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\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqar{\begin{eqnarray}}
\def\eeqar{\end{eqnarray}}
\def\Ga{\Gamma}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\I{{\cal I}}
\def\L{{\cal L}}
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\def\O{{\cal O}}
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\def\hT{\widehat T}
\def\cT{\check T}
\def\NT{{\bf N}_{T}}
\def\HT{{\bf H}_{T}}
\def\GNT{{^{G}\bf N}_{T}}
\def\GHT{{^{G}\bf H}_{T}}
\def\Cinf{C^\infty}
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\hfill DSMA-TS QM2 256
\hfill February, 1992
\vspace{.5cm}
\begin{center}
{\Large \bf REMARKS ON THE COMPLETE INTEGRABILITY \\
OF DYNAMICAL SYSTEMS WITH \\
FERMIONIC VARIABLES}
\footnote{Supported in part by the italian Ministero
dell' Universit\`a e della Ricerca Scientifica e Tecnologica.}
\end{center}
\vspace{.5cm}
\begin{center}
{\Large G. Landi$^{1,4}$, G. Marmo$^{2,4}$ and G. Vilasi$^{3,4}$}
\end{center}
\bigskip
\begin{center}
$^{1}$
Dipartimento di Scienze Matematiche - Universit\`a di Trieste,\\
P.le Europa, 1 - I-34100 Trieste --- Italy.\\
\smallskip
$^{2}$
Dipartimento di Scienze Fisiche - Universit\`a di Napoli,\\
Mostra d'Oltremare, Pad.19 - I-80125 Napoli --- Italy.\\
\smallskip
$^{3}$
Dipartimento di Fisica Teorica e S.M.S.A. - Universit\`a di Salerno,\\
Via S. Allende, I-84081 Baronissi (SA) --- Italy.\\
\smallskip
$^{4}$
Istituto Nazionale di Fisica Nucleare - Sezione di Napoli\\
Mostra d'Oltremare, Pad.20 - I-80125 Napoli --- Italy.
\end{center}
\vspace{1.0cm}
{\bf Abstract.} We study the r\^{o}le of $(1,1)$ graded tensor field
$T$ in the analysis of complete integrability of dynamical systems
with fermionic variables. We find that such a tensor $T$ can be a
recursion operator if and only if $T$ is even as a graded map,
namely, if and only if $p(T) = 0$.
We clarify this fact by
constructing an odd tensor for two examples, a supersymmetric Toda
chain and a supersymmetric harmonic oscillator. We explicitly show
that it cannot be a recursion operator not allowing then to build new
constants of the motion out of the first two ones in contrast to what
usually
happens with ordinary, i.e. non graded systems.
\pagebreak
\noindent
{\bf 1. Introduction.}
In recent years there has been a renewed interest in
completely integrable Hamiltonian systems, specially
in connection with the study of integrable quantum field theory,
Yang-Baxter algebras and, more recently, quantum groups.
Integrability criteria available both in finite and infinite
dimensions have been established by methods directly related to group theory
and to familiar procedures in classical mechanics [AM], [Ar],
by looking at soliton
equations as dynamical systems on (infinite-dimensional)
phase manifold [DMSV 1-2], [Vi], [GD], [Ma], [KM].
This last approach was also suggested by the occurrence in the
Inverse Scattering Transform of a peculiar operator,
the so called recursion operator [La],
which naturaly fits in this geometrical
setting as a mixed tensor field on the phase manifold. This tensor
has to satisfies few requirements, the most important being that its
Nijenhuis torsion [FN], [Ni] vanishes.
There have been several attempts to analyse
integrability of fermionic dynamical systems (see for instance, [Ku],
[MR], [DR]) and to extend to such systems [DHR], in
algorithimic sense at least, results and techniques
used for bosonic dynamics and based on the r\^{o}le of recursion
operators. In particular, one would like to define a graded Nijenhuis
torsion.
In this paper we address this issues. We show that a mixed $(1,1)$
graded
tensor field $T$ can act as a recursion operator if and only if $T$ is
an even map.
There are dynamical systems, like supersymmetric Witten's dynamics
[Wi] which allow a bi-Hamiltonian description with an even and odd
Hamiltonian function and in term of an even and an odd
Poisson structure respectively (so that the dynamical vector
field is always even) [VPST], [So]. This allows to construct an odd
tensor field which could be a good candidate as a recursion operator.
We explicitely show that this is not possible.
The paper is organized as follows. First, we fix notation and
recall the
formulation of complete integrability in term of a $(1,1)$ tensor
field available in the
bosonic case.
After a r\'{e}sum\'{e} of graded differential calculus and graded Poisson
structures, we analyse a
supersymmetric harmonic oscillator and a supersymmetric Toda chain
(both are examples of Witten's supersymmetric dynamics).
We then prove that for an odd $(1,1)$ tensor,
a $(2,1)$ tensor correspondig to its
torsion (graded-Nijenhuis torsion) cannot be defined and that a graded
$(1,1)$ tensor cannot be a recursion operator unless it is even.
Finally, we present some conclusions.
\bigskip
\bigskip
\noindent
{\bf 2. Complete Integrability and recursion operators in the bosonic
case.}
Complete integrability of Hamiltonian systems with finitely
many degrees of freedom is exhaustively characterized by the
Liouville-Arnold
theorem [AM], [Ar].
Here we briefly recall an alternative characterization in term of an
invariant (under the dynamical evolution) (1,1) tensor field $T$.
Example of such tensors can be constructed also for
systems with infinitely many degrees of freedom so that the approach
described could be of use in the latter cases as well.
We shall deal only with smooth, i.e. $\Cinf$ objects, and notations will
follows as close as possible those of [AM] and [MSSV]. In particular
if $M$ is a (finite dimensional) ordinary manifold we denote by $\func$ the
ring of real valued functions on $M$, by $\vect$ the Lie
algebra of vector fields, by $\form$ its dual of forms and by
$\tens$ the mixed $(1,1)$ tensor fields.
Associated with every $T \in \tens$ there are two endomorphisms
of $\vect$ and $\form$ which are defined by
\beqar
& & \hT : \vect \longrightarrow \vect ~, ~~~
\cT : \form \longrightarrow \form ~, \nonumber \\
& & T (X, \alpha) =: \hT X \inner \alpha =:
X \inner \cT \alpha~,~~~~
\forall ~X \in \vect ~~,~~\alpha \in \form~. \label{AA}
\eeqar
The {\it Nijenhuis tensor} (or {\it torsion})
of $T$ is the $(2,1)$ tensor field $\NT$
defined by [FN], [Ni]
\beq
\NT(X,Y; \alpha) =: \HT(X,Y) \inner \alpha ~, \label{AB}
\eeq
\noindent
where $\HT : \vect \times \vect \rightarrow \vect $ is the
$\func$-linear map given by
\beq
\HT(X,Y) =: \hT^{2} [X,Y] + [\hT X,\hT Y]
- \hT [\hT X,Y] - \hT [X,\hT Y]~. \label{AC}
\eeq
Equivalently, this can be written as
\beq
\HT(X,Y) = [ \wh{L_{\hT X} T} - \hT \circ \wh{L_{X} T} ](Y)
~~,~~\forall ~X~,~Y \in \vect~. \label{AD}
\eeq
To simplify our notation, in the following, when no confusion arises,
we shall denote both the endomorphisms $\hT$ and $\cT$ with the same
symbol, namely $T$.
>From the very definition (\ref{AB}) it is clear that the vanishing of
the tensor $\NT$ is equivalent to the vanishing of $\HT$, namely $\NT
\equiv 0$ iff $\HT \equiv 0$.
The following proposition has been proved in [DMSV 2]
\prop 1.
A dynamical vector field $\Ga$ which admits a mixed tensor field $T$,
which is invariant ($L_{\Ga} T = 0$), with vanishing Nijenhuis torsion,
diagonalizable with doubly
degenerate eigenvalues $\lambda$, without stationary points
$(d\lambda \neq 0)$ is separable integrable and Hamiltonian, i.e. is
a separable completely integrable Hamiltonian system.
\bigskip
The proof is given observing that:
$\NT = 0$ implies the integrability, in the Frobenius sense, of
the eigenspaces of T;
$L_{\Ga} T = 0$ implies the separability of $\Ga$ along the eigenmanifolds,
in dynamics with 1-degree of freedom, each of which having a constant
of motion.
\bigskip
A $(1,1)$ tensor field with the previously stated properties,
acts as a `recursion operator' [Ma], [GD], i.e.
when iteratively applied to $\Ga$ one produces
symmetries $\Ga_k = \hT^k \Ga$ or constants of motion
$H_k$ by $dH_k = \cT^k dH$.
\bigskip
The main property of the tensor field $T$ in the analysis of complete
integrability
of its infinitesimal
automorphisms is the vanishing of its Nijenhuis tensor
$\NT = 0$.
It is then, plausible that
a suitable generalization of such a condition could play an important
r\^{o}le in analysing
the integrability of dynamical systems with fermionic degrees of
freedom.
Moreover,
it seems natural to think that such a generalization could come from a
graded generalization of some of the following
relations which are available in the bosonic case :
\begin{description}
\item[a.] $\NT = 0 \Longrightarrow ImT$ is a Lie algebra.
\item[b.] $\NT = 0$ , $d(TdH) = 0 \Longrightarrow d(T^k dH) = 0$~.
\item[c.] $\NT = 0 \Longleftrightarrow d_T \circ d_T = 0$~; here $d_T$
is a suitable generalization of the exterior derivative associated
with any (1,1) tensor field [MFLMR].
\item[d.] $T =: \Lambda_1^{-1} \circ \Lambda_2$~, $\NT = 0 \Longleftrightarrow
\Lambda_1 + \Lambda_2$ satisfies the Jacobi identity.
Here $\Lambda_1$ and $\Lambda_2$ are two Poisson structures.
\item[e.] $\omega (X,Y) =: [TX,Y] + [X,TY] - T[X,Y]$~;
$T \omega(X,Y) = [TX,TY]$ (this is the same as $\NT = 0$)
$\Longleftrightarrow [X,Y]_{\lambda} =: [X,Y] + \lambda \omega(X,Y)$
satisfies the Jacobi identity for any value of the real parameter
$\lambda$.
\end{description}
\noindent
One could expect that some, if not all, of the previous relations do
not hold true in the graded situation.
Before we proceed with the analysis of the graded Nijenhuis condition
we shall give
a brief review of the graded
differential calculus on supermanifolds which will be followed
by the study of some simple examples.
\bigskip
\bigskip
\noindent
{\bf 3. Graded differential calculus.}
We review some fundamentals of supermanifold theory
[DW],[Rog] while refering to the literature for a mathematically
coherent definition [Rot], [BB]. In the following, by graded we shall
always mean ${\bf Z}_2$-graded.
The basic algebraic object is a real exterior algebra
$B_L=(B_L)_0\oplus(B_L)_1$ with $L$ generators.
An $(m,n)$ dimensional supermanifold is a topological
manifold $S$ modelled over the ``vector superspace''
\beq
B_L^{m,n}=(B_L)_0^m\times(B_L)_1^n \label{BA}
\eeq
by means of an atlas whose
transition functions fulfil a suitable ``supersmoothness'' condition.
A supersmooth function $f:U\subset B_L^{m,n}\to B_L$ has the usual
superfield expansion
\beq
f(x^1\dots x^m,\theta^1\dots \theta^n)=
f_0(x)+\sum_{\alpha=1}^nf_\alpha(x)\,\theta^\alpha
+\dots+f_{1\dots n}(x)\,\theta^1\dots \theta^n \label{BB}
\eeq
where the $x$'s are the even (Grassmann)
coordinates, the $\theta$'s are the odd ones,
and the dependence of the coefficient functions $f_{\dots}(x)$
on the even variables is fixed by their values for real arguments.
We shall denote by $\sfunc$ and $\G (U)$ the graded ring of supersmooth
$B_L$-valued functions on $S$ and $U \subset S$ respectively.
The class of supermanifolds which, up to now, turns out to be relevant
for applications
in physics is given by the
De Witt supermanifolds. They are defined in terms
of a coarse topology on $B_L^{m,n}$, called the De Witt topology, whose
open sets are the counterimages of open sets in ${\bf R}^m$ through the
body map $\sigma^{m,n}: B_L^{m,n} \to {\bf R}^m$.
An $(m,n)$ supermanifold
is De Witt if it has an atlas such that the images of the coordinate
maps are open in the De Witt topology.
A De Witt $(m,n)$ supermanifold
is a locally trivial fibre bundle over an ordinary $m$-manifold
$S_0$ (called the body of $S$) with a vector fibre [Rog].
This make not a surprise the fact
that, modulo some technicalities, a De Witt supermanifold
can be identified with a
Berezin-Konstant supermanifold [Be], [Ko].
\smallskip
The graded tangent space $TS$ is constructed in the following manner.
For each $x\in S$, let $\G (x)$ be the germs of functions at $x$ and
denote by $T_xS$ the space of
graded $B_L$-linear maps $X : \G (x) \to B_L$ which satisfy Leibnitz
rule. Then,
$T_xS$ is a free graded $B_L$-module of dimension $(m,n)$, and
the disjoint union $\bigcup_{x\in S}T_xS$
can be given the structure of a rank $(m,n)$ super vector bundle
over $S$, denoted
by $TS$. The sections $\svect$ of $TS$ are a graded $\sfunc$-module
and are identified with the graded Lie algebra $Der\sfunc$ of
derivations of $\sfunc$.
Derivations (or vector fields) are
said to be even (or odd) if they are even (or odd) as maps
(satisfying in addition a graded Leibnitz rule)
from $\sfunc \to \sfunc$. A local basis is given by
\beq
\pd{},{x^1}~, \dots , \pd{},{x^m}~,
\pd{},{\theta^1}~, \dots , \pd{},{\theta^n}~. \label{BD}
\eeq
\bigskip
\noindent {\bf Remark.} Unless explicitely stated, by using a partial
derivative we shall always mean a left derivative, namely a derivative
acting from left. In general, if $z^i = (x^j, \theta^k)$, when acting
on any homogeneous function $f \in \sfunc$, left and
right derivative are related by
\beq
\lpd{},{z^i} f = (-1)^{p(z^i)[p(f)+1]} f \rpd{},{z^i}~,~~~
i \in \{1, \dots, m+n \}~. \label{BE}
\eeq
\bigskip
In a similar way one defines the cotangent space and bundle.
$T_x^{\star}S$ is the space of
graded $B_L$-linear maps from $ T_{x}(S) \to B_L$
and $T^{\star}S = \bigcup_{x\in S}T_{x}^{\star}S$.
$T_x^{\star}S$ is a free graded $B_L$-module of dimension $(m,n)$, while
$T^{\star}S$
is a rank $(m,n)$ super vector bundle over $S$.
The sections $\sform$ of $T^{\star}S$ are a graded $\sfunc$-module
and are identified with the graded $\sfunc$-linear maps from $Der\sfunc
\to \sfunc$. They are the 1-forms on $S$.
Forms are
said to be even (or odd) if they are even (or odd) as maps
$\vect \to \sfunc$.
In general, a $p$ covariant and $q$ contravariant
graded tensor is any graded $\sfunc$-multilinear map
$\alpha :\svect \times
\cdots \times \svect \times \sform \times \cdots
\times \sform \longrightarrow \sfunc$
($p~~\svect$ factors and $q~~\sform $ factors). The collection of all rank
$(p,q)$ tensors is a graded $\sfunc $-module.
A graded {\it p-form} is a skew-symmetric
covariant graded tensors of rank $p$.
We denote by $\Omega^{p}(S)$ the collection of all p-forms.
The {\it exterior differential} on $S$
is defined by letting $X\inner df=X(f)\ \forall f\in\sfunc,\,X\in
\svect$
and is extended to maps $\Omega^p(S)\to\Omega^{p+1}(S),\ p\geq 0$,
in the usual way, so that $d^2=0$.
If $X_i\in \svect$ are homogeneous elements,
\beqar
& & X_1\wedge\dots\wedge X_{p+1}\inner d\varphi =:
\sum_{i=1}^{p+1}(-1)^{a(i)}\,
X_i(X_1\wedge \stackrel{\stackrel{i}{\surd}}{\ldots \ldots}
\wedge X_{p+1}\inner\varphi)
\nonumber \\
& &+\sum_{1\leq iFrom definition one has that $p(d) = 0$.
The Lie derivative $L_{(\cdot)}$ of forms is defined by
\beqar
&& L_{(\cdot)} : \svect \times \Omega^p(S) \to \Omega^p(S)~, \nonumber
\\
&& L_{X} = X \inner \circ d + d \circ X\inner~,~~\forall X \in \svect~.
\label{BH}
\eeqar
Clearly, $p(L_{X}) = p(X)$.
The Lie derivative of any tensor product can be
defined in an obvious manner by requiring the Leibnitz
rule and can be extended to any tensor by using linearity.
\bigskip
Suppose now that we have a tensor $T \in \stens$ which is homogeneous
of degree $p(T)$. Again we can define
two graded endomorphisms
of $\svect$ and $\sform$ by the formul{\ae} (in the following two
formul{\ae} $X, Y$ are homogeneous elements in $\svect$ while $\alpha$ is
any element in $\sform$)
\beqar
& & \hT : \svect \longrightarrow \svect ~, ~~~
\cT : \sform \longrightarrow \sform ~, \nonumber \\
& & T (X, \alpha) =: \hT X \inner \alpha =:
(-1)^{p(X)p(T)} X \inner \cT \alpha~. \label{BL}
\eeqar
We could be tempted to define a graded
Nijenhuis torsion of $T$ by a relation analogous to (\ref{AB})
\beqar
\GNT(X,Y; \alpha) &=:& \GHT(X,Y) \inner \alpha ~, \nonumber \\
\GHT(X,Y) &=:& \hT^{2} [X,Y] + (-1)^{p(T)p(X)}[\hT X,\hT Y]
- \hT [\hT X,Y] \nonumber
\\
&&~~ - (-1)^{p(T)p(X)} \hT [X,\hT Y]~. \label{BM}
\eeqar
\prop 2.
The map $\GHT : \svect \times \svect \rightarrow \svect $ defined in
(\ref{BM}) is
$\sfunc$-linear and graded antisymmetric if and only if $p(T) = 0$.
\proof Just compute.
\bigskip
\noindent {\bf Remark.}
When $p(T)=1$, the map defined in (\ref{BM}) is not antisymmetric nor
linear also over even function, also when it is restrict to even vector
fields.
Therefore eqs. (\ref{BL}) and (\ref{BM}) define a graded tensor (which
is in addition graded antisymmetric) if and only if $p(T) = 0$.
\bigskip
\bigskip
\noindent
{\bf 4. Poisson supermanifold.}
We briefly describe how to introduce super Poisson structures on a
$(m,n)$-dimensional
supermanifold $S$ [Be], [Le]. For additional results see also [CI].
As before, we shall denote by
$z^i = (x^j, \theta^k)~,~~ i\in \{1, \dots, m+n \}$ the local
coordinates on $S$. The following proposition is in [Be] and can be
proved by direct calculations
\newpage
\prop 3.
Let $\vert \vert \omega^{i j} \vert \vert $
be a $(m+n) \times (m+n)$ matrix (depending upon the point
$z \in S$) with the following properties:
\begin{description}
\item[1.] the elements
$\omega^{i j}$ are homogeneous with parity
$p(\omega^{i j}) = p(z^i) + p(z^j) + p(\omega)$ and
$p(\omega)$ not depending on the indices $i$ and $j$~;
\item[2.]
\beq \omega^{j i}
= - (-1)^{[p(z^i) + p(\omega)][p(z^j) + p(\omega)]} \omega^{i j}~;
\label{CA}
\eeq
\item[3.]
\beqar
(-1)^{[p(z^i) + p(\omega)][p(z^l) + p(\omega)]}
\omega^{i s} \lpd{},{z^s} \omega^{j l} +
(-1)^{[p(z^l) + p(\omega)][p(z^j) + p(\omega)]}
\omega^{l s} \lpd{},{z^s} \omega^{i j} \nonumber \\ +
(-1)^{[p(z^j) + p(\omega)][p(z^i) + p(\omega)]}
\omega^{j s} \lpd{},{z^s} \omega^{l i} = 0~. \label{CB}
\eeqar
\end{description}
Then, the following bracket
\beq
\{ F, G \} =: F \rpd{},{z^i} \omega^{i j} \lpd{},{z^j} G \label{CC}
\eeq
\noindent
makes $\sfunc$ a Lie superalgebras (Poisson superstructure).
\bigskip
\bigskip
We have two different kind of structures according to the fact that
$p(\omega) = 0$ (even Poisson structure) or
$p(\omega) = 1$ (odd Poisson structure). Indeed, one can check that
the bracket (\ref{CC}) has properties
\beqar
&& \{ F, G \}
= - (-1)^{[p(F) + p(\omega)][p(G) + p(\omega)]}\{ G, F \}~;
\label{CD} \\
&& ~~\nonumber \\
&& (-1)^{[p(F) + p(\omega)][p(H) + p(\omega)]} \{ \{ F, G \}, H \} +
(-1)^{[p(G) + p(\omega)][p(F) + p(\omega)]} \{ \{ G, H \}, F \}
\nonumber \\
&& ~~~~~~~~~~~ + (-1)^{[p(H) + p(\omega)][p(G)
+ p(\omega)]} \{ \{ H, F \}, G \} = 0~.
\label{CE}
\eeqar
We infer from (\ref{CD}) and (\ref{CE}) that, when thought of as
elements of the Poisson superalgebra, homogeneous elements of $\sfunc$
preserve their parity if $p(\omega) = 0$, while they change it if
$p(\omega) = 1$.
\bigskip
If the matrix $\vert \vert \omega^{i j} \vert \vert$ is regular, then
its inverse $\vert \vert \omega_{i j} \vert \vert~,~ \omega_{ij}
\omega^{jk} = \delta_i^k~, $
gives the components
of a symplectic structure
$\omega = \fraz1,2 dz^{i} \wedge dz^{j} \omega_{j i}$ ,
namely, $\omega$ is closed and nondegenerate with the
properties
\beqar
&& p(\omega_{i j}) = p(z^i) + p(z^j) + p(\omega) \nonumber \\
&& \omega_{j i}
= - (-1)^{p(z^i) p(z^j)} \omega_{i j}~, \label{CF}
\eeqar
and $\omega$ is homogeneous with parity just equal to $p(\omega)$.
There is also a Darboux theorem [Le]
\prop 4.
Let $(S, \omega)$ be a (m, n)-dimensional symplectic manifold
with $\omega$ homogeneous. Then
\begin{description}
\item[1.] If $p(\omega) = 0$~, then dim $S = (2r,n)$ and there exist
local coordinates such that
\beq
\omega = d q^{i} \wedge d p^{i} + d \xi^{j} \wedge d \xi^{j}~;~~~
~~~~~~\omega_{i j} =
{\scriptstyle % All this junk is to make a smaller matrix
\addtolength{\arraycolsep}{-.5\arraycolsep}
\renewcommand{\arraystretch}{0.5}
\left( \begin{array}{ccc}
\scriptstyle 0 & \scriptstyle {\bf I}_r & \scriptstyle 0 \\
\scriptstyle -{\bf I}_r & \scriptstyle 0 & \scriptstyle0 \\
\scriptstyle 0 & \scriptstyle 0 & \scriptstyle {\bf I}_n \end{array}
\scriptstyle\right)}~.
\label{CG}
\eeq
\item[2.] If $p(\omega) = 1$~, then dim $S = (m,m)$ and there exist
local coordinates such that
\beq
\omega = d u^{i} \wedge d\xi^{i}~;~~~~~~
\omega_{i j} =
{\scriptstyle % All this junk is to make a smaller matrix
\addtolength{\arraycolsep}{-.5\arraycolsep}
\renewcommand{\arraystretch}{0.5}
\left( \begin{array}{cc}
\scriptstyle 0 & \scriptstyle {\bf I}_m \\
\scriptstyle -{\bf I}_m & \scriptstyle 0 \end{array} \scriptstyle\right)}~,
\label{CH}
\eeq
\end{description}
\bigskip
Having a Poisson structure we can deal with Hamilton equations. From
(\ref{CC}), if $H$ is the hamiltonian, the corresponding equations are
\beq
{\d z}^i = \omega^{i j} \lpd{},{z^j} H \label{CI}.
\eeq
Now we would like to maintain the possibility of explicitly
constructing the flow of (\ref{CI}). This requires that the dynamical
evolution be an even vector field. In turn this implies that the
Poisson structure and the Hamiltonian function
should have the same parity so
that in particular, with an odd Poisson structure we need an odd
Hamiltonian function.
\bigskip
\bigskip
\newpage
\noindent {\bf 5. Examples.}
Before we analyze the graded Nijenhuis condition we study few examples.
\bigskip
\noindent {\bf 5.1 Mixed bosonic-fermionic harmonic oscillator.}
The mixed bosonic-fermionic harmonic oscilator in $(2,2)$ dimensions is
described with coordinates $(q, p, \eta, \xi)$ and has the following
equations of motion
\beqar
&& \d{q} = p~, \nonumber \\
&& \d{p} = -q~, \nonumber \\
&& \d{\eta} = \xi~, \nonumber \\
&& \d{\xi} = -\eta~. \label{DA}
\eeqar
Equations (\ref{DA}) can be given two Hamiltonian descriptions. The
Hamiltonians are: the usual even one
\beq
H = \fraz1,2 (p^2 + q^2) +i\xi \eta~, \label{DB}
\eeq
and an odd one
\beq
K = p\xi + q\eta ~, \label{DC}
\eeq
\noindent
while the two Poisson structures are respectively
\beq
\Lambda_H = \left( \col 0,-1,0,0 \col1,0,0,0
\col 0,0,i,0 \col 0,0,0,i \right)~,~~~~~
\omega_H = \left( \col 0,1,0,0 \col-1,0,0,0
\col 0,0,-i,0 \col 0,0,0,-i \right)~,
\label{DD}
\eeq
and
\beq
\Lambda_K = \left( \col 0,0,0,-1 \col 0,0,1,0
\col 0,-1,0,0 \col 1,0,0,0 \right)~,~~~~~
\omega_K = \left( \col 0,0,0,1 \col 0,0,-1,0
\col 0,1,0,0 \col -1,0,0,0 \right)~.
\label{DE}
\eeq
\bigskip
\noindent
We can construct a a mixed invariant tensor field T by
\beq
T =: \omega_H \circ \Lambda_K =
\left( \col 0,0,0,i \col 0,0,-i,0
\col 1,0,0,0 \col 0,1,0,0 \right)
\label{DF}
\eeq
\noindent
However, this odd tensor field ($p(T) = 1$) is not a recursion operator.
One can easly find that
\beqar
&& T dK = dH~, \nonumber \\
&& T dH = -i(dq) \xi + i (dp) \eta -i(d\eta) p +i(d\xi) q~,~~~
d( T dH) \neq 0~.
\label{DG}
\eeqar
If we evaluate the Poisson brackets of the coordinate variables with
the two symplectic structure (\ref{DD}) and (\ref{DE}) we find that
\beq
\{q, p\}_H = 1~,~~~\{p, q\}_H = -1~,~~~
\{\eta, \eta\}_H = i~,~~~\{\xi, \xi\}_H = i~,~~~ \label{DH}
\eeq
and
\beq
\{q, \xi\}_K = 1~,~~~\{\xi, q\}_K = -1~,~~~
\{p, \eta\}_K = -1~,~~~\{\eta, p\}_K = 1~,~~~ \label{DI}
\eeq
the remaining ones being identically zero.
We see that the sum $\{\cdot, \cdot \}_+$ of the two structures is
itself a Poisson structure with the property
\beq
\{ F, G \}_{+}
= - (-1)^{p(F) p(G)}\{ G, F \}_{+}~,
\label{DL} \\
\eeq
but it has not definite parity.
Moreover $\{\cdot, \cdot \}_+$ is
degenerate.
\bigskip
\bigskip
\noindent {\bf 5.2 Witten Dynamics [Wi].}
Interesting examples come from supersimmetric dynamics.
It has been shown [VPST], [So] that the dynamics of Witten's Hamilton
systems [Wi]
can be given a bi-Hamiltonian description with an even Poisson
bracket and Grassmann even Hamiltonian
or with an odd bracket and Grassmann odd Hamiltonians.
Instead of considering the general case we shall study a
supersymmetric Toda chain with coordinates $(q,p,\eta,\xi)$.
The even Hamiltonian is given by
\beq
H = \fraz1,2 (p^2 + e^q) + \fraz1,2 i \xi \eta e^{\fraz q,2}~. \label{DM}
\eeq
With the even Poisson structure
\beq
\Lambda_H = \left( \col 0,-1,0,0 \col1,0,0,0
\col 0,0,i,0 \col 0,0,0,i \right)~,~~~~~
\omega_H = \left( \col 0,1,0,0 \col-1,0,0,0
\col 0,0,-i,0 \col 0,0,0,-i \right)~,
\label{DN}
\eeq
the equations of motion read
\beqar
&& \d{q} = p~, \nonumber \\
&& \d{p} = - \fraz1,2 e^q~ - \fraz1,4 i\xi \eta e^{\fraz q,2}, \nonumber \\
&& \d{\eta} = \fraz1,2 \xi e^{\fraz q,2}~, \nonumber \\
&& \d{\xi} = - \fraz1,2 \eta e^{\fraz q,2}~. \label{DO}
\eeqar
Then the following functions are constants of the motion
\beqar
&& K = p \xi + e^{\fraz q,2} \eta~, \nonumber \\
&& L = p \eta - e^{\fraz q,2} \xi~, \nonumber \\
&& F = i \xi \eta~. \label{DP}
\eeqar
We can use $K$ in (\ref{DP}) (or $L$) as an alternative Hamiltonian.
The corresponding symplectic structure can be written as
\beqar
\omega_K &=& dq \wedge d \xi + dp \wedge dq ( e^{- \fraz {q},2} \eta)
+ dp \wedge d\eta (-2 e^{- \fraz {q},2}) + df \wedge dH \nonumber \\
&=& d \{ dq (- \xi) + dp (2 e^{- \fraz {q},2} \eta) + f dH \}~, \label{DQ}
\eeqar
where $f(q,p,\eta,\xi)$ is a function explicitly given by
\beqar
&& f = A \xi + B \eta \nonumber \\
&& A(q,p) = {\fraz{1},{p^2 + e^q}}
\left( {\fraz{2p},{\sqrt{p^2 + e^q}}} log (
{\fraz{e^{\fraz{q},{2}}},{p + \sqrt{p^2 + e^q}}} ) +
{\fraz{2e^{\fraz{q},{2}}},{\sqrt{p^2 + e^q}}} - 2 \right) \nonumber \\
&& B(q,p) = {\fraz{1},{p^2 + e^q}}
\left( {\fraz{2e^{\fraz{q},{2}}},{\sqrt{p^2 + e^q}}} log (
{\fraz{e^{\fraz{q},2}},{p + \sqrt{p^2 + e^q}}} ) -
{\fraz{2p},{\sqrt{p^2 + e^q}}} - 2 p e^{-\fraz{q},{2}} \right)~. \label{DQA}
\eeqar
\bigskip
\noindent
If $\Gamma$ is the dynamical vector
field of the Toda system, as given by (\ref{DO}), than,
the function $f$ is such that
~$i_{\Gamma}df = e^{-\fraz{q},{2}} \eta$~ and this, in turn, assures that
~$i_{\Gamma}\omega_{K} = dK$.
\bigskip
It take some algebra to check that
the $(1,1)$ tensor field
\beq
T = \omega_K \circ \Lambda_H ~, \label{DR}
\eeq
is such that
\beqar
&& T dH = dK~, \nonumber \\
&& d (T^2 dH) \neq 0~. \label{DS}
\eeqar
\bigskip \noindent
Again, $T$ in (\ref{DR}) is not a recursion operator.
\bigskip
\bigskip
\noindent
{\bf 6. Super Nijenhuis torsion.}
One of the most relevant consequences deriving from a (not graded)
(1-1) tensor field $T$
with vanishing Nijenhuis torsion is the possibility to generate
sequences of exact 1-forms according to
\prop 5.
\beq
\NT = 0,~~ d(T dF) = 0 \Longrightarrow d(T^k dF) = 0~. \label{EA}
\eeq
\proof
Let $\alpha$ be any 1-form. By using the expression of the exterior
derivative, after some algebra one finds that
\beqar
{X \wedge Y} \inner d(T^2 \alpha) &=&
\{X \wedge TY + TX \wedge Y\} \inner d(T \alpha)
- \{TX \wedge TY\} \inner d\alpha \nonumber \\
&&~~- {\HT (X, Y)}\inner \alpha~. \label{EB}
\eeqar
Where $\HT$ is defined in (\ref{AC}).
Assume now that both $\alpha$ and $T \alpha$ are closed.
>From (\ref{EB}) we see that $T^2 \alpha$ is closed if and only if
$\HT = 0$, namely if and only if the Nijenhuis torsion of $T$
vanishes.
\bigskip
Let us analize now the graded situation.
Suppose $T$ is a graded $(1,1)$ tensor field which is homogeneous of
parity $p(T)$. Then, if $\alpha$ is any 1-form, by using the
definition (\ref{BF}), after some (graded) algebra, the analogue of
(\ref{EB}) reads
\beqar
{X \wedge Y} \inner d(T^2 \alpha) &=&
{ \{(-1)^{p(T)p(Y)} X \wedge TY
+ (-1)^{p(T)[p(X)+p(Y)]} TX \wedge Y\} } \inner d(T \alpha)
\nonumber \\
&&~~- (-1)^{p(T)[p(X)+p(T)]} {TX \wedge TY} \inner d\alpha \nonumber \\
&&~~- (-1)^{p(T)}~ \GHT (X,Y) \inner \alpha
\nonumber \\
&&~~+ (-1)^{p(T)p(X)} [ 1 - (-1)^{p(T)}] L_{TX} ( {TY}\inner \alpha)~.
\label{EC}
\eeqar
Where $\GHT$ is defined in (\ref{BM}).
\bigskip
It is clear then, that for an (1,1) odd tensor a (2,1) tensor
corresponding to its
torsion (super-Nijenhuis torsion) can
be defined only when $p(T) = 0$.
The same result is attained with the use of the general
approach $d_T \circ d_T = 0$ .
\bigskip
\bigskip
\noindent {\bf 7. Conclusions.}
Summing up, we have shown that there are examples of dynamical
systems whose dynamical vector field $\Gamma$
admits two Hamiltonian descriptions,
odd and even respectively, and that the tensor field T , constructed
out of the corresponding Poisson structures is not
a recursion operator since it cannot generate new
integrals of motion after the first two ones.
We have also shown that this fact is general and that
for a generic graded $(1,1)$ tensor field $T$
a graded Nijenhuis torsion cannot be defined unless $T$ is even.
>From the nature of the proof it seems plausible that a similar
theorem should hold true also in infinite dimensions.
The `no go' theorem we have proved in our paper does not exhausts,
obviously, the analysis of complete integrability for graded
Hamiltonian systems. Much more attention must be paid, however, in
generalizing to the graded case geometrical structures which play a
relevent and natural r\^{o}le in the non graded situation.
\bigskip
\bigskip
\noindent {\bf Acknowledgement}
We thank U. Bruzzo for useful remarks. We are very grateful to D. Del
Santo and P. Omari for their invaluable help in solving a system of
partial differential equations which lead to the second symplectic
structure for the Toda system.
\bigskip
\pagebreak
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\end{document}
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