Real Hamiltonian forms of affine Toda field
theories: spectral aspects

Vladimir S. Gerdjikova,b,111E-mail<EMAIL_ADDRESS>Georgi G.
Grahovskic,222E-mail<EMAIL_ADDRESS>and Alexander A. Stefanova,e,333E-mail:
<EMAIL_ADDRESS>
a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8
Acad. G. Bonchev str., 1113 Sofia, Bulgaria
bInstitute for Advanced Physical Studies, 111 Tsarigradsko chaussee, Sofia
1784, Bulgaria
c Department of Mathematical Sciences, University of Essex, Wivenhoe Park,
Colchester, UK
d Faculty of Mathematics and Informatics, Sofia University ”St. Kliment
Ohridski” 5 James Bourchier Blvd., 1164 Sofia, Bulgaria

###### Abstract

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field
theories related to exceptional simple Lie algebras, and to the spectral
theory of the associated Lax operators. Real Hamiltonian forms are a special
type of “reductions” of Hamiltonian systems, similar to real forms of semi-
simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field
theories related to exceptional complex untwisted affine Kac-Moody algebras
are studied. Along with the associated Lax representations, we also formulate
the relevant Riemann-Hilbert problems and derive the minimal sets of
scattering data that determine uniquely the scattering matrices and the
potentials of the Lax operators.

##  1 Introduction

Affine Toda field theories [45, 8, 9, 47, 48] received considerable attention
of the Mathematical physics community in the past three decades and are one of
the best understood integrable massive field theories at classical and quantum
levels in $1+1$ dimensions [4]. The interest in Toda field theories was
inspired by the work of A. Zamolodchikov [54] on deformation of conformal
field theories preserving integrabilty. It was shown in [54] that the
resultant theory is characterised by eight masses related to the Cartan matrix
of $E_{8}$, and by integrals of motion with spins given by the exponents of
$E_{8}$ modulo its Coxeter number.

Affine Toda field theories (ATFT) are integrable models of real scalar field
${\bf q}=(q_{1},\dots q_{n})$, in one space dimension with exponential
interactions. The Lagrangian of the theory is given by

${\cal L}[{\bf q}]={1\over 2}(\partial_{\mu}{\bf q}\cdot\partial^{\mu}{\bf
q})-{m^{2}\over\beta^{2}}\sum_{k=0}^{n}n_{k}\left({\rm
e}^{\beta(\alpha_{k}\cdot{\bf q})}-1\right),$ (1)

where the field ${\bf q}(x,t)$ is an $n$-dimensional vector. This corresponds
to equations of motion of the form

${\partial^{2}{\bf q}\over\partial x\partial
t}=\sum_{j=0}^{r}n_{j}\alpha_{j}e^{-(\alpha_{j},{\bf q}(x,t))},$ (2)

Each ATFT is associated to a (finite) simple Lie algebra ${\mathfrak{g}}$ [3,
4]. Here $n$ is the rank of ${\mathfrak{g}}$, $\alpha_{k}$s ($k=1,\dots,n$)
are the simple roots of ${\mathfrak{g}}$ and $\alpha_{0}$ is the minimal root.
Thus, (1) are described by the extended Dynkin diagrams of the associated
affine algebra $\hat{\mathfrak{g}}$. The fields $q_{k}$ can be rescaled so
that $\beta$ only appears in ${\cal L}$ through a common factor $1/\beta^{2}$
(expanding in powers of $\beta^{2}$ is equivalent in Quantum theory to
expanding in $\hbar$) [8, 9].

Non-simply laced theories are obtained from the affine Toda theories based on
simply laced algebras by folding the corresponding Dynkin diagrams. The same
process, called classical “reduction”, provides solutions of a non-simply
laced theory from the classical solutions with special symmetries of the
parent simply laced theory.

One of the important features of affine Toda field theories is their
integrability by the inverse scattering method (ISM) [18, 46]. The starting
point is the existence of the so-called Lax operator (see (4) below). The
interpretation of the ISM as a generalized Fourier transforms [2, 19, 34]
allows one to study all the fundamental properties of the corresponding
nonlinear evolutionary equations (NLEE’s): i) the description of the class of
NLEE related to a given Lax operator $L(\lambda)$ and solvable by the ISM; ii)
derivation of the infinite family of integrals of motion; iii) their hierarchy
of Hamiltonian structures [33]; and iv) description of the gauge equivalent
systems [25, 38, 39].

Real Hamiltonian forms (RHF) are another type of “reductions” of Hamiltonian
systems. The extraction of RHFs is similar to the obtaining a real forms of a
semi-simple Lie algebra. The Killing form for the later is indefinite in
general (it is negatively- definite for the compact real forms). So one should
not be surprised of getting RHF’s with indefinite kinetic energy quadratic
form. Of course this is an obstacle for their quantization.

The purpose of this paper is to outline the spectral theory of the Lax
operators of real Hamiltonian forms of affine Toda field theories related to
complex untwisted exceptional affine Lie algebras.

The structure of the paper is as follows: In Section 2 we provide a brief
summary of the Lax representation of ATFTs and all necessary Lie algebraic
background knowledge. As to the root systems of the exceptional algebras we
are following the conventions in [7]. In Section 3 first we describe the
general method for constructing RHFs. Then we briefly describe the sets of
admissible roots of the exceptional Lie algebras, and the explicit
constructions of their RHFs. Section 4 is devoted to the spectral theory of
the Lax operators of ATFTs with ${\mathbb{Z}}_{h}$-reductions, where $h$ is
the Coxeter number of ${\mathfrak{g}}$. We first provide the general
construction of the fundamental analytic solutions $\xi_{\nu}(x,t,\lambda)$
(FAS) of the Lax operator and show their relevance to the Riemann-Hilbert
problem (RHP) on a set of $2h_{g}$ rays $l_{\nu}$ closing angles $\pi/h_{g}$.
Then we introduce the asymptotics of the FAS $\xi_{\nu}(x,t,\lambda)$ for
$x\to\pm\infty$ and $\lambda\in l_{\nu}$ which determine the scattering data
of $L$. This Section ends with a theorem specifying the minimal sets of
scattering data for generic choice of the simple Lie algebra $\mathfrak{g}$.
Section 5 contains the specific data, concerning each of the exceptional Lie
algebras that allows one to determine the minimal set os scattering data for
each case. The paper ends up with Conclusions.

##  2 Affine Toda field theories: preliminaries

To each simple Lie algebra $\mathfrak{g}$ one can relate a Toda field theory
in $1+1$ dimensions. It allows a Lax representation in a zero-curvature form:

$[L,M]=0,$ (3)

where $L$ and $M$ are first order ordinary differential (Lax) operators:

$\displaystyle L\psi\equiv\left(i{d\over dx}-iq_{x}(x,t)-\lambda
J_{0}\right)\psi(x,t,\lambda)=0,$ (4) $\displaystyle M\psi\equiv\left(i{d\over
dt}-{1\over\lambda}I(x,t)\right)\psi(x,t,\lambda)=0.$ (5)

whose potentials take values in $\mathfrak{g}$. Here also
$q(x,t)\in\mathfrak{h}$ \- the Cartan subalgebra of $\mathfrak{g}$,
${\boldsymbol{q}}(x,t)=(q_{1},\dots,q_{r})$ is its dual $r$-component vector
($r=\mbox{rank}\,\mathfrak{g}$). The potentials of the Lax operators are
chosen as follows

$J_{0}=\sum_{\alpha\in\pi}E_{\alpha},\qquad
I(x,t)=\sum_{\alpha\in\pi}e^{-(\alpha,{\boldsymbol{q}}(x,t))}E_{-\alpha}.$ (6)

Here $\pi_{\mathfrak{g}}$ stands for the set of admissible roots of
$\mathfrak{g}$, i.e.
$\pi_{\mathfrak{g}}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{r}\\}$, with
$\alpha_{1},\dots,\alpha_{r}$ being the simple roots of $\mathfrak{g}$ and
$\alpha_{0}$ being the minimal root of $\mathfrak{g}$. The corresponding Toda
field theory is known as affine Toda field theory (ATFT). The Dynkin graph
corresponding to the set of admissible roots
$\pi_{\mathfrak{g}}=\\{\alpha_{0},\alpha_{1},\dots,\alpha_{r}\\}$ of
${\mathfrak{g}}$ is called extended Dynkin diagram (EDD). The equations of
motion are of the form:

${\partial^{2}{\boldsymbol{q}}\over\partial x\partial
t}=\sum_{j=0}^{r}n_{j}\alpha_{j}e^{-(\alpha_{j},{\boldsymbol{q}}(x,t))},$ (7)

where $n_{j}$ are the minimal positive integer coefficients $n_{k}$ that
provide the decomposition of the $\alpha_{0}$ over the simple roots of
$\mathfrak{g}$:

$-\alpha_{0}=\sum_{k=1}^{r}n_{k}\alpha_{k}.$ (8)

It is well known that ATFT models are an infinite-dimensional Hamiltonian
system. The (canonical) Hamiltonian structure is given by:

$\displaystyle H_{\mathfrak{g}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx\,\mathcal{H}_{\mathfrak{g}}(x,t),\qquad\mathcal{H}_{\mathfrak{g}}(x,t)={1\over
2}({\bf p}(x,t),{\bf p}(x,t))+\sum_{k=0}^{r}n_{k}(e^{-({\bf
q}(x,t),\alpha_{k})}-1),$ (9) $\displaystyle\Omega_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dx\,\omega_{\mathfrak{g}}(x,t),\qquad\omega_{\mathfrak{g}}(x,t)=(\delta{\bf
p}(x,t)\wedge\delta{\bf q}(x,t)),$ (10)

where $H_{\mathfrak{g}}$ is the canonical Hamilton function and
$\Omega_{\mathfrak{g}}$ is the canonical symplectic structure. Here also ${\bf
p}=d{\bf q}/dt$ are the canonical momenta and coordinates satisfying canonical
Poisson brackets:

$\\{q_{k}(x,t),p_{j}(y,t)\\}=\delta_{jk}\delta(x-y).$ (11)

The infinite-dimensional phase space $\mathcal{M}=\\{{\bf q}(x,t),{\bf
p}(x,t)\\}$ is spanned by the canonical coordinates and momenta.

##  3 Real Hamiltonian forms and affine Toda field theories

The Lax representations of the ATFT models widely discussed in the literature
(see e.g. [44, 45, 47, 43] and the references therein) are related mostly to
the normal real form of the Lie algebra $\mathfrak{g}$, see [40]. Here we will
study real Hamiltonian forms of ATFT models. The notion of real Hamiltonian
forms was introduced in [26] and used to study reductions of ATFTs in [24].
After a brief outline of the basic theory (following [26]), in this Section we
will describe the real Hamiltonian forms for ATFT models related to
exceptional untwisted complex Kac-Moody algebras [42, 49, 50].

###  3.1 Real Hamiltonian forms

The starting point in the construction of real Hamiltonian forms (RHF) is the
complexification of the field involved in the Hamiltonian of the model. First
for dynamical variables we consider complex-valued fields

${\bf q}^{\mathbb{C}}={\bf q}^{0}+i{\bf q}^{1},\qquad{\bf p}^{\mathbb{C}}={\bf
p}^{0}+i{\bf p}^{1}.$

Next we introduce an involution $\mathcal{C}$ acting on the phase space
$\mathcal{M}\equiv\\{q_{k}(x),p_{k}(x)\\}_{k=1}^{n}$ as follows:

$\displaystyle\mbox{1)}\qquad\mathcal{C}(F(p_{k},q_{k}))=F(\mathcal{C}(p_{k}),\mathcal{C}(q_{k})),$
$\displaystyle\mbox{2)}\qquad\mathcal{C}\left(\\{F(p_{k},q_{k}),G(p_{k},q_{k})\\}\right)=\left\\{\mathcal{C}(F),\mathcal{C}(G)\right\\},$
(12) $\displaystyle\mbox{3)}\qquad\mathcal{C}(H(p_{k},q_{k}))=H(p_{k},q_{k}).$

Here $F(p_{k},q_{k})$, $G(p_{k},q_{k})$ and the Hamiltonian $H(p_{k},q_{k})$
are functionals on $\mathcal{M}$ depending analytically on the fields
$q_{k}(x,t)$ and $p_{k}(x,t)$.

The complexification of the ATFT is rather straightforward. The resulting
complex ATFT (CATFT) can be written down as standard Hamiltonian system with
twice as many fields ${\bf q}^{a}(x,t)$, ${\bf p}^{a}(x,t)$, $a=0,1$:

${\bf p}^{\mathbb{C}}(x,t)={\bf p}^{0}(x,t)+i{\bf p}^{1}(x,t),\qquad{\bf
q}^{\mathbb{C}}(x,t)={\bf q}^{0}(x,t)+i{\bf q}^{1}(x,t),$ (13)
$\\{{q}_{k}^{0}(x,t),{p}_{j}^{0}(y,t)\\}=-\\{{q}_{k}^{1}(x,t),{p}_{j}^{1}(y,t)\\}=\delta_{kj}\delta(x-y).$
(14)

The densities of the corresponding Hamiltonian and symplectic form equal

$\displaystyle\mathcal{H}_{\rm ATFT}^{\mathbb{C}}$ $\displaystyle\equiv$
$\displaystyle\mathrm{Re\,}\mathcal{H}_{\rm ATFT}({\bf p}^{0}+i{\bf
p}^{1},{\bf q}^{0}+i{\bf q}^{1})$ (15) $\displaystyle=$ $\displaystyle{1\over
2}({\bf p}^{0},{\bf p}^{0})-{1\over 2}({\bf p}^{1},{\bf
p}^{1})+\sum_{k=0}^{r}e^{-({\bf q}^{0},\alpha_{k})}\cos(({\bf
q}^{1},\alpha_{k})),$ $\displaystyle\omega^{\mathbb{C}}$ $\displaystyle=$
$\displaystyle(d{\bf p}^{0}\wedge id{\bf q}^{0})-(d{\bf p}^{1}\wedge d{\bf
q}^{1}).$ (16)

The family of RHF then are obtained from the CATFT by imposing an invariance
condition with respect to the involution
$\tilde{\mathcal{C}}\equiv\mathcal{C}\circ\ast$ where by $\ast$ we denote the
complex conjugation. The involution $\tilde{\mathcal{C}}$ splits the phase
space $\mathcal{M}^{\mathbb{C}}$ into a direct sum
$\mathcal{M}^{\mathbb{C}}\equiv{\cal
M}_{+}^{\mathbb{C}}\oplus\mathcal{M}_{-}^{\mathbb{C}}$ where

$\displaystyle\mathcal{M}_{+}^{\mathbb{C}}$
$\displaystyle=\mathcal{M}_{0}\oplus i\mathcal{M}_{1},$
$\displaystyle\quad\mathcal{M}_{-}^{\mathbb{C}}$
$\displaystyle=\mathcal{M}_{0}\oplus-i\mathcal{M}_{1},$ (17)
$\displaystyle\mathcal{C}({\boldsymbol{q}}^{+}+i{\boldsymbol{q}}^{-})$
$\displaystyle=({\boldsymbol{q}}^{+}-i{\boldsymbol{q}}^{-}),$
$\displaystyle\quad\mathcal{C}({\boldsymbol{p}}^{+}+i{\boldsymbol{p}}^{-})$
$\displaystyle=({\boldsymbol{p}}^{+}-i{\boldsymbol{p}}^{-}).$

Each involution $\mathcal{C}$ induces an involution (involutive automorphism)
$\mathcal{C}^{\\#}$ also of $\mathfrak{g}$. Below we will choose
$\mathcal{C}^{\\#}$ in a special way, which is related to the Coxeter
automorphism defined below in its dihedral form (26), see [7, 40, 10]. Indeed,
we will define $\mathcal{C}^{\\#}$ using the reflection $S_{1}$ with respect
to the set of black roots of the Dinkin dyagram (see Figure 1). More
precisely:

$\displaystyle\mathcal{C}^{\\#}\alpha_{j}$
$\displaystyle=\alpha_{j}\quad\mbox{for}\quad\alpha_{j}\in W_{g},$
$\displaystyle\qquad\mathcal{C}^{\\#}\beta_{j}$
$\displaystyle=-\beta_{j}\quad\mbox{for}\quad\beta_{j}\in B_{g},$ (18)
$\displaystyle{\bf p}^{0}(x,t)$ $\displaystyle=\sum_{\alpha_{j}\in W_{g}}{\bf
p}_{j}(x,t)\alpha_{j},$ $\displaystyle\qquad{\bf p}^{1}(x,t)$
$\displaystyle=\sum_{\beta_{j}\in B_{g}}{\bf p}_{j}(x,t)\beta_{j},$
$\displaystyle{\bf q}^{0}(x,t)$ $\displaystyle=\sum_{\alpha_{j}\in W_{g}}{\bf
q}_{j}(x,t)\alpha_{j},$ $\displaystyle\qquad{\bf q}^{1}(x,t)$
$\displaystyle=\sum_{\beta_{j}\in B_{g}}{\bf q}_{j}(x,t)\beta_{j},$

where $W_{g}$ (resp. $B_{g}$) is the set of white roots (resp. black roots) of
the Dinkin diagram.

Thus to each involution $\mathcal{C}$ one can relate a RHF of the ATFT. Note
that $\mathcal{C}^{\\#}$ preserves the system of admissible roots of
$\mathfrak{g}$ and the extended Dynkin diagrams of $\mathfrak{g}$ studied in
[43]. For a sake of brevity below we will skip the dependence of ${\bf p}$ and
${\bf q}$ on $x$ and $t$. Indeed, the condition 3) in (3.1) requires that:

$(\mathcal{C}({\boldsymbol{q}}),\alpha)=({\bf
q},\mathcal{C}^{\\#}(\alpha)),\qquad\alpha\in\pi_{\mathfrak{g}},$ (19)

and therefore we must have
$\mathcal{C}(\pi_{\mathfrak{g}})=\pi_{\mathfrak{g}}$. As a result:

$\displaystyle\mathcal{C}({\bf p}^{0}(x,t),\alpha_{j})$ $\displaystyle=({\bf
p}^{0}(x,t),\alpha_{j}),$ $\displaystyle\qquad\mathcal{C}({\bf
p}^{1}(x,t),\beta_{j})$ $\displaystyle=-({\bf p}^{1}(x,t),\beta_{j}),$ (20)
$\displaystyle({\bf p}^{0}(x,t),\beta_{j})$ $\displaystyle=0,$
$\displaystyle\qquad({\bf p}^{1}(x,t),\alpha_{j})$ $\displaystyle=0,$
$\displaystyle\mathcal{C}({\bf q}^{0}(x,t),\alpha_{j})$ $\displaystyle=({\bf
q}^{0}(x,t),\alpha_{j}),$ $\displaystyle\qquad\mathcal{C}({\bf
q}^{1}(x,t),\beta_{j})$ $\displaystyle=-({\bf q}^{1}(x,t),\beta_{j}),$
$\displaystyle({\bf q}^{0}(x,t),\beta_{j})$ $\displaystyle=0,$
$\displaystyle\qquad({\bf q}^{1}(x,t),\alpha_{j})$ $\displaystyle=0,$

where $\alpha_{j}\in W_{g}$, and $\beta_{j}\in B_{g}$. Then applying the ideas
of [44] we obtain the following result for the RHF of the ATFT related to
$\mathfrak{g}$:

$\displaystyle\mathcal{H}^{{\mathbb{R}}}_{\mathfrak{g}}$
$\displaystyle={1\over 2}({\bf p}^{+},{\bf p}^{+})-{1\over 2}({\bf p}^{-},{\bf
p}^{-})+\sum_{\alpha_{k}\in W_{g}}n_{k}^{\prime}(e^{-({\bf
q}^{+},\alpha_{k})}-1)+\sum_{\beta_{k}\in B_{g}}n_{k}^{\prime\prime}(\cos({\bf
q}^{-},\beta_{k})-1),$ (21)
$\begin{split}\omega^{{\mathbb{R}}}_{\mathfrak{g}}=(\delta{\bf
p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf p}^{-}\wedge\delta{\bf
q}^{-}),\end{split}$ (22)

The Hamiltonian along with the terms related to the simple roots, contains
also the minimal root $-\alpha_{0}$. It is well known that the maximal root
$\alpha_{0}$ can be expanded as sum of the simple roots with integer
nonnegative coefficients:

$\begin{split}\alpha_{0}=\sum_{j=1}^{r}n_{j}\alpha_{j}=\sum_{\alpha_{k}\in
W_{g}}n_{k}^{\prime}\alpha_{k}+\sum_{\beta_{j}\in
B_{g}}n_{k}^{\prime\prime}\beta_{j}.\end{split}$ (23)

In the second expression above we have separated the terms with the white and
black roots.

The RHF of ATFT are more general integrable systems than the models described
in [15, 16, 17, 43] which involve only the fields ${\bf q}^{+}$, ${\bf p}^{+}$
invariant with respect to $\mathcal{C}$.

###  3.2 Affine Toda field theories related to $E_{6}^{(1)}$

The set of admissible roots for this algebra is

$\displaystyle\alpha_{1}$ $\displaystyle={1\over
2}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}-e_{7}+e_{8}),$
$\displaystyle\quad\alpha_{2}$ $\displaystyle=e_{1}+e_{2},$ (24)
$\displaystyle\alpha_{3}$
$\displaystyle=e_{2}-e_{1},\qquad\alpha_{4}=e_{3}-e_{2},\qquad\alpha_{5}=e_{4}-e_{3},$
$\displaystyle\quad\alpha_{6}$ $\displaystyle=e_{5}-e_{4},$
$\displaystyle\alpha_{0}$ $\displaystyle=-{1\over
2}(e_{1}+e_{2}+e_{3}+e_{4}+e_{5}-e_{6}-e_{7}+e_{8}).$

Here $\alpha_{1},\dots,\alpha_{6}$ form the set of simple roots of $E_{6}$ and
$\alpha_{0}$ is the minimal root of the algebra. The extended Dynkin diagram
of $E_{6}^{(1)}$ is shown on Figure 1. This is the standard definition of the
root system of $E_{6}$ embedded into the 8-dimensional Euclidean space
${\mathbb{E}}^{8}$. The root space ${\mathbb{E}}_{6}$ of the algebra $E_{6}$
is the 6-dimensional subspace of ${\mathbb{E}}^{8}$ orthogonal to the vectors
$e_{7}+e_{8}$ and $e_{6}+e_{7}+2e_{8}$. Thus any vector ${\bf q}$ belonging to
${\mathbb{E}}_{6}$ has only 6 independent coordinates and can be written as:

$\displaystyle{\bf q}=\sum_{k=1}^{5}q_{k}e_{k}+q_{6}e^{\prime}_{6},\qquad
e^{\prime}_{6}={1\over\sqrt{3}}(e_{6}+e_{7}-e_{8}).$ (25)

The fundamental weights of $E_{6}$ are

$\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle{2\over
3}(e_{8}-e_{7}-e_{6}),\qquad\omega_{2}={1\over
2}(e_{1}+e_{2}+e_{3}+e_{4}+e_{5}-e_{6}-e_{7}+e_{8});$
$\displaystyle\omega_{3}$ $\displaystyle=$ $\displaystyle{1\over
2}(-e_{1}+e_{2}+e_{3}+e_{4}+e_{5})+{5\over
6}(e_{8}-e_{7}-e_{6}),\qquad\omega_{4}=e_{3}+e_{4}+e_{5}-e_{6}-e_{7}+e_{8},$
$\displaystyle\omega_{5}$ $\displaystyle=$ $\displaystyle e_{4}+e_{5}+{2\over
3}(e_{8}-e_{7}-e_{6}),\qquad\omega_{6}=e_{5}+{1\over 3}(e_{8}-e_{7}-e_{6}).$

$\alpha_{1}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$\alpha_{2}$$\alpha_{0}$$E_{6}^{(1)}$
Figure 1: The extended Dynkin diagram of the complex untwisted affine Kac-
Moody algebra $E_{6}^{(1)}$. The white roots are invariant with respect to the
automorphism $S_{2}$ in (26).

We will use the dihedral realization of the Coxeter automorphism for
$E_{6}^{(1)}$:

${\rm Cox}\,(E_{6}^{(1)})=S_{1}\circ S_{2},\qquad S_{1}=S_{\alpha_{2}}\circ
S_{\alpha_{3}}\circ S_{\alpha_{5}},\qquad S_{2}=S_{\alpha_{1}}\circ
S_{\alpha_{4}}\circ S_{\alpha_{6}}.$ (26)

If we require an invariance of the Hamiltonian and the symplectic form (21)
with respect to $S_{1}$, and restrict on the set of admissible roots
$\beta_{k}$, then we get a real Hamiltonian form of the $E_{6}^{(1)}$ ATFT,
described by:

$\begin{split}\mathcal{H}^{{\mathbb{R}}}_{E_{6}}&={1\over 2}({\bf p}^{+},{\bf
p}^{+})-{1\over 2}({\bf p}^{-},{\bf
p}^{-})+\sum_{k=0,1,4,6}n_{k}^{\prime}(e^{-({\bf
q}^{+},\alpha_{k})}-1)+\sum_{k=2,3,5}n_{k}^{\prime\prime}(\cos({\bf
q}^{-},\beta_{k})-1),\end{split}$ (27)

where $n_{1}^{\prime}=n_{6}^{\prime}=1$, $n_{4}^{\prime}=3$ and
$n_{2}^{\prime\prime}=n_{3}^{\prime\prime}=n_{5}^{\prime\prime}=2$ [7], and

$\begin{split}\omega^{{\mathbb{R}}}_{E_{6}^{(1)}}=(\delta{\bf
p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf p}^{-}\wedge\delta{\bf
q}^{-}).\end{split}$ (28)

###  3.3 Affine Toda field theories related to $E_{7}^{(1)}$

The extended root system of this algebra is given by

$\displaystyle\alpha_{1}$ $\displaystyle={1\over
2}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}-e_{7}+e_{8}),$
$\displaystyle\quad\alpha_{2}$ $\displaystyle=e_{1}+e_{2},$ (29)
$\displaystyle\alpha_{3}$
$\displaystyle=e_{2}-e_{1},\qquad\alpha_{4}=e_{3}-e_{2},$
$\displaystyle\quad\alpha_{5}$ $\displaystyle=e_{4}-e_{3},$
$\displaystyle\alpha_{6}$
$\displaystyle=e_{5}-e_{4},\qquad\alpha_{7}=e_{6}-e_{5},$
$\displaystyle\quad\alpha_{0}$ $\displaystyle=e_{7}-e_{8},$

where $\alpha_{1},\dots,\alpha_{7}$ form the set of simple roots of $E_{7}$
and $\alpha_{0}$ is the minimal root of the algebra. The extended Dynkin
diagram for $E_{7}^{(1)}$ is shown on Figure 2. This is the standard
definition of the root system of $E_{7}$ embedded into the 8-dimensional
Euclidean space ${\mathbb{E}}^{8}$. The root space ${\mathbb{E}}_{7}$ of the
algebra $E_{7}$ is the 7-dimensional subspace of ${\mathbb{E}}^{8}$ orthogonal
to the vector $e_{7}+e_{8}$. Thus any vector $\vec{q}$ belonging to
${\mathbb{E}}_{7}$ has 7 independent coordinates and can be written as:

$\displaystyle\vec{q}=\sum_{k=1}^{6}q_{k}e_{k}+q_{7}e^{\prime}_{7},\qquad
e^{\prime}_{7}={1\over\sqrt{2}}(e_{7}-e_{8}).$ (30)

$\alpha_{0}$$\alpha_{1}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$\alpha_{7}$$\alpha_{2}$$E_{7}^{(1)}$
Figure 2: The extended Dynkin diagram of the complex untwisted affine Kac-
Moody algebra $E_{7}^{(1)}$. The white roots are invariant with respect to the
automorphism $S_{1}$ in (31).

Using the dihedral realization of the Coxeter automorphism for $E_{7}^{(1)}$:

${\rm Cox}\,(E_{7}^{(1)})=S_{1}\circ S_{2},\qquad S_{1}=S_{\alpha_{1}}\circ
S_{\alpha_{4}}\circ S_{\alpha_{6}},\qquad S_{2}=S_{\alpha_{2}}\circ
S_{\alpha_{3}}\circ S_{\alpha_{5}}\circ S_{\alpha_{7}},$ (31)

we impose an invariance of the Hamiltonian (21) with respect to the
automorphism $S_{1}$ in (31). Then the root space ${\mathbb{E}}_{7}$ will
split into direct sums:
${\mathbb{E}}_{7}={\mathbb{E}}_{7,+}\oplus{\mathbb{E}}_{7,-}$, with

$\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{7}\in{\mathbb{E}}^{7,+},\qquad\alpha_{1},\alpha_{4},\alpha_{6}\in{\mathbb{E}}^{7,-}.$

If we again require an invariance of the Hamiltonian and the symplectic form
(21) with respect to $S_{1}$ from (31), and restrict on the set of admissible
roots $\beta_{k}$, then we get a real Hamiltonian form of the $E_{7}^{(1)}$
ATFT, described by:

$\begin{split}\mathcal{H}^{{\mathbb{R}}}_{E_{7}}&={1\over 2}({\bf p}^{+},{\bf
p}^{+})-{1\over 2}({\bf p}^{-},{\bf
p}^{-})+\sum_{k=0,2,3,5,7}n_{k}^{\prime}(e^{-({\bf
q}^{+},\alpha_{k})}-1)+\sum_{k=1,4,6}n_{k}^{\prime\prime}(\cos({\bf
q}^{-},\beta_{k})-1),\end{split}$ (32)

where $n_{1}^{\prime}=n_{6}^{\prime}=2$, $n_{4}^{\prime}=4$ and
$n_{2}^{\prime\prime}=2$, $n_{3}^{\prime\prime}=n_{5}^{\prime\prime}=3$,
$n_{7}^{\prime\prime}=1$ [7], and

$\begin{split}\omega^{{\mathbb{R}}}_{E_{7}^{(1)}}=(\delta{\bf
p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf p}^{-}\wedge\delta{\bf
q}^{-}).\end{split}$ (33)

###  3.4 Affine Toda field theories related to $E_{8}^{(1)}$

The set of admissible roots for this algebra is

$\displaystyle\alpha_{0}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+e_{3}+e_{4}+e_{5}-e_{6}-e_{7}+e_{8})\quad\alpha_{1}=\frac{1}{2}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}-e_{7}+e_{8}),$
(34) $\displaystyle\alpha_{2}$
$\displaystyle=e_{1}+e_{2},\quad\alpha_{3}=e_{2}-e_{1},\quad\alpha_{4}=e_{3}-e_{2},$
$\displaystyle\alpha_{5}$
$\displaystyle=e_{4}-e_{3},\quad\alpha_{6}=e_{5}-e_{4},\quad\alpha_{7}=e_{6}-e_{5}\quad\alpha_{8}=e_{7}-e_{6}.$

where $\alpha_{1},\dots,\alpha_{8}$ form the set of simple roots of $E_{8}$
and $\alpha_{0}$ is the minimal root of the algebra. If we exclude
$\alpha_{8}$ from (34), we will get the set of positive roots (29) form
$E_{7}$. The extended Dynkin diagram for $E_{8}^{(1)}$ is shown on Figure 3.

The fundamental weights of $E_{8}$ are

$\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle
2e_{8},\qquad\omega_{2}={1\over
2}(e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+e_{6}+e_{7}+5e_{8}),$
$\displaystyle\omega_{3}$ $\displaystyle=$ $\displaystyle{1\over
2}(-e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+e_{6}+e_{7}+7e_{8}),\qquad\omega_{4}=e_{3}+e_{4}+e_{5}+e_{6}+e_{7}+5e_{8},$
$\displaystyle\omega_{5}$ $\displaystyle=$ $\displaystyle
e_{4}+e_{5}+e_{6}+e_{7}+4e_{8},\qquad\omega_{6}=e_{5}+e_{6}+e_{7}+3e_{8},$
$\displaystyle\omega_{7}$ $\displaystyle=$ $\displaystyle
e_{6}+e_{7}+2e_{8},\qquad\omega_{8}=e_{7}+e_{8}=-\alpha_{0}.$

$\alpha_{1}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$\alpha_{7}$$\alpha_{8}$$\alpha_{0}$$\alpha_{2}$$E_{8}^{(1)}$
Figure 3: The extended Dynkin diagram of the complex untwisted affine Kac-
Moody algebra $E_{8}^{(1)}$. The white roots are invariant with respect to the
automorphism $S_{2}$ in (35).

We will use the dihedral realization of the Coxeter automorphism for
$E_{8}^{(1)}$:

${\rm Cox}\,(E_{8}^{(1)})=S_{1}\circ S_{2},\qquad S_{1}=S_{\alpha_{1}}\circ
S_{\alpha_{4}}\circ S_{\alpha_{6}}\circ S_{\alpha_{8}},\qquad
S_{2}=S_{\alpha_{2}}\circ S_{\alpha_{3}}\circ S_{\alpha_{5}}\circ
S_{\alpha_{7}}.$ (35)

We impose an invariance of the Hamiltonian (21) with respect to the
automorphism $S_{1}$ in (31). Then the root space ${\mathbb{E}}_{8}$ will
split into direct sums:
${\mathbb{E}}_{8}={\mathbb{E}}_{8,+}\oplus{\mathbb{E}}_{8,-}$, with

$\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{7}\in{\mathbb{E}}_{8,+},\qquad\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{8}\in{\mathbb{E}}_{8,-}.$

The invariance of the Hamiltonian and the symplectic form (21) with respect to
$S_{1}$ from (35), and restricting on the set of admissible roots $\beta_{k}$,
will result in a real Hamiltonian form of the $E_{8}^{(1)}$ ATFT, described
by:

$\begin{split}\mathcal{H}^{{\mathbb{R}}}_{E_{8}}&={1\over 2}({\bf p}^{+},{\bf
p}^{+})-{1\over 2}({\bf p}^{-},{\bf
p}^{-})+\sum_{k=0,2,3,5,7}n_{k}^{\prime}(e^{-({\bf
q}^{+},\alpha_{k})}-1)+\sum_{k=1,4,6,8}n_{k}^{\prime\prime}(\cos({\bf
q}^{-},\beta_{k})-1),\end{split}$ (36)

where $n_{1}^{\prime\prime}=n_{8}^{\prime\prime}=2$,
$n_{4}^{\prime\prime}=6,n_{6}^{\prime\prime}=4$ and
$n_{2}^{\prime}=n_{7}^{\prime}=3$, $n_{3}^{\prime}=4$, $n_{5}^{\prime}=5$ [7],
and

$\begin{split}\omega^{{\mathbb{R}}}_{E_{8}^{(1)}}=(\delta{\bf
p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf p}^{-}\wedge\delta{\bf
q}^{-}).\end{split}$ (37)

###  3.5 Affine Toda field theories related to $F_{4}^{(1)}$

The extended roots of this algebra are

$\displaystyle\alpha_{1}$
$\displaystyle=e_{2}-e_{3},\qquad\alpha_{2}=e_{3}-e_{4},$
$\displaystyle\quad\alpha_{3}$ $\displaystyle=e_{4},$ (38)
$\displaystyle\alpha_{4}$ $\displaystyle={1\over 2}(e_{1}-e_{2}-e_{3}-e_{4}),$
$\displaystyle\quad\alpha_{0}$ $\displaystyle=-e_{1}-e_{2},$

with $\alpha_{0}$ being the minimal root. The extended Dynkin diagram of
$F_{4}^{(1)}$ is shown on Figure 4. The fundamental weights of $F_{4}$ are

$\displaystyle\omega_{1}$ $\displaystyle=e_{1}+e_{2},$
$\displaystyle\quad\omega_{2}$ $\displaystyle=2e_{1}+e_{2}+e_{3},$ (39)
$\displaystyle\omega_{3}$ $\displaystyle={1\over
2}(3e_{1}+e_{2}+e_{3}+e_{4}),$ $\displaystyle\quad\omega_{4}$
$\displaystyle=e_{1}.$

$\alpha_{0}$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$F_{4}^{(1)}$
Figure 4: The extended Dynkin diagram of the complex untwisted affine Kac-
Moody algebra $F_{4}^{(1)}$. The white roots are invariant with respect to the
automorphism $S_{2}$ in (40).

Using the dihedral realisation of the Coxeter automorphism for $F_{4}^{(1)}$:

${\rm Cox}\,(F_{4}^{(1)})=S_{1}\circ S_{2},\qquad S_{1}=S_{\alpha_{1}}\circ
S_{\alpha_{3}},\qquad S_{2}=S_{\alpha_{2}}\circ S_{\alpha_{4}},$ (40)

and imposing an invariance of the Hamiltonian (21) with respect to the
automorphism $S_{1}$ in (40), will extract a real Hamiltonian form of
$F_{4}^{(1)}$ ATFT of the form

$\begin{split}\mathcal{H}^{{\mathbb{R}}}_{F_{4}^{(1)}}={1\over
2}\sum_{k=1}^{4}\left((p_{k}^{+})^{2}-(p_{k}^{-})^{2}\right)+\sum_{k=0,2,4}n_{k}^{\prime}(e^{-({\bf
q}^{+},\alpha_{k})}-1)+\sum_{k=1,3}n_{k}^{\prime\prime}(\cos({\bf
q}^{-},\beta_{k})-1),\end{split}$ (41)

where $n_{2}^{\prime}=3$, $n_{4}^{\prime}=2$ and $n_{1}^{\prime\prime}=2$,
$n_{3}^{\prime\prime}=4$. In addition

$\begin{split}\omega^{{\mathbb{R}}}_{F_{4}^{(1)}}=(\delta{\bf
p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf p}^{-}\wedge\delta{\bf
q}^{-}).\end{split}$ (42)

Then the root space ${\mathbb{E}}_{4}$ will split into direct sums:
${\mathbb{E}}_{4}={\mathbb{E}}_{4,+}\oplus{\mathbb{E}}_{4,-}$, with
$\alpha_{2},\alpha_{4}\in{\mathbb{E}}_{4,+}$ and
$\alpha_{1},\alpha_{3}\in{\mathbb{E}}_{4,-}$.

###  3.6 Affine Toda field theories related to $G_{2}^{(1)}$

The set of admissible roots for this algebra is given by

$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
e_{1}-e_{2},\qquad\alpha_{2}=-e_{1}+e_{2}+e_{3},\qquad\alpha_{0}=-e_{1}-e_{2}+2e_{3}.$
(43)

The fundamental weights of $G_{2}$ are

$\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle
e_{1}-e_{2}+2e_{3},\qquad\omega_{2}=-e_{1}-e_{2}+2e_{3}.$

The extended Dynkin diagram of $G_{2}^{(1)}$ is shown on Figure 5.

Let us take $C_{1}=S_{\alpha_{2}}$. The invariance of the Hamiltonian (21)
with respect to the automorphism $C_{1}$ in will give a real Hamiltonian form
of $G_{2}^{(1)}$ ATFT described by

$\displaystyle\mathcal{H}^{{\mathbb{R}}}_{G_{2}^{(1)}}$ $\displaystyle=$
$\displaystyle{1\over
2}\sum_{k=1}^{r}\left((p_{k}^{+})^{2}-(p_{k}^{-})^{2}\right)+(e^{-({\bf
q}^{+},\alpha_{0})}-1)+3(e^{-({\bf q}^{+},\alpha_{1})}-1)+2(\cos({\bf
q}^{-},\alpha_{2})-1),$ (44)
$\displaystyle\omega^{{\mathbb{R}}}_{G_{2}^{(1)}}$ $\displaystyle=$
$\displaystyle(\delta{\bf p}^{+}\wedge\delta{\bf q}^{+})-(\delta{\bf
p}^{-}\wedge\delta{\bf q}^{-}),$ (45)

The root space ${\mathbb{E}}_{2}$ will split into direct sums:
${\mathbb{E}}_{2}={\mathbb{E}}_{2,+}\oplus{\mathbb{E}}_{2,-}$, with
$\alpha_{0},\alpha_{1}\in{\mathbb{E}}_{2,+}$ and
$\alpha_{2}\in{\mathbb{E}}_{2,-}$.

$\alpha_{1}$$\alpha_{2}$$\alpha_{0}$$G_{2}^{(1)}$ Figure 5: The extended
Dynkin diagram of the complex untwisted affine Kac-Moody algebra
$G_{2}^{(1)}$.

##  4 On the spectral properties of the Lax operators with
$\mathbb{Z}_{h}$-reduction

###  4.1 General theory

The operator $L$ (4) is very convenient for deriving the 2-dimensional TFT.
However this formulation is not convenient for constructing the fundamental
analytic solutions (FAS) of $L$. The first step we will do will be to apply on
$L$ a similarity transformation

$\begin{split}\tilde{L}\tilde{\psi}=g_{0}^{-1}Lg_{0}\tilde{\psi}=i\frac{\partial\tilde{\psi}}{\partial
x}-\left(i\sum_{k=0}^{6}q_{k}\mathcal{E}_{k}+\lambda\tilde{J}_{0}\right)\tilde{\psi}=0\end{split}$
(46)

where $\tilde{J}_{0}\in\mathfrak{h}$; i.e.

$\tilde{J}_{0}=\sum_{k=1}^{6}y_{j}H_{\alpha_{j}}=\sum_{k=1}^{6}z_{j}H_{\omega_{j}}$

where $y_{j}$ and $z_{j}$ must be expressed in terms of $\omega=\exp(2\pi
i/h_{g})$.

Let us somewhat simplify the notations in the two Lax operators

$\begin{split}L\psi&\equiv\frac{\partial\psi}{\partial x}+(\vec{q}_{x}-\lambda
J_{0})\psi(x,t,\lambda)=0,\\\
\tilde{L}\tilde{\psi}&\equiv\frac{\partial\tilde{\psi}}{\partial
x}+(Q(x,t)-\lambda\tilde{J})\tilde{\psi}(x,t,\lambda)=0,\end{split}$ (47)

where $J_{0}=\sum_{\alpha\in\delta_{0}}E_{\alpha}$, $\delta_{0}$ is the set of
simple roots $\alpha_{j}$ and the minimal root $\alpha_{0}$, and
$\tilde{J}\in\mathfrak{h}$ and $\omega=\exp(2\pi i/h)$.

The spectral theory of the Lax operators with $\mathbb{Z}_{h}$ Mikhailov
reduction groups [44] for the classical series of simple Lie algebras are by
now well developed, see [34, 35, 36] Here we will formulate them taking into
account when necessary the peculiarities of the exceptional algebras.

In [23] the problem for the interrelation between $L$ and $\tilde{L}$ was
solved by using the Chevallie basis for $A_{5}^{(1)}$ [14]. Indeed, for the
classical series Chevallie basis for the typical representations are well
known so it is not difficult to diagonalize $J_{0}$ and then evaluate
$\alpha_{j}(\tilde{J}_{0})$. The same procedure for the exceptional algebras
(except $G_{2}$) is rather more complicated. Indeed, the bases for the typical
representations of $F_{4}$, $E_{6}$ and $E_{7}$ are well known, see [41] where
one can find all root-vectors of $F_{4}$ as $26\times 26$ matrices. For
$E_{6}$ and $E_{7}$ the simple roots have been given in [41]; it is possible
that all root vectors for all exceptional algebras have been evaluated in the
framework of MAGMA project. So in general one can find $J_{0}$ for the typical
representations of all exceptional algebras. Then the construction of
$\tilde{J}_{0}$ becomes the next task. The eigenvalues of $J_{0}$ and the
eigenvectors may be calculated, though the latter would be rather involved
expressions. The next challenge would be to constructing out of the properly
normalized eigenvectors $g_{0}$ and to ensure that it is an element of the
corresponding exceptional group.

The easiest way to get a realization of the Coxeter automorphism $C_{g}$ is to
represent it as an element of the Cartan subgroup:

$\begin{split}C_{g}=\exp\left(\frac{2\pi
i}{h_{g}}\sum_{j=1}^{r}H_{\omega_{j}}\right),\end{split}$ (48)

where $h_{g}$ is the Coxeter number and $\omega_{j}$ are the fundamental
weights of the algebra $\mathfrak{g}$. Indeed, using the Cartan-Weyl
commutation relations [40] one can easily check that

$\begin{split}C_{g}J_{0}C_{g}^{-1}=\omega
J_{0},\qquad\omega=\exp\left(\frac{2\pi i}{h_{g}}\right).\end{split}$ (49)

This construction is compatible with the Lax operator $L$, since $C_{g}$
commutes with $\vec{q}_{x}$ and therefore

$\begin{split}C_{g}L(\lambda)C_{g}^{-1}=L(\omega\lambda).\end{split}$ (50)

Treating the operator $\tilde{L}$ requires the use of the Coxeter automorphism
$\tilde{C}_{g}$ realized as element of the Weyl group $\mathcal{W}_{g}$. We
already mentioned the difficulties involved with the standard approach based
on the use of the Chevallie basis. Therefore, treating the exceptional
algebras we will use a different approach. Our idea is to work only in the
Euclidean space $\mathbb{E}_{r}$ in which the root systems are embedded.
Indeed, there we can construct $\tilde{C}_{g}$ using its dihedral form [40, 7,
10]:

$\begin{split}\tilde{C}_{g}=S_{w}\circ S_{b},\qquad S_{w}=\prod_{\alpha\in
W_{g}}S_{\alpha},\qquad S_{b}=\prod_{\beta\in B_{g}}S_{\beta},\end{split}$
(51)

where $W_{g}$ (respectively $B_{g}$) are the sets of ‘white‘ roots (resp.
‘black‘ roots) of the Dinkin diagram of the algebra $\mathfrak{g}$. Note that
all elements of $W_{g}$ (resp. $B_{g}$) are all orthogonal, so $S_{w}$ and
$S_{b}$ are Weyl reflections, i.e. $S_{w}^{2}=\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1}$ and $S_{b}^{2}=\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1}$. Then $C_{g}^{h_{g}}=\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1}$, where $h_{g}$ is the Coxeter number of the
algebra $\mathfrak{g}$. Thus we will find the realization of the Coxeter
automorphism as element of the Weyl group $\mathcal{W}_{g}$ of the algebra.

The next step should be to construct the Cartan subalgebra element
$\tilde{J}_{0}$. In fact we will replace this step, and instead of
constructing $\tilde{J}_{0}$ we will construct the vector
$\vec{v}_{g}\in\mathbb{E}_{r}$ which is dual to $\tilde{J}_{0}$. Then we will
us the fact that $\alpha_{j}(\tilde{J}_{0})=(\alpha_{j},\vec{v}_{g})$. This is
all we need in order to construct the continuous spectrum and the spectral
data for $\tilde{L}$. On the other hand, since $\tilde{L}$ is related to $L$
by the similarity transformation $\tilde{L}=g_{0}Lg_{0}^{-1}$ then both
operators will have the same spectral data.

The construction of the vector $\vec{v}_{g}$ is based on its basic property of
invariance under the $\mathbb{Z}_{h}$ Mikhailov reduction group; like in (50)
we must have:

$\begin{split}\tilde{C}_{g}\tilde{L}(\lambda)\tilde{C}_{g}^{-1}=\tilde{L}(\omega\lambda),\qquad\mbox{i.e.}\qquad\tilde{C}_{g}(\vec{v}_{g})=\omega\vec{v}_{g}.\end{split}$
(52)

Such invariant vectors can be obtained by taking weighted averaged action by
$\tilde{C}_{g}$:

$\begin{split}\vec{v}_{0}=\sum_{s=0}^{h_{g}-1}\omega^{-s}\tilde{C}_{g}^{s}(\vec{v}_{g}).\end{split}$
(53)

Obviously there is an arbitrariness in the choice of $\vec{v}_{g}$: it can be
a root, or fundamental weight of the algebra $\mathfrak{g}$. At the same time
we can use the fact that the spectral parameter $\lambda$ can be redefined, so
this arbitrariness can be taken care of.

###  4.2 The FAS of the Lax operators $L$

Here we just outline the procedure of constructing the fundamental analytic
solutions (FAS) of $L$ [5, 6, 35, 34]. First we have to determine the regions
of analyticity. For smooth potentials $Q(x)$ that fall off fast enough for
$x\to\pm\infty$ these regions are the $2h$ sectors $\Omega_{\nu}$ separated by
the rays $l_{\nu}$ on which $\mathrm{Re\,}\lambda(\alpha,\vec{v}_{g})=0$,
where by $\alpha$ is a root of $\mathfrak{g}$. The rays $l_{\nu}$ and the
sectors $\Omega_{\nu}$ are given by:

$\displaystyle l_{\nu}\colon\arg(\lambda)={\pi(\nu-1)\over
h_{g}},\qquad\Omega_{\nu}\colon\frac{\pi(\nu-1)}{h_{g}}\leq\arg(\lambda)\leq\frac{\nu}{h_{g}},\qquad\nu=1,\dots,2h,$
(54)

and close angles equal to $\pi/h$.

The main result is that in each of the sectors $\Omega_{\nu}$ we can construct
fundamental analytic solutions (FAS) $\xi_{\nu}(x,t,\lambda)$ of the Lax
operator $\mathcal{L}$:

$\begin{split}\mathcal{L}\xi_{\nu}(x,t,\lambda)\equiv
i\frac{\partial\xi_{\nu}}{\partial
x}+Q_{x}\xi_{\nu}(x,\lambda)-\lambda[\tilde{J},\xi_{\nu}]=0,\\\
Q_{x}=\sum_{j=1}^{r}q_{j,x}\mathcal{O}_{j},\qquad\mathcal{O}_{j}=\sum_{s=0}^{h_{g}-1}C_{g}^{s}(E_{\alpha_{j}}).\end{split}$
(55)

The asymptotics of $\xi^{\nu}(x,\lambda)$ and $\xi^{\nu-1}(x,\lambda)$ along
the ray $l_{\nu}$ can be written in the form [21, 34]:

$\displaystyle\lim_{x\to-\infty}e^{-\lambda\tilde{J}x}\xi^{\nu}(x,\lambda
e^{i0})e^{\lambda\tilde{J}x}$ $\displaystyle=S_{\nu}^{+}(\lambda),$
$\displaystyle\quad\lambda\in l_{\nu},$ (56)
$\displaystyle\lim_{x\to-\infty}e^{-\lambda\tilde{J}x}\xi^{\nu-1}(x,\lambda
e^{-i0})e^{\lambda\tilde{J}x}$ $\displaystyle=S_{\nu}^{-}(\lambda),$
$\displaystyle\quad\lambda\in l_{\nu},$
$\displaystyle\lim_{x\to\infty}e^{-\lambda\tilde{J}x}\xi^{\nu}(x,\lambda
e^{i0})e^{\lambda\tilde{J}x}$ $\displaystyle=T_{\nu}^{-}D_{\nu}^{+}(\lambda),$
$\displaystyle\quad\lambda\in l_{\nu},$
$\displaystyle\lim_{x\to\infty}e^{-\lambda\tilde{J}x}\xi^{\nu-1}(x,\lambda
e^{-i0})e^{\lambda\tilde{J}x}$
$\displaystyle=T_{\nu}^{+}D_{\nu}^{-}(\lambda),$ $\displaystyle\quad\lambda\in
l_{\nu},$

where the matrices $S_{\nu}^{+}$, $T_{\nu}^{+}$ (resp. $S_{\nu}^{-}$,
$T_{\nu}^{-}$) are upper-triangular (resp. lower-triangular) with respect to
the $\nu$-ordering. They provide the Gauss decomposition of the scattering
matrix with respect to the $\nu$-ordering, i.e.:

$\begin{split}T_{\nu}(\lambda)&=T_{\nu}^{-}(\lambda)D_{\nu}^{+}(\lambda)\hat{S}_{\nu}^{+}(\lambda),\qquad\lambda\in
l_{\nu}e^{i0},\\\
&=T_{\nu}^{+}(\lambda)D_{\nu}^{-}(\lambda)\hat{S}_{\nu}^{-}(\lambda),\qquad\lambda\in
l_{\nu}e^{-i0}.\end{split}$ (57)

More careful analysis shows [34] that in fact $T_{\nu}(\lambda)$ belongs to a
subgroup ${\cal G}_{\nu}$ of $SL(N,\mathbb{C})$. In particular we can choose
$\lambda$ in such a way that the simple roots of $\mathfrak{g}$ will be
related to the rays $l_{0}$ and $l_{1}$; more precisely the ‘black‘ roots will
be related to $l_{0}$ while the ‘white‘ roots will be related to $l_{1}$. In
particular this means that with $l_{0}$ and $l_{1}$ we can relate subalgebras
of $\mathfrak{g}$ which will be direct sums of several copies of $sl(2)$. In
addition we can introduce the analogs of the reflection coefficients
$\rho^{\pm}_{\nu,j}$ and $\tau^{\pm}_{\nu,j}$ with $\nu=0,1$ as follows:

$\displaystyle S_{0}^{\pm}(\lambda)$ $\displaystyle=\exp\left(\sum_{\alpha\in
B_{g}}\tau_{0,\alpha}^{\pm}(\lambda)E_{\pm\alpha}\right),$ $\displaystyle\quad
T_{0}^{\mp}(\lambda)$ $\displaystyle=\exp\left(\sum_{\alpha\in
B_{g}}\rho_{0,\alpha}^{\pm}(\lambda)E_{\mp\alpha}\right),$ (58) $\displaystyle
S_{1}^{\pm}(\lambda)$ $\displaystyle=\exp\left(\sum_{\alpha\in
W_{g}}\tau_{0,\alpha}^{\pm}(\lambda)E_{\pm\alpha}\right),$ $\displaystyle\quad
T_{1}^{\mp}(\lambda)$ $\displaystyle=\exp\left(\sum_{\alpha\in
W_{g}}\rho_{0,\alpha}^{\pm}(\lambda)E_{\mp\alpha}\right),$ $\displaystyle
D_{0}^{\pm}(\lambda)$
$\displaystyle=\exp\left(\sum_{j=1}^{r}\frac{2d_{0,j}^{\pm}(\lambda)}{(\alpha_{j},\alpha_{j})}H_{\alpha_{j}}\right),$
$\displaystyle\quad D_{1}^{\pm}(\lambda)$
$\displaystyle=\exp\left(\sum_{j=1}^{r}\frac{2d_{0,j}^{\pm}(\lambda)}{(\alpha_{j},\alpha_{j})}H_{\alpha_{j}}\right).$

Next the $\mathbb{Z}_{h}$-symmetry allows us to extend these scattering data
to the other rays as well. Indeed applying the Coxeter automorphism we get
first the relations between FAS:

$\displaystyle C_{g}(\xi^{\nu}(x,\lambda))$
$\displaystyle=\xi^{\nu+2}(x,\lambda\omega),$ $\displaystyle\quad
C_{g}(T_{\nu}^{\pm}(\lambda))$ $\displaystyle=T_{\nu+2}(\lambda\omega),$ (59)
$\displaystyle C_{g}(S^{\pm}_{\nu}(\lambda))$
$\displaystyle=S^{\pm}_{\nu+2}(\lambda\omega),$ $\displaystyle\quad
C_{g}(D^{\pm}_{\nu}(\lambda))$ $\displaystyle=D^{\pm}_{\nu+2}(\lambda\omega),$

where the index $\nu+2$ should be taken modulo $2h$. Consequently we can view
as independent only the data on two of the rays, e.g. on $l_{1}$ and $l_{0}$;
all the rest will be recovered using (59).

$\lambda$$l_{0}$$l_{6}$$l_{12}$$l_{18}$$l_{1}$$l_{2}$$l_{3}$$l_{4}$$l_{5}$$l_{7}$$l_{8}$$l_{9}$$l_{10}$$l_{11}$$l_{13}$$l_{14}$$l_{15}$$l_{16}$$l_{17}$$l_{19}$$l_{20}$$l_{21}$$l_{22}$$l_{23}$$\Omega_{0}$$\Omega_{1}$$\Omega_{2}$$\Omega_{3}$$\Omega_{4}$$\Omega_{5}$$\Omega_{6}$$\Omega_{7}$$\Omega_{8}$$\Omega_{9}$$\Omega_{10}$$\Omega_{11}$$\Omega_{12}$$\Omega_{13}$$\Omega_{14}$$\Omega_{15}$$\Omega_{16}$$\Omega_{17}$$\Omega_{18}$$\Omega_{19}$$\Omega_{20}$$\Omega_{21}$$\Omega_{22}$$\Omega_{23}$
Figure 6: The continuous spectrum of the Lax operators related to the algebras
$E_{6}$ and $F_{4}$ whose Coxeter nubers equal 12. Obviously the continuous
spectra of for $E_{8}$, $E_{7}$ and $G_{2}$ will have different number of rays
equal to 60, 36 and 12 respectively.

###  4.3 The inverse scattering problem and the Riemann-Hilbert problem

The next important step is the possibility to reduce the solution of the
inverse scattering problem (ISP) for the generalized Zakharov-Shabat system to
a (local) RHP. More precisely, we have:

$\displaystyle\xi^{\nu}(x,t,\lambda)$
$\displaystyle=\xi^{\nu-1}(x,t,\lambda)G_{\nu}(x,t,\lambda),$
$\displaystyle\;\lambda$ $\displaystyle\in l_{\nu},$ (60) $\displaystyle
G_{\nu}(x,t,\lambda)$
$\displaystyle=e^{\lambda\tilde{J}x-\lambda^{-1}\tilde{I}t}G_{0,\nu}(\lambda)e^{-\lambda\tilde{J}x+\lambda^{-1}\tilde{I}t},$
$\displaystyle\;G_{0,\nu}(\lambda)$
$\displaystyle=\left.\hat{S}_{\nu}^{-}S_{\nu}^{+}(\lambda)\right|_{t=0}.$

where $\tilde{I}=g_{0}^{-1}I(x,t)g_{0}$ The collection of all relations (60)
for $\nu=1,2,\dots,2h$ together with

$\lim_{\lambda\to\infty}\xi^{\nu}(x,t,\lambda)=\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1},$ (61)

can be viewed as a local RHP posed on the collection of rays
$\Sigma\equiv\\{l_{\nu}\\}_{\nu=1}^{2h}$ with canonical normalization. Rather
straightforwardly we can prove that if $\xi^{\nu}(x,\lambda)$ is a solution of
the RHP (60), (61) then $\chi^{\nu}(x,\lambda)=\xi^{\nu}(x,\lambda)e^{-\lambda
U_{1}x}$ is a FAS of $L$ with potential

$Q_{x}(x,t)=\lim_{\lambda\to\infty}\lambda\left(\tilde{J}-\xi^{\nu}(x,t,\lambda)\tilde{J}\hat{\xi}^{\nu}(x,t,\lambda)\right).$
(62)

For the classical Lie algebras the analyticity properties of
$d_{j}^{\pm}(\lambda)$ allow one to reconstruct them from the sewing function
$G(\lambda)$ (60) and from the and from the locations of their zeroes and
poles [34, 33]. The idea was to consider the relations (57) in the $j$-th
fundamental representation of $\mathfrak{g}$ and take the matrix elements
between the highest $|\omega_{j}^{+}\rangle$ (resp. lowest
$|\omega_{j}^{-}\rangle$) weight vectors. This can be done also for
exceptional groups with the results

$\displaystyle d_{\nu,j}^{+}(\lambda)-d_{\nu,j}^{-}(\lambda)$
$\displaystyle=-\ln\langle\omega_{j}^{+}|\hat{T}_{\nu}^{+}T_{\nu}^{-}(\lambda)|\omega_{j}^{+}\rangle$
(63)
$\displaystyle=-\ln\langle\omega_{j}^{-}|\hat{S}_{\nu}^{-}S_{\nu}^{+}(\lambda)|\omega_{j}^{+}\rangle.$

If $\mathfrak{g}$ is one of the classical Lie algebras, then the right hand
sides of (63) can be related to the principal upper and lower minors of the
scattering matric $T(\lambda)$ in the typical representation. For the
exceptional algebras the fundamental representations can not so easy be
related to the typical (or lowest dimensional) one. So recovering the
generating functionals of integrals of motion from their analyticity
properties is an open problem.

##  5 The minimal sets of scattering data

###  5.1 The minimal sets of scattering data from the RHP

As a consequence of the results in Section 4.3 we conclude that the following
theorem holds true:

######  Theorem 1.

Let us assume that the Lax operator $\mathcal{L}$ (55) is constructed along
the ideas in Section 4 above, i.e. it is related to a simple Lie algebra
$\mathfrak{g}$ and possesses $\mathbb{Z}_{h}$ symmetry where $h$ is the
Coxeter number of $\mathfrak{g}$. Let us also assume that its potential
$Q_{x}$ is such that $\mathcal{L}$ has no discrete eigenvalues. Then its FAS
$\xi^{\nu}(x,t,\lambda)$ in the sector $\Omega_{\nu}$ will provide a regular
solution of the Riemann-Hilbert Problem (60). Let us denote by $\delta_{0}$
and $\delta_{1}$ the subsets of roots of the algebra $\mathfrak{g}$ which
satisfy the equations:

$\displaystyle\alpha\in\delta_{0}\qquad\mbox{if}\qquad{\mbox{Im}}\;\lambda(\alpha,\vec{v}_{g})=0\qquad\mbox{for}\qquad\lambda\in
l_{0},$ (64)
$\displaystyle\alpha\in\delta_{1}\qquad\mbox{if}\qquad{\mbox{Im}}\;\lambda(\alpha,\vec{v}_{g})=0\qquad\mbox{for}\qquad\lambda\in
l_{1}.$

Let also

$\begin{split}\mathcal{T}_{1}&=\\{\tau^{\pm}_{0,\alpha}(\lambda),\quad\alpha\in\delta_{0},\;\lambda\in
l_{0}\\}\cup\\{\tau^{\pm}_{1,\alpha}(\lambda),\quad\alpha\in\delta_{1},\;\lambda\in
l_{1}\\};\\\
\mathcal{T}_{2}&=\\{\rho^{\pm}_{0,\alpha}(\lambda),\quad\alpha\in\delta_{0},\;\lambda\in
l_{0}\\}\cup\\{\rho^{\pm}_{1,\alpha}(\lambda),\quad\alpha\in\delta_{1},\;\lambda\in
l_{1}\\}.\end{split}$ (65)

Then: A) $\mathcal{T}_{1}$ (resp. $\mathcal{T}_{2}$) provide a minimal set of
scattering data which allow one to recover:

  1. 1.

the Gauss factors $S_{\nu}^{\pm}(\lambda)$ for $\nu=0,1$ (resp.
$T_{\nu}^{\pm}(\lambda)$ for $\nu=0,1$;

  2. 2.

the Gauss factors $D_{\nu}^{\pm}(\lambda)$ for $\nu=0,1$ (resp.
$T_{\nu}^{\pm}(\lambda)$ for $\nu=0,1$;

  3. 3.

the sewing functions $G_{\nu}(x,t,\lambda)$ and the scattering matrices
$T_{\nu}(\lambda)$ for each ray $l_{\nu}$, $\nu=0,\dots,2h_{g}-1$;

B) $\mathcal{T}_{1}$ (resp. $\mathcal{T}_{2}$) allows one to construct the
regular solution $\xi^{\nu}(x,t,\lambda)$ of the RHP;

C) $\mathcal{T}_{1}$ (resp. $\mathcal{T}_{2}$) allows one to recover the
potential $Q_{x}$ of the Lax operator.

###### Proof.

A) Given $\mathcal{T}_{1}$ (respectively $\mathcal{T}_{2}$) and using (59) we
recover the Gauss factors, i.e., the group-valued functions
$S_{\nu}^{\pm}(\lambda,t)$ (respectively $T_{\nu}^{\pm}(\lambda,t)$) for
$\nu=0,1$. Next, using the $\mathbb{Z}_{h}$-symmetry we construct
$S_{\nu}^{\pm}(\lambda,t)$ (respectively $T_{\nu}^{\pm}(\lambda,t)$) for
$\nu=2,\dots,2h_{g}-1$, see eq. (59).

The next step consists in recovering the functionals of the integrals of
motion $d_{\nu,j}^{\pm}(\lambda)$. The relations (63) provide in principle
this possibility. Indeed, the sets $\mathcal{T}_{1}$ and $\mathcal{T}_{1}$
then we know the Gauss factors $S_{\nu}^{\pm}(\lambda,t)$ and
$T_{\nu}^{\pm}(\lambda,t)$ not only in the typical representations, but also
in all fundamental representations. Of course there are serious difficulties
in evaluating the right hand sides of (63). However there is no doubt that
they are determined uniquely by any of the sets $\mathcal{T}_{1}$ and
$\mathcal{T}_{1}$. As a result we have determined all Gauss factors in eq.
(58), i.e. we obtained all sewing functions $G_{\nu}(x,t,\lambda)$ and the
scattering matrices $T_{\nu}(t,\lambda)$.

B) Since the operator $\mathcal{L}$ has no discrete eigenvalues, then the
corresponding FAS satisfy a regular RHP. Therefore the sewing functions
$G_{\nu}(x,t,\lambda)$ determine uniquely the solution
$\xi_{\nu}^{\pm}(\lambda,t)$ of the RHP.

C) Since the solution $\xi_{\nu}^{\pm}(\lambda,t)$ of the RHP is uniquely
determined then the corresponding potential of $\mathcal{L}$ is determined
from eq. (62).

∎

###  5.2 Minimal sets of scattering data for the exceptional algebras

Here we will formulate the results about the spectral data for the Lax
operators related to the exceptional algebras. The calculations were performed
by Maple. The allowed us to calculate the corresponding vectors $\vec{v}_{g}$
as well as the scalar products $(\vec{v}_{g},\alpha_{j})$ and to establish for
each of the exceptional algebras the results in Subsection 4.2. In particular
we found that the subsets of roots $\delta_{0}$ and $\delta_{1}$ are given by
the sets of ‘black‘ and ‘white‘ roots, i.e:

$\begin{split}\delta_{0}\equiv B_{g},\qquad\delta_{1}\equiv W_{g}.\end{split}$
(66)

Below we list, besides the sets of roots $\delta_{0}$ and $\delta_{1}$, also
Coxeter automorphism as a linear operator (matrix) acting on the Euclidean
space ${\mathbb{E}}^{r}$.

####  5.2.1 $\mathfrak{g}\simeq E_{6}^{(1)}$

Let $\varepsilon_{6}=\frac{1}{\sqrt{3}}(-e_{6}-e_{7}+e_{8})$. Then the set of
admissible roots (24) ca be rewritten as:

$\displaystyle\alpha_{0}$
$\displaystyle=\frac{1}{2}(e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+\sqrt{3}\varepsilon_{6}),\quad\alpha_{1}=\frac{1}{2}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}+\sqrt{3}\varepsilon_{6}),$
(67) $\displaystyle\alpha_{2}$
$\displaystyle=e_{1}+e_{2},\quad\alpha_{3}=e_{2}-e_{1},\quad\alpha_{4}=e_{3}-e_{2},\quad\alpha_{5}=e_{4}-e_{3},\quad\alpha_{6}=e_{5}-e_{4}.$

There are two rays (54), corresponding tho the Coxeter automorphism of
$E_{6}^{(1)}$. Roots related to the rays $l_{0}$ and $l_{1}$:

$\displaystyle l_{0}$ $\displaystyle=\arg\lambda=0,$ $\displaystyle\qquad
B_{g}$ $\displaystyle\equiv\\{\alpha_{1},\alpha_{4},\alpha_{6}\\};$ (68)
$\displaystyle l_{1}$ $\displaystyle=\arg\lambda=\frac{\pi}{12},$
$\displaystyle\qquad W_{g}$
$\displaystyle\equiv\\{-\alpha_{2},-\alpha_{3},-\alpha_{5}\\};$

The Coxeter number of $E_{6}^{(1)}$ is $h=12$ and the exponents are
$1,4,5,7,8,11$.

The Coxeter automorphism as a linear operator on the dual Euclidean space
$\mathbb{E}_{6}$:

$\begin{split}C_{E_{6}}=\frac{1}{4}\left(\begin{array}[]{ccccccc}-3&-1&-1&-1&-1&\sqrt{3}\\\
-1&1&-3&1&1&-\sqrt{3}\\\ 1&-1&-1&-1&3&\sqrt{3}\\\ 1&3&-1&-1&-1&\sqrt{3}\\\
1&-1&-1&3&-1&\sqrt{3}\\\
-\sqrt{3}&\sqrt{3}&\sqrt{3}&\sqrt{3}&\sqrt{3}&1\end{array}\right),\quad
C_{E_{6}}^{12}=\leavevmode\hbox{\small 1\kern-3.3pt\normalsize
1}_{6},\quad\det C_{E_{6}}=1.\end{split}$ (69)

Here $e_{j}$, $j=1,\dots,5$ and $\varepsilon_{6}$ is the basis of
$\mathbb{E}_{6}$. The characteristic polynomial of $C_{E_{6}}$ reads:

$\begin{split}P_{E_{6}}=z^{6}+z^{5}-z^{3}+z+1.\end{split}$ (70)

####  5.2.2 $\mathfrak{g}\simeq E_{7}^{(1)}$

Introducing the notation

$\varepsilon_{7}=-{e_{7}-e_{8}\over\sqrt{2}},$

the set of admissible roots (29) can be rewritten as:

$\displaystyle\alpha_{0}$
$\displaystyle=-\sqrt{2}\varepsilon_{7}=e_{7}-e_{8},$
$\displaystyle\quad\alpha_{1}$
$\displaystyle=\frac{1}{2}(e_{1}-e_{2}-e_{3}-e_{4}-e_{5}-e_{6}+\sqrt{2}\varepsilon_{7}),$
(71) $\displaystyle\alpha_{2}$
$\displaystyle=e_{1}+e_{2},\qquad\alpha_{2}=e_{1}+e_{2},$
$\displaystyle\quad\alpha_{3}$
$\displaystyle=e_{2}-e_{1},\quad\alpha_{4}=e_{3}-e_{2},$
$\displaystyle\alpha_{5}$
$\displaystyle=e_{4}-e_{3},\qquad\alpha_{6}=e_{5}-e_{4},$
$\displaystyle\quad\alpha_{3}$ $\displaystyle=e_{6}-e_{5}.$

Roots related to the rays $l_{0}$ and $l_{1}$ are respectively:

$\displaystyle l_{0}$ $\displaystyle=\arg\lambda=0,$ $\displaystyle\qquad
B_{g}$ $\displaystyle\equiv\\{\alpha_{1},\alpha_{4},\alpha_{6}\\};$ (72)
$\displaystyle l_{1}$ $\displaystyle=\arg\lambda=\frac{\pi}{18},$
$\displaystyle\qquad W_{g}$
$\displaystyle\equiv\\{-\alpha_{2},-\alpha_{3},-\alpha_{5},-\alpha_{7}\\};$

The Coxeter number for $E_{7}^{(1)}$ is $h=18$ and the exponents are
$1,5,7,9,11,13,17$. The Coxeter automorphism acts as a linear operator on the
space $\mathbb{E}_{7}$ as:

$\begin{split}C_{E_{7}}=\frac{1}{4}\left(\begin{array}[]{ccccccc}-3&-1&1&1&1&1&\sqrt{2}\\\
-1&1&-1&3&-1&-1&-\sqrt{2}\\\ -1&-1&-3&-1&-1&-1&-\sqrt{2}\\\
-1&1&-1&-1&-1&3&-\sqrt{2}\\\ -1&1&-1&-1&3&-1&-\sqrt{2}\\\
-\sqrt{2}&-\sqrt{2}&-\sqrt{2}&-\sqrt{2}&-\sqrt{2}&-\sqrt{2}&2\end{array}\right),\quad
C_{E_{7}}^{9}=-\leavevmode\hbox{\small 1\kern-3.3pt\normalsize
1}_{7},\quad\det C_{E_{7}}=-1.\end{split}$ (73)

Here $e_{j}$, $j=1,\dots,6$ and $\varepsilon_{7}$ forms the basis in the dual
Euclidean space ${\mathbb{E}}_{7}$. The characteristic polynomial of
$C_{E_{7}}$ is:

$\begin{split}P_{E_{7}}=z^{7}+z^{8}-z^{4}-z^{3}+z+1\end{split}$ (74)

####  5.2.3 $\mathfrak{g}\simeq E_{8}^{(1)}$

The set of admissible roots is given by (34). Roots related to the rays
$l_{0}$ and $l_{1}$ are:

$\displaystyle l_{0}$ $\displaystyle=\arg\lambda=0,$ $\displaystyle\qquad
B_{g}$ $\displaystyle\equiv\\{\alpha_{1},\alpha_{4},\alpha_{6},\alpha_{8}\\};$
(75) $\displaystyle l_{1}$ $\displaystyle=\arg\lambda=\frac{\pi}{30},$
$\displaystyle\qquad W_{g}$
$\displaystyle\equiv\\{-\alpha_{2},-\alpha_{3},-\alpha_{5},-\alpha_{7}\\};$

The Coxeter number for $E_{8}^{(1)}$ is $h=30$ and the exponents are
$1,7,11,13,17,19,23,29$. The action of the Coxeter automorphism as a linear
operator on the dual Euclidean space $\mathbb{E}_{7}$ is given by

$\begin{split}C_{E_{8}}=\frac{1}{4}\left(\begin{array}[]{cccccccc}-3&-1&1&1&1&1&1&-1\\\
-1&1&-1&3&-1&-1&-1&1\\\ -1&-3&-1&-1&-1&-1&-1&1\\\ -1&1&-1&-1&-1&3&-1&1\\\
-1&1&3&-1&-1&-1&-1&1\\\ -1&1&-1&-1&-1&-1&3&1\\\ -1&1&-1&-1&3&-1&-1&1\\\
1&-1&1&1&1&1&1&3\end{array}\right),\quad
C_{E_{8}}^{15}=-\leavevmode\hbox{\small 1\kern-3.3pt\normalsize
1}_{8},\quad\det C_{E_{8}}=-1.\end{split}$ (76)

Here $e_{j}$, $j=1,\dots,6$ and $\varepsilon_{7}$ form up the basis of
${\mathbb{E}}_{7}$. The characteristic polynomial of $C_{E_{8}}$:

$\begin{split}P_{E_{8}}=z^{8}+z^{7}-z^{5}-z^{4}-z^{3}+z+1.\end{split}$ (77)

####  5.2.4 $\mathfrak{g}\simeq F_{4}^{(1)}$

The set of admissible roots is given by (38). Roots related to the rays
$l_{0}$ and $l_{1}$ are as follows:

$\displaystyle l_{0}$ $\displaystyle=\arg\lambda=0,$ $\displaystyle\qquad
B_{g}$ $\displaystyle\equiv\\{\alpha_{1},\alpha_{3}\\};$ (78) $\displaystyle
l_{1}$ $\displaystyle=\arg\lambda=\frac{\pi}{12},$ $\displaystyle\qquad W_{g}$
$\displaystyle\equiv\\{-\alpha_{2},-\alpha_{4}\\};$

The Coxeter number of $F_{4}^{(1)}$ is $h=12$ and the exponents are
$1,5,7,11$. The Coxeter automorphism acts as a linear operator on the dual
space $\mathbb{E}_{4}$ via:

$\begin{split}C_{F_{4}}=\frac{1}{2}\left(\begin{array}[]{cccccccc}1&-1&1&-1\\\
-1&-1&1&1\\\ 1&-1&-1&1\\\ 1&1&1&1\end{array}\right),\quad
C_{F_{4}}^{6}=-\leavevmode\hbox{\small 1\kern-3.3pt\normalsize
1}_{4},\quad\det C_{F_{4}}=1.\end{split}$ (79)

Here $e_{j}$, $j=1,\dots,4$ is the basis of ${\mathbb{E}}_{4}$. The
characteristic polynomial of $C_{F_{4}}$:

$\begin{split}P_{F_{4}}=z^{4}-z^{2}+1.\end{split}$ (80)

####  5.2.5 $\mathfrak{g}\simeq G_{2}^{(1)}$

This comes to be both the simplest and the most peculiar case. The peculiarity
is in the fact, that though the rank of $G_{2}$ equals 2, its root system is
traditionally written as a set of vectors in the 3-dimensional space
$\mathbb{E}_{3}$ which are all orthogonal to the vector $e_{1}+e_{2}+e_{3}$.
The Coxeter automorphism here is the composition of the Weyl reflection
$C_{G_{2}}=S_{\alpha_{1}}S_{\alpha_{2}}$. One can check that $C_{G_{2}}$ is
expressed as the following linear operator in $\mathbb{E}_{3}$:

$\begin{split}C_{G_{2}}=\frac{1}{3}\left(\begin{array}[]{ccc}2&-1&2\\\
2&2&-1\\\ -1&2&2\end{array}\right),\end{split}$ (81)

which has the following properties:

$\begin{split}\qquad C_{G_{2}}^{2}=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\
0&1&0\end{array}\right),\qquad
C_{G_{2}}^{4}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right),\qquad C_{G_{2}}^{6}=\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1}_{3}.\end{split}$ (82)

The characteristic polynomial is

$\begin{split}P_{G_{2}}=z^{3}-2z^{2}+2z-1=(z-1)(z^{2}-z+1).\end{split}$ (83)

Another way to approach the problem is to project the root system of $G_{2}$
onto the 2-dimensional plane orthogonal to the vector $e_{1}+e_{2}+e_{3}$.
Indeed, let

$\varepsilon_{1}=\frac{1}{\sqrt{2}}(e_{2}-e_{1}),\qquad\varepsilon_{2}=\frac{1}{\sqrt{2}}(e_{3}-e_{2}).$

Then the set of admissible roots (43) can be rewritten as:

$\displaystyle\alpha_{0}$
$\displaystyle=\sqrt{2}(\varepsilon_{1}+2\varepsilon_{2})=-e_{1}-e_{2}+2e_{3},\quad\alpha_{1}=\sqrt{2}\varepsilon_{1}=e_{2}-e_{1},$
(84) $\displaystyle\alpha_{2}$
$\displaystyle=\sqrt{2}(\varepsilon_{2}-\varepsilon_{1})=e_{1}-2e_{2}+e_{3},$

Roots related to the rays $l_{0}$ and $l_{1}$ are:

$\displaystyle l_{0}$ $\displaystyle=\arg\lambda=0,$ $\displaystyle\qquad
B_{g}$ $\displaystyle\equiv\\{\alpha_{1}\\};$ (85) $\displaystyle l_{1}$
$\displaystyle=\arg\lambda=\frac{\pi}{12},$ $\displaystyle\qquad W_{g}$
$\displaystyle\equiv\\{-\alpha_{2}\\};$

The Coxeter number for $G_{2}^{(1)}$ is $h=6$ and the exponents are $1,5$.
This Coxeter automorphism induces an action on the dual space $\mathbb{E}_{2}$
with a matrix:

$\begin{split}C^{\prime}_{G_{2}}=\left(\begin{array}[]{cccccccc}0&1\\\
-1&1\end{array}\right),\quad C^{\prime}_{G_{2}}{}^{3}=-\leavevmode\hbox{\small
1\kern-3.3pt\normalsize 1}_{2},\quad\det C^{\prime}_{G_{2}}=1.\end{split}$
(86)

Here $\\{\varepsilon_{1},\varepsilon_{2}\\}$ form up a basis in
${\mathbb{E}}_{2}$. Finally, the characteristic polynomial of $C_{G_{2}}$
reads:

$\begin{split}P_{G_{2}}=z^{2}-z+1.\end{split}$ (87)

##  6 Conclusions

We presented here real Hamiltonian forms of affine Toda field theories related
exceptional untwisted complex Kac-Moody algebras. We established that the
special properties of these models allow us to relate the construction of the
RHF to the study of ${\mathbb{Z}}_{h}$ symmetries of the extended Dynkin
diagrams (with $h$ being the Coxeter number of ${\mathfrak{g}}$). The general
construction is illustrated by several examples of such models related to the
exceptional Kac-Moody algebras $E_{6}^{(1)}$, $E_{7}^{(1)}$,
$E_{8}^{(1)}$,$D_{4}^{(1)}$ and $G_{2}^{(1)}$. We used the “dihedral
realisation” of the Coxeter automorphism ${\rm
Cox}\,({\mathfrak{g}})=S_{1}\circ S_{2}$, where $S_{1},S_{2}$ are
${\mathbb{Z}}_{2}$-automorphisms of ${\mathfrak{g}}$. One of these
automorphisms extracts the RHF and the other one acts as additional
${\mathbb{Z}_{2}}$-reduction on it.

Each of the real Hamiltonian forms obtained above has its dual one. Indeed, we
could define the involution $\mathcal{C}$ using the Weyl reflection $S_{2}$.
The consideration is quite analogous: simply we interchange the black roots
with the white ones. The reformulation of all the above results is rather
obvious. Note that the dual real Hamiltonian forms of 2DTFT are not equivalent
to the ones obtained above. Indeed, it is easy to check that in all cases
treated above the number of the white roots is different from the number of
black roots.

Some additional problems are natural extensions to the results presented here:

  * •

The complete classification of all nonequivalent RHF of ATFT.

  * •

The description of the hierarchy of Hamiltonian structures of RHF of ATFT (for
a review of the infinite-dimensional cases see e.g. [14, 21] and the
references therein) It is also an open problem to construct the RHF for ATFT
using some of its higher Hamiltonian structures.

  * •

The extension of the dressing Zakharov-Shabat method [53] to the above classes
of Lax operators is also an open problem. One of the difficulties is due to
the fact that the $\mathbb{Z}_{h}$ reductions requires dressing factors with
$2h$ pole singularities [20]. This makes the relevant linear algebraic
equations rather involved [55].

  * •

Another open problem is to study types of boundary conditions and boundary
effects of ATFT’s and their RHF [11, 12]. This includes types of boundary
defects [13].

  * •

The last and technically more involved problem is to solve the inverse
scattering problem for the Lax operator and thus prove the complete
integrability of all these models. The ideas of [2, 33] about the
interpretation of the inverse scattering method as a generalized Fourier
transform holds true also for the $\mathbb{Z}_{h}$ reduces Lax operators [35,
34, 51, 36, 52]. This may allow one to derive the action-angle variables for
these classes of NLEE.

## Acknowledgements

This work is supported by the Bulgarian National Science Fund, grant
KP-06N42-2.

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