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\begin{document}

\begin{titlepage}
{\hbox to\hsize{April  2001 \hfill
{Bicocca--FT--01--11}}}
{\hbox to\hsize{${~}$ \hfill
{IFUM--686--FT}}}
\begin{center}
\vglue .06in
{\Large\bf Non(anti)commutative Superspace}
\footnote{Supported in 
part by INFN, MURST and the European Commission RTN program
HPRN--CT--2000--00131, in which S.P. is associated to
the University of Padova and D.~K.~is associated to the University of
Torino. }\\[.45in]
Dietmar Klemm \footnote{dietmar.klemm@mi.infn.it}\\
{\it Dipartimento di Fisica, Universit\`a degli studi di Milano \\
and INFN, Sezione di Milano, via Celoria 16, 20133 Milano, Italy}
\\
[.2in]
Silvia Penati\footnote{silvia.penati@mib.infn.it}\\
{\it Dipartimento di Fisica dell'Universit\`a degli studi di
Milano-Bicocca\\ 
and INFN, Sezione di Milano, piazza della Scienza 3, I-20126 Milano, 
Italy}
\\
[.2in]
Laura Tamassia\footnote{tamlaura6@hotmail.com}\\
{\it Dipartimento di Fisica dell'Universit\`a degli studi di
Milano-Bicocca,\\ 
piazza della Scienza 3, I-20126 Milano, 
Italy}\\[.8in]


{\bf ABSTRACT}\\[.0015in]
\end{center}

We investigate the most general non(anti)commutative geometry in $N=1$ 
four dimensional superspace.
We find that a nontrivial
non(anti)commutative superspace geometry compatible with 
supertranslations exists with non(anti)commutation parameters which may
depend on the spinorial coordinates. The algebra is in general 
nonassociative. Imposing associativity
introduces additional constraints which however allow 
for nontrivial commutation relations involving fermionic 
coordinates. We define an associative $\star$--product 
by extending to superspace the Kontsevich procedure.  
In a string theory contest we discuss the connection between  
non(anti)\-com\-mutat\-ive grassmannian geometry in superspace and
string propagation in curved backgrounds. Finally, $N=2$ euclidean  
superspace is also discussed.

${~~~}$ \newline
PACS: 03.70.+k, 11.15.-q, 11.10.-z, 11.30.Pb, 11.30.Rd  \\[.01in]  
Keywords: Noncommutative geometry, Supersymmetry, Supergravity.

\end{titlepage}



\sect{Introduction}

During the past two years a clear connection between string theory
and noncommutative geometry has emerged. In the presence of a
constant magnetic Neveu-Schwarz field, the low energy dynamics of D3 brane
excitations is described by noncommutative ${\cal N}=4$ super Yang-Mills
theory \cite{seibwitt}. The result by Seiberg and Witten followed
earlier work \cite{connes}, where it was found that noncommutative
geometry arises very naturally in the framework of M(atrix) theory.
Apart from the string theory context, noncommutative field theories
are interesting in their own right. This interest is motivated
by many intriguing features of field theories on noncommutative
spaces, like the UV/IR mixing \cite{minwalla} or the Morita equivalence
\cite{schwarz} between Yang-Mills theories on noncommutative tori.
Noncommutative field theories also play a role
in solid state physics, e.~g.~noncommutative Chern Simons theory was
recently proposed by Susskind \cite{susskind} to provide a description
of the fractional quantum Hall effect. Also in the physics of black
holes, noncommutativity of spacetime naturally emerges from
`t Hooft's S-matrix ansatz \cite{thooft} (Cf.~also \cite{li} for a recent
review). The hope is that this may eventually lead to a better
understanding of some of the puzzles of black hole physics, such
as the information loss paradox.

Up to now, the concept of noncommutativity has been limited
essentially to the bosonic coordinates, i.~e.~one has
\beq
[x^{\mu}, x^{\nu}] = i\Theta^{\mu\nu}
\label{bosonic}
\eeq
where $\Theta^{\mu\nu}(x)$ is antisymmetric. In view of the fact
that supersymmetry plays a fundamental role in string theory,
it seems natural and
compelling to ask what happens if we deform also the anticommutators
between fermionic coordinates of superspace, or the commutators
between bosonic and fermionic coordinates. To investigate the most
general deformations compatible with supersymmetry is the main purpose
of this paper. First steps in this direction were undertaken
in \cite{kosinski}, where quantum deformations of the Poincar\'{e}
supergroup were considered, and in \cite{ferrara}, where it was
shown that in general chiral superfields are not closed under
star products that involve also deformations of fermionic coordinates.
Here we will be mainly concerned with the conditions imposed
on the possible deformations of superspace by requirements such
as covariance under translations and supertranslations, Jacobi identities,
associativity of the star product, and closure of chiral superfields
under the star product.

The paper is organized as follows: In Section 2 we
determine the most general deformation of four dimensional, $N=1$ 
Minkowski superspace that is covariant
under supertranslations and discuss the deformation of the 
supersymmetry algebra which follows. In Section 3 we study
the restriction imposed on this general structure by the Jacobi identities,
i.~e.~by the requirement to have a super Poisson structure on superspace.
We will see that these additional constraints necessarily impose the 
spinorial coordinates to be anticommuting, but allow for possible nontrivial
commutation relations among bosonic and fermionic coordinates. In this 
case the standard supersymmetry algebra is restored. 
In Section 4 it is shown that the violation of the
Jacobi identities, implied by the deformation of the fermionic coordinates,
is equivalent to the nonvanishing of a super three-form field strength.
This suggests that a deformation of fermionic variables may arise
in string theory with backgrounds that involve nonvanishing
Neveu-Schwarz super field strengths. In the following section
we construct the star product which, due to the violation of
the Jacobi identities, will be in general nonassociative. In the
cases where the Jacobi identities are satisfied, we show that our
star product is associative up to quadratic order in a deformation
parameter $\hbar$. Finally, in the last Section we discuss possible
non(anti)commutative deformations for superspaces with euclidean signature.
In the simplest $N=2$ case, we find that deformations involving nontrivial
anticommutation relations among spinorial variables are in this case 
allowed by the request of consistency with supercovariance and associativity. 
We conclude with some final remarks. The Appendix contains our 
notations and conventions in superspace.


\sect{Covariant non(anti)commutative geometry}

We consider a four dimensional $N=1$ Minkowski superspace. 
The set of superspace coordinates are $Z^A = (x^{\a \ad}, \th^\a, \thb^\ad)$,
where $x^{\a \ad}$ are the four real bosonic coordinates and $\th^\a$, 
$\thb^\ad$ two--component complex Weyl fermions.  
The conjugation rule $\thb^\ad = (\th^\a)^{\dag}$ follows from the
requirement to have a four component Majorana fermion  
(we use conventions of {\em Superspace} \cite{super}. See also Appendix).

In the standard (anti)commutative superspace the algebra of the coordinates is
\bea
&& \{ \th^\a , \th^\b \} ~=~\{ \thb^\ad , \thb^\bd \} ~=~
\{ \th^\a , \thb^\ad \} ~=~ 0 \non\\
&& [ x^{\a \ad} , \th^\b ] ~=~ [ x^{\a \ad} , \thb^\bd ] ~=~ 0 \non\\
&& [ x^{\a \ad} , x^{\b \bd} ] ~=~ 0 
\label{coord}
\eea
and it is trivially covariant under the superpoincar\'e group. 
The subgroup of the (super)translations (spacetime translations and 
supersymmetry transformations) 
\bea
&& \th'^\a ~=~ \th^\a ~+~ \epsilon^\a \non \\
&& \thb'^\ad ~=~ \thb^\ad ~+~ \bar{\epsilon}^\ad \non \\
&& x'^{\a \ad} ~=~ x^{\a \ad} ~+~ a^{\a \ad}
~-~\frac{i}{2} \left( \epsilon^\a \thb^\ad ~+~ 
\bar{\epsilon}^\ad \th^\a \right) 
\label{transf1}
\eea  
is generated by two
complex charges $Q_\a$ ($\bar{Q}_\ad = Q_\a^\dag$) and the four--momentum
$P_{\a \ad}$ subjected to
\beq
\{ Q_\a , Q_\b \} ~=~ \{ \bar{Q}_\ad , \bar{Q}_\bd \} ~=~ 0  \quad , \quad
\{ Q_\a , \bar{Q}_\ad \} ~=~ P_{\a \ad}
\eeq

Representations of supersymmetry are given by superfields 
$V(x^{\a \ad}, \th^\a, \thb^{\ad})$ whose components are obtained by expanding
$V$ in powers of the spinorial coordinates. The set of superfields is closed  
under the standard product of functions. The product of two superfields 
is (anti)commutative, $V \cdot W = (-1)^{deg(V) \cdot deg(W)} W \cdot V$, 
and associative, $(K \cdot V) \cdot W = K \cdot (V \cdot W)$.

In order to define a non(anti)commutative superspace, 
we consider the most 
general structure of the algebra for a set of four bosonic real coordinates 
and a complex two--component Weyl spinor with $(\th^\a)^\dag = \thb^\ad$  
\bea
&&\left\{\th^{\a},\th^{\b}\right\} ~=~{\cal A}^{\a\b}(x,\th,
\bar{\th}) \qquad , \qquad
\left\{\bar{\th}^{\dot{\a}},\bar{\th}^{\dot{\b}}\right\} ~=~
\bar{\cal A}^{\dot{\a}\bd}(x,\th,\bar{\th})
\non\\  
&&\left\{\th^{\a},\bar{\th}^{\dot{\a}}\right\}
~=~{\cal B}^{\a\dot{\a}}
(x,\th,\bar{\th})\non\\
&&\left[x^{\underline{a}},\th^{\b}\right]~=~i{\cal C}^{\underline{a}\b}
(x,\th,\bar{\th})\qquad , \qquad
\left[x^{\underline{a}},\bar{\th}^{\dot{\b}}\right]~=~
i\bar{\cal C}^{\underline{a} \dot{\b}}(x,\th,\bar{\th})\non \\
&&\left[x^{\underline{a}},x^{\underline{b}}\right]~=~
i{\cal D}^{\underline{a} \underline{b}}
(x,\th,\bar{\th})
\label{coord2}
\ena
Here, ${\cal A}, {\cal B}, {\cal C}, {\cal D}$ are local functions of the 
superspace variables and we have defined $\bar{\cal A}^{\ad \bd} 
\equiv ({\cal A}^{\a \b})^\dag$, 
$\bar{\cal C}^{\underline{a} \bd} \equiv ({\cal C}^{\underline{a} \b})^\dag$.
From the conjugation rules on the coordinates it follows also 
$\left({\cal B}^{\a\dot{\a}}\right)^{\dag}={\cal B}^{\a\dot{\a}}$ and
$\left({\cal D}^{\underline{a} \underline{b}}\right)^{\dag} =
{\cal D}^{\underline{a} \underline{b}}$. 

To implement (\ref{coord2}) to be the algebra of the coordinates
of a non(anti)commutative $N=1$ superspace we require its covariance 
under the group of space translations and supertranslations (\ref{transf1}). 
We do not require covariance under Lorentz group which is in general broken
in a noncommutative geometry.
We restrict our analysis to the case of a
non--deformed group where the parameters $a^{\a\ad}$, $\epsilon^\a$ 
and $\bar{\epsilon}^\ad$ in (\ref{transf1}) are kept 
(anti)commuting \footnote{More general 
constructions of non(anti)commutative geometries in grassmannian spaces 
have been
considered, where also the algebra of the parameters is deformed
\cite{kosinski}.}. 

Imposing covariance amounts to ask the functional dependence of the 
${\cal A} , {\cal B}, {\cal C} , {\cal D}$ in (\ref{coord2}) to be the 
same at {\em any} point of the supermanifold. 
To work out explicitly the constraints which follow, 
we perform a (super)translation (\ref{transf1}) on the coordinates
and compute the algebra of the new coordinates in terms of
the old ones. We find
\bea
&& \left\{\th'^{\a},\th'^{\b}\right\}~=~ \left\{\th^{\a},\th^{\b}\right\}
\quad , \quad 
\left\{\th'^{\a},\thb'^{\ad}\right\}~=~ \left\{\th^{\a},\thb^{\ad}\right\}
\non\\
&& \left[x'^{\a\dot{\a}},\th'^{\b}\right]~=~
\left[x^{\a\dot{\a}},\th^{\b}\right]-\frac{i}{2}
\e^{\a}\left\{\bar{\th}^{\dot{\a}},\th^{\b}\right\} ~-~
\frac{i}{2}\bar{\e}^{\dot{\a}}\left\{\th^{\a},\th^{\b}\right\}
\non\\
&& \left[x'^{\a\dot{\a}},x'^{\b\dot{\b}}\right] ~=~
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right]
\non\\
&&~~~~~~~~~~
-~\frac{i}{2}\left(\e^{\b}\left[x^{\a\dot{\a}},\thb^{\dot{\b}}\right]
~+~\bar{\e}^{\dot{\b}}\left[x^{\a\dot{\a}},\th^{\b}\right]
~+~\e^{\a}\left[\bar{\th}^{\dot{\a}},x^{\b\dot{\b}}\right]
~+~\bar{\e}^{\dot{\a}}\left[\th^{\a}, x^{\b \bd} \right]\right) \non\\
&&~~~~~~~~~~
+~\frac{1}{4}\left(\e^{\a}\left\{\bar{\th}^{\dot{\a}},\bar{\th}^{\dot{\b}}
\right\}\e^{\b}+\e^{\a}\left\{\bar{\th}^{\dot{\a}},\th^{\b}\right\}
\bar{\e}^{\dot{\b}}+\bar{\e}^{\dot{\a}}\left\{\th^{\a},\bar{\th}^{\dot{\b}}
\right\}\e^{\b}+\bar{\e}^{\dot{\a}}\left\{\th^{\a},\th^{\b}\right\}
\bar{\e}^{\dot{\b}}\right)
\non\\
&&~~~~~~~~~~
\eea
and the corresponding hermitian conjugates. Substituting the 
functions as given in (\ref{coord2}), we obtain that the most general
non(anti)commutative geometry in superspace is described by the algebra 
(\ref{coord2}) where the functions are constrained by the following 
set of independent equations
\beq
{\cal A}^{\a \b} (x',\th',\thb' ) ~=~ 
{\cal A}^{\a \b} (x,\th,\thb ) \quad , \quad
{\cal B}^{\a \ad} (x',\th',\thb' ) ~=~ 
{\cal B}^{\a \ad} (x,\th,\thb )
\label{first} 
\eeq
\beq
{\cal C}^{\a\dot{\a}\b}(x',\th',\thb') ~=~
{\cal C}^{\a\dot{\a}\b}(x,\th,\thb )~-~\frac{1}{2}
\e^{\a}{\cal B}^{\b \ad}(x,\th,\thb )~-~\frac{1}{2}
\bar{\e}^{\ad}{\cal A}^{\a\b}(x,\th,\thb )
\label{second}
\eeq
\bea
&& {\cal D}^{\a\dot{\a}\b\dot{\b}}(x',\th',\thb') ~=~
{\cal D}^{\a\dot{\a}\b\dot{\b}}(x,\th,\overline{\th}) 
\non\\
&&-~
\frac{i}{2}\left(\e^{\b}\bar{\cal C}^{\a\dot{\a}\dot{\b}}
(x,\th,\thb )
+\bar{\e}^{\dot{\b}}{\cal C}^{\a\dot{\a}\b}(x,\th,\thb)
-\e^{\a}\bar{\cal C}^{\b\dot{\b}\ad}(x,\th,\thb )
-\bar{\e}^{\dot{\a}}{\cal C}^{\b\dot{\b}\a}(x,\th,\thb)\right)
\non\\
&& -\frac{i}{4}\left(\e^{\a}\bar{\cal A}^{\dot{\a}\dot{\b}}
(x,\th,\thb )\e^{\b}~+~\e^{\a} {\cal B}^{\b\dot{\a}}
(x,\th,\thb )\bar{\e}^{\dot{\b}} \right.
\non\\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left. +\bar{\e}^{\dot{\a}}
{\cal B}^{\a\dot{\b}}(x,\th,\thb )\e^{\b}+
\bar{\e}^{\dot{\a}}{\cal A}^{\a\b}(x,\th,\thb )
\bar{\e}^{\dot{\b}}\right)
\non\\
&&~~~~~~~~
\label{third}
\eea
The rest of the functions are then determined by hermitian conjugation. 

We now concentrate on the last set of equations in order to establish 
the existence of possible nontrivial local solutions. The trivial solution
corresponding to standard (anti)commutative superspace always exists.  
Covariance under spacetime translations imposes the functions
${\cal A}, {\cal B}, {\cal C}, {\cal D}$ to be independent of the bosonic
coordinates. We then look for the most general 
dependence on the spinorial coordinates allowed by these constraints. 
From (\ref{first}) it immediately follows that the functions ${\cal A},
{\cal B}$, and consequently $\bar{\cal A}$, must be also independent
of $(\th, \thb)$. Therefore, we set 
\bea
&&{\cal A}^{\a\b}(x,\th,\thb) ~\equiv~ A^{\a\b}
\quad , \quad  
\bar{\cal A}^{\ad\bd}(x,\th,\thb) ~\equiv~ \bar{A}^{\ad\bd}
\non\\
&&{\cal B}^{\a \bd}(x,\th,\thb) ~\equiv~
B^{\a \bd} 
\label{sol1}
\eea 
where $B^{\a \bd}$ is real and $A^{\a\b} = A^{\b\a}$.
Using these results the constraint (\ref{second}) can be rewritten as
\beq
{\cal C}^{\a\ad \b}(\th',\thb') ~=~
{\cal C}^{\a \ad \b}(\th,\thb ) ~-~
\frac{1}{2}\e^{\a} B^{\b \ad} ~-~ \frac{1}{2} \bar{\e}^{\ad} A^{\a\b}
\eeq
Since this equation is linear in the supersymmetry parameters 
$\e^{\a}=\d\th^{\a}$ e $\bar{\e}^{\dot{\a}}=\d \thb^{\dot{\a}}$, 
it is easy to convince that the most general local solution  
is 
\bea
&& {\cal C}^{\a\dot{\a}\b}(\th,\thb ) ~=~ 
C^{\a\dot{\a}\b} ~-~\frac{1}{2}\th^{\a} B^{\b\dot{\a}}
~-~ \frac{1}{2}\thb^{\dot{\a}} A^{\a\b}
\non\\
&& \bar{\cal C}^{\a\dot{\a}\bd}(\th,\thb ) ~=~ 
\bar{C}^{\a\dot{\a}\bd} ~-~\frac{1}{2}\thb^{\ad} B^{\a\dot{\b}}
~-~ \frac{1}{2}\th^{\a} \bar{A}^{\ad\bd}
\label{sol2}
\eea
where $C$ is a constant complex spinor and we have defined 
$\bar{C}^{\a\dot{\a}\dot{\b}}= \left(C^{\a\dot{\a}\b}\right)^{\dag}$.
 
Finally, using the results (\ref{sol1}, \ref{sol2}) in (\ref{third})
we obtain
\bea
{\cal D}^{\a\dot{\a}\b\dot{\b}}(\th',\thb' ) &&=
~{\cal D}^{\a\dot{\a}\b\dot{\b}}( \th, \thb )
\non\\
&&-~\frac{i}{2}\e^{\b}\left(\bar{C}^{\a\dot{\a}\bd}
~-~ \frac{1}{2}\th^{\a}\bar{A}^{\dot{\a}\dot{\b}}
~-~\frac{1}{2}\thb^{\dot{\a}} B^{\a\dot{\b}}\right)
\non\\
&&-~\frac{i}{2}\bar{\e}^{\dot{\b}}\left(C^{\a\dot{\a}\b}
~-~\frac{1}{2}\th^{\a}B^{\b\dot{\a}}
~-~\frac{1}{2}\thb^{\dot{\a}} A^{\a\b}\right)
\non\\
&&+~\frac{i}{2}\e^{\a}\left(\bar{C}^{\b\dot{\b}\dot{\a}}
~-~\frac{1}{2}\th^{\b} \bar{A}^{\dot{\b}\dot{\a}}
~-~\frac{1}{2}\thb^{\dot{\b}} B^{\b\dot{\a}}\right)
\non\\
&&+~\frac{i}{2}\bar{\e}^{\dot{\a}}\left(C^{\b\dot{\b}\a}
~-~\frac{1}{2}\th^{\b} B^{\a\dot{\b}}
~-~\frac{1}{2}\thb^{\dot{\b}} A^{\b\a} \right)
\non\\
&&-~\frac{i}{4}\left(\e^{\a} \bar{A}^{\dot{\a}\dot{\b}} \e^{\b}
~+~ \e^{\a} B^{\b\dot{\a}} \bar{\e}^{\dot{\b}}
~+~\bar{\e}^{\dot{\a}} B^{\a\dot{\b}}\e^{\b}
~+~\bar{\e}^{\dot{\a}} A^{\a\b} \bar{\e}^{\dot{\b}}\right)
\eea
A direct inspection reveals that the terms linear and quadratic in the
supersymmetry parameters combine to reproduce the variation of
\bea
{\cal D}^{\a\dot{\a}\b\dot{\b}}(\th,\thb )
&&=~ D^{\a\dot{\a}\b\dot{\b}}
~-~\frac{i}{2}\left(\th^{\b} \bar{C}^{\a \dot{\a}\dot{\b}}
~-~\thb^{\dot{\a}} C^{\b\dot{\b}\a}
~-~ \th^{\a} \bar{C}^{\b\dot{\b}\dot{\a}}
~+~ \thb^{\dot{\b}} C^{\a\dot{\a}\b}\right)
\non\\
&&-~\frac{i}{4}\left(\th^{\a} \bar{A}^{\dot{\a}\dot{\b}} \th^{\b}
~+~\th^{\a} B^{\b\dot{\a}} \thb^{\dot{\b}}
~+~ \thb^{\dot{\a}} B^{\a\dot{\b}}\th^{\b}
~+~ \thb^{\dot{\a}} A^{\a\b} \thb^{\dot{\b}} \right)
\label{sol3}
\eea
where the constant matrix $D$ is antisymmetric and hermitian.

To summarize, we have found that the most general non(anti)commutative
geometry in Minkowski superspace which is consistent with translations and
supertranslations is 
\bea
\left\{\th^{\a},\th^{\b}\right\} &&=~ A^{\a\b} \quad , \quad
\left\{ \thb^{\ad} , \thb^{\bd} \right\} ~=~ \bar{A}^{\ad \bd} 
\quad , \quad
\left\{ \th^{\a},\thb^{\dot{\a}} \right\} ~=~ B^{\a\dot{\a}}
\non\\
\left[ x^{\a\dot{\a}},\th^{\b} \right] &&=~ 
i {\cal C}^{\a\dot{\a}\b} (\th, \thb)
\non\\
\left[x^{\a\dot{\a}},\thb^{\dot{\b}}\right] &&=~
i \bar{{\cal C}}^{\a\dot{\a}\dot{\b}}(\th, \thb) 
\non\\
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right] 
&&=~ i {\cal D}^{\a\dot{\a}\b\dot{\b}}(\th,\thb)
\label{nonanti}
\eea
where the functions ${\cal C}$ and ${\cal D}$ are given in (\ref{sol2}, 
\ref{sol3}).
We notice that, while covariance under spacetime 
translations necessarily requires the non(anti)commutation functions to be
independent of the $x$ coordinates, the covariance under supersymmetry is 
less restrictive and allows for a particular dependence on the spinorial 
coordinates. 

On the algebra of smooth functions of the superspace variables we can formally
define a graded bracket which reproduces the fundamental 
algebra (\ref{nonanti}) when applied to the coordinates. 
In the case of bosonic Minkowski spacetime,
the noncommutative algebra (\ref{bosonic}) can be obtained by 
interpreting the l.h.s. of this relation as the Poisson bracket of classical
commuting variables, where, for generic functions of spacetime, the Poisson 
bracket is defined as $\{ f , g \}_{P} = i \Theta^{\m \n}\pa_\m f \pa_\n g$. 
Generalizing to Minkowski superspace, the graded bracket 
must be constructed as a bidifferential operator with respect to the
superspace variables. In this case we
have two sets of spinorial derivatives, the noncovariant derivatives 
$\pa_\a , \bar{\pa}_\ad$ and the derivatives (\ref{covariant}) covariant with 
respect to supersymmetry transformations. 
Using covariant derivatives $D_A \equiv (D_\a, \bar{D}_\ad,
\pa_{\a \ad})$, for generic functions $\Phi$ and $\Psi$ of the superspace 
coordinates we define (conventions on left and right derivatives are given
in Appendix) 
\beq
\{ \Phi , \Psi \}_P ~=~ \Phi \overleftarrow{D}_A \, P^{AB} \, 
\overrightarrow{D}_B \Psi 
\label{poisson1}
\eeq
where 
\beq
P^{AB} ~\equiv~
\pmatrix{
P^{\a \b}  & P^{\a \bd} & P^{\a \underline{b}} \cr
P^{\ad \b} & P^{\ad \bd} & P^{\ad \underline{b}}  \cr
P^{\underline{a} \b} & P^{\underline{a} \bd} & 
P^{\underline{a} \underline{b}} \cr }
~=~ \pmatrix{
-A^{\a \b}  & -B^{\a \bd} & iC^{\b \bd \a} \cr
-B^{\ad \b} & -\bar{A}^{\ad \bd} & i\bar{C}^{\b \bd \ad}  \cr
iC^{\a \ad \b} & i\bar{C}^{\a \ad \bd} & iD^{\a \ad \b \bd} \cr }
\label{supermatrix}
\eeq
is a constant graded symplectic supermatrix satisfying 
$P^{BA} = (-1)^{(a+1)(b+1)} P^{AB}$, where $a$ denotes the grading of $A$. 
It is easy to verify that applying this 
operator to the superspace coordinates we obtain (\ref{nonanti}).

Alternatively, we can express the graded brackets (\ref{poisson1}) 
in terms of noncovariant spinorial derivatives
$(\pa_\a, \bar{\pa}_\ad, \pa_{\a \ad})$  
\beq
\{ \Phi , \Psi \}_P ~=~ \Phi \overleftarrow{\pa}_A \, \Tilde{P}^{AB} \, 
\overrightarrow{\pa}_B \Psi
\label{poisson2} 
\eeq
where now the symplectic supermatrix $\Tilde{P}^{AB}$ has an explicit 
dependence on the grassmannian coordinates
\beq
\Tilde{P}^{AB}(\th, \thb) ~=~ \pmatrix{
-A^{\a \b}  & -B^{\a \bd} & i{\cal C}^{\b \bd \a}(\th, \thb) \cr
-B^{\ad \b} & -\bar{A}^{\ad \bd} & i\bar{{\cal C}}^{\b \bd \ad}(\th, \thb) \cr
i{\cal C}^{\a \ad \b}(\th,\thb) & i\bar{{\cal C}}^{\a \ad \bd}(\th, \thb) & 
i{\cal D}^{\a \ad \b \bd}(\th, \thb) \cr }
\label{matrix2}
\eeq
with the ${\cal C}$ and ${\cal D}$ functions given in
(\ref{sol2},\ref{sol3}). Its graded symmetry is
$\Tilde{P}^{BA} = (-1)^{(a+1)(b+1)} \Tilde{P}^{AB}$. 
Now, it is easy to verify that the
bidifferential operator $\{ \, , \}_P$ satisfies 
\bea
&& \{ \Phi, \Psi \}_P ~=~ (-1)^{1+ deg(\Phi) \cdot deg(\Psi)} \, \{ \Psi ,
\Phi \}_P 
\non \\
&& \{ c \Phi ,\Psi \}_P ~=~ c \, \{ \Phi, \Psi \}_P  \quad , \quad
\{ \Phi ,c \Psi \}_P ~=~ (-1)^{deg(c) \cdot deg(\Phi)} \,c \, \{ \Phi,
\Psi \}_P 
\non\\
&& \{ \Phi + \Psi , \Omega \}_P ~=~ \{ \Phi , \Omega \}_P ~+~ \{\Psi,
\Omega\}_P
\eea  
The operator $\{ \, , \}_P$ will be promoted to a graded Poisson structure 
on superspace if and only if it satisfies the Jacobi identities. 
This property is not in general satisfied
as a consequence of possible lack of 
associativity in the fundamental algebra (\ref{nonanti}). The investigation
of the additional constraints imposed by associativity will be the subject 
of the next Section.  
 
It is important to notice that the non(anti)commutative extension given in 
(\ref{nonanti}) in general deforms the supersymmetry algebra. In the standard
case, defining $Q_A \equiv (Q_\a, \bar{Q}_\ad, -i\pa_{\a\ad})$, the 
supersymmetry algebra can be written as 
\bea
&& [Q_A,Q_B\} ~=~ i {T_{AB}}^C Q_C \quad , \quad 
[D_A,D_B\} ~=~ {T_{AB}}^C D_C 
\non\\
&& [Q_A,D_B\} ~=~ 0
\label{standard}  
\eea 
where ${T_{AB}}^C$ is the torsion of the flat superspace
(${T_{\a \bd}}^{\underline{c}} = {T_{\bd \a}}^{\underline{c}} = i \d^{~\g}_\a 
\d_\bd^{~\dot{\g}}$ are the only nonzero components) and we have introduced
the notation $[F_A,G_B\} \equiv F_A G_B - (-1)^{ab} G_B F_A$.
Turning on non(anti)commutation in superspace leads instead to
\bea
&& [Q_A,Q_B\} ~=~ i {T_{AB}}^C Q_C ~+~ {R_{AB}}^{CD} Q_C Q_D
\non\\
&& [D_A,D_B\} ~=~ {T_{AB}}^C D_C ~+~ {R_{AB}}^{CD} D_C D_D
\non\\
&& [Q_A,D_B\} ~=~ {R_{AB}}^{CD} Q_C D_D
\label{modalg}
\eea
where ${T_{AB}}^C$ is still the torsion of the flat superspace, while
the presence of the curvature tensor ${R_{AB}}^{CD}$ is a direct consequence 
of the non(anti)commutation of the grassmannian coordinates. Its nonvanishing 
components are
\bea
&& {R_{\a \b}}^{\underline{c} \underline{d}} ~=~ \frac{1}{8} 
P^{\dot{\gamma} \dot{\delta}} \d_{(\a}^{~\g} \d_{\b)}^{~\d}
\quad , \quad
{R_{\ad \bd}}^{\underline{c} \underline{d}} ~=~ \frac{1}{8} 
P^{\gamma \delta} \d_{(\ad}^{~\dot{\g}} \d_{\bd)}^{~\dot{\d}}
\non\\
&& {R_{\a \bd}}^{\underline{c} \underline{d}} ~=~ 
{R_{\bd \a}}^{\underline{c} \underline{d}} ~=~ \frac{1}{8} 
\left( P^{\gamma \dot{\d}} \d_\a^{~\d} \d_\bd^{~\dot{\g}} 
~+~ P^{\d \dot{\g}} \d_\a^{~\g} \d_\bd^{~\dot{\d}} \right)
\label{rtensor}
\eea
It can be expressed in terms of the torsion as
\beq
{R_{AB}}^{CD} ~=~ -\frac{1}{8} \, P^{MN} {T_{M[A}}^C {T_{B)N}}^D
\label{rtensor2}
\eeq
where $[ab)$ means antisymmetrization when at least one of the indices
is a vector index, symmetrization otherwise.
  
Since the terms proportional to the curvature in the algebra (\ref{modalg}) 
are quadratic in the supersymmetry charges and covariant derivatives, 
we can define new graded brackets
\bea
&& [ Q_A , Q_B \}_q ~\equiv~ Q_A Q_B ~-~ (-1)^{ab} [ {\d_B}^C {\d_A}^D ~+~ 
(-1)^{ab} {R_{AB}}^{CD} ] Q_C Q_D
\non\\
&& [ D_A , D_B \}_q ~\equiv~ D_A D_B ~-~ (-1)^{ab} [ {\d_B}^C {\d_A}^D ~+~ 
(-1)^{ab} {R_{AB}}^{CD} ] D_C D_D
\non\\
&& [ Q_A , D_B \}_q ~\equiv~ Q_A D_B ~-~ (-1)^{ab} [ {\d_B}^C {\d_A}^D ~+~ 
(-1)^{ab} {R_{AB}}^{CD} ] D_C Q_D
\non\\
&&~~~~~~~~~~~
\eea
which satisfy the standard algebra (\ref{standard}).  
The new brackets can be interpreted as a quantum deformation 
associated to a $q$--parameter which in this case is a rank--four tensor
\beq
{q_{AB}}^{CD} ~\equiv~ {\d_B}^C {\d_A}^D ~+~ (-1)^{ab} {R_{AB}}^{CD}
\eeq
This issue, which certainly deserves a deeper investigation, is out of the
target of the present paper and will be discussed elsewhere.
 
 
\sect{Requiring associativity}

Given the most general non(anti)commutative geometry consistent with 
superspace translations, the next step is to investigate whether it is
associative, or equivalently, whether the 
graded brackets defined in the previous section satisfy the Jacobi identities 
\bea
&& \{ \Phi, \{ \Psi , \Omega \}_P \}_P + 
(-1)^{deg(\Phi)\cdot [deg(\Psi) + deg(\Omega)]}
\{ \Psi, \{ \Phi, \Omega \}_P \}_P 
\non\\
&& ~~~~~~~~~~~~~~~~~~~~~~+ (-1)^{deg(\Omega)\cdot [deg(\Phi) + deg(\Psi)]}
\{ \Omega, \{ \Phi, \Psi \}_P \}_P 
~=~ 0
\eea   
We explicitly compute this expression for any
set of bosonic and/or spinorial coordinates and study the additional
constraints which come from requiring the vanishing 
of these expressions. 

It is easy to check that the Jacobi identities involving only spinorial 
coordinates are trivially satisfied. Instead, the rest of the identities
involving mixed bosonic and fermionic coordinates or only bosonic ones
give rise to the following new set of independent nontrivial conditions:
From the $x,\th,\th$ and $x,\th,\thb$ identities we obtain 
\bea
&& B^{\a\dot{\b}} A^{\b\g} ~+~ A^{\b\a}B^{\g\dot{\b}} ~=~ 0
\non\\
&& B^{\a\dot{\b}}B^{\b\dot{\g}} ~+~ A^{\b\a} \bar{A}^{\dot{\b}\dot{\g}}
~=~ 0
\label{jacobi1}
\ena
from the $x,x,\th$ identity 
\bea
&& \left( \bar{C}^{\b\dot{\b}\dot{\a}} A^{\a\g}
~+~ C^{\b\dot{\b}\a} B^{\g\dot{\a}} ~-~
\bar{C}^{\a\dot{\a}\dot{\b}} A^{\b\g}
~-~ C^{\a\dot{\a}\b} B^{\g\dot{\b}} \right)
\non\\
&&+~\frac{1}{2} \th^{\a}\left( \bar{A}^{\dot{\a}\dot{\b}} A^{\b\g} ~+~
B^{\b\dot{\a}} B^{\g\dot{\b}}\right) ~-~ \frac12 \th^{\b}\left( 
\bar{A}^{\dot{\a}\dot{\b}} A^{\a\g}
~+~ B^{\a\dot{\b}} B^{\g\dot{\a}}\right) 
\non\\
&&+~ \frac{1}{2} \thb^{\dot{\a}} \left( B^{\a\dot{\b}} A^{\b\g} ~+~
A^{\a\b} B^{\g\dot{\b}}\right) ~-~ \frac12 
\thb^{\dot{\b}} \left( B^{\b\dot{\a}}
A^{\a\g} ~+~ A^{\a\b} B^{\g\dot{\a}} \right)  ~=~ 0
\non\\
&&~~~~~~~~~~~~
\label{jacobi2}
\ena
Finally, from the Jacobi identity for three bosonic coordinates we have
\bea
&~& \left[\frac{1}{2} \bar{C}^{\b\dot{\b}\dot{\a}}
~-~\frac{1}{4}\left(\th^{\b} \bar{A}^{\dot{\a}\dot{\b}}
~+~ \thb^{\dot{\b}} B^{\b\dot{\a}}\right) \right]
\left( C^{\g\dot{\g}\a} ~-~\frac{1}{2} \th^{\g} B^{\a\dot{\g}}
~-~ \frac{1}{2} \thb^{\dot{\g}} A^{\a\g} \right)
\non\\
&&+~  \left[\frac{1}{2} C^{\b\dot{\b}\a} ~-~ \frac{1}{4}\left(\th^{\b}
B^{\a\dot{\b}} ~+~ \thb^{\dot{\b}} A^{\a\b}\right) \right]
\left( \bar{C}^{\g\dot{\g}\dot{\a}} 
~-~ \frac{1}{2}\th^{\g} \bar{A}^{\dot{\a}\dot{\g}}
~-~ \frac{1}{2} \thb^{\dot{\g}} B^{\g\dot{\a}} \right)
\non\\
&&-~  \left[ \frac{1}{2}\bar{C}^{\a\dot{\a}\dot{\b}}
~-~\frac{1}{4}\left(\th^{\a} \bar{A}^{\dot{\a}\dot{\b}}
~+~ \thb^{\dot{\a}} B^{\a\dot{\b}} \right) \right]
\left( C^{\g\dot{\g}\b} ~-~ \frac{1}{2}\th^{\g} B^{\b\dot{\g}}
~-~ \frac{1}{2} \thb^{\dot{\g}} A^{\b\g} \right)
\non\\
&&-~ \left[ \frac{1}{2} C^{\a\dot{\a}\b}
~-~ \frac{1}{4}\left(\th^{\a} B^{\b\dot{\a}}
~+~ \thb^{\dot{\a}} A^{\a\b}\right) \right]
\left( \bar{C}^{\g\dot{\g}\dot{\b}}
~-~ \frac{1}{2}\th^{\g} \bar{A}^{\dot{\b}\dot{\g}}
~-~ \frac{1}{2} \thb^{\dot{\g}} B^{\g\dot{\b}} \right) 
\non\\
&&+~\rm{permutations ~of } ~(\a\ad, ~\b\bd, ~\g\dot{\g}) ~=~ 0
\non\\
&&~~~~~~~~~~~~
\label{jacobi4}
\ena 
The first two conditions (\ref{jacobi1}) necessarily imply the vanishing
of the constants $A$ and $B$.  
Inserting these results in the rest of the constraints we immediately realize
that (\ref{jacobi2}) is identically satisfied. 
We are then left with the only extra condition coming from (\ref{jacobi4})
which eventually can be written as
\beq
{\cal I}{\rm m} 
\left( \bar{C}^{\a\dot{\a}\dot{\b}} C^{\g\dot{\g}\b} ~+~
\bar{C}^{\g\dot{\g}\dot{\a}} C^{\b\dot{\b}\a}
~+~ \bar{C}^{\b\dot{\b}\dot{\g}} C^{\a\dot{\a}\g}\right) ~=~ 0
\label{jacobi4simple}
\eeq
This equation has nontrivial solutions. For example, the matrix 
\beq
C^{\a \ad \b} ~=~
\pmatrix{
\psi^\b & \psi^\b \cr
\psi^\b & \psi^\b \cr }
\eeq
for any spinor $\psi^\b$, is a solution. 
It would correspond to assume the
same commutations rules among any bosonic coordinate and the spinorial
variables. 

We conclude that the most general {\em associative} and non(anti)commuting 
algebra in Minkowski superspace has the form
\bea
\left\{\th^{\a},\th^{\b}\right\} &&=~  
\left\{ \thb^{\ad} , \thb^{\bd} \right\} 
~=~ \left\{ \th^{\a},\thb^{\dot{\b}} \right\} ~=~ 0
\non\\
\left[ x^{\a\dot{\a}},\th^{\b} \right] &&=~ i C^{\a\dot{\a}\b} 
\non\\
\left[x^{\a\dot{\a}},\thb^{\dot{\b}}\right] &&=~ i 
\bar{C}^{\a\dot{\a}\dot{\b}} 
\non\\
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right] &&=~
i D^{\a\dot{\a}\b\dot{\b}} 
~+~\frac{1}{2} \left( \bar{C}^{\b\dot{\b}\dot{\a}}
\th^{\a} ~-~ \bar{C}^{\a\dot{\a}\dot{\b}} \th^\b
~+~ C^{\b\dot{\b}\a} \thb^{\dot{\a}} 
~-~ C^{\a\dot{\a}\b} \thb^\bd \right) 
\non\\
&&~~~~~~~~~~~~~~
\label{assnonanti}
\eea
Setting $C^{\a \ad \b}=0$ we recover the usual noncommuting
superspace considered so far in literature \cite{ferrara,chuzamora,all}. 

We notice that a non(anti)commutative but associative geometry always 
mantains the standard algebra for the covariant derivatives. In fact, 
in this case, from (\ref{rtensor}) it follows ${R_{AB}}^{CD} =0$.

The graded brackets obtained from 
(\ref{poisson1}--\ref{matrix2}) by setting $A= B=0$ and 
$C$ solution of (\ref{jacobi4simple}), define a super Poisson 
structure on superspace. In fact, they satisfy the Jacobi identities
$\{ \Phi , \{ \Psi, \Omega \}_P \}_P + {\rm cyclic} = 0$ as can be easily 
proved by expanding the three functions in power series. 


\sect{Violation of the Jacobi identities}

In this section we study in more details the violation
of the Jacobi identities, and relate this violation to the presence of a
nonvanishing super three--form field strength. We first consider the
graded bracket (\ref{poisson1})
\begin{equation}
\{ \Phi , \Psi \}_P ~=~ \Phi\overleftarrow{\partial}_A \, \Tilde{P}^{AB} \, 
\overrightarrow{\partial}_B \Psi 
\label{bracket_picc}
\end{equation}
where the symplectic supermatrix $\Tilde{P}^{AB}$ is given in
(\ref{matrix2}) as a function of the spinorial coordinates. 
Imposing the Jacobi identity $\{\{ \Phi , \Psi \}_P, \Omega \}_P +
{\mbox{cycl.}} = 0$ yields
\begin{eqnarray}
0 &=& \Tilde{P}^{AD}\overrightarrow{\partial}_D \Tilde{P}^{BC} +
(-1)^{c+a(c+b+1)}\tilde{P}^{BD}\overrightarrow{\partial}_D \Tilde{P}^{CA}
\nonumber \\
& & + \,
(-1)^{b+c(b+a+1)}\tilde{P}^{CD}\overrightarrow{\partial}_D \Tilde{P}^{AB}
\label{Jac_picc}
\end{eqnarray}
When (\ref{Jac_picc}) are satisfied,
(\ref{bracket_picc}) defines a super Poisson structure. If $\Tilde{P}^{AB}$
is invertible, i.~e.~if there exists a supermatrix $\Tilde{P}_{AB}$
with $\Tilde{P}_{AB}\Tilde{P}^{BC} = \delta_A^C$, then (\ref{Jac_picc})
is equivalent to
\begin{equation}
\Tilde{H}_{ABC} ~\equiv~ (-1)^{ac}\overrightarrow{\partial}_A \Tilde{P}_{BC} +
(-1)^{ba}\overrightarrow{\partial}_B \Tilde{P}_{CA} +
(-1)^{cb}\overrightarrow{\partial}_C \Tilde{P}_{AB} = 0
\end{equation}
i.~e.~the super two-form potential $\Tilde{P}_{AB}$ has vanishing
field strength.

In the covariant formalism, where the graded brackets are written
as in (\ref{poisson1}) the Jacobi identities are satisfied iff
\begin{eqnarray}
0 &=& P^{AR} P^{BS}T_{SR}^{\;\;\;\;C}(-1)^{c+b(c+a+r)} +
P^{BR}P^{CS}T_{SR}^{\;\;\;\;A}(-1)^{a+c(a+b+r)} \nonumber \\
& & + \, P^{CR}P^{AS}T_{SR}^{\;\;\;\;B}(-1)^{b+a(b+c+r)} 
\label{Jac_cov}
\end{eqnarray}
and
\beq
(-1)^{bm} P^{AM} P^{BN} {R_{MN}}^{CD} ~=~ 0
\label{Jac_cov2}
\eeq
where the torsion ${T_{AB}}^C$ and the curvature ${R_{AB}}^{CD}$ have been 
introduced in (\ref{standard}, \ref{modalg}).

As a check, we explicitly write the conditions (\ref{Jac_cov}) for all
possible choices of external $ABC$ indices 
\bea
&& B^{\a\dot{\b}} A^{\b\g} ~+~ A^{\b\a}B^{\g\dot{\b}} ~=~ 0
\non\\
&& B^{\a\dot{\b}}B^{\b\dot{\g}} ~+~ A^{\b\a} \bar{A}^{\dot{\b}\dot{\g}}
~=~ 0
\non\\
&& \left( \bar{C}^{\b\dot{\b}\dot{\a}} A^{\a\g}
~+~ C^{\b\dot{\b}\a} B^{\g\dot{\a}} ~-~
\bar{C}^{\a\dot{\a}\dot{\b}} A^{\b\g}
~-~ C^{\a\dot{\a}\b} B^{\g\dot{\b}} \right) ~=~ 0
\non\\
&& {\cal I}{\rm m} 
\left( \bar{C}^{\a\dot{\a}\dot{\b}} C^{\g\dot{\g}\b} ~+~
\bar{C}^{\g\dot{\g}\dot{\a}} C^{\b\dot{\b}\a}
~+~ \bar{C}^{\b\dot{\b}\dot{\g}} C^{\a\dot{\a}\g}\right) ~=~ 0
\eea
These equations coincide with the independent constraints 
(\ref{jacobi1}--\ref{jacobi4simple}) already obtained in the previous Section
by requiring the associativity of the fundamental algebra. 
From the condition (\ref{Jac_cov2})
instead, we do not get any additional constraint since this equation is
automatically satisfied once (\ref{Jac_cov}) is valid.

Again, if $P^{AB}$ is invertible, (\ref{Jac_cov}) is equivalent to
the vanishing of the contorsion tensor $H_{ABC}$ defined by
\begin{equation}
H_{ABC} = T_{AB}^{\;\;\;\;D}P_{DC}(-1)^{ac} +
T_{CA}^{\;\;\;\;D}P_{DB}(-1)^{cb} + T_{BC}^{\;\;\;\;D}P_{DA}(-1)^{ba},
\end{equation}
where $P_{AB}P^{BC}=\delta_A^C$.

In view of the fact that the violation of the Jacobi identities is
equivalent to having a nonvanishing three--form field strength,
it is natural to ask whether the considered deformations of fermionic
variables arise in string theory with backgrounds that involve
Neveu-Schwarz super field--strengths $H$ different from zero.
In \cite{chuzamora}, open superstrings ending
on a D-brane in the presence of a constant NS-NS $B$ field in flat
spacetime were considered,
and it was found that the anticommutation relations for the fermionic
variables of superspace remain unmodified. It would be interesting to
extend these calculations to more general backgrounds to see whether
in this way also a deformation of superspace can arise.


\sect{Construction of the star product}

In this section we construct a star product on the class of
general superfields.
We begin by considering the Moyal--deformed product defined
in the usual way
\begin{equation}
\Phi \ast \Psi \equiv \Phi \exp(\hbar\overleftarrow{D}_A P^{AB}
                       \overrightarrow{D}_B) \Psi, \label{deformprod}
\end{equation}
where $\Phi$ and $\Psi$ are arbitrary superfields, and
$\hbar$ denotes a deformation parameter. In general, due to 
the lack of (anti)commutativity among covariant derivatives (see eq. 
(\ref{modalg})), it is easy to prove that the $\ast$--product is
not associative even when the Poisson brackets are.
However, by extending the Kontsevich procedure \cite{kontsevich} to 
superspace we can construct a modified star product $\star$ with the property to
be associative up to second order in $\hbar$ when the Jacobi identities
are satisfied, i.~e.~, when (\ref{Jac_cov}) holds. 
We define
\begin{eqnarray}
\Phi \star \Psi &\equiv & \Phi\Psi ~+~ \hbar \Phi\overleftarrow{D}_A P^{AB}
\overrightarrow{D}_B \Psi ~+~ \frac{\hbar^2}{2}\Phi(\overleftarrow{D}_A P^{AB}
\overrightarrow{D}_B)(\overleftarrow{D}_C P^{CD}\overrightarrow{D}_D) \Psi
\nonumber \\
&& -\frac{\hbar^2}{3}\left(\overrightarrow{D}_A \Phi \, {\cal M}^{ABC}
\, \overrightarrow{D}_B\overrightarrow{D}_C \Psi - (-1)^c
\overrightarrow{D}_C\overrightarrow{D}_A \Phi \, {\cal M}^{ABC} \,
\overrightarrow{D}_B \Psi\right) \nonumber \\
&& + {\cal O}(\hbar^3)\,, \label{konts}
\end{eqnarray}
where
\begin{eqnarray}
{\cal M}^{ABC} &=& P^{AD}{T_{DE}}^C P^{EB} (-1)^{ce} +
            \frac12 P^{BD}{T_{DE}}^A P^{EC} (-1)^{ae + a + b + ab + bc}
            \nonumber \\
        & & + \frac12 P^{CD}{T_{DE}}^B P^{EA} (-1)^{be + a + c + ac + ab}.
\end{eqnarray}
It is straightforward to show that
\begin{eqnarray}
\lefteqn{(\Phi \star \Psi)\star \Omega ~-~ \Phi \star (\Psi \star \Omega) 
~=~}
\nonumber \\
& & -\frac 23 \hbar^2(-1)^{(c+b)(e+1)+eg+cf}\overrightarrow{D}_A \Phi \,
    P^{AE}P^{BF}P^{CG} \, H_{GFE} \, \overrightarrow{D}_C \Omega \,
    \overrightarrow{D}_B \Psi \\
& & ~+~ {\cal O}(\hbar^3). \nonumber
\end{eqnarray}
We note that at this order only the torsion enters the breaking 
of associativity, since the curvature tensor $R$ is already of order $\hbar$.

The previous formula shows that the modified $\star$--product is associative 
up to the second order in $\hbar$ whenever the 
fundamental algebra is associative, i.e. the contorsion tensor $H$ 
vanishes. 


One can also write the star product in terms of ordinary derivatives
$\partial_A$. In this case it reads
\begin{eqnarray}
\Phi \star \Psi &\equiv & \Phi\Psi ~+~ \hbar \Phi\overleftarrow{\partial}_A
\Tilde{P}^{AB}\overrightarrow{\partial}_B \Psi ~+~
\frac{\hbar^2}{2} \Phi \left(\overleftarrow{\partial}_A \Tilde{P}^{AB}
\overrightarrow{\partial}_{B}\right)\left(\overleftarrow{\partial}_C
\Tilde{P}^{CD}\overrightarrow{\partial}_D\right)\Psi \nonumber \\
&& ~-~ \frac{\hbar^2}{3}\left((-1)^b\overrightarrow{\partial}_A \Phi
\Tilde{\cal M}^{ADB}\overrightarrow{\partial}_D \overrightarrow{\partial}_B 
\Psi
~-~ \overrightarrow{\partial}_B \overrightarrow{\partial}_A \Phi
\Tilde{\cal M}^{ADB}\overrightarrow{\partial}_D \Psi\right) \nonumber \\
&& ~+~ {\cal O}(\hbar^3)\,, \label{kontsord}
\end{eqnarray}
where
\begin{eqnarray}
\Tilde{\cal M}^{ADB} &=& \frac12\Tilde{P}^{AC} \overrightarrow{\partial}_C
\Tilde{P}^{BD} +
\frac12\Tilde{P}^{DC} \overrightarrow{\partial}_C
\Tilde{P}^{AB} (-1)^{b + d(b + a + 1)} \nonumber \\
&& + \Tilde{P}^{BC} \overrightarrow{\partial}_C \Tilde{P}^{DA}
(-1)^{d + a(d + b + 1)}.
\end{eqnarray}
It follows that
\begin{eqnarray}
\lefteqn{(\Phi \star \Psi)\star \Omega ~-~ \Phi \star (\Psi \star \Omega) ~=~}
\nonumber \\
& & \frac 23 \hbar^2 (-1)^{(c+b)(e+1)+eg+cf}
\overrightarrow{\partial}_A \Phi \, \Tilde{P}^{AE}\Tilde{P}^{BF}\Tilde{P}^{CG}
\, \Tilde{H}_{GFE} \, 
\overrightarrow{\partial}_C \Omega \, \overrightarrow{\partial}_B \Psi
\\
& & ~+~ {\cal O}(\hbar^3). \nonumber
\end{eqnarray}
Again, this $\star$--product is associative up to $\hbar^2$--order whenever
the three--form field strength $\Tilde{H}$ is zero.

We now discuss the closure of the class of chiral superfields under the
deformed products we have introduced. For a generic choice 
of the supermatrix $P^{AB}$ the star product of two chiral superfields
(satisfying $\bar{D}_{\dot{\a}}\Phi = 0$)
is {\em not} a chiral superfield, both for associative and nonassociative 
products.   
However, in the particular case where the only nonvanishing components of
the symplectic supermatrix $P^{AB}$ are $P^{\a\dot{\b}}$ and
$P^{\underline{a}\underline{b}}$, chiral superfields 
are closed both under
the deformed product defined in (\ref{deformprod}) and under the Kontsevich
star product (\ref{konts}) (for the latter up to terms of order
${\cal O}(\hbar^3)$).
Clearly for $P^{\a\dot{\b}} \neq 0$ the above star products
are no more associative.
For chiral superfields however, they become commutative\footnote{Generalized
star products that are commutative but nonassociative have been considered
in a different context in \cite{dastrivedi}.}.
This commutativity implies that there is no ambiguity in putting
the parenthesis e.~g.~in the cubic interaction term of a deformed
Wess--Zumino model, with action
\begin{equation}
S = \int d^4x d^2\theta d^2\bar{\theta}\,\Phi\star\bar{\Phi}
+ \int d^4x \left[\int d^2\theta \left(\frac m2 \Phi\star\Phi + \frac g3
\Phi\star\Phi\star\Phi\right) + {\mbox{c.~c.}} \right].
\label{WZ}
\end{equation}
Notice that in this case the $\star$--product in the kinetic action cannot
be simply substituted with the standard product as it happens in superspace
geometries where grassmannian coordinates anticommute \cite{ferrara, all}.


\sect{Non(anti)commutative $N=2$ Euclidean superspace}

The main difference in the description of euclidean superspace 
with respect to Minkowski relies on the reality conditions 
satisfied by the spinorial variables. 
As it is well known \cite{PVP}, in euclidean signature 
a reality condition on spinors is applicable only in the presence of 
extended supersymmetry. We concentrate on
the simplest case, the $N=2$ euclidean superspace even if our analysis
can be easily extended to more general cases. In a chiral description
the two--component Weyl spinors satisfy a symplectic Majorana condition
\beq
(\th^\a_i )^{\ast} ~=~ \th_\a^i ~\equiv~ C^{ij} \, \th^\b_j \, C_{\b \a}
\quad , \quad
(\thb^{\ad, i} )^{\ast} ~=~ \thb_{\ad, i} ~\equiv~  
\thb^{\bd, j} \, C_{\bd \ad} \, C_{ji}
\eeq
where $C^{12} = -C_{12} = i$. 
This implies that the most general non(anti)commutative
algebra can be written as  
\bea
&&\left\{ \th^\a_i,\th^\b_j \right\} ~=~{{\cal A}_1}^{\a\b,}_{~~~ij}
\left(x,\th,
\bar{\th}\right) \qquad , \qquad
\left\{\bar{\th}^{\dot{\a},i},\bar{\th}^{\dot{\b},j}\right\} ~=~
{\cal A}_2^{\dot{\a}\bd,ij}\left(x,\th,\bar{\th}\right)
\non\\  
&&\left\{\th^\a_i ,\bar{\th}^{\dot{\a},j}\right\}
~=~{\cal B}^{\a\dot{\a},~~j}_{~~~~i}
\left(x,\th,\bar{\th}\right)\non\\
&&\left[x^{\underline{a}},\th^\b_i \right]~=~
i{{\cal C}_1}^{\underline{a}\b,}_{~~~ i}
\left(x,\th,\bar{\th}\right)\qquad , \qquad
\left[x^{\underline{a}},\bar{\th}^{\dot{\b},i}\right]~=~
i{\cal C}_2^{\underline{a} \dot{\b},i}\left(x,\th,\bar{\th}\right)\non \\
&&\left[x^{\underline{a}},x^{\underline{b}}\right]~=~
i{\cal D}^{\underline{a} \underline{b}}
\left(x,\th,\bar{\th}\right)
\label{coord3}
\ena
where ${\cal D}$ is real and
${\cal A}_1,{\cal A}_2$, ${\cal B}$, ${\cal C}_1$ and ${\cal C}_2$ 
satisfy symplectic Majorana conditions with respect to each spinorial 
index. When imposing covariance under (super)translations 
\bea
&& \th'^{\a}_i ~~=~ \th^{\a}_i ~+~ \epsilon^{\a}_i 
\non\\
&& \thb'^{\ad,i} ~=~ \thb^{\ad,i} ~+~ \bar{\epsilon}^{\ad,i} 
\non\\
&& x'^{\a\ad} ~=~ x^{\a\ad} ~+~ a^{\a\ad} ~-~ \frac12 
\left( \epsilon^{\a}_i
\thb^{\ad, i} ~+~ \bar{\epsilon}^{\ad,i} \th^{\a}_i \right)
\eea
we obtain 
that the most general non(anti)commutative geometry in euclidean 
superspace is 
\bea
\left\{\th^{\a}_i,\th^{\b}_j\right\} &&=~ {A_1}^{\a\b,}_{~~~ij} \quad , \quad
\left\{ \thb^{\ad,i} , \thb^{\bd,j} \right\} ~=~ A_2^{\ad \bd,ij} 
\quad , \quad
\left\{ \th^{\a}_i,\thb^{\dot{\a},j} \right\} ~=~ B^{\a\dot{\a},~~j}_{~~~~i}
\non\\
\left[ x^{\a\dot{\a}},\th^{\b}_i \right] &&=~ 
i{{\cal C}_1}^{\underline{a}\b,}_{~~~ i} (\th, \thb)
\non\\
\left[x^{\a\dot{\a}},\thb^{\dot{\b},i}\right] &&=~
i {\cal C}_2^{\a\dot{\a}\dot{\b},i}(\th, \thb)  
\non\\
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right] &&=~
i {\cal D}^{\a\dot{\a}\b\dot{\b}}(\th,\thb)
\label{nonanti2}
\eea
where we have defined
\bea
&& {{\cal C}_1}^{\a\dot{\a}\b,}_{~~~~~i} (\th, \thb) ~\equiv~
{C_1}^{\a\dot{\a}\b,}_{~~~~~i} 
~+~\frac{i}{2} \th^{\a}_j B^{\b \dot{\a},~~j}_{~~~~i} 
~+~ \frac{i}{2} \thb^{\dot{\a}, j} {A_1}^{\a\b,}_{~~~ji}
\non\\
&& {\cal C}_2^{\a\dot{\a}\dot{\b},i}(\th, \thb) ~\equiv~
C_2^{\a\dot{\a}\dot{\b},i}
~+~\frac{i}{2} \th^{\a}_j A_2^{\ad \bd,ji}
~+~\frac{i}{2} \thb^{\ad, j} B^{\a\dot{\b},~~i}_{~~~~j} 
\non\\ 
&& {\cal D}^{\a\dot{\a}\b\dot{\b}}(\th,\thb) ~\equiv~
D^{\a\dot{\a}\b\dot{\b}} 
\non\\
&&~~~~~~~
+~\frac{1}{2} \left( \th^{\a}_i C_2^{\b\dot{\b}\dot{\a},i}
~-~ \th^\b_i C_2^{\a\dot{\a}\dot{\b},i} 
~+~ \thb^{\dot{\a}, i} {C_1}^{\b\bd\a,}_{~~~~~ i}
~-~ \thb^{\bd,i}  {C_1}^{\a\ad\b,}_{~~~~~ i} \right) 
\non\\
&&~~~~~~~
+~\frac{i}{4}\left( \th^{\a}_i \, A_2^{\dot{\a}\dot{\b},ij} \, \th^{\b}_j
~+~ \th^{\a}_i \, B^{\b \dot{\a},~~i}_{~~~~j} \, \thb^{\dot{\b},j}
~+~\thb^{\dot{\a},i} \, B^{\a \dot{\b},~~j}_{~~~~i} \, \th^{\b}_j
~+~\thb^{\dot{\a},i} \, {A_1}^{\a\b,}_{~~~ij} \, \thb^{\dot{\b},j} \right)
\non\\
&&~~~~~~~~~
\label{cal2}
\eea
with $A_1$, $A_2$, $B$, $C_1$, $C_2$ and $D$ constant.

We now discuss the associativity of the algebra.
Starting from the supersymmetric non(anti)commutative algebra
(\ref{nonanti2}) we again impose the validity of the Jacobi identities.
Without going into details, we list the independent constraints 
one obtains in this case
\bea
&& B^{\a\dot{\b},~~k}_{~~~~i} \, {A_1}^{\b\g,}_{~~~kj} ~+~ 
{A_1}^{\b\a,}_{~~~ki} \, B^{\g\dot{\b},~~k}_{~~~~j} ~=~ 0
\non\\
&& B^{\b\ad,~~i}_{~~~~k} \, A_2^{\bd \dot{\g},kj} ~+~ A_2^{\bd\ad,ki} 
\, B^{\b\dot{\g},~~j}_{~~~~k}  ~=~ 0
\non\\
&& B^{\a\dot{\b},~~k}_{~~~~i} \, B^{\b\dot{\g},~~j}_{~~~~k} ~+~ 
{A_1}^{\b\a,}_{~~~ki} \, A_2^{\dot{\b}\dot{\g},kj} 
~=~ 0
\label{jacobie1}
\eea
\bea
&& C_2^{\b\dot{\b}\dot{\a},i} {A_1}^{\a\g,}_{~~~ij}
~+~ {C_1^{\b\dot{\b}\a,}}_i B^{\g\dot{\a},~~i}_{~~~~j} ~-~
C_2^{\a\dot{\a}\dot{\b},i} {A_1}^{\b\g,}_{~~~ij}
~-~ {C_1^{\a\dot{\a}\b,}}_i B^{\g\dot{\b},~~i}_{~~~~j} ~=~ 0 
\non\\
&& C_2^{\b\dot{\b}\dot{\a},i} B^{\a\dot{\g},~~j}_{~~~~i}
~+~ {C_1^{\b\dot{\b}\a,}}_i A_2^{\dot{\g}\dot{\a},ji}
~-~ C_2^{\a\dot{\a}\dot{\b},i} B^{\b\dot{\g},~~j}_{~~~~i}
~-~ {C_1^{\a\dot{\a}\b,}}_i A_2^{\dot{\g} \dot{\b},ji}  ~=~ 0
\label{jacobie2}
\eea
\bea 
&& C_2^{\a\dot{\a}\dot{\b},i} {C_1^{\g\dot{\g}\b,}}_i ~+~
C_2^{\g\dot{\g}\dot{\a},i} {C_1^{\b\dot{\b}\a,}}_i
~+~ C_2^{\b\dot{\b}\dot{\g},i} {C_1^{\a\dot{\a}\g,}}_i
\non\\
&& -~ C_2^{\b\dot{\b}\dot{\a},i} {C_1^{\g\dot{\g}\a,}}_i ~-~
C_2^{\g\dot{\g}\dot{\b},i} {C_1^{\a\dot{\a}\b,}}_i
~-~ C_2^{\a\dot{\a}\dot{\g},i} {C_1^{\b\dot{\b}\g,}}_i ~=~ 0
\label{jacobie4}
\eea
We notice that the first three constraints are less restrictive than the
corresponding ones in Minkowski (see eq. (\ref{jacobi1})). In fact, 
since in this case there is no conjugation relation between $A_1$ and $A_2$, 
besides the vanishing solution $A_1 = A_2 = B=0$ analogous to
the solution in Minkowski, we can also have $B=0$ and $A_1 = 0$ {\em or}
$B=0$ and $A_2=0$. Taking for instance $A_2 =0$, from the constraint
(\ref{jacobie2}) it follows $C_2 =0 $. The remaining constraint
is then identically satisfied without any additional condition on $C_1$. 
On the other hand, the solution with $A_1=0$ implies $C_1=0$ and no extra 
conditions on
$C_2$ which can be generically nonvanishing.
Therefore, the most general {\em associative} non(anti)commuting 
geometries in euclidean superspace are
\bea
\left\{\th^{\a}_i ,\th^{\b}_j \right\} &&=~ {A_1}^{\a\b,}_{~~~ij} \quad ,\quad 
\left\{ \thb^{\ad,i} , \thb^{\bd,j} \right\} ~=~ 0 \quad , \quad
\left\{ \th^{\a}_i,\thb^{\dot{\b},j} \right\} ~=~ 0
\non\\
\left[ x^{\a\dot{\a}},\th^{\b}_i \right] &&=~ i{C_1}^{\a\dot{\a}\b,}_{~~~~~i} 
~-~ \frac12 \thb^{\ad,j} {A_1}^{\a \b,}_{~~~ji} 
\non\\
\left[x^{\a\dot{\a}},\thb^{\dot{\b},i}\right] &&=~ 0 
\\
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right] &&=~
iD^{\a\dot{\a}\b\dot{\b}} 
~+~ \frac{i}{2} \left(\thb^{\dot{\a},i}  {C_1}^{\b\dot{\b}\a,}_{~~~~~i}  
~-~ \thb^{\bd,i} {C_1}^{\a\dot{\a}\b,}_{~~~~~i}  \right) ~-~ 
\frac14 \thb^{\ad,i} \, {A_1}^{\a \b,}_{~~~ij} \, \thb^{\bd,j}
\non
\label{assnonantie1}
\eea 
or
\bea
\left\{\th^{\a}_i,\th^{\b}_j\right\} &&=~ 0 \quad , \quad 
\left\{ \thb^{\ad,i} , \thb^{\bd,j} \right\} 
~=~ A_2^{\ad \bd,ij} \quad ,\quad 
\left\{ \th^{\a}_i,\thb^{\dot{\b},j} \right\} ~=~ 0
\non\\
\left[ x^{\a\dot{\a}},\th^{\b}_i \right] &&=~ 0 
\non\\
\left[x^{\a\dot{\a}},\thb^{\dot{\b},i}\right] &&=~ iC_2^{\a\dot{\a}\dot{\b},i}
~-~ \frac12 \th^\a_j A_2^{\ad \bd , ji} 
\\
\left[x^{\a\dot{\a}},x^{\b\dot{\b}}\right] &&=~
iD^{\a\dot{\a}\b\dot{\b}} 
~+~\frac{i}{2} \left( \th^{\a}_i C_2^{\b\dot{\b}\dot{\a},i}
~-~ \th^\b_i C_2^{\a\dot{\a}\dot{\b},i}  \right) 
~-~ \frac14 \th^\a_i \, A_2^{\ad \bd , ij} \, \th^\b_j
\non
\label{assnonantie2}
\eea
We observe that the R--symmetry group of the $N=2$ euclidean superalgebra is
broken only by the constant terms $C_1$ and $C_2$. Setting these terms equal 
to zero leads to nontrivial (anti)commutation relations preserving $R$--symmetry.

Again, explicit expressions for the corresponding graded brackets 
can be obtained as an obvious generalization of (\ref{poisson1}--
\ref{matrix2}). In this case 
they define a super Poisson structure on the euclidean superspace.
A simple example of super Poisson structure is 
\beq
\{ \Phi , \Psi \}_P ~=~ - ~\Phi \overleftarrow{D}_{\a}^i \, 
{A_1}^{\a \b,}_{~~~ij} \, \overrightarrow{D}_{\b}^j \Psi
\label{example2}
\eeq
We notice that this extension is allowed {\em only} in euclidean superspace, 
where it is consistent with the reality conditions on the spinorial 
variables.  



\section{Final remarks}
 
In this paper we have studied the most general non(anti)commutative
geometry in $N=1$ four dimensional Minkowski superspace that is compatible
with supertranslations. We have shown that nonanticommutation relations 
among spinorial variables are allowed if the commutation relations of
bosonic coordinates with the spinorial ones and bosonic coordinates among
themselves acquire a particular dependence on the $\th$--variables.
In a geometric framework we can interpret the supermatrix $P^{AB}$ 
defining the non(anti)commutative algebra (see eq. (\ref{supermatrix})) as a 
nontrivial metric in superspace. The geometry is in general nonassociative 
and deforms the supersymmetry algebra through a curvature term. This
deformation can be interpreted as a quantum deformation associated to 
a four--rank tensor $q$. The geometric properties of the $q$--deformed
superspace haven't been considered in this paper but certainly deserve a 
deeper investigation.  

We have further showed that imposing associativity, i.e. the validity
of the Jacobi identities,
implies additional conditions on the (anti)com\-mutat\-ors
of superspace coordinates which nevertheless allow for nontrivial
deformations involving also fermionic variables. In particular, the spinorial
variables are required to anticommute, but they can have nonzero commutation
relations with the bosonic coordinates. 
As a consequence, in the associative case
the supersymmetry algebra is not $q$--deformed. 
Based upon the Kontsevich procedure, we have been able to construct
a star product that is associative up to second order in the
deformation parameter, if the Jacobi identities are satisfied.
For the general case, the deviation from associativity
has been shown to be proportional to a super
three--form field strength which, in a complete covariant formalism, has
the interpretation of the contorsion of the non(anti)commutative superspace.
For the bosonic case, similar results were obtained
in \cite{cornalba}, where D--branes in curved backgrounds,
with $H = d(B+F) \neq 0$, were considered, leading to a nonassociative
star product deformation of the brane world--volume.

We have extended our analysis to the case of $N=2$ euclidean 
superspace. Due to the different hermiticity conditions on the spinorial 
variables, the non(anti)commutat\-ive euclidean superspace manifests quite
different features from Minkowski. In fact, in this case nonzero 
anticommutation relations among spinorial variables are allowed by 
covariance and associativity. This implies that, even in the case of
non(anti)com\-mutat\-ive but associative geometry, the supersymmetry algebra 
is $q$--deformed. The results obtained for the $N=2$ case can be easily 
extended to generic $N >1$. It is however important to stress that in general
the R--symmetry group is broken by the noncommutativity among fermionic
and bosonic coordinates.   

An interesting continuation of our work would be to
generalize the results of \cite{chuzamora}, in order to see
whether deformations of fermionic variables can arise for open
superstrings ending on a D--brane in the presence of an NS-NS
super field strength different from zero.
It would be also interesting to study field theories defined on a
non(anti)commutative superspace, such as the extended Wess--Zumino model 
proposed in (\ref{WZ}). As already noticed, when the spinorial variables
satisfy a nontrivial algebra, also the kinetic action contains interaction 
terms. 



\newpage

\appendix
\section{Superspace conventions}

In four dimensions, $N=1$ Minkowski superspace is described by a set of 
coordinates  
$Z^A \equiv\left(x^{\a \ad},\th^{\a},\bar{\th}^{\ad}\right)$, where 
$x^{\a \ad} \equiv x^{\underline{a}}$ are the four bosonic real coordinates, 
while $\theta^\a$ (and $\bar{\th}^\ad = (\theta^\a)^{\dag}$) are complex 
two--component Weyl fermions. We use conventions of {\em Superspace} 
\cite{super}, with $(\psi^\a)^{\dag} = \bar{\psi}^{\ad}$,   
$(\psi_\a)^{\dag} = - \bar{\psi}_{\ad}$.

The supersymmetry algebra
\bea
&&\left\{Q_{\a},\bar{Q}_{\dot{\a}}\right\}=P_{\a\dot{\a}} \non\\
&&\left\{Q_{\a},Q_{\b}\right\}=\left\{\bar{Q}_{\dot{\a}},
\bar{Q}_{\dot{\b}}\right\}=0 \non\\
&&\left[P_{\underline{a}},P_{\underline{b}}\right]=0
\label{algebra}
\ena
with $\bar{Q}_{\ad} = Q_\a^{\dag}$, is realized by
\bea
&&Q_{\a}=i\left(\pa_{\a}-\frac{i}{2}\bar{\th}^{\dot{\a}}
\pa_{\a\dot{\a}}\right)\non\\
&&\bar{Q}_{\dot{\a}}=i\left(\bar{\pa}_{\dot{\a}}-
\frac{i}{2}\th^{\a}\pa_{\a\dot{\a}}\right)\non\\
&&P_{\a\dot{\a}}=i\pa_{\a\dot{\a}}
\label{charges}
\ena
Under supersymmetry transformations a generic superfield $V$ transforms
according to $\d V = -i \left(\e^{\a}Q_{\a}+\bar{\e}^{\dot{\a}}
\bar{Q}_{\dot{\a}}\right) V$. In particular, the action on the superspace 
coordinates defines supertranslations 
\bea
&&\d x^{\underline{b}} \equiv 
-i\left(\e^{\a}Q_{\a}+\bar{\e}^{\dot{\a}}\bar{Q}_{\dot{\a}}\right)
x^{\underline{b}}= -\frac{i}{2}\left(\e^{\b}\bar{\th}^{\dot{\b}}+
\bar{\e}^{\dot{\b}}\th^{\b}\right)\non\\
&&\d\th^{\b} \equiv -i\left(\e^{\a}Q_{\a}+\bar{\e}^{\dot{\a}}
\bar{Q}_{\dot{\a}}\right)\th^{\b}=\e^{\b}\non\\
&&\d\bar{\th}^{\dot{\b}} \equiv -i\left(\e^{\a}Q_{\a}+\bar{\e}^{\dot{\a}}
\bar{Q}_{\dot{\a}}\right)\bar{\th}^{\dot{\b}}=\bar{\e}^{\dot{\b}}
\label{supert}
\ena
Covariant derivatives with respect to (\ref{supert}) are 
$D_A \equiv \left(D_{\a},\bar{D}_{\dot{\a}}, \pa_{\a\dot{\a}}\right)$ 
where
\bea
D_{\a}=\pa_{\a}+\frac{i}{2}\bar{\th}^{\dot{\a}}\pa_{\a\dot{\a}} 
\quad &,& \quad 
\bar{D}_{\dot{\a}} ~=~ \bar{\pa}_{\dot{\a}}+\frac{i}{2}\th^{\a}\pa_{\a\dot{\a}}
\non\\
\pa_{\a\dot{\a}} &=& -i \left\{D_{\a},\bar{D}_{\dot{\a}}\right\} 
\label{covariant}
\eea
Moreover, they satisfy $\{ D_\a , D_\b \} = \{ \bar{D}_\ad , \bar{D}_\bd \}
=0$ and anticommute with the generators of supersymmetry transformations.
 
We can define left and right grassmannian derivatives 
according to the following rules
\bea
&& (\pa_L)_\a \th^\b ~\equiv~ \overrightarrow{\pa}_\a \th^\b ~=~ \d^{~\b}_\a 
~~~~~~\quad ,\quad 
(\bar{\pa}_L)_\ad \bar{\th}^\bd \equiv \vec{\bar{\pa}}_\ad \bar{\th}^\bd 
~=~ \d^{~\bd}_\ad \non\\
&& (\pa_R)_\a \th^\b ~\equiv~ \th^{\b}\overleftarrow{\pa}_{\a} ~=~
-\d_{\a}^{~\b} \quad , \quad
(\bar{\pa}_R)_\ad \thb^\bd ~\equiv~
\bar{\th}^{\dot{\b}}\overleftarrow{\bar{\pa}}_{\dot{\a}} ~=~
-\d_{\dot{\a}}^{~\dot{\b}}
\label{rule1}
\ena
Their action on a generic superfield is defined as 
\beq
\pa_L V ~\equiv~ \overrightarrow{\pa} V \qquad , \qquad 
\pa_R V ~\equiv~ V \overleftarrow{\pa}
\label{rule2}
\eeq
Notice that these definitions hold independently of the nature of the 
superfield V. In particular, in the case of a spinorial superfield $V_\b$
we have $(\pa_R)_A V_\b \equiv V_\b \overleftarrow{\pa}_A$. As a consequence of
the general definitions (\ref{rule1}, \ref{rule2}) we immediately obtain
$\pa_L V = \pa_R V$ for any tensorial superfield, whereas 
$\pa_L V_\b = -\pa_R V_\b$ for any spinorial superfield.  

From the identities
\bea
&& \left(\overrightarrow{\pa}_{\a}\th^{\b}\right)^{\dag} ~=~
-\bar{\th}^{\dot{\b}}\overleftarrow{\bar{\pa}}_{\dot{\a}} \quad , \quad
\left(\vec{\bar{\pa}}_{\dot{\a}}
\bar{\th}^{\dot{\b}}\right)^{\dag} ~=~
-\th^{\b}\overleftarrow{\pa}_{\a} \non\\
&& \left(\overrightarrow{\pa}^{\a}\th^{\b}\right)^{\dag} ~=~
\bar{\th}^{\dot{\b}}\overleftarrow{\bar{\pa}}^{\dot{\a}} ~~~\quad , \quad
\left(\vec{\bar{\pa}}^{\dot{\a}}
\bar{\th}^{\dot{\b}}\right)^{\dag} ~=~
\th^{\b}\overleftarrow{\pa}^{\a} 
\label{rule3}
\eea
the hermitian conjugation rules for left and right derivatives follow
\beq
((\pa_L)_\a)^{\dag} = -(\bar{\pa}_R)_\ad \qquad , \qquad 
(\pa_L^\a)^{\dag} = (\bar{\pa}_R)^\ad
\eeq

We can also introduce left and right bosonic derivatives which are
simply given by
\beq
(\pa_L)_{\a \ad} x^{\b\dot{\b}} ~\equiv~ \pa_{\a \ad} x^{\b \bd}
~=~ \d_{\a}^{~\b}\d_{\dot{\a}}^{~\dot{\b}} \quad , \quad
(\pa_R)_{\a \ad} x^{\b\dot{\b}} ~\equiv~ 
x^{\b\dot{\b}}\overleftarrow{\pa}_{\a\dot{\a}} 
~=~ \d_{\a}^{~\b}\d_{\dot{\a}}^{~\dot{\b}}
\label{rule4}
\eeq
Their action on a superfield $V$ is defined as $\pa_L V \equiv 
\overrightarrow{\pa} V$
and $\pa_R V \equiv V \overleftarrow{\pa} $. 
Therefore, from (\ref{rule4}), it easily follows that  
$(\pa_L)_{\a \ad} V = (\pa_R)_{\a \ad} V$ for {\em any} superfield. 

As a consequence of the previous identities, left and right covariant 
derivatives can be equally defined. 
Left covariant derivatives act on a generic superfield 
from the left as 
\beq
(D_L)_\a V \equiv \overrightarrow{D}_\a V  \qquad , 
\qquad (\bar{D}_L)_\ad V \equiv 
\vec{\bar{D}}_\ad V
\eeq
where $D_\a$ and $\bar{D}_\ad$ are explicitly given in (\ref{covariant}).
Right covariant derivatives are defined as acting from the
right 
\bea
&& (D_R)_\a V ~\equiv~ V \overleftarrow{D}_{\a}
~=~ 
V \left( \overleftarrow{\pa}_{\a}+\frac{i}{2}
\overleftarrow{\pa}_{\a\dot{\a}} \bar{\th}^{\dot{\a}} \right)
\non\\
&& (\bar{D}_R)_\ad V ~\equiv~ V \overleftarrow{\bar{D}}_{\ad}
~=~ 
V \left(
\overleftarrow{\bar{D}}_{\dot{\a}}=\overleftarrow{\bar{\pa}}_{\dot{\a}}
+\frac{i}{2}\overleftarrow{\pa}_{\a\dot{\a}}\th^{\a} \right)
\eea
It is easy to check that $D_L V = D_R V$ on any tensorial superfield,
whereas $D_L V_\b = -D_R V_\b$. Moreover, left and
right derivatives are related by hermitian conjugation $((D_L)_\a)^{\dag} =
-(\bar{D}_R)_\ad$ and $((D_L)^\a)^{\dag} = (\bar{D}_R)^\ad$.

Defining the right momentum operator as 
$(P_R)_{\a \ad} V =- i V \overleftarrow{\pa}_{\a \ad}$, 
it is easy to show that the algebra 
of right derivatives is the standard one  
$\{ D_R , D_R \} = \{ \bar{D}_R , \bar{D}_R \} =0$ and 
$\{ (D_R)_\a , (\bar{D}_R)_\ad \} = (P_R)_{\a \ad}$. 
Moreover
\bea
&&\left[ D_{\a}^R , D_{\b}^L \right] ~=~ 0 ~=~
\left[\overline{D}_{\dot{\a}}^R,\overline{D}_{\dot{\b}}^L\right]  \non\\
&&\left[D_{\a}^R,\overline{D}_{\dot{\a}}^L\right] V 
~=~ (P_R)_{\a \ad} V \non\\
&&\left[D_{\a}^L , \overline{D}_{\dot{\a}}^R \right] V 
~=~ (P_L)_{\a \ad} V 
\label{B}
\ena
for any tensor superfield. 
When the commutators are applied to a spinor $V_\b$,
a minus sign appears on the r.h.s. of the last two identities due 
to the anticommutation of the spinorial derivatives with $V_\b$.

Following the same procedure one can equally define left and right 
supersymmetry generators as $(Q_L)_A V \equiv \vec{Q}_A V$ and 
\bea
&& (Q_R)_\a V ~\equiv~ V \overleftarrow{Q}_{\a} ~=~ 
V \left[ -i \left(\overleftarrow{\pa}_{\a}-\frac{i}{2}
\overleftarrow{\pa}_{\a\dot{\a}} \bar{\th}^{\dot{\a}} \right) \right]
\non\\
&& (\bar{Q}_R)_\ad V ~\equiv~ 
V \overleftarrow{\bar{Q}}_{\dot{\a}} ~=~
V \left[-i \left( \overleftarrow{\bar{\pa}}_{\dot{\a}}
-\frac{i}{2}\overleftarrow{\pa}_{\a\dot{\a}}\th^{\a} \right) \right]
\eea
The algebra of right generators is again given by (\ref{algebra}).  
The algebra of the commutators on a tensor superfield is
\bea
&&\left[ Q_{\a}^R , Q_{\b}^L \right] ~=~ 0 ~=~
\left[\overline{Q}_{\dot{\a}}^R,\overline{Q}_{\dot{\b}}^L\right]  \non\\
&&\left[Q_{\a}^R,\overline{Q}_{\dot{\a}}^L\right] V
~=~ (P_L)_{\a \ad} V \non\\
&&\left[Q_{\a}^L , \overline{Q}_{\dot{\a}}^R \right] V 
~=~ (P_R)_{\a \ad} V 
\ena 
Instead, when the commutators act on spinor objects 
we get a change of sign on the r.h.s. of the last two equalities.  



\newpage

\begin{thebibliography}{100}

%\cite{Seiberg:1999vs}
\bibitem{seibwitt}
N.~Seiberg and E.~Witten,
``String theory and noncommutative geometry,''
JHEP {\bf 9909} (1999) 032
[hep-th/9908142].
%%CITATION = HEP-TH 9908142;%%

%\cite{connes}
\bibitem{connes}
A.~Connes, M.~R.~Douglas and A.~Schwarz,
``Noncommutative geometry and matrix theory: Compactification on tori,''
JHEP {\bf 9802} (1998) 003
[hep-th/9711162].
%%CITATION = HEP-TH 9711162;%%

%\cite{Minwalla:2000px}
\bibitem{minwalla}
S.~Minwalla, M.~Van Raamsdonk and N.~Seiberg,
``Noncommutative perturbative dynamics,''
JHEP {\bf 0002} (2000) 020
[hep-th/9912072].
%%CITATION = HEP-TH 9912072;%%

%\cite{Schwarz:1998qj}
\bibitem{schwarz}
A.~Schwarz,
``Morita equivalence and duality,''
Nucl.\ Phys.\ B {\bf 534} (1998) 720
[hep-th/9805034].
%%CITATION = HEP-TH 9805034;%%

%\cite{Susskind:2001fb}
\bibitem{susskind}
L.~Susskind,
``The quantum Hall fluid and non-commutative Chern Simons theory,''
hep-th/0101029.
%%CITATION = HEP-TH 0101029;%%

%\cite{'tHooft:1985re}
\bibitem{thooft}
G.~'t Hooft,
``On the quantum structure of a black hole,''
Nucl.\ Phys.\ B {\bf 256} (1985) 727;
%%CITATION = NUPHA,B256,727;%%
%\cite{'tHooft:1996tq}
``The scattering matrix approach for the quantum black hole: An overview,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 11} (1996) 4623
[gr-qc/9607022].
%%CITATION = GR-QC 9607022;%%

%\cite{Li:2000ww}
\bibitem{li}
M.~Li,
``Black holes and spacetimes physics in string/M theory,''
hep-th/0006024.
%%CITATION = HEP-TH 0006024;%%

%\cite{Kosinski:1994vc}
\bibitem{kosinski}
P.~Kosi\'{n}ski, J.~Lukierski, P.~Ma\'{s}lanka and J.~Sobczyk,
``Quantum deformation of the Poincare supergroup and kappa
deformed superspace,''
J.\ Phys.\ A {\bf 27} (1994) 6827
[hep-th/9405076];
%%CITATION = HEP-TH 9405076;%%
P.~Kosi\'{n}ski, J.~Lukierski and P.~Ma\'{s}lanka,
``Quantum deformations of space-time SUSY and noncommutative
superfield  theory,''
hep-th/0011053.
%%CITATION = HEP-TH 0011053;%%

%\cite{Ferrara:2000mm}
\bibitem{ferrara}
S.~Ferrara and M.~A.~Lled\'{o},
``Some aspects of deformations of supersymmetric field theories,''
JHEP {\bf 0005} (2000) 008
[hep-th/0002084].
%%CITATION = HEP-TH 0002084;%%

\bibitem{super}{S.J. Gates, Jr., M.T. Grisaru, M. Ro\v cek and W. Siegel,
{\em Superspace}, Benjamin Cummings, (1983) Reading, MA.}

%\cite{Chu:2000ij}
\bibitem{chuzamora}
C.~Chu and F.~Zamora,
``Manifest supersymmetry in non-commutative geometry,''
JHEP {\bf 0002} (2000) 022
[hep-th/9912153].
%%CITATION = HEP-TH 9912153;%%

%\cite{Terashima:2000xq}
\bibitem{all}
S.~Terashima,
``A note on superfields and noncommutative geometry,''
Phys.\ Lett.\ B {\bf 482} (2000) 276
[hep-th/0002119];\\
%%CITATION = HEP-TH 0002119;%%
N.~Grandi, R.~L.~Pakman and F.~A.~Schaposnik,
``Supersymmetric Dirac-Born-Infeld theory in noncommutative space,''
Nucl.\ Phys.\ B {\bf 588} (2000) 508
[hep-th/0004104];\\
%%CITATION = HEP-TH 0004104;%%
D.~Zanon,
``Noncommutative perturbation in superspace,''
Phys.\ Lett.\ B {\bf 504} (2001) 101
[hep-th/0009196];\\
%%CITATION = HEP-TH 0009196;%%
M.~T.~Grisaru and S.~Penati,
``Noncommutative supersymmetric gauge anomaly,''
Phys.\ Lett.\ B {\bf 504} (2001) 89
[hep-th/0010177];\\
%%CITATION = HEP-TH 0010177;%%
A.~Santambrogio and D.~Zanon,
``One-loop four-point function in noncommutative N=4 Yang-Mills theory,''
JHEP {\bf 0101} (2001) 024
[hep-th/0010275];\\
%%CITATION = HEP-TH 0010275;%%
M.~Pernici, A.~Santambrogio and D.~Zanon,
``The one-loop effective action of noncommutative N = 4 super Yang-Mills
is gauge invariant,''
Phys.\ Lett.\ B {\bf 504} (2001) 131
[hep-th/0011140].
%%CITATION = HEP-TH 0011140;%%

%\cite{Kontsevich:1997vb}
\bibitem{kontsevich}
M.~Kontsevich,
``Deformation quantization of Poisson manifolds, I,''
q-alg/9709040.
%%CITATION = Q-ALG 9709040;%%

%\cite{Mehen:2000vs}
\bibitem{dastrivedi}
T.~Mehen and M.~B.~Wise,
``Generalized *-products, Wilson lines and the solution of the
Seiberg-Witten equations,''
JHEP {\bf 0012} (2000) 008
[hep-th/0010204];\\
%%CITATION = HEP-TH 0010204;%%
S.~R.~Das and S.~P.~Trivedi,
``Supergravity couplings to noncommutative branes, open Wilson lines and
generalized star products,''
JHEP {\bf 0102} (2001) 046
[hep-th/0011131].
%%CITATION = HEP-TH 0011131;%%

%\cite{VanProeyen:1999ni}
\bibitem{PVP}
A.~Van Proeyen,
``Tools for supersymmetry,''
hep-th/9910030.
%%CITATION = HEP-TH 9910030;%%

%\cite{Cornalba:2001sm}
\bibitem{cornalba}
L.~Cornalba and R.~Schiappa,
``Nonassociative star product deformations for D-brane worldvolumes in
curved backgrounds,''
hep-th/0101219.
%%CITATION = HEP-TH 0101219;%%

\end{thebibliography}   

\end{document}

