Generalized fermionic discrete Toda hierarchy

0.37 MB·

by  V. V. Gribanov

nlin.SI/0311030
Generalized fermionic discrete
Toda hierarchy
4
0
0
2 V.V. Gribanova,1, V.G. Kadyshevskyb,2 and A.S. Sorinb,3

n
a (a)Dzhelepov Laboratory of Nuclear Problems,
J
6 (b)Bogoliubov Laboratory of Theoretical Physics,
1 Joint Institute for Nuclear Research,

141980 Dubna, Moscow Region, Russia
]
I
S
.
n
i Dedicated to the memory of Professor I. Prigogine
l
n
[

2
v
0 Abstract
3
Bi-Hamiltonian structure and Lax pair formulation with the spectral parameter of the gen-
0
1 eralized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for
1
different (including periodic) boundary conditions are described. Its two reductions —N = 4
3
0 and N = 2 supersymmetric Toda lattice hierarchies— in different (including canonical) bases
/
n areinvestigated. Itsr-matrixdescription, monodromymatrix,andspectralcurvesarediscussed.
i
l
n
:
v
i
X
r
PACS: 02.20.Sv; 02.30.Jr; 11.30.Pb
a
Keywords: Completely integrable systems; Toda field theory; Supersymmetry; Discrete
symmetries
E-mail:
1) [email protected]
2) [email protected]
3) [email protected]
1 Introduction
At present, two different non-trivial supersymmetric extensions of the two-dimensional (2D)
infinite bosonic Toda lattice hierarchy are known. They are the N = (2|2) [1, 2, 3, 4, 5] and
N = (0|2) [5] supersymmetric Toda lattice hierarchies. Actually, besides a different number of
supersymmetries they have different bosonic limits which are decoupled systems of two infinite
bosonic Toda lattice hierarchies and single infinite bosonic Toda lattice hierarchy, respectively.
One-dimensional(1D)reductions ofthesehierarchies—N = 4andN = 2supersymmetric Toda
latticehierarchies—werestudiedin[6,7],whiletheirfinitereductionscorrespondingtodifferent
boundary conditions (e.g., fixed ends, periodic boundary conditions, etc.) were investigated in
[8, 9, 10, 11, 12]. Quite recently, a dispersionless limit of the N = (1|1) supersymmetric Toda
lattice hierarchy was constructed in [13, 14].
The present paper continues studies of the above-mentioned hierarchies and is addressed to
yet unsolved problems of constructing their periodic counterparts, bi-Hamiltonian structure in
different (including canonical) bases, (2m×2m)-matrix and 4×4-matrix (3×3-matrix) Lax
pair descriptions with the spectral parameter, r-matrix approach, and spectral curves.
The structure of this paper is as follows. In section 2.1, starting with the zero-curvature
representation we introduce the 2D generalized fermionic Toda lattice equations and describe
their two reductions related to the N = (2|2) and N = (0|2) supersymmetric Toda lattice
equations. Then, insection2.2, weconstruct thebi-Hamiltonianstructureofthe1Dgeneralized
fermionic Toda lattice hierarchy, and its fermionic and bosonic Hamiltonians.
Sections 3 and 4 are devoted to the 1D N = 4 and N = 2 supersymmetric Toda lattice
hierarchies, respectively. We construct their bi-Hamiltonian structure in sections 3.1 and 4.1,
fermionic symmetries in section 3.2, and in sections 3.3 and 4.2, we investigate a transition to
the canonical basis which spoils a number of supersymmetries.
In section 5, we consider periodic supersymmetric Toda lattice hierarchies. Thus, in sec-
tion 5.1, we construct the (2m × 2m)-matrix zero-curvature representation with the spectral
parameter for the periodic 2D generalized fermionic Toda lattice hierarchy. Then, in section
5.2, we obtain the bi-Hamiltonian structure of its one-dimensional reduction. In section 5.3, we
construct the (4 × 4)-matrix Lax pair representation of this hierarchy, calculate its r-matrix,
and analyze monodromy matrix. We next calculate its spectral curves in section 5.4. In section
5.5, we give a short summary of the (3×3)-matrix Lax pair representation and the r-matrix
formalism for the periodic 1D N = 2 Toda lattice hierarchy, and calculate spectral curves of
the latter. In section 5.6, we discuss periodic Toda lattice equations in the canonical basis and
their fermionic symmetries.
2 Generalized fermionic Toda lattice hierarchy
2.1 2D generalized fermionic Toda lattice equations
In this subsection we define two-dimensional generalized fermionic Toda lattice equations and
describe their two different representations which being reduced relate them with the N = (2|2)
[2, 12] and N = (0|2) [12, 5] supersymmetric Toda lattice equations.
1
Our starting point is the following zero-curvature representation:
[∂ +L−,∂ −L+] = 0 (2.1)
1 2
for the infinite matrices
(L−) = ρ δ +d δ , (L+) = δ +γ δ +c δ , (2.2)
i,j i i,j+1 i i,j+2 i,j i,j−2 i i,j−1 i i,j
... ... ... ...
... 0 0 0 0 0 0 ...
 
ρ 0 0 0 0 0
j+1
 d ρ 0 0 0 0 
L− =  j+2 j+2 ,
 0 d ρ 0 0 0 
j+3 j+3
 
 0 0 d ρ 0 0 
j+4 j+4
 
 ... 0 0 0 d ρ 0 ... 
 j+5 j+5 
 ... ... ... ... 
 
 
... ... ... ...
... c γ 1 0 0 0 ...
j j
 
0 c γ 1 0 0
j+1 j+1
 0 0 c γ 1 0 
L+ =  j+2 j+2 .
 0 0 0 c γ 1 
j+3 j+3
 
 0 0 0 0 c γ 
 j+4 j+4 
 ... 0 0 0 0 0 c ... 
 j+5 
 ... ... ... ... 
 
 
Here, z and z are the bosonic coordinates (∂ ≡ ∂ ); the matrix entries d ,c (ρ ,γ ) are
1 2 1,2 ∂z1,2 j j j j
the bosonic (fermionic) fields with Grassmann parity 0 (1) and length dimensions [d ] = −2,
j
[c ] = −1, [ρ ] = −3/2 and [γ ] = −1/2. The zero-curvature representation (2.1) leads to the
j j j
following system of evolution equations with respect to the bosonic evolution derivatives ∂ :
1,2
∂ d = d (c −c ), ∂ c = d −d +γ ρ +γ ρ ,
2 j j j j−2 1 j j+2 j j j+1 j−1 j
∂ γ = ρ −ρ , ∂ ρ = ρ (c −c )+d γ −d γ . (2.3)
1 j j+2 j 2 j j j j−1 j+1 j j j−2
Keeping in mind that in the bosonic limit (i.e., when all fermionic fields are put equal to zero)
these equations describe a system of two decoupled bosonic 2D Toda lattices, we call equations
(2.3) the 2D generalized fermionic Toda lattice equations.
Our next goal is to describe fermionic symmetries of the 2D generalized fermionic Toda
lattice equations (2.3). Before doing so let us first supply the fields (d ,c ,γ ,ρ ) with boundary
j j j j
conditions. In what follows we consider the boundary conditions of the following four types:
I). lim d = 0, lim c = 0, lim γ = 0, lim ρ = 0;
j j j j
j→±∞ j→±∞ j→±∞ j→±∞
II). lim d = 1, lim c = 0, lim γ = 0, lim ρ = 0;
j j j j
j→±∞ j→±∞ j→±∞ j→±∞
III). lim d = 1, lim d = 0, lim c = 0, lim γ = 0, lim ρ = 0;
2j+1 2j j j j
j→±∞ j→±∞ j→±∞ j→±∞ j→±∞
IV). d = d , c = c , γ = γ , ρ = ρ , n ∈ Z. (2.4)
j j+n j j+n j j+n j j+n
2
The first three types specify the behavior of the fields at the lattice points at infinity while the
boundary condition of the fourth type is periodic and corresponds to the closed 2D generalized
fermionic Toda lattice.
For the boundary conditions I) and II) (2.4) the above described equations (2.3) possess
the N = (2|2) supersymmetry. Indeed, in this case there exist four fermionic symmetries of
equations (2.3)
D1d = g ρ +g ρ , D1d = (−1)j(g ρ −g ρ ),
1 j j−1 j j j−1 2 j j−1 j j j−1
D1c = g γ +g γ , D1c = (−1)j(g γ −g γ ),
1 j j j−1 j+1 j 2 j j+1 j j j−1 (2.5)
D1ρ = −∂ g , D1ρ = (−1)j∂ g ,
1 j 1 j 2 j 1 j
D1γ = g −g , D1γ = (−1)j(g −g )
1 j j j+2 2 j j+2 j
D2d = d (γ +γ ), D2d = (−1)jd (γ −γ ),
3 j j j−1 j−2 4 j j j−1 j−2
j−1 j−1
D2c = ∂ γ , D2c = −∂ (−1)kγ ,
3 j 2 k 4 j 2 k (2.6)
k=−∞ k=−∞
D2ρ = d P−d −ρ γ , D2ρ = (−1)jP(d −d −ρ γ ),
3 j j+1 j j j−1 4 j j+1 j j j−1
D2γ = c −c , D2γ = (−1)j(c −c )
3 j j+1 j 4 j j+1 j
where D1,D1,D2 and D2 arethe fermionic evolution derivatives; g denotes the infinite product
1 2 3 4 j
∞
d
j−2k
g ≡ (2.7)
j
d
j−2k−1
k=0
Y
with the properties g g = d and
j j−1 j
D1g = ρ , D1g = (−1)jρ , D2g = g γ , D2g = (−1)jg γ ,∂ g = g (c −c ).
1 j j 2 j j 3 j j j−1 4 j j j−1 2 j j j j−1
Now using eqs. (2.3) and (2.5)–(2.6) one can easily check that the bosonic and fermionic
evolution derivatives satisfy the algebra of the N = (2|2) supersymmetry
[∂ ,∂ ] = [∂ ,Db] = 0, {D1,D1} = (−1)s2δ ∂ , {D2,D2} = −(−1)s2δ ∂ (2.8)
a b a s s p s,p 1 s p s,p 2
which can be realized via
∂ ∂ ∂ ∂ ∂
∂ = , D1 = +(−1)sθ , D2 = −(−1)pθ (2.9)
a ∂z s ∂ s∂z p ∂ p∂z
a θs 1 θp 2
where z (a = 1,2) and θ ,θ (s = 1,2; p = 3,4) are the bosonic and fermionic evolution times
a s p
of the N = (2|2) superspace, respectively.
Looking at equations (2.5)–(2.6) one can see that they are not consistent with the boundary
conditions III) (2.4). Thus, it is impossible to simultaneously satisfy the boundary conditions
for the fields g entering into eqs. (2.5)
j
lim d = lim g g = 0, lim d = lim g g = 1, (2.10)
2j 2j 2j−1 2j+1 2j+1 2j
j→±∞ j→±∞ j→±∞ j→±∞
while eqs. (2.6) contain a contradiction at infinity in the equation for the field ρ . Thus, one
j
can conclude that the boundary conditions strictly restrict the symmetries of eqs.(2.3). The
periodic boundary conditions will be considered in section 5.
3
Now we present other two related representations of the 2D generalized fermionic Toda
lattice equations (2.3) which will be useful in what follows.
The first representation can easily be derived if one introduces a new basis {g ,c ,γ+,γ−}
j j j j
in the space of the fields {d ,c ,γ ,ρ }
j j j j
d = g g , ρ = g γ−, γ = γ+ (2.11)
j j j−1 j j j j j+1
and eliminate the fields c from eq. (2.3) in order to get the conventional form of the 2D
j
N = (2|2) supersymmetric Toda lattice equations [12]
∂ ∂ lng = g g −g (g +g )+g g +g γ+ γ− −g γ+ γ− ,
1 2 j j+1 j+2 j j+1 j−1 j−1 j−2 j+1 j+1 j+1 j−1 j−1 j−1
∂ γ+ = g γ− −g γ− , ∂ γ− = g γ+ −g γ+ . (2.12)
1 j j+1 j+1 j−1 j−1 2 j j+1 j+1 j−1 j−1
together with their fermionic N = (2|2) symmetries
D1g = g γ−, D1g = (−1)jg γ−,
1 j j j 2 j j j
D1γ− = −∂ lng , D1γ− = (−1)j∂ lng ,
1 j 1 j 2 j 1 j
D1γ+ = g −g , D1γ+ = (−1)j(g −g ),
1 j j−1 j+1 2 j j−1 j+1
D2g = g γ+, D2g = (−1)jg γ+,
3 j j j 4 j j j
D2γ− = g −g , D2γ− = (−1)j(g −g ), (2.13)
3 j j+1 j−1 4 j j+1 j−1
D2γ+ = ∂ lng , D2γ+ = −(−1)j∂ lng .
3 j 2 j 4 j 2 j
In order to derive the second representation, let us introduce a new notation for the fields
at odd and even values of the lattice coordinate j
a ≡ c , b ≡ d , α ≡ γ , β ≡ ρ ,
j 2j+1 j 2j+1 j 2j−1 j 2j+1
¯ ¯
a¯ ≡ c , b ≡ d , α¯ ≡ −γ , β ≡ ρ (2.14)
j 2j j 2j j 2j j 2j
and rewrite eqs. (2.3), (2.5–2.6) in the following form:
¯
∂ b = b (a −a ), ∂ a = b −b +β α¯ +α β ,
2 j j j j−1 1 j j+1 j j j j+1 j+1
¯ ¯ ¯ ¯ ¯
∂ b = b (a¯ −a¯ ), ∂ a¯ = b −b +β α¯ +α β ,
2 j j j j−1 1 j j+1 j j j j j
¯
∂ α = β −β , ∂ β = (a −a¯ )β −b α +b α ,
1 j j j−1 2 j j j j j j j+1 j+1
¯ ¯ ¯ ¯ ¯
∂ α¯ = β −β , ∂ β = (a¯ −a )β −b α¯ +b α , (2.15)
1 j j j+1 2 j j j−1 j j j j j−1
D1b = e β¯ +e¯ β , D1b = −e β¯ −e¯ β ,
1 j j j j j 2 j j j j j
D1¯b = e β¯ +e¯ β , D1¯b = e β¯ −e¯ β ,
1 j j−1 j j j−1 2 j j−1 j j j−1
D1a = e¯ α −e α¯ , D1a = −e¯ α −e α¯ ,
1 j j+1 j+1 j j 2 j j+1 j+1 j j
D1a¯ = e¯ α −e α¯ , D1a¯ = −e¯ α −e α¯ ,
1 j j j j j 2 j j j j j (2.16)
D1β = −∂ e¯ , D1β = −∂ e¯ ,
1 j 1 j 2 j 1 j
D1β¯ = −∂ e¯ , D1β¯ = ∂ e¯ ,
1 j 1 j 2 j 1 j
D1α = e −e , D1α = e −e ,
1 j j−1 j 2 j j−1 j
D1α¯ = e¯ −e¯ , D1α¯ = e¯ −e¯ ,
1 j j+1 j 2 j j j+1
4
D2b = b (α −α¯ ), D2b = b (α +α¯ ),
3 j j j j 4 j j j j
D2¯b =¯b (α −α¯ ), D2¯b =¯b (α +α¯ ),
3 j j j j−1 4 j j j j−1
j j
D2a = ∂ (α −α¯ ), D2a = ∂ (α +α¯ ),
3 j 2 k k 4 j 2 k k
k=−∞ k=−∞
Pj Pj
D2a¯ = ∂ (α −α¯ ), D2a¯ = ∂ (α +α¯ ), (2.17)
3 j 2 k k−1 4 j 2 k k−1
k=−∞ k=−∞
D2β =¯b −Pb −¯b +β α¯ , D21β = b −P¯b −β α¯ ,
3 j j j i j j 4 j j j+1 j j
D2β¯ = b −¯b −β¯ α , D2β¯ = b −¯b −β¯ α ,
3 j j j j j 4 j j j j j
D2α = a¯ −a , D2α = a −a¯ ,
3 j j j−1 4 j j−1 j
D2α¯ = a¯ −a , D2α¯ = a¯ −a
3 j j j 4 j j j
where e ,e¯ are the composite fields
j j
∞ ∞ ¯
b b
j−k j−k
e ≡ g ≡ , e¯ ≡ g ≡ (2.18)
j 2j+1 ¯b j 2j b
j−k j−k−1
k=0 k=0
Y Y
which obey the equations
∂ e = e (a −a¯ ), ∂ e¯ = e¯ (a¯ −a ),
2 j j j j 2 j j j j−1
D1e = β , D1e = −β , D2e = e α , D2e = −e α ,
1 j j 2 j j 3 j j j 4 j j j
D1e¯ = β¯ , D1e¯ = β¯ , D2e¯ = −e¯ α¯ , D2e¯ = −e¯ α¯ . (2.19)
1 j j 2 j j 3 j j j 4 j j j
The reduction
¯
b = 0 (2.20)
j
of eqs. (2.15) leads to the 2D N = (0|2) supersymmetric Toda lattice equations [12, 5]. One
can easily see that fermionic symmetries (2.16) are not consistent with this reduction, while
fermionic symmetries (2.17) are consistent and form the algebra of the N = (0|2) supersymme-
try.
2.2 Bi-Hamiltonian structure of the 1D generalized fermionic Toda
lattice hierarchy
Our further purpose is to construct a bi-Hamiltonian structure of the generalized fermionic
Toda lattice equations (2.3) (and, consequently, originating from them eqs. (2.12) and (2.15))
in one-dimensional space when all the fields depend on only one bosonic coordinate z = z +z .
1 2
This task was solved in [6] for the 1D N = 2 Toda lattice hierarchy obtained by reduction
(2.20) of the 1D generalized fermionic Toda lattice hierarchy. Here we solve this task for the
original 1D generalized fermionic Toda lattice hierarchy.
At the reduction to one-dimensional space,
∂ = ∂ ≡ ∂, (2.21)
1 2
5
the zero-curvature representation (2.1) can identically be rewritten in the form of the Lax-pair
representation
∂L = [L,L−], L ≡ L+ +L−, (2.22)
... ... ... ...
... c γ 1 0 0 0 ...
j j
 
ρ c γ 1 0 0
j+1 j+1 j+1
 d ρ c γ 1 0 
L =  j+2 j+2 j+2 j+2 .
 0 d ρ c γ 1 
j+3 j+3 j+3 j+3
 
 0 0 d ρ c γ 
j+4 j+4 j+4 j+4
 
 ... 0 0 0 d ρ c ... 
 j+5 j+5 j+5 
 ... ... ... ... 
 
 
Using the Lax pair representation (2.22), it is easy to derive the general expression for bosonic
Hamiltonians which are in involution via the standard formula
∞
1 1
H = strLk ≡ (−1)p(Lk) . (2.23)
k pp
k k
p=1
X
The first two of them have the following explicit form:
∞ ∞
1
H = (−1)ic , H = (−1)i( c2 +d +ρ γ ). (2.24)
1 i 2 2 i i i i−1
i=−∞ i=−∞
X X
A bi-Hamiltonian system of evolution equations can be represented in the following general
form:
∂
q = {H ,q } = {H ,q } (2.25)
i k+1 i 1 k i 2
∂t
Hk
where t are the evolution times, q denotes any field from the set q = {d ,c ,ρ ,γ } and the
Hk j i i i i i
brackets {,} are appropriate Poisson brackets corresponding to the first (second) Hamilto-
1(2)
nian structure. Using eqs. (2.25) and the 2D generalized fermionic Toda lattice equations (2.3)
at the reduction to one-dimensional space (2.21) – the 1D generalized fermionic Toda lattice
equations
∂d = d (c −c ), ∂c = d −d +γ ρ +γ ρ ,
i i i i−2 i i+2 i i i+1 i−1 i
∂γ = ρ −ρ , ∂ρ = ρ (c −c )+d γ −d γ (2.26)
i i+2 i i i i i−1 i+1 i i i−2
as well as Hamiltonians (2.24), we have found the first two Hamiltonian structures of the
hierarchy. As the result, we have the following explicit expressions:
{d ,c } = (−1)jd (δ −δ ),
i j 1 i i,j+2 i,j
{c ,ρ } = (−1)jρ (δ +δ ),
i j 1 j i,j−1 i,j
{ρ ,ρ } = (−1)j(d δ −d δ ),
i j 1 i i,j+1 j i,j−1
{γ ,γ } = (−1)j(δ −δ ) (2.27)
i j 1 i,j+1 i,j−1
6
for the first and
{d ,d } = (−1)jd d (δ −δ ),
i j 2 i j i,j+2 i,j−2
{d ,c } = (−1)jd c (δ −δ ),
i j 2 i j i,j+2 i,j
{c ,c } = (−1)j(d δ −d δ −γ ρ δ −γ ρ δ ),
i j 2 i i,j+2 j i,j−2 j i i,j+1 i j i,j−1
{d ,ρ } = (−1)jd ρ (δ +δ ),
i j 2 i j i,j+2 i,j−1
{d ,γ } = (−1)jd γ (δ +δ ),
i j 2 i j i,j+2 i,j+1
{c ,ρ } = (−1)j(c ρ (δ +δ )−d γ δ −d γ δ ),
i j 2 i j i,j i,j−1 j i i,j−2 i j i,j+1
{c ,γ } = (−1)j(ρ δ +ρ δ ),
i j 2 i i,j+2 j i,j−1
{ρ ,γ } = (−1)j(ρ γ δ +d δ −d δ ),
i j 2 i j i,j+1 i i,j+3 j i,j−1
{ρ ,ρ } = (−1)j((ρ ρ −d c )δ +(ρ ρ +d c ) δ ),
i j 2 i j j i i,j−1 i j i j i,j+1
{γ ,γ } = (−1)j(c δ −c δ ) (2.28)
i j 2 i i,j+1 j i,j−1
for the second Hamiltonian structures, where only nonzero brackets are written down.
Note that the first {,} (2.27) and the second {,} (2.28) Hamiltonian structures are obvi-
1 2
ously compatible: the deformation of the fields c → c +ν, where ν is an arbitrary constant,
j j
transforms {,} into the Hamiltonian structure which is their sum
2
{,} → {,} +ν {,} .
2 2 1
Thus, one concludes that the corresponding recursion operator
R = {,} {,}−1
2 1
is hereditary like the operator obtained from the compatible pair of Hamiltonian structures.
We have checked that the one-dimensional reduction (2.21) of the fermionic symmetries
(2.5)–(2.6)
D d = g ρ +g ρ , D d = (−1)i(g ρ −g ρ ),
1 i i−1 i i i−1 2 i i−1 i i i−1
D c = g γ +g γ , D c = (−1)i(g γ −g γ ),
1 i i i−1 i+1 i 2 i i+1 i i i−1
D ρ = g (c −c ), D ρ = (−1)ig (c −c ),
1 i i i−1 i 2 i i i i−1
D γ = g −g , D γ = (−1)i(g −g )
1 i i i+2 2 i i+2 i
D d = d (γ +γ ), D d = (−1)id (γ −γ ),
3 i i i−1 i−2 4 i i i−1 i−2
D c = ρ +ρ , D c = (−1)i(ρ −ρ ),
3 i i+1 i 4 i i+1 i (2.29)
D ρ = d −d −ρ γ , D ρ = (−1)i(d −d −ρ γ ),
3 i i+1 i i i−1 4 i i+1 i i i−1
D γ = c −c , D γ = (−1)i(c −c )
3 i i+1 i 4 i i+1 i
and the equations for the composite fields g (2.7)
i
∂g = g (c −c ), D g = ρ , D g = (−1)jρ , D g = g γ , D g = (−1)jg γ (2.30)
j j j j−1 1 j j 2 j j 3 j j j−1 4 j j j−1
can also be represented in a bi-Hamiltonian form with fermionic Hamiltonians S and Hamil-
s,k
tonian structures (2.27) and (2.28)
D q = {S ,q } = {S ,q } (2.31)
tSs,k i s,k+1 i 1 s,k i 2
7
where D are the fermionic evolution derivatives. In section 3.2 we show how fermionic
tSs,k
Hamiltonians can be derived in an algorithmic way, but now let us only mention that there are
four infinite towers of fermionic Hamiltonians S (s = 1,2,3,4; k ∈ N) and present without
s,k
any comments only explicit expressions for the first few of them
∞ ∞ i−1
S = (−1)iρ g−1, S = − (−1)i g γ +ρ g−1 (−1)jc ,
1,1 i i 1,2 i i−1 i i j
!
i=−∞ i=−∞ j=−∞
X X X
∞ ∞ i−1
S = ρ g−1, S = g γ −(−1)iρ g−1 (−1)jc ,
2,1 i i 2,2 i i−1 i i j
!
i=−∞ i=−∞ j=−∞
X X X
∞ ∞ i−1
S = − (−1)iγ , S = − (−1)iρ +γ (−1)j c ,
3,1 i 3,2 i i−1 j
!
i=−∞ i=−∞ j=−∞
X X X
∞ ∞ i−1
S = γ , S = ρ −(−1)iγ (−1)j c . (2.32)
4,1 i 4,2 i i−1 j
!
i=−∞ i=−∞ j=−∞
X X X
For completeness we also present the nonzero Poisson brackets of the composite field g (2.7)
i
with other fields of the hierarchy which are useful when producing fermionic Hamiltonian flows
{g ,c } = (−1)jg (δ −δ ),
i j 1 i i,j+1 i,j
{g ,γ } = (−1)jg γ δ ,
i j 2 i j i,j+1
{g ,c } = (−1)jg c (δ −δ ),
i j 2 i j i,j+1 i,j
{g ,ρ } = (−1)jg ρ (δ −δ +δ ),
i j 2 i j i,j+1 i,j i,j−1
{g ,d } = (−1)jg d (δ −δ +δ −δ ),
i j 2 i j i,j+1 i,j i,j−1 i,j−2
{g ,g } = (−1)jg g (δ +δ ). (2.33)
i j 2 i j i,j+1 i,j−1
Now we have all necessary ingredients to derive Hamiltonian flows of the 1D generalized
Toda lattice hierarchy. Let us end this section with a few remarks.
First, the Hamiltonians H (2.24) and S (2.32) give trivial flows via the first Hamiltonian
1 s,1
structure (2.27) because they belong to the center of the algebra (2.27)
{H ,q } = {S ,q } = 0. (2.34)
1 j 1 s,1 j 1
Second, while the densities corresponding to the fermionic Hamiltonians S (2.32) have a
p,k
nonlocalcharacterwithrespect tothelatticeindices, thefermionicflows(2.29)havenononlocal
terms.
Finally, the algebras of the first and second Hamiltonian structures (2.27)–(2.28) together
with eqs. (2.33) possess a discrete inner automorphism f which transforms nontrivially only
fermionic fields
γ 7−f→ (−1)jρ g−1 , ρ 7−f→ (−1)jγ g . (2.35)
j j+1 j+1 j j−1 j
8
Using eqs. (2.30) one can easily check that the automorphism f transforms eqs. (2.26), (2.29)
and Hamiltonians (2.24), (2.32) according to the following rule:
f
{∂,D ,D ,D ,D } 7−→ {∂,D ,D ,−D ,−D },
1 2 3 4 4 3 2 1
f
{H ,S ,S ,S ,S } 7−→ {H ,−S ,−S ,−S ,−S }. (2.36)
k 1,k 2,k 3,k 4,k k 4,k 3,k 2,k 1,k
3 Reduction: 1D N=4 supersymmetric Toda lattice hi-
erarchy
3.1 Bi-Hamiltonian structure of the 1D N=4 Toda lattice hierarchy
In this section we consider the bi-Hamiltonian formulation of the one-dimensional reduction
(2.21) of the 2D N = (2|2) supersymmetric Toda lattice equations (2.12) and their fermionic
symmetries (2.13). Starting with Hamiltonians (2.24), (2.32) and Hamiltonian structures
(2.27)–(2.28) as well as using relations (2.11) it is easy to represent eqs. (2.12) in one-
dimensional space as a bi-Hamiltonian system of first order evolution equations.
Thus, we obtain the following bosonic and fermionic Hamiltonians:
∞ ∞
1
HN=4 = (−1)ic , HN=4 = (−1)i c2 +g g +g γ−γ+ ,
1 i 2 2 i i i−1 i i i
i=−∞ i=−∞ (cid:18) (cid:19)
X X
∞ ∞ i−1
SN=4 = (−1)iγ−, SN=4 = − (−1)ig γ+ +γ− (−1)kc ,
1,1 i 1,2 i i i k
!
i=−∞ i=−∞ k=−∞
X X X
∞ ∞ i−1
SN=4 = γ−, SN=4 = g γ+ −(−1)iγ− (−1)kc ,
2,1 i 2,2 i i i k
!
i=−∞ i=−∞ k=−∞
X X X
∞ ∞ i−1
SN=4 = (−1)iγ+, SN=4 = − (−1)ig γ− +γ+ (−1)kc ,
3,1 i 3,2 i i i k
!
i=−∞ i=−∞ k=−∞
X X X
∞ ∞ i−1
SN=4 = γ+, SN=4 = g γ− −(−1)iγ+ (−1)kc , (3.1)
4,1 i 4,2 i i i k
!
i=−∞ i=−∞ k=−∞
X X X
and the first
{γ±,γ±} = ±(−1)j(δ −δ ),
i j 1 i,j−1 i,j+1
{g ,c } = (−1)jg (δ −δ ) (3.2)
i j 1 i i,j+1 i,j
and the second
{g ,g } = (−1)jg g (δ +δ ),
i j 2 i j i,j+1 i,j−1
{g ,γ±} = −(−1)j g γ± δ ,
i j 2 i j i,j
9