}({\mathbb{C}})$ | $\mu_{2}$
Spin${}_{4n}({\mathbb{C}})$ | $\mu_{2}\oplus\mu_{2}$
Spin${}_{4n+2}({\mathbb{C}})$ | $\mu_{4}$
$E_{6}({\mathbb{C}})$ | $\mu_{3}$
$E_{7}({\mathbb{C}})$ | $\mu_{2}$

Table 4
Centers of universal covering groups

($\mu_{N}={\mathbb{Z}}/N{\mathbb{Z}}$)

The factor-group $P^{\vee}/Q^{\vee}$ is isomorphic to the center
$\mathcal{Z}(\bar{G})$ of simply-connected group $\bar{G}$. It is a cyclic
group except ${\mathfrak{g}}=D_{4l}$. The adjoint group $G^{ad}$ is the factor
group

$G^{ad}=G^{\mathbb{C}}/\mathcal{Z}(\bar{G})\,.$ (B.4)

Let $H^{\mathbb{C}}$ be a Cartan subgroup
(Lie$(H^{\mathbb{C}})={\mathfrak{h}}^{\mathbb{C}})$ and $N^{\pm}$ be the
unipotent subgroups corresponding to the Lie algebras
${\mathfrak{n}}^{\pm}=\sum_{\beta\in R^{\pm}}c_{\beta}E_{\beta}$. Thus
${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{n}}^{-}+{\mathfrak{h}}^{\mathbb{C}}+{\mathfrak{n}}^{+}$.
At the group level there is the Bruhat decomposition

$G^{\mathbb{C}}=N^{-}\cup_{w\in
W}B^{\mathbb{C}}\,,~{}~{}({\mathfrak{n}}^{-}=Lie(N^{-}))\,,$ (B.5)

where $B^{\mathbb{C}}$ is the Borel subgroup

$B^{\mathbb{C}}=H^{\mathbb{C}}N^{+}\,,~{}~{}({\mathfrak{n}}^{+}=Lie(N^{+}))\,,$
(B.6)

and $W$ is the Weyl group.

##### Real forms.

Let $\varsigma$ be the involutive automorphism of the group $G^{\mathbb{C}}$,
i.e. $\varsigma^{2}=Id$ and $\varsigma\neq Id$. The action of $d\varsigma$ on
the Lie algebra ${\mathfrak{g}}^{\mathbb{C}}$ defines the decomposition

${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{k}}+{\mathfrak{x}}\,,~{}~{}d\varsigma({\mathfrak{k}})={\mathfrak{k}}\,,~{}d\varsigma({\mathfrak{x}})=-{\mathfrak{x}}\,.$
(B.7)

Here ${\mathfrak{k}}$ is the invariant Lie subalgebra
$[{\mathfrak{k}},{\mathfrak{k}}]={\mathfrak{k}}$. Let $K$ be the corresponding
Lie groups. Then $K$ is the fixed set subgroup with respect to the involution
$\varsigma$. The coset space $\mathcal{X}=G/K$ is the symmetric space. [4,
15]. If $g^{*}=\varsigma(g^{-1}$ then

$\mathcal{Q}=g^{*}g$ (B.8)

belongs to $\mathcal{X}$.

We define three involutive automorphisms and the corresponding three types of
symmetric spaces.
I. Let $d\rho$ be the involutive automorphism with the fixed point subalgebra
${\mathfrak{c}}$ is the maximal compact subalgebra. It can be defined by the
action on
$x=\sum_{\alpha}x_{\alpha}E_{\alpha}+\sum_{j}x_{j}e_{j}\in{\mathfrak{g}}^{\mathbb{C}}$
(B.1) as

$d\rho\,:\,x_{\alpha}\to-\bar{x}_{-\alpha}\,,~{}x_{j}\to-\bar{x}_{j}\,.$ (B.9)

The corresponding basis of ${\mathfrak{c}}$ is (B.2). The algebra
${\mathfrak{c}}$ is real and coefficients of $x\in{\mathfrak{c}}$ in the
expansion in these basis are real. The corresponding subgroup of
$G^{\mathbb{C}}$ is the maximal compact subgroup $C$ and the quotient
$\mathcal{X}_{I}=G^{\mathbb{C}}/C$ is noncompact Riemannian symmetric space.
II. Let ${\mathfrak{g}}^{\mathbb{R}}$ be the normal real form of
${\mathfrak{g}}^{\mathbb{C}}$. It has the same basis as
${\mathfrak{g}}^{\mathbb{C}}$ (B.1) with real coefficients. This subalgebra is
invariant under the action

$d\sigma(x)=\bar{x}\,.$ (B.10)

The symmetric space $\mathcal{X}_{II}=G^{\mathbb{C}}/G^{\mathbb{R}}$ is non-
compact with pseudo-Riemannian metric.
V. The involutive automorphisms $\sigma$ and $\rho$ commutes. The automorphism
$d\theta=d\sigma\circ d\rho$ acts on ${\mathfrak{g}}^{\mathbb{C}}$ as

$d\theta\,:\,e_{j}\to-e_{j}\,,~{}E_{\alpha}\to-E_{-\alpha}\,.$

The invariant subalgebra ${\mathfrak{k}}$ in (B.7) is the complex subalgebra
${\mathfrak{u}}^{\mathbb{C}}$

${\mathfrak{k}}={\mathfrak{u}}^{\mathbb{C}}=\left\\{\sum_{\alpha\in
R^{+}}c_{\alpha}(E_{\alpha}-E_{-\alpha})\,,~{}c_{\alpha}\in{\mathbb{C}}\right\\}$
(B.11)

and

${\mathfrak{x}}=\left\\{\sum c_{j}e_{j}+\sum_{\alpha\in
R^{+}}c_{\alpha}(E_{\alpha}+E_{-\alpha})\,,~{}c_{j}\,,\,c_{\alpha}\in{\mathbb{C}}\right\\}\,.$
(B.12)

Let $U^{\mathbb{C}}$ be the invariant subgroup of $G^{\mathbb{C}}$
($\theta(U^{\mathbb{C}})=U^{\mathbb{C}}$). The quotient
$\mathcal{X}_{V}=G^{\mathbb{C}}/U^{\mathbb{C}}$ is the complex pseudo-
Riemannian symmetric space. Let ${\mathfrak{h}}^{\mathbb{C}}$ be a Cartan
subalgebra ${\mathfrak{h}}^{\mathbb{C}}\subset{\mathfrak{g}}^{\mathbb{C}}$. It
follows from (B.12) that ${\mathfrak{h}}^{\mathbb{C}}\subset{\mathfrak{x}}$.
Similar the Cartan subgroup $H^{\mathbb{C}}$ of $G^{\mathbb{C}}$ is

$H^{\mathbb{C}}\subset\mathcal{X}_{V}\,.$ (B.13)

In addition, consider another two types symmetric spaces.
III. Consider the defined above maximal compact subalgebra ${\mathfrak{c}}$.
The involutive automorphism $d\sigma$ acts on ${\mathfrak{c}}$ as
($d\sigma(x)=\bar{x}$). It follows from (B.3) that the invariant subalgebra
$gu$ has the basis (B.11), where the coefficients $x_{\alpha}$ are real. Let
$U$ be the corresponding Lie group. Then the quotient $\mathcal{X}_{III}=C/U$
is compact Riemannian symmetric space.
IV. Consider action of the involutive automorphism $d\rho$ on the subalgebra
${\mathfrak{g}}^{\mathbb{R}}$ The invariant subalgebra coincides with
${\mathfrak{u}}$. It is the maximal compact subalgebra of
${\mathfrak{g}}^{\mathbb{R}}$. The quotient $G^{\mathbb{R}}/U$ is a real non-
compact Riemannian symmetric space $\mathcal{X}_{IV}=G^{\mathbb{R}}/U$. The
spaces $\mathcal{X}_{III}$ and $\mathcal{X}_{IV}$ are Cartan dual symmetric
spaces. The considered symmetric spaces are arranged into the Table:

$\mathcal{X}$ | $G$ | $K$
---|---|---
$\mathcal{X}_{I}$ | $G^{\mathbb{C}}$ | $C$
$\mathcal{X}_{II}$ | $G^{\mathbb{C}}$ | $G^{\mathbb{R}}$
$\mathcal{X}_{III}$ | $C$ | $U$
$\mathcal{X}_{IV}$ | $G^{\mathbb{R}}$ | $U$
$\mathcal{X}_{V}$ | $G^{\mathbb{C}}$ | $U^{\mathbb{C}}$

Table 5
Symmetric spaces

The concrete forms of the invariant subgroups are presented in Table 6.

$G^{\mathbb{C}}$ | $G^{\mathbb{R}}$ | $C$ | $U$ | $U^{\mathbb{C}}$
---|---|---|---|---
${\rm SL}(N,{\mathbb{C}})$ | SL$(N,{\mathbb{R}})$ | SU$(N)$ | SO$(N)$ | SO$(N,{\mathbb{C}})$
SO$(2N\\!+\\!1,{\mathbb{C}})$ | SO$(N\\!+\\!1,N)$ | SO$(2N\\!+\\!1)$ | SO$(N\\!+\\!1)\times$SO$(N)$ | SO$(N\\!+\\!1,{\mathbb{C}})\times$SO$(N,{\mathbb{C}})$
Sp$(N,{\mathbb{C}})$ | Sp$(N,{\mathbb{R}})$ | Sp$(N)$ | U$(N)$ | GL$(N,{\mathbb{C}})$
SO$(2N,{\mathbb{C}})$ | SO$(N,N)$ | SO$(2N)$ | SO$(N)\times$SO$(N)$ | SO$(N,{\mathbb{C}})\times$SO$(N,{\mathbb{C}})$
G${}^{\mathbb{C}}_{2}$ | G${}^{\mathbb{R}}_{2}$ | G2 | SU$(2)\times$SU$(2)$ | SL$(2,{\mathbb{C}})\times$SL$(2,{\mathbb{C}})$
F${}_{4}^{\mathbb{C}}$ | F${}_{4}^{\mathbb{R}}$ | F4 | Sp$(3)\times$SU$(2)$ | Sp$(3,{\mathbb{C}})\times$SL$(2,{\mathbb{C}})$
E${}_{6}^{\mathbb{C}}$ | E${}_{6}^{\mathbb{R}}$ | E6 | Sp$(4)$ | Sp$(4,{\mathbb{C}})$
E${}_{7}^{\mathbb{C}}$ | E${}_{7}^{\mathbb{R}}$ | E7 | SU$(8)$ | SL$(8,{\mathbb{C}})$
E${}_{8}^{\mathbb{C}}$ | E${}_{8}^{\mathbb{R}}$ | E8 | SO$(16)$ | SO$(16,{\mathbb{C}})$

Table 6
Groups $G^{\mathbb{C}}$, $G^{\mathbb{R}}$, $C$, $U$, $U^{\mathbb{C}}$

These construction can be arranged in the following diagrams.

$\textstyle{G^{\mathbb{C}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G^{\mathbb{R}}}$$\scriptstyle{\sigma}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\rho}$
(B.14)

and

$G^{\mathbb{C}}\stackrel{{\scriptstyle\theta}}{{\longleftarrow}}U^{\mathbb{C}}\,.$
(B.15)

Here arrows mean embeddings as the fixed point sets of the involutive
automorphisms of the source objects to the target objects. There are the
similar relations between symmetric spaces

$\begin{array}[]{ccccc}\mathcal{X}_{II}=G^{\mathbb{R}}\setminus
G^{\mathbb{C}}&&\mathcal{X}_{V}=U^{\mathbb{C}}\setminus
G^{\mathbb{C}}&&\mathcal{X}_{I}=C\setminus G^{\mathbb{C}}\\\
\qquad\qquad\nwarrow\rho&&\nearrow\rho\qquad\qquad\sigma\nwarrow&&\nearrow\sigma\\\
&\mathcal{X}_{III}=U\setminus C&&\mathcal{X}_{IV}=U\setminus
G^{\mathbb{R}}\end{array}$ (B.16)

##### Dimensions of algebras.

Let $d_{j}$ be the order of the invariants of algebra
${\mathfrak{g}}^{\mathbb{C}}$ and ${\rm rank}({\mathfrak{g}}^{\mathbb{C}})=l$

$d_{1}=2,\ldots,d_{l}=h\,,~{}~{}j=1,\ldots,l\,,$ (B.17)

where $h$ is the Coxeter number. In terms of invariants

$\dim_{{\mathbb{C}}}\,G^{\mathbb{C}}=\sum_{j=1}^{l}(2d_{j}-1)=2\sum_{j=1}^{l}d_{j}-l\,.$
(B.18)

It coincides with the real dimension of the maximal compact subgroup $C$

$\dim_{{\mathbb{R}}}\,C=\sum_{j=1}^{l}(2d_{j}-1)\,,$ (B.19)

and the normal form $G^{\mathbb{R}}$

$\dim_{{\mathbb{R}}}\,G^{\mathbb{R}}=\dim_{{\mathbb{R}}}\,C=\sum_{j=1}^{l}(2d_{j}-1)\,.$
(B.20)

For $\mathcal{X}_{I}=G^{\mathbb{C}}/C$ from (B.18) and (B.19) we have

$\dim_{{\mathbb{R}}}\,\mathcal{X}_{I}=\sum_{j=1}^{l}(2d_{j}-1)=\dim_{{\mathbb{R}}}\,C\,.$
(B.21)

The same dimension we have for $\mathcal{X}_{II}=G^{\mathbb{R}}/C$

$\dim_{{\mathbb{R}}}\,\mathcal{X}_{II}=\sum_{j=1}^{l}(2d_{j}-1)=\dim_{{\mathbb{R}}}\,C\,.$
(B.22)

The dimension of the compact form $U$ is

$\dim_{\mathbb{R}}\,U=\frac{1}{2}(\dim_{\mathbb{R}}\,C-l)=\sum_{j=1}^{l}(d_{j}-1)\,.$
(B.23)

Similarly,

$\dim_{\mathbb{C}}\,U^{\mathbb{C}}=\sum_{j=1}^{l}(d_{j}-1)\,.$ (B.24)

Then

$\dim_{{\mathbb{C}}}\,(\mathcal{X}_{V})=\dim_{\mathbb{C}}\,(G^{\mathbb{C}})-\dim_{\mathbb{C}}\,(U^{\mathbb{C}})=\sum_{j=1}^{l}d_{j}$
(B.25)

From (B.20) and (B.23) for the symmetric spaces is
$\mathcal{X}_{IV}=G^{\mathbb{R}}/U$ and $\mathcal{X}_{III}=C/U$ we have

$\dim_{{\mathbb{R}}}\,\mathcal{X}_{III}=\dim_{{\mathbb{R}}}\,\mathcal{X}_{IV}=\sum_{j=1}^{l}d_{j}=\frac{1}{2}(\dim_{{\mathbb{R}}}\,\mathcal{X}_{I}+l)\,.$
(B.26)

Let $R^{+}$ be a subset of positive roots and $B^{\mathbb{C}}$ is the
corresponding Borel subgroup of $G^{\mathbb{C}}$. The flag variety ${\rm
Fl}^{\mathbb{C}}$ is the homogeneous space ${\rm
Fl}^{\mathbb{C}}(G^{\mathbb{C}})=G^{\mathbb{C}}/B$. It has dimension

$\dim_{{\mathbb{C}}}\,{\rm
Fl}^{\mathbb{C}}(G^{\mathbb{C}})=\sum_{j=1}^{l}(d_{j}-1)\,.$ (B.27)

From (B.21) we have

$\dim_{{\mathbb{C}}}\,\mathcal{X}_{V}=\dim_{{\mathbb{C}}}\,{\rm
Fl}^{\mathbb{C}}+l\,.$ (B.28)

##  Appendix C Cotangent bundles and coadjoint orbits

Let $G$ be a real simple Lie group and ${\mathfrak{g}}$ its Lie algebra. We
identify ${\mathfrak{g}}$ with the Lie coalgebra ${\mathfrak{g}}^{*}$ by means
the invariant metric on ${\mathfrak{g}}$.

The cotangent bundle $T^{*}G\sim T\,G$ to the group $G$ is identified with the
pair

$T^{*}G={\mathfrak{g}}\times G=\\{\zeta\in{\mathfrak{g}}\,,\,g\in G\\}\,.$
(C.1)

using its trivialization. It is equipped with the symplectic form

$\omega=D(\zeta,Dgg^{-1})\,,~{}~{}\zeta\in{\mathfrak{g}}\,,~{}g\in G\,.$ (C.2)

The form is invariant under the left action of $G$

$\zeta\to f\zeta f^{-1}\,,~{}~{}g\to fg$ (C.3)

producing the left moment

$\mu^{L}(\zeta,g)=\zeta\,.$ (C.4)

The right action

$\zeta\to\zeta\,,~{}~{}g\to gf^{-1}\,,~{}~{}f\in G$ (C.5)

leads to the moment

$\mu^{R}(\zeta,g)=g^{-1}\zeta g\,.$ (C.6)

Co-adjoint orbits of complex groups
There are two ways to construct the co-adjoint $G^{\mathbb{C}}$ orbits from
the cotangent bundle $T^{*}G^{\mathbb{C}}$.

1\. Let $G=G^{\mathbb{C}}$ be the complex simple Lie group. Assume that in
(C.3) $f\in B^{\mathbb{C}}$ \- the Borel subgroup (B.6). Remind that the
quotient $B^{\mathbb{C}}\setminus G^{\mathbb{C}}$ is the flag variety
$Fl^{\mathbb{C}}$. The moment $\mu^{L}$ (C.6) of the left
$B^{\mathbb{C}}$-action takes value in the subalgebra
${\mathfrak{b}}^{*}={\mathfrak{h}}^{\mathbb{C}}\oplus{\mathfrak{n}}^{-}$ and
is equal to

$\mu^{L}(\zeta,g)=Pr|_{{\mathfrak{b}}^{*}}\zeta\,.$ (C.7)

Fix its value as

$Pr|_{{\mathfrak{b}}^{*}}(\zeta)=\nu\in{\mathfrak{h}}^{\mathbb{C}}\,.$ (C.8)

It means that solution of the moment equation is

$\zeta=\nu+\xi\,,~{}~{}\xi\in{\mathfrak{n}}^{+}\,,$ (C.9)

where $\nu$ is fixed and $\xi$ is an arbitrary element of
${\mathfrak{n}}^{+}$. It follows from (C.3) and (C.9) that the symplectic
quotient $B^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}G^{\mathbb{C}}$ is
defined by the pairs

$B^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}G^{\mathbb{C}}=\\{(g,\zeta)\,|\,1.\,g\sim
fg\,,\,f\in B^{\mathbb{C}}\,,~{}~{}2.\,\zeta~{}{\rm
satisfy}~{}(\ref{sme})\\}\,.$ (C.10)

The reduced symplectic manifold is

$B^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}G^{\mathbb{C}}=B^{\mathbb{C}}\setminus(\mu^{L})^{-1}(\nu)\,.$
(C.11)

Following (B.5) fix the gauge by taking $g=z\in N^{-}$.

Notice, that the coadjoint action of $B^{\mathbb{C}}$ on
$\xi\in{\mathfrak{n}}^{+}$ in (C.9) is the affine action due to the $\nu$
term. The group $B^{\mathbb{C}}$ acts free on ${\mathfrak{n}}^{+}$. It means
that the symplectic quotient
$B^{\mathbb{C}}\setminus\setminus_{\nu}T^{*}G^{\mathbb{C}}$ is the principal
homogeneous space $PH/T^{*}(B^{\mathbb{C}}\setminus G^{\mathbb{C}}/)$ over the
cotangent bundle to the flag variety ${\rm
Fl}^{\mathbb{C}}=B^{\mathbb{C}}\setminus G^{\mathbb{C}}$

$B^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}G^{\mathbb{C}}/\sim
PH/T^{*}(B^{\mathbb{C}}\setminus G^{\mathbb{C}})\,.$ (C.12)

The cotangent bundle $T^{*}{\rm Fl}^{\mathbb{C}}$ corresponds to the choice
$\nu=0$.

The symplectic form on $PH/T^{*}(G^{\mathbb{C}}/B^{\mathbb{C}})$ is

$D(\zeta,Dgg^{-1})\,,~{}~{}(\zeta=\nu+\xi)\,.$ (C.13)

In the coordinates $(z\in N^{-},\xi)$ it takes the form

$D(\zeta,Dzz^{-1})\,.$

Then $(\xi\in{\mathfrak{n}}^{+},\,\log g\in{\mathfrak{n}}^{-})$ are Darboux
coordinates on $T^{*}G^{\mathbb{C}}//B^{\mathbb{C}}$.

The form is invariant under the right $G^{\mathbb{C}}$-action $g\to gf^{-1}$
and $\zeta\to\zeta$. The corresponding moment (C.6) is

$\mu^{R}=g^{-1}\zeta g=g^{-1}(\nu+\xi)g\,.$ (C.14)

2\. Assume now that in (C.3) $f\in G^{\mathbb{C}}$ and consider the left
symplectic quotient
$G^{\mathbb{C}}\setminus\setminus_{\nu}T^{*}(G^{\mathbb{C}})$. with the moment
taking value in the Cartan subalgebra ${\mathfrak{h}}^{\mathbb{C}}$

$\mu^{L}(\zeta,g)=\zeta=\nu\in({\mathfrak{h}}^{{\mathbb{C}}})^{*}\sim{\mathfrak{h}}^{{\mathbb{C}}}\,.$
(C.15)

Here $\nu$ is a fixed regular element of ${\mathfrak{h}}^{{\mathbb{C}}}$. The
subgroup preserving this value is the Cartan subgroup $H^{\mathbb{C}}$. Thus
the symplectic quotient is of the form:

$G^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}(G^{\mathbb{C}})=\\{(\zeta,g)\,|\,\zeta=\nu\in{\mathfrak{h}}^{\mathbb{C}}\,,~{}g\sim
fg\,,\,f\in H^{\mathbb{C}}\\}\,.$ (C.16)

It is the co-adjoint orbit

$G^{\mathbb{C}}\setminus\\!\setminus_{\nu}T^{*}(G^{\mathbb{C}})\sim\mathcal{O}_{\nu}=H^{\mathbb{C}}\setminus
G^{\mathbb{C}}=\\{{\bf S}=g^{-1}\nu g\\}\,.$ (C.17)

After substituting $\zeta=\nu$ (C.15) in (C.2) the form $\omega$ on
$T^{*}G^{\mathbb{C}}$ becomes the Kirillov-Kostant on $\mathcal{O}_{\nu}$ form

$\omega^{KK}=D(\nu,Dgg^{-1})\,.$ (C.18)

The form is invariant under the right $G^{\mathbb{C}}$ action (C.3). This
transformation generates the moment (C.14)

$\mu^{R}=g^{-1}\nu g$ (C.19)

(compare with (C.14)).

The representation of the orbit (C.17) is isomorphic to (C.12). To prove it
fix the gauge action (C.3) by the choice $\xi=0$ in (C.9). The left action of
the Cartan subgroup $H^{\mathbb{C}}$ preserves the gauge. It means that the