Symmetry, Integrability and Geometry: Methods and Applications
Vol. 2 (2006), Paper 022, 11 pages
Real Hamiltonian Forms of Af f ine Toda Models
Related to Exceptional Lie Algebras
Vladimir S. GERDJIKOV
†
and Georgi G. GRAHOVSKI
†‡
†
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
72 Tsarigradsko Chaussee, 1784 Sof ia, Bulgaria
E-mail: gerjikov@inrne.bas.bg, grah@inrne.bas.bg
‡
Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise,
2 Avenue Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France
Received December 19, 2005, in final form February 05, 2006; Published online February 17, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper022/
Abstract. The construction of a family of real Hamiltonian forms (RHF) for the special
class of affine 1 + 1-dimensional Toda field theories (ATFT) is reported. Thus the method,
proposed in [1] for systems with finite number of degrees of freedom is generalized to infinitedimensional Hamiltonian systems. The construction method is illustrated on the explicit
nontrivial example of RHF of ATFT related to the exceptional algebras E6 and E7 . The
involutions of the local integrals of motion are proved by means of the classical R-matrix
approach.
Key words: solitons; affine Toda field theories; Hamiltonian systems
2000 Mathematics Subject Classification: 37K15; 17B70; 37K10; 17B80
1
Introduction
To each simple Lie algebra rank g one can relate Toda field theory (TFT) in 1 + 1 dimensions.
It allows Lax representation: [L, M ] = 0, where L and M are first order ordinary differential
operators, see e.g. [2, 3, 4, 5, 6, 7]:
d
− iqx (x, t) − λJ0 ψ(x, t, λ) = 0,
dx
d
1
M ψ ≡ i − I(x, t) ψ(x, t, λ) = 0.
dt λ
Lψ ≡
i
qx (x, t) =
r
X
qk,x Hk ,
(1)
k=1
whose potentials take values in g. Here q(x, t) ∈ h is the Cartan subalgebra of g, ~q(x, t) =
r
P
qk~ek
k=1
is its dual r-component vector, r = rank g. Hk are the Cartan generators dual to the orthonormal
basis elements ~ek in the root space, and
J0 =
X
α∈π
Eα ,
I(x, t) =
X
e−(α,~q(x,t)) E−α .
α∈π
By πg we denote the set of admissible roots of g, i.e. πg = {α0 , α1 , . . . , αr } where α1 , . . . , αr are
the simple roots of g and α0 is the minimal root of g. The corresponding TFT is known as the
affine TFT. The Dynkin graph that corresponds to the set of admissible roots of g is called
2
V.S. Gerdjikov and G.G. Grahovski
extended Dynkin diagrams (EDD). The equations of motion are of the form:
r
X
∂ 2 ~q
=
αj e−(αj ,~q(x,t)) .
∂x∂t
j=0
The present paper extends the ideas of [8] and [9] to the ATFT related to the exceptional simple
Lie algebra E6 ; for finite Toda chains see [10, 11, 1, 12].
2
The reduction group
The operators L and M are invariant with respect to the reduction group GR ' Dh where h
is the Coxeter number of g. It is generated by two elements satisfying g1h = g22 = (g1 g2 )2 = 11
which allow realizations both as elements in Aut g and in Conf C. The invariance condition has
the form [2]:
C1 (U (x, t, κ1 (λ))) = U (x, t, λ),
†
Ck (U (x, t, κk (λ))) = U (x, t, λ),
C1 (V (x, t, κ1 (λ))) = V (x, t, λ),
Ck (V † (x, t, κk (λ))) = V (x, t, λ),
k = 2, 3.
(2)
where U (x, t, λ) = −iqx (x, t) − λJ0 and V (x, t, λ) = − λ1 I(x, t). Here Ck are automorphisms
of finite order of g, i.e. C1h = C22 = (C1 C2 )2 = 11 while κk (λ) are conformal mappings of the
complex λ-plane:
κ1 (λ) = ωλ,
κ2 (λ) = λ∗ ,
κ3 (λ) = (ωλ)∗ ,
where ω = exp(2πi/h). The algebraic constraints (2) are automatically compatible with the
evolution. A number of nontrivial reductions of nonlinear evolution equations can be found
in [13, 14].
3
Spectral properties of the Lax operator
The reduction conditions (2) lead to rather special properties of the operator L. Along with L
we will use also the equivalent system:
dm
e
− iqx m(x, t, λ) − λ[J0 , m(x, t, λ)] = 0,
Lm(x,
t, λ) ≡ i
dx
(3)
where m(x, t, λ) = ψ(x, t, λ)eiJ0 xλ . Here ~qx is the potential which we choose to be a Schwartztype function taking values in the complexified Cartan subalgebra hC ⊂ g. This means that the
boundary conditions for ~q are determined up to the constant vector ρ
~0 :
ρ
~0 = lim ~q(x, t).
x→±∞
e
The change of the variables ~q 0 = ~q − ρ
~0 does not affect the potential and the spectral data of L,
but they change the right hand side of the ATFT equations into:
r
X
∂ 2 ~q 0
0
=
sj αj e−(αj ,~q (x,t)) ,
∂x∂t
j=0
where sj = exp[−(αj , ρ
~0 )] obviously satisfy the consistency conditions:


r
r
Y
X
n
s j = exp −
(nj αj , ρ
~0 ) = 1.
j
j=0
j=0
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
where nj are the minimal positive integers for which
r
P
3
nj αj = 0. In the literature there are
j=0
two canonical ways of fixing up the vector ρ
~0 . The first one is to put sj = 1 for all j, the second
is to have sj = nj for all j.
The spectral properties of the Lax operator are rather involved due to the fact that both J0
and I(x, t) have complex-valued eigenvalues. In fact all these eigenvalues are constant and are
proportional to ω k , k = 0, 1, . . . , h − 1, where ω = e2πi/h and h is the Coxeter number of g. As
a result one finds that the continuous spectrum of L(λ) fills up 2h rays passing through the
origin:
λ ∈ lν : arg λ =
(ν − 1)π
.
h
For such operators one can introduce Jost solutions only for potentials on a compact support [15, 16]:
lim Ψ(x, t, λ)eiλJ0 x+(i/λ)I0 t = 11,
x→∞
lim Φ(x, t, λ)eiλJ0 x+(i/λ)I0 t = 11,
x→−∞
where
I0 = lim I(x, t) =
x→∞
r
X
E−αk ,
αk ∈ π(g).
k=0
Then the corresponding scattering matrix T (t, λ) is defined by
T (t, λ) = (Ψ(x, t, λ))−1 Φ(x, t, λ).
The compactness of the potential ensures that the Jost solutions Φ(x, t, λ), Ψ(x, t, λ) and the
scattering matrix T (t, λ) are meromorphic functions of λ.
The fact that Φ(x, t, λ) and Ψ(x, t, λ) are also Jost solutions of M (λ) means that T (t, λ)
evolves according to:
i
1
dT
− [I0 , T (t, λ)] = 0.
dt
λ
(4)
The limiting procedure to non-compact potentials can be done only for the so-called fundamental
analytical solutions (FAS) mν (x, t, λ), λ ∈ Ων , i.e. (ν − 1)π/h ≤ arg λ ≤ νπ/h. Their construction is outlined in [15, 16] for generic complex-valued J0 . Our Lax pair is special, because it
satisfies the reduction condition (2) with the Coxeter automorphism [17]:
C̄1 (X) ≡ C1−1 XC1 ,
C1 = e2πiHρ /h ,
ρ=
1X
α;
2
α>0
obviously C1h = 11 and C̄1 (J0 ) = ω −1 J0 . The FAS mν (x, t, λ) of (3) are defined only in the
sector Ω3 and satisfy:
C̄1 (mν (x, t, ωλ)) = mν−2 (x, t, λ),
λ ∈ lν−2 .
Thus the inverse scattering problem for L(λ) canSbe formulated as a Riemann–Hilbert problem
for mν (x, t, λ) on the continuous spectrum Σ ≡ 2h
ν=1 lν .
Skipping the details, we remark that equation (4) allows one to show that ATFT possess
generating functionals of the integrals of motion. This can be shown easily if the potential q(x, t)
is such that T (t, λ) can be diagonalized:
T (t, λ) = u−1
0 (t, λ)D(λ)u0 (t, λ).
(5)
4
V.S. Gerdjikov and G.G. Grahovski
All factors in the above equation take values in the Lie group G with the Lie algebra g and D(λ)
is a diagonal matrix. Then there exist a set of functions τk (λ), k = 1, . . . , r such that:
!
r
X
2τk (λ)
D(λ) = exp
(6)
Hα ,
(αk , αk ) k
k=1
where Hαk are the Cartan generators of g corresponding to the simple root αk . Inserting (5)
and (6) into (4) one finds that
dD
= 0,
dt
dτk
= 0,
dt
i.e.
for all k = 1, . . . , r. Obviously all eigenvalues of the scattering matrix T (t, λ) will be expressed
in terms of τk . Each τk (λ) can be expanded over the negative powers of λ:
τk (λ) =
∞
X
Is(k) λ−s ,
s=0
(k)
and all the coefficients Is
4
will be integrals of motion.
Real Hamiltonian forms
The Lax representations of the ATFT models (see e.g. [2, 3, 4, 18, 6] and the references therein)
are related mostly to the normal real form of the Lie algebra g, see [20].
Our aim here is to:
1) generalize the ATFT to complex-valued fields ~q C = ~q 0 + i~q 1 , and to
2) describe the family of RHF of these ATFT models.
We also provide a tool generalizing of the one in [1] for the construction of new inequivalent
RHF’s of the ATFT. The ATFT for the algebra sl(n) can be written down as an infinitedimensional Hamiltonian system as follows:
dqk
dpk
= {qk , HATFT },
= {pk , HATFT },
dt
dt
!
Z ∞
r
X
1
−(~
q (x,t),αk )
(~
p(x, t), p~(x, t)) +
e
,
HATFT =
dx
2
−∞
k=0
where ~q(x, t) and p~ = ∂~q/∂x are the canonical coordinates and momenta satisfying canonical
Poisson brackets:
{pk (x), qj (y)} = δjk δ(x − y).
(7)
Next we define the involution C acting on the phase space M as follows:
1) C(F (pk , qk )) = F (C(pk ), C(qk )),
2) C ({F (pk , qk ), G(pk , qk )}) = {C(F ), C(G)} ,
3) C(H(pk , qk )) = H(pk , qk ).
Here F (pk , qk ), G(pk , qk ) and the Hamiltonian H(pk , qk ) are functionals on M depending analytically on the fields qk (x, t) and pk (x, t).
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
5
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
~q a (x, t), p~ a (x, t), a = 0, 1:
p~ C (x, t) = p~ 0 (x, t) + i~
p 1 (x, t),
~q C (x, t) = ~q 0 (x, t) + i~q 1 (x, t),
0
pk (x, t), qj0 (y, t) = − p1k (x, t), qj1 (y, t) = δkj δ(x − y).
The densities of the corresponding Hamiltonian and symplectic form equal
C
HATFT
≡ Re HATFT (~
p 0 + i~
p 1 , ~q 0 + i~q 1 )
r
X
1 0 0
1 1 1
0
= (~
p , p~ ) − (~
p , p~ ) +
e−(~q ,αk ) cos((~q 1 , αk )),
2
2
k=0
ω C = (d~
p 0 ∧0 d~q 0 ) − (d~
p 1 ∧0 d~q 1 ).
The family of RHF then are obtained from the CATFT by imposing an invariance condition
with respect to the involution C˜ ≡ C ◦ ∗ where by ∗ we denote the complex conjugation. The
involution C˜ splits the phase space MC into a direct sum MC ≡ MC ⊕ MC where
+
MC
+ = M0 ⊕ iM1 ,
−
MC
− = iM0 ⊕ M1 ,
The phase space of the RHF is MR ≡ MC
+ . By M0 and M1 we denote the eigensubspaces of C,
i.e. C(ua ) = (−1)a ua for any ua ∈ Ma .
Then extracting of Real Hamiltonian forms (RHF’s) 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 negativelydefinite 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.
Thus to each involution C satisfying 1)–3) one can relate a RHF of the ATFT. Due to
the condition 3) C must preserve the system of admissible roots of g; such involutions can be
constructed from the Z2 -symmetries of the extended Dynkin diagrams of g studied in [6].
5
Examples
In this Section we provide several examples of RHF of ATFT related to exceptional Kac–Moody
algebras with height 1. We show that the involutions C in all these cases are dual to Z2 symmetries of the extended Dynkin diagrams derived in [6].
The examples below illustrate the procedure outlined above and display new types of ATFT.
5.1
(1)
E6
Toda f ield theories
The set of admissible roots for this algebra is
1
α1 = (e1 − e2 − e3 − e4 − e5 − e6 − e7 + e8 ),
α2 = e1 + e2 ,
2
α3 = e2 − e1 ,
α4 = e3 − e2 ,
α5 = e4 − e3 ,
α6 = e5 − e4 ,
1
α0 = − (e1 + e2 + e3 + e4 + e5 − e6 − e7 + e8 ),
2
where α1 , . . . , α6 form the set of simple roots of E6 and α0 is the minimal root of the algebra. This
is the standard definition of the root system of E6 embedded into the 8-dimensional Euclidean
space E8 . The root space E6 of the algebra E6 is the 6-dimensional subspace of E8 orthogonal to
6
V.S. Gerdjikov and G.G. Grahovski
the vectors e7 + e8 and e6 + e7 + 2e8 . Thus any vector ~q belonging to E6 has only 6 independent
coordinates and can be written as:
~q =
5
X
1
e06 = √ (e6 + e7 − e8 ).
3
qk ek + q6 e06 ,
k=1
(8)
Let us fix up the action of the involution C on a generic vector ~q in E8 by:
C(qk ) = −q5−k +
= q13−k −
4
1X
qm ,
2
1
2
for k = 1, . . . , 4,
m=1
8
X
qm ,
for k = 5, . . . , 8.
(9)
m=5
This action is compatible with the Z2 -symmetry C # of the extended Dynkin diagram (see Fig. 1)
(1)
and reflects an involution of the Kac–Moody algebra E6 , see [21]. It acts on the root space as
follows:
C # ek = −e5−k +
= e13−k −
C # α1 = α6 ,
α1
4
1X
em ,
2
1
2
for k = 1, . . . , 4,
m=1
8
X
em ,
m=5
C # α3 = α5 ,
◦
α0
◦
α2
α3
◦ PiP ◦PiP ◦
α4
for k = 5, . . . , 8,
C # αk = αk ,
k = 0, 2, 4.
⇒ ◦
β0
α5
◦
1
◦
β2
(10)
◦ i◦
β4
β3
◦
β1
α6
◦
1
(1)
(1)
Figure 1. E6 → F4 .
The involution C # splits the root space E6 into a direct sum of its eigensubspaces: E6 =
E+ ⊕ E− with dim E+ = 4, dim E− = 2. The vectors:
√
1
1
1
ee2 = (e1 + e2 + e3 + e4 ),
ee3 = (−e1 − e2 + e3 + e4 ),
ee1 = (e5 − 3e06 ),
2
2
2
1
1
1 √
ee4 = (−e1 + e2 − e3 + e4 ), ee5 = (−e1 + e2 + e3 − e4 ), ee6 = ( 3e5 + e06 ).
2
2
2
form an orthonormal basis in E6 . The first four satisfy C # eek = eek , k = 1, . . . , 4, so they span E+ ;
the last two span E− because C # eej = −e
ej , j = 5, 6. In terms of eek the admissible root system
(1)
of F4 takes the standard form:
β0 = −e
e2 − ee1 ,
β2 = ee2 − ee3 ,
1
e1 − ee2 − ee3 − ee4 ),
β1 = (e
2
β3 = ee4 ,
β4 = ee3 − ee4 .
satisfying β0 + 2β1 + 2β2 + 4β3 + 3β4 = 0.
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
7
Let us take the complex vector ~q(x, t) = ~q 0 (x, t) + i~q 1 (x, t) ∈ E6 (i.e., of the form (8)) and
let p~(x, t) = ∂~q/∂x. Let us denote their projections onto E± by ~q± and p~± respectively. Then
the densities H1R , ω1R for the RHF of AFTF equal:
H1R =
0
1
(~
p 0+ (x, t), p~ 0+ (x, t)) − (~
p 0− (x, t), p~ 0− (x, t)) + e−(~q + (x,t),β0 )
2
√
0
0
e6 )) + 2e−(~q + (x,t),β2 )
+ 2e−(~q + (x,t),β1 ) cos((~q 1− (x, t), ee5 + 3e
0
0
+ 4e−(~q + (x,t),β3 ) cos((~q 1− (x, t), ee5 )) + 3e−(~q + (x,t),β4 ) ,
ω1R = δ~
p+ (x) ∧0 δ~q+ (x) − δ~
p− (x) ∧0 δ~q− (x) ,
If we put ~q− (x, t) = 0 then also p~− (x, t) = 0 and we get the reduced ATFT related to the
(1)
Kac–Moody algebra F4 [6].
5.2
(1)
E7
Toda f ield theories
The set of admissible roots for this algebra is
1
α2 = e1 + e2 ,
α1 = (e1 − e2 − e3 − e4 − e5 − e6 − e7 + e8 ),
2
α3 = e2 − e1 ,
α4 = e3 − e2 ,
α5 = e4 − e3 ,
α6 = e5 − e4 ,
α7 = e6 − e5 ,
α0 = e7 − e8 ,
where α1 , . . . , α7 form the set of simple roots of E7 and α0 is the minimal root of the algebra. This
is the standard definition of the root system of E7 embedded into the 8-dimensional Euclidean
space E8 . The root space E7 of the algebra E7 is the 7-dimensional subspace of E8 orthogonal
to the vector e7 + e8 . Thus any vector ~q belonging to E7 has 7 independent coordinates and can
be written as:
~q =
6
X
1
e07 = √ (e7 − e8 ).
2
qk ek + q7 e07 ,
k=1
(11)
Let us fix up the action of the involution C on a generic vector ~q in E8 by equation (9). This
action is compatible with the Z2 -symmetry C # of the extended Dynkin diagram (see Fig. 2) and
(1)
reflects an involution of the Kac–Moody algebra E7 , see [21]. It acts on the root space as in
equation (10) above.
◦
α0
α1
α3
◦Pi ◦ Pi ◦ Pi ◦
α4
α2
α5
◦
1
α6
◦
1
(1)
Figure 2. The E7
α7
◦
1
(2)
→ E6
⇒
β2
◦
β4
β3
◦ i◦
β1
◦
β0
◦
RHF of affine TFT.
However its action on the root space E7 is different. It splits E7 into a direct sum of its
eigensubspaces: E7 = E+ ⊕ E− with dim E+ = 4, dim E− = 3. The vectors:
√
1
ee1 = √ (e1 + e2 + e3 + e4 + e5 − e6 − 2e07 ),
2 2
√
1
ee2 = √ (−e1 − e2 − e3 − e4 + e5 − e6 − 2e07 ),
2 2
1
ee3 = √ (−e1 + e4 ),
2
1
ee4 = √ (−e2 + e3 ),
2
8
V.S. Gerdjikov and G.G. Grahovski
form an orthonormal basis in E+ . Indeed, they satisfy (e
ej , eek ) = δjk and C # eek = eek , k = 1, . . . , 4.
(2)
In terms of eek the admissible root system of E6 takes the standard form:
β0 = −e
e2 − ee1 ,
β1 = ee2 − ee3 ,
β2 = ee3 − ee4 ,
β4 = ee1 − ee2 − ee3 − ee4 ,
β3 = 2e
e4 ,
satisfying β0 + 2β1 + β2 + 3β3 + 2β4 = 0.
The vectors
1
ee6 = √ (e5 + e6 ),
2
1
ee5 = (−e1 + e2 + e3 − e4 ),
2
√
1
ee7 = (−e5 + e6 − 2e07 ),
2
span E− because C # eej = −e
ej , j = 5, 6, 7.
Let us take the complex vector ~q(x, t) = ~q 0 (x, t) + i~q 1 (x, t) ∈ E7 (i.e., of the form (11)) and
let p~(x, t) = ∂~q/∂x. Let us denote their projections onto E± by ~q± and p~± respectively. Then
the densities H1R , ω1R for the RHF of AFTF equal:
H2R =
0
1
(~
p 0+ (x, t), p~ 0+ (x, t)) − (~
p 0− (x, t), p~ 0− (x, t)) + 2e−(~q + (x,t),β0 ) cos((~q 1− (x, t), ee7 ))
2
√
0
1
1
−(~
q 0+ (x,t),β1 )
cos (~q − (x, t), (e
+ 4e
e5 + 2e
e6 − ee7 ) + 2e−(~q + (x,t),β2 )
2
0
0
+ 6e−(~q + (x,t),β3 ) cos((~q 1− (x, t), ee5 )) + 4e−(~q + (x,t),β4 ) ,
ω2R = δ~
p+ (x) ∧0 δ~q+ (x) − δ~
p− (x) ∧0 δ~q− (x) .
Again, if we put ~q− (x, t) = 0 then also p~− (x, t) = 0 and we get the reduced ATFT related to
(2)
the Kac–Moody algebra E6 [6].
6
Classical R-matrix method and ATFT
There are several methods to approach the Hamiltonian properties of the ATFT, see e.g. [10,
18, 13, 11, 19]. One of the effective methods is based on the well known classical R-matrix [22]
which is introduced by:
U (x, λ) ⊗ U (y, µ) = [R(λ, µ), U (x, λ) ⊗ 11 + 11 ⊗ U (y, µ)] δ(x − y),
0
(12)
where U (x, λ) ⊗ U (y, µ) ij,kl = {Uij (x, λ), Ukl (y, µ)}. In order to apply this definition effecti0
vely we make use of another Lax operator for the ATFT:
e
dψe
ee
ee
e
e ψ(x,
+ U (x, λ)ψ(x,
λ) = 0,
L
λ) ≡ i
dx
r
r
X
iX
U (x, λ) = −
qj,x Hj − λ
e−(αj ,~q)/2 Eαj ,
2
j=1
j=0
which is gauge equivalent to L (1). Since the Poisson brackets are introduced by (7) the left
hand side of (12) takes the form:
r
i X −(αk ,~q)/2
Uij (x, λ) ⊗ Ukl (y, µ) =
e
(µHαk ⊗ Eαk − λEαk ⊗ Hαk ) δ(x − y).
0
4
k=0
Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
9
Thus equation (12) becomes an over-determined set of equations for R(λ, µ) which is solved
by [22]:
r
1 λh + µh X
1 X λ p(α) µh Eα ⊗ E−α
R(λ, µ) =
Hk ⊗ Hk +
4i λh − µh
2i
µ
λh − µh
k=1
1
=
4i sinh(hη/2)
cosh(hη/2)
α∈∆
r
X
X
k=1
α∈∆
!
Hk ⊗ Hk +
(p(α)−h/2)η
e
Eα ⊗ E−α
,
where h is the Coxeter number of g, η = ln(λ/µ) and p(α) is the height of the root α modulo h.
r
r
P
P
If α =
nα,j αj where nα,j are integers, then p(α) =
nα,j .
j=1
j=1
The relation (12) allows one to derive the Poisson brackets between the matrix elements of
the fundamental solutions Tx+0 (x, λ) and Tx−0 (x, λ) (or the scattering matrix Tx0 (λ)) of L which
are defined by:
LTx±0 (x, λ) = 0,
Tx0 (λ) =
lim T ± (x, λ)e−iλxJ0
x→±x0 x0
(Tx+0 (x, λ))−1 Tx−0 (x, λ).
= 11,
Then
Tx±0 (x, λ) ⊗ Tx±0 (y, µ) = R(λ, µ), Tx±0 (x, λ) ⊗ Tx±0 (y, µ) ,
0
and
Tx0 (λ) ⊗ Tx0 (µ) = [R(λ, µ), Tx0 (λ) ⊗ Tx0 (µ)] ,
(13)
0
These results hold true for potentials on compact support provided we choose x0 large enough
so that ~qx = 0 for |x| > x0 .
With Tx0 (t, λ) we can associate a set of generating functionals τk (x0 , λ) of the integrals of
motion
!
r
X
2τ
(x
,
λ)
0
k
Hα k .
Tx0 (t, λ) = u−1
Dx0 (λ) = exp
0 (x0 , t, λ)Dx0 (λ)u0 (x0 , t, λ),
(αk , αk )
k=1
Again we can write the expansion
τk (x0 , λ) =
∞
X
(s)
Ix0 ,k λ−s .
s=0
An important consequence of (13) is that the functions τk (x0 , λ) are in involution. Indeed
from (13) there follows that:
{tr Txk0 (λ), tr Txp0 (µ)} = 0,
(14)
for any pair of integers k, p. Obviously tr T k (λ) can be expressed in terms of the invariants τk (λ) (6) of the scattering matrix T (λ) ∈ G. Therefore from (14) we find that
{τk (x0 , λ), τm (x0 , µ)} = 0,
(s)
i.e. the integrals of motion Ik
(s)
{Ix0 ,k , Ix(n)
} = 0,
0 ,m
1 ≤ k, m ≤ r,
are all in involution:
0 ≤ k, m ≤ r,
s, n ≥ 0.
10
V.S. Gerdjikov and G.G. Grahovski
In order to derive the corresponding results for the RHF of the ATFT we have to use the fact
that the automorphism C induces an automorphism C ∨ on the Lie algebra g and on the Lax
operator as follows [23]:
C ∨ (Hj ) = HC # (ej ) ,
C ∨ Eα = EC # (α) ,
L(C(~q∗ ), x, λ∗ ) = C ∨ (L(~q, x, λ))∗ .
These relations allow one to prove that the fundamental solution Tx0 (x, λ) and the scattering
matrix Tx0 (λ) for the corresponding RHF model has the properties:
(Tx0 (x, λ∗ ))∗ = C ∨ (Tx0 (x, λ)),
(Tx0 (λ∗ ))∗ = C ∨ (Tx0 (λ)),
and as a consequence: (τk (λ∗ ))∗ = τk̄ (λ)), where k̄ is defined through C # (αk ) = αk̄ .
7
Conclusions
(1)
(1)
The RHF of the ATFT models related to the exceptional Kac–Moody algebras E6 and E7
are constructed. These models generalize the ones in [6] since they contain two types of fields
~q+ (x, t) and ~q− (x, t) with different properties with respect to the involution C. The models in [6]
contain only fields invariant with respect to C.
We outlined the derivation of the Hamiltonian properties through the classical R-matrix
approach.
Acknowledgments
One of us (GGG) thanks the organizing committee of the Sixth International Conference “Symmetry in Nonlinear Mathematical Physics” for the scholarship and for the warm hospitality in
Kyiv. The present paper is the written version of the talk delivered by GGG at this conference.
The work of GGG is supported by the Bulgarian National Scientific Foundation Young Scientists
Scholarship for the project “Solitons, Differential Geometry and Biophysical Models”. We also
acknowledge support by the National Science Foundation of Bulgaria, contract No. F-1410. We
also thank an anonymous referee for careful reading of the manuscript and for useful suggestions.
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