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    %        LaTeX document      October, 2001                           %
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    %        PAPER:                                                      %
    %                 Superfield algorithms for                          %
    %                 topological field theories                         %    
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    %        BY                                                          %
    %                I A Batalin and R Marnelius                         %
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\documentstyle[sprocl,twoside]{article}
\pagestyle{myheadings} \markboth{\small \em \quad I.A.~Batalin 
and R.~Marnelius \hfill}{\hfill \small
\em  Superfield algorithms for topological field theories \quad}

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\begin{document}
\sloppy
\medskip
\rightline{\sl In respectful memory of Professor Michael  Marinov}
\medskip

\title{SUPERFIELD ALGORITHMS FOR TOPOLOGICAL FIELD THEORIES}
 
\author{IGOR BATALIN$^\dagger$   and ROBERT MARNELIUS}

\footnotetext {$^\dagger$ On leave of absence from
P.N.~Lebedev Physical Institute, 117924  Moscow, Russia}

\address{Department of Theoretical Physics,\\ Chalmers University of Technology,
G\"{o}teborg University,\\
S-412 96  G\"{o}teborg, Sweden}
%.\hfill{\em In respectful memory of Professor Michael Marinov}\\ \\
\maketitle

\vspace*{1mm}

\abstracts{ A superfield algorithm for master actions of a class of
  gauge field theories including topological ones in arbitrary
  dimensions is presented generalizing a previous treatment in two
  dimensions.  General forms for master actions in superspace are
  given, and possible theories are determined by means of a ghost
  number prescription and the master equations. The resulting master
  actions determine the original actions together with their gauge
  invariances.  Generalized Poisson sigma models in arbitrary
  dimensions are constructed by means of this algorithm, and other
  simple applications in low dimensions are given including a
  derivation of the non-abelian Chern-Simon model.}  \vspace{0.2cm}
\tableofcontents
\newpage
\section{The basic formulation}
In our paper\,\cite{Sigma} we gave
 a superfield algorithm for a class of master actions in two dimensions  by
means
of which we derived generalized Poisson sigma models. (A superfield form of
the master action for the ordinary Poisson sigma models\,\cite{Ikeda,SS}
was given
by Cattaneo and Felder.\cite{CF})  In this paper we generalize
this algorithm  to arbitrary dimensions. This provides  then  for a general
framework for
 topological field
theories and generalizations. All these models  share the following
properties: i)  The equations of motion are of first order in the derivatives.
ii) They are directly defined in terms of their corresponding master actions. 
iii) There are general simple rules for how the original actions are obtained 
from these master actions.
iv) The  quantum theory of the models  are obtained by a gauge fixing of
the master actions. For other related works see
also Refs.\,\protect\citebk{AKSZ}$-$\protect\citebk{BBD}.
%Refs.\cite{AKSZ,GD,CR,Park,CF1,CF2,BBD}.

 The master actions corresponding to an $n$-dimensional
 field theory will be entirely expressed in terms of fields on a
supermanifold of dimension $(n, n)$, \ie half of the coordinates are
bosonic, even
ones, and half are fermionic, odd ones. To begin with we consider a class of
master actions expressed in terms of pairs of superfields. They are
\be
&&\Sigma[\Phi, \Phi^*]=\int d^{n}u d^{n}\tau \cL_n(u,\tau)\,,
\e{1}
where the Lagrangian densities $\cL_n(u,\tau)$ are given by
\be
&\cL_n(u,\tau)&=\Phi^*_A(u,\tau)D\Phi^A(u,\tau)(-1)^{\ve_A+n}-
S(\Phi(u,\tau),\Phi^*(u,\tau))\,,
\e{2}
where $u^a$, $a=1,2,\ldots,n$ are bosonic coordinates on the base space and
$\tau^a$,
$a=1,2,\ldots,n$, the corresponding fermionic ones.    $D$ is
the odd de Rham differential
\be
&&D\equiv \tau^a\dif_a, \quad \dif_a\equiv{\dif\over\dif u^a}\quad \Rightarrow
\quad D^2=0\,.
\e{3}
Since $\Sigma$ is required to be even and since the measure $d^n\tau$  is
even or odd
depending on whether $n$ is even or odd, it follows that the Grassmann
parity of the
Lagrangian density is $n$, \ie $\ve(\cL_n(u,\tau))=n$. This implies that
$\ve(S)\!=\!n$ and $\ve(\Phi^*_A)\!=\!\ve_A\!+1\!+\!n$, where
$\ve_A\!\equiv\!\ve(\Phi^A)$. (For even $n$ the form of $\cL_n$ in
\r{2} is a generalization of the two dimensional treatment 
in Ref.\,\protect\citebk{Sigma}, and for odd $n$ it is a generalization of the
one-dimensional  expression given in
Refs.\,\protect\citebk{GD}\,and\,\protect\citebk{BBD}. Such superfield
forms for master actions for topological field theories were also
proposed.\cite{AKSZ,GD})

The master actions
\r{1}  are required to satisfy the classical master equation
\be
&&(\Sigma,\Sigma)=0\,,
\e{4}
where the antibracket is defined by
\be
(F, G)\!\!\!\!&\equiv&\!\!\!\!\int\!\!
F\,{\ldel\over\del\Phi^A(u,\tau)}\,(-1)^{n\ve_A}d^{n}u
d^{n}\tau\,{\stackrel{\rightarrow}{\del}\over\del\Phi^*_A(u,\tau)}\,G\nn\\[2mm]
&-&\!\!\!\!(F\leftrightarrow G)(-1)^{\big(\ve(F)+1\big)\big(\ve(G)+1\big)}\,,
\e{5}
which in turn can be defined by\,\cite{EW}
\be
&&(F,G)=\Delta(FG)(-1)^{\ve_F}-(\Delta F)G(-1)^{\ve_F}-F(\Delta G)
\e{102}
in terms of the basic  odd, nilpotent $\Delta$-operator
\be
&&\Delta\equiv\int d^{n}u d^{n}\tau\,
(-1)^{(n+1)\ve_A}\,{\del\over\del\Phi^A(u,\tau)}\ 
{\del\over\del\Phi^*_A(u,\tau)}\ .
\e{6}
This $\Delta$-operator also differentiates the antibracket \r{5} according 
to the Leibnitz rule, \ie
\be
&&\Delta(F,G)=(\Delta F, G)+(F, \Delta G)(-1)^{\ve_F+1}.
\e{601}
 The functional derivatives in
\r{5} and
\r{6} are defined by the properties
\be
&&{\del\over\del\Phi^B(u,\tau)}\,\Phi^A(u',\tau')=
\del^A_B\,\del^n(u-u')\del^n(\tau-\tau')=
\Phi^A(u,\tau)\,
{\stackrel{\leftarrow}{\del}\over\del\Phi^B(u',\tau')}\ ,\nn\\[2mm]
&&\ve\left({\del\over\del\Phi^A}\right)\!=
\ve\bigg({\ldel\over\del\Phi^A}\bigg)\!=\ve_A+n\,,\\[2mm]
&&F\,{\stackrel{\leftarrow}{\del}\over
\del\Phi^A(u,\tau)}\,d^n\tau=(-1)^{\ve_A(\ve(F)+1)}d^n\tau\,
{\del\over\del\Phi^A(u,\tau)}\,F\,,\nn
\e{7}
\\
where the delta-function in the odd coordinates $\tau^a$ satisfies\\
\be
&&\int\!\!f(\tau')\,\del^n(\tau-\tau')\,d^n\tau'=f(\tau)=\!\!\int\!\!
d^n\tau'\,\del^n(\tau'-\tau)f(\tau')\,.
\e{101}

 An explicit calculation of the antibracket of the master actions \r{1} yields
\be
&&\half(\Sigma,\Sigma)=\int d^{n}u
d^{n}\tau\left(D\cL_{n}(u,\tau)+\half(S,S)_n(u,\tau)\right),
\e{11}
where we have introduced the local $n$-bracket
\be
&(f, g)_n\equiv&\!\!\!\!
f{\ldif\over\dif\Phi^A}\,
{\stackrel{\rightarrow}{\dif}\over\dif\Phi^*_A}\,
g-(f\leftrightarrow g)(-1)^{(\ve(f)+1+n)(\ve(g)+1+n)},
\e{13}
where $f$ and $g$ are local functions of $\Phi^A(u,\tau)$ and
$\Phi^*_A(u,\tau)$.
For even $n$ this is an antibracket and for odd $n$ it is a Poisson
bracket. In other
words $\Phi^A$ and
$\Phi_A^*$ are canonical conjugate field variables on an antisymplectic
manifold for
even  $n$ and on a symplectic manifold for odd $n$. The condition  \r{4}
combined
with \r{11} yields
\be
&&\int d^nud^n\tau D\cL_n(u,\tau)=0\,,
\e{141}
which determines the allowed boundary conditions,
and
\be
&&(S,S)_n=0\,.
\e{15}
Furthermore, we have from \r{1} and \r{6}
\be
&&\Delta \Sigma =0\,,
\e{16}
since the $\tau$-part yields a factor zero. (As usual we believe that the
bosonic part can be regularized appropriately.) Therefore, $\Sigma$ also 
satisfies the
quantum master equation
\be
&&\half(\Sigma,\Sigma)=i \hbar \Delta \Sigma\,,
\e{17}
when \r{141} and \r{15} are satisfied,
which means that no quantum corrections of the measure in the path integral are
required for these models.

If one treats the master action $\Sigma$ in \r{1} as an ordinary action,
then the
equations of motion are
\be
&&D\Phi^A=(S,\Phi^A)_n\,,\quad D\Phi^*_A=(S,\Phi^*_A)_n\,,
\e{18}
the consistency of which again requires \r{15}. Notice that the equation of
motion
for $S$ then becomes $DS=0$.
 Notice also the relations 
\be
&&\left(\Sigma, \Phi^A\right)=(-1)^n\left(D\Phi^A-(S, \Phi^A)_n\right),\nn\\
&&\big(\Sigma, \Phi^*_A\big)=(-1)^n\big(D\Phi^*_A-(S, \Phi^*_A)_n\big)\,.
\e{181}

The superfields above can be Taylor expanded in the odd $\tau$-coordinates
in such a
fashion that the coefficients are fields and antifields in the
ordinary sense
on the base manifold. Or in other words, such that the antibracket \r{5} and
the $\Delta$-operator \r{6} have their conventional forms:
\be
(F, G)\!\!\!\!&=&\!\!\!\!
\sum_r\!\int
F\,{\ldel\over\del\Phi^{rA}(u)}\,d^nu\,
{\stackrel{\rightarrow}{\del}\over\del\Phi^*_{rA}(u)}\,G -
(F\!\leftrightarrow\!G)(-1)^{\big(\ve(F)+1\big)\big(\ve(G)+1\big)}\,,\nn\\[2mm]
\Delta\!\!\!\!&\equiv&\!\!\!\!
\sum_r\!\int\!\!d^nu\,(-1)^{\ve_{rA}}\,{\del\over\del\Phi^{rA}(u)}\,
{\del\over\del\Phi^*_{rA}(u)}\ ,\qquad\ve_{rA}\!\equiv\ve(\Phi^{rA})\,,
\e{20}
where $\Phi^{rA}(u)$ and $\Phi^*_{rA}(u)$ are the coefficient fields ($r$
denotes
antisymmetric $u$-indices). The antibracket in \r{20} must yield
\be
&&(\Phi^A(u,\tau),
\Phi^*_B(u',\tau'))=(-1)^{n\ve_A}\del^A_B\del^{n}(u-u')\del^{n}(\tau-\tau')\,,
\e{21}
which trivially follows from Eq.\,\r{5}. An explicit construction is given
in Appendix B.

So far we have given superfield actions in arbitrary dimensions which
under certain conditions satisfy the master equations. In the
following we will always require $\Sigma$ to actually satisfy the
master equation, \ie we require the boundary conditions to be
consistent with \r{141} and the local equation \r{15} to be satisfied.
In order to determine the class of gauge field theories which have
such master actions we need also to prescribe ghost numbers to the
field. We generalize then the prescription given in
Ref.\,\protect\citebk{Sigma} (such ghost numbers were also considered in
Refs.\,\protect\citebk{CF},\,\protect\citebk{GD},\,\protect\citebk{Park}
and\,\protect\citebk{BBD}).
We choose the odd coordinates $\tau^a$ to have ghost number plus one,
which implies that $D$ in \r{3} has ghost number plus one.  Since we
require the master action $\Sigma$ to have ghost number zero and since
the measure $d^n\tau$ has ghost number $-n$ the Lagrangian density
$\cL_n$ in \r{2} must have ghost number $n$, \ie
\be
gh(\Sigma)=0\,,\quad gh(d^n\tau)=-\,n\,,\qquad\Rightarrow\qquad(\cL_n)=n\,.
\e{3311}
The form \r{2} of $\cL_n$  leads us to the following
general rule for the superfields and the local function
$S$:
\be
gh(\Phi^A)+gh(\Phi^*_A)=n-1, \quad gh(S)=n\,.
\e{34}
We assume that $S$ is given by a power expansion in the superfields 
$\Phi^A$ and
$\Phi^*_A$, in which case this ghost number prescription will restrict the
possible terms in $S$. In such an analysis it is convenient to use the
following convention:
\be
gh(\Phi^*_A)\geq gh(\Phi^A)\,,
\e{36}
\pagebreak[3]
which in itself does not impose any restriction.
Notice that $\Phi^A$ and
$\Phi^*_A$  can only have equal ghost number in odd dimensions. (More
precisely for
$n=2m+1$ we can have the ghost numbers
$gh(\Phi^A)=gh(\Phi^*_A)=m$.)
Since the local bracket \r{13} satisfies
\be
\left(\Phi^A(u,\tau),\Phi^*_B(u,\tau)\right)_n=\del^A_B\,,
\e{361}
it follows from the ghost number prescription \r{34} that the local bracket
$(\;,\;)_n$ in itself carries ghost number $1-n$. From \r{21} and the fact that
$\del^n(\tau-\tau')$ carries ghost number $n$, it follows that the
antibracket \r{5} or, equivalently, \r{20} carries ghost number plus one
which is the standard ghost number prescription for the antibracket in the
field-antifield formalism.

We have a consistent master action $\Sigma$ given by \r{1} if
$S$ satisfies $gh(S)=n$ and  $(S,S)_n=0$  and the set
of superfields satisfies \r{34} and  the boundary conditions compatible 
with \r{141}.
%When we find $S$ that satisfies $gh(S)=n$ and  $(S,S)_n=0$ for a given set
%of superfields, which satisfy \r{34} and  boundary conditions consistent with
%\r{141}, then we have a consistent master action $\Sigma$ given by \r{1}.
From such a  master action we can then extract the original model according
to the following rules. First, remove the integration measure.
Then perform the following replacements:
\be
D&\!\longrightarrow\!&{\rm exterior}\;{\rm derivative}\;d\,,\nn\\
\Phi^A\!,\ gh(\Phi^A)=k\geq0\,,&\!\longrightarrow\!&k{\rm -form}\;{\rm
field}\;\phi^A,\;\ve(\phi^A)=\ve_A+k\,,\nn\\
\Phi^*_A,\,gh(\Phi^*_A)\!=(n\!-\!1\!-\!k)\geq0\,,&\!
\longrightarrow\!&\!(n\!-\!1\!-\!k)
{\rm -form}\;{\rm field}\;\phi^*_A,\ \ve(\phi^*_A)=\ve_A\!+\!k\,,\nn\\
\Phi^A\!,\,gh(\Phi^A)<0\,,&\!\longrightarrow\!&0\nn\\
&{\rm or}&\nn\\
%\qquad\qquad
\Phi^*_A,\;gh(\Phi^*_A)<0\,,&\!\longrightarrow\!&0\,,\nn\\
{\rm ordinary}\;{\rm multiplication}&\!\longrightarrow\!&{\rm wedge}\;{\rm
products},
\e{37}
where the form fields, $\phi^A$ and $\phi^*_A$, are fields on the
$n$-dimensional $u$-space. These rules are easily extracted from the
fact that the original fields, $\phi^A(u)$ and $\phi^*_A(u)$, are
component fields with ghost number zero of the superfields,
$\Phi^A(u,\tau)$ and $\Phi^*_A(u,\tau)$. (Signs depend on how the
coefficient fields are defined precisely.)



The superfield master actions determine also the gauge invariance of
the original action. The appropriate gauge transformations can be
obtained by following steps. First identify the components of the
superfields that are antifields to the original fields. These
antifields have ghost number minus one and if $\Phi^A$ contains the
field then $\Phi^*_A$ contains the corresponding antifield or vice
versa.  Identify then the terms in the master Lagrangian $\int
d^n\tau\cL_n$ which are linear in these antifields. The terms in the
coefficients of these antifields, which contain no antifields,
represent then the gauge transformations of the field. The ghost
fields with ghost number plus one are to be interpreted then as gauge
parameters.

%\setcounter{equation}{0}
\section{Master actions in terms of general superfields}
We have generalized\,\cite{Sigma} the two-dimensional treatment along the
lines of the generalized antisymplectic formulation.\cite{BT} Here we give
the corresponding treatment in arbitrary dimensions. Let the superfields
$Z^I(u,\tau)$, $\ve(Z^I)\equiv\ve_I$, represent arbitrary coordinates on
an (anti)symplectic manifold. The local brackets \r{13} are then defined by
\be
&&(f, g)_n\equiv
f(Z(u,\tau))\,{\ldif\over\dif Z^I}E^{IK}(Z(u,\tau))
{\stackrel{\rightarrow}{\dif}\over\dif Z^K}
\,g\,(Z(u,\tau))\,,
\e{38}
which implies
\be
&&(Z^I(u,\tau), Z^K(u,\tau))_n=E^{IK}(Z(u,\tau))\,.
\e{39}
The functions $E^{IK}$ have the properties
\be
\ve\,\left(E^{IK}\right)\!\!\!&=&\!\!\!\ve_I+\ve_K+1+n,\nn\\[2mm]
E^{KI}\!\!\!\!\!&=&\!\!\!-\,E^{IK}(-1)^{(\ve_I+1+n)(\ve_K+1+n)}\,.
\e{40}
These properties follow from the requirements
\be
\ve\,\big((f,g)_n\big)\!\!\!\!&=&\!\!\!\ve_f+\ve_g+1+n\,,\nn\\
(f,g)_n\!\!\!\!\!&=&\!\!\!\!-\,(g,f)_n\,(-1)^{(\ve_f+1+n)(\ve_g+1+n)}\,.
\e{41}
The Jacobi identities of the bracket \r{38}, namely
\be
&&((f,g)_n,h)_n\,(-1)^{(\ve_f+1+n)(\ve_h+1+n)}+cycle\,(f,g,h)=0\,,
\e{411}
require in turn
\be
&&E^{IL}\,\dif_LE^{JK}(-1)^{(\ve_I+1+n)(\ve_K+1+n)}+cycle\,(I,J,K)=0\,.
\e{42}
We assume that $Z^I$ span the bracket \r{38} which implies that $E^{IK}$ is
invertible. The inverse, $E_{IK}$, is defined by
\be
&&E_{IL}E^{LK}=\del^K_I=E^{KL}E_{LI}\,.
\e{43}
It satisfies
\be
\ve(E_{IK})\!\!\!\!\!&&\!\!\!\!\!=\ve_I+\ve_K+1+n\,,\qquad\qquad
E_{KI}=-E_{IK}(-1)^{(\ve_I+n)(\ve_K+n)+n}\,,\nn\\[2mm]
&&\dif_IE_{JK}(Z)(-1)^{(\ve_I+n)\ve_K}+cycle(I,J,K)=0\,.
\e{44}
\pagebreak[3]
The last relation follows  from \r{43} and the Jacobi identities \r{42}.

We define the generalized antibracket by
\be
&&\hspace{-8mm}(F, G)\equiv
%\nn
%\\&&
F\!\int \!\! {\ldel\over\del Z^I(u,\tau)}\,d^{n}u
d^{n}\tau
\left(Z^I(u,\tau), Z^K(u',\tau')\right)d^{n}u'
d^{n}\tau'\,{\stackrel{\rightarrow}{\del}\over\del Z^K(u',\tau')}\,
G\,,\nn\\[1.8mm]
&&\hspace{-8mm}{\rm where}\e{45}
\vspace*{-8mm}
%where
\be
\hspace{-10mm}
\left(Z^I(u,\tau), Z^K(u',\tau')\right)\!\!\!\!\!&=&\!\!\!\!
E^{IK}(Z(u,\tau))(-1)^{n\ve_K}\del^n(u-u')\,\del^n(\tau-\tau')\nn\\[1.5mm]
\!\!\!&=&\!\!\!
\del^n(u-u')\,\del^n(\tau-\tau')(-1)^{n\ve_I}E^{IK}(Z(u,\tau))\,.
\e{46}
Here the Jacobi identities are
\be
&&\left(Z^I(u,\tau), (Z^J(u',\tau'),
Z^K(u'',\tau''))\right)(-1)^{(\ve_I+1)(\ve_K+1)}\nn\\[1.5mm]
&&+\,cycle\,(I,u,\tau;J,u',\tau';K,u'',\tau'')=0\,,
\e{462}
which are satisfied due to \r{42}. In fact, the properties of $E^{IK}$
imply that
the antibracket \r{45}, \r{46}
satisfies all required properties. Inserting \r{46} into \r{45} yields the
expression
\be
&&(F, G)\equiv\!
F\!\int\!{\ldel\over\del Z^I(u,\tau)}\,d^{n}u\,d^{n}\tau\,
E^{IK}(-1)^{n\ve_K}{\stackrel{\rightarrow}{\del}\over\del Z^K(u,\tau)}\,G\,.
\e{463}
This antibracket follows also from the relation \r{102} and the
$\Delta$-operator
\be
\hspace{-4mm}\Delta\equiv
%\nn\\&&
\half\!\int\!\!d^nu\,d^n\tau\,
(-1)^{\ve_I}\rho^{-1}{{\del}\over\del Z^I(u,\tau)}\,\rho
E^{IK}(Z(u,\tau))(-1)^{n\ve_K}{{\del}\over\del Z^K(u,\tau)}\,,
%\nn\\[1mm]
\e{461}
where $\rho(Z(u,\tau))$ is a measure density on the field-antifield space. This
$\Delta$-operator differentiates the antibracket \r{463} according to the rules
\r{601}.

A master action $\Sigma$ in terms of the superfields $Z^I(u,\tau)$ may have,
\eg, the form \r{1} with a Lagrangian density given by
\be
\cL_n(u,\tau)=V_I(Z(u,\tau))D Z^I(u,\tau)(-1)^{\ve_I+n}\!\!-S(Z(u,\tau))\,,
\e{47}
where the
 (anti)symplectic potential $V_I(Z(u,\tau))$ has the Grassmann parity
$\ve_I\!+\!n\!+\!1$.
Here we obtain the left functional derivative of $\Sigma$ in the form
\be
{\del\Sigma\over\del Z^I}=(-1)^{n\ve_I}
\left(E_{IK}DZ^K(-1)^{\ve_K+n}-\dif_IS\right),
\e{471}
where 
$E_{IK}$ is defined by
\be
&&E_{IK}(Z)=\dif_IV_K(Z)-\dif_KV_I(Z)(-1)^{(\ve_I+n)(\ve_K+n)+n}\ .
\e{49}
The equations of motion are therefore
\be
&&DZ^I=(S, Z^I)_n\,.
\e{48}
Then, consistency requires $(S,S)_n=0$ again.
A still more general form for the master action $\Sigma$ is obtained from the
Lagrangian density
\be
%&
\cL_n(u,\tau)=Z^K(u,\tau)\,\overline{E}_{KI}(Z(u,\tau))\,DZ^I(u,\tau)
(-1)^{\ve_I+n}\!-S(Z(u,\tau))\,,
\e{491}
where \vspace{-4mm}
\be
&&\overline{E}_{KI}(Z)\equiv (Z^J\dif_J+2)^{-1}E_{KI}(Z)=\!
\int_0^1\!\!\!d\al\,\al E_{KI}(\al Z)\,.
\e{492}
(Such an expression  was first given in the symplectic case in
Ref.\,\protect\citebk{BF}).
Notice also the general form of the expression \r{181}:
\be
&&(\Sigma, Z^I)=(-1)^n\left(DZ^I-(S,Z^I)_n\right).
\e{4921}

When calculating $(\Sigma,\Sigma)$ in terms of the antibracket \r{45} or
\r{463},
where
$\Sigma$ is expressed in terms of \r{47} or \r{491}, we again find the relation
\r{11}. This means that the master equation
$(\Sigma,\Sigma)=0$  also here requires  the conditions \r{141}, \ie $\int
d^nud^n\tau D\cL_n=0$, which determines the allowed boundary conditions, and
$(S,S)_n=0$ in terms of the bracket \r{38} which together with the ghost
number prescription determine the allowed form of
$S$.

Here the ghost number prescriptions for Lagrangians \r{47} and \r{491} are
\be
&&gh(S)=n, \quad gh(V_I)+gh(Z^I)=n-1,\; {\rm each}\; I,
\e{493}
\vspace{-7mm}
\begin{center} and \end{center}\vspace{-2mm}
\be
gh(S)=n\,,\ gh(Z^K)+gh(Z^I)+gh(\overline{E}_{IK})=n-1\quad
{\rm (for~each~}I{\rm ~and~}K)\,,
\e{494}
%for the Lagrangian \r{491}
respectively. The relation \r{494} together with \r{492}, or
\r{493} and
\r{49} imply
\be
&&gh(E_{IK}(Z))=n-1-gh(Z^I)-gh(Z^K),\nn\\&&
gh(E^{IK}(Z))=1-n+gh(Z^I)+gh(Z^K),
\e{495}
where the last relation follows from \r{43}. The last equality implies then
that the
local bracket $(\;,\;)_n$ carries the extra ghost number $1-n$
(see Eq.\,\r{39}), exactly
what we had in the previous section. Only for
$n=1$ do we have the  standard Poisson bracket.

%\setcounter{equation}{0}
\section{Further generalizations}
1. As in Ref.\,\protect\citebk{Sigma}, we can choose arbitrary coordinates
$u^a$ on the surface. For the measure
\be
&&d^nu{\left(\det h^b_a(u)\right)}\!^{-1}\,,
\e{50}
we have the nilpotent $D$-operator
\be
D=\tau^aT_a+\half\tau^b\tau^aU^c_{ab}(u){\dif\over\dif\tau^c}\ ,
\e{51}
where
\be
T_a\!\!\!\!&\equiv&\!\!\!\!
h^b_a(u){\dif\over\dif u^b}\,, \nn\\
U^c_{ab}(u)\!\!\!\!&\equiv&\!\!\!\!-h^f_a(u)h^d_b(u){\dif\over\dif
u^f}\left(h^{-1}\right)^c_d\!(u)-(a\leftrightarrow b)\,.
\e{52}
Notice that
\be
&&D^2=0\;\Rightarrow\;[T_a, T_b]=U^c_{ab}(u)T_c\,.
\e{53}\\
2. If the original field theory is a superfield theory on a space with
coordinates
$u^a$ having Grassmann parities $\ve_a$, then the $\tau^a$-coordinates have
Grassmann
parities
$\ve_a+1$. In this case we must assume that the superfields $\Phi^A(u,\tau)$
are still possible to expand as a power series in $\tau^a$ which then is
an infinite expansion for the bosonic $\tau^a$-coordinates. (The fact that
this implies an infinite number of component fields with arbitrarily low ghost
numbers suggests that infinite reducibility is a generic feature in the
superfield case.) If the original base manifold has dimension
$(n,m)$ then the supermanifold has dimension  $(n+m,n+m)$ and we get a
subdivision into the two cases $n+m$ odd or even. This means that the
dimension of the base space still determines the odd and even cases.\\\\
3. The local functions $S$ can also have explicit $\tau$-dependence, since
the master actions above  still satisfy the master equations provided $S$
satisfies \r{15}. However, such master actions are not of the same geometric
nature as the previous ones since they do not lead to topological field
theories due to the terms with explicit $\tau$-dependence. (This
generalization was first considered in the one-dimensional treatments 
in Refs.\,\protect\citebk{GD}, \protect\citebk{BBD}.)\\\\
4. A natural extension of the formalism in Sec.\,2 is to let $E^{IK}(Z)$ be
singular. It may be, \eg, a Dirac bracket in which case we have 
\be
E^{IK}_{({\cD})}\equiv(Z^I, Z^K)_{n(\cD)}\equiv(Z^I, Z^K)_n-
(Z^I, \Theta^{\mu}(Z))_nC_{\mu\nu}(\Theta^{\nu}(Z), Z^K)_n\,,
\e{5000}
where $\Theta^{\mu}(Z)$ are constraint variables such that
$C^{\mu\nu}\equiv(\Theta^{\mu}, \Theta^{\nu})_n$ is invertible with the
inverse $C_{\mu\nu}$. Then, according to \r{46}, the basic Dirac antibracket is
given by
\be
\left(Z^I(u,\tau), Z^K(u',\tau')\right)_{(\cD)}=
E^{IK}_{(\cD)}(Z(u,\tau))\,(-1)^{n\ve_K}\del^n(u-u')\,\del^n(\tau-\tau')\,.
\e{5100}
The local action $S$ should be replaced by
\be&&S(Z(u,\tau))\;\longrightarrow\;S(Z(u,\tau))+\Pi_{\mu}(u,\tau)\,
\Theta^{\mu}(Z(u,\tau))\,,
\e{5111}
where $\Pi_{\mu}$ is a Lagrange multiplier superfield.



\section{Generalized Poisson sigma models in any dimension}
Consider dimension $n$ and consider first superfield pairs $X^i$ and $X^*_i$
with the properties \vspace{-2mm}
\be
\ve(X^*_i)\!\!\!\!&=&\!\!\!\!\ve_i+1+n\,,\qquad\qquad\ve_i\equiv\ve(X^i)\,,
\nn\\
gh(X^*_i)\!\!\!\!&=&\!\!\!\!n-1-gh(X^i)\,,
\e{54}
which are required by our general rules in Sec.\,1. Let us now consider the
local function \vspace{-2mm}
\be
&&S=\half X^*_jX^*_i\om^{ij}(X)(-1)^{\ve_j+n}.
\e{55}
From the general prescription \vspace{-2mm}
\be
\ve(S)=n\,,\quad gh(S)=n\,,
\e{56}
it follows that the functions $\om^{ij}(X)$ must satisfy the properties
\be
\ve(\om^{ij})\!\!\!&=&\!\!\!\!n+\ve_i+\ve_j\,,\nn\\
gh(\om^{ij})\!\!\!\!&=&\!\!\!\!2-n+gh(X^i)+gh(X^j)\,.
\e{57}
The expression \r{55} implies furthermore that
\be
\om^{ij}=-\,\om^{ji}(-1)^{(\ve_i+n)(\ve_j+n)}.
\e{58}
The master equation $(S,S)_n=0$ requires then
\be
&&\om^{il}\dif_l\om^{jk}(-1)^{(\ve_i+n)(\ve_k+n)}+cycle\,(i,j,k)=0\,.
\e{59}
From these results it follows that $\om^{ij}(X)$ has exactly the same
properties as
the local bracket
\r{38} for $n-1$. Therefore,  we can make the identification
\be
\om^{ij}(X)=(X^i, X^j)_{n-1}\,,
\e{591}
where $(\;,\;)_0$ can be identified with the conventional antibracket in the
field-antifield formalism. For even $n$ $\om^{ij}$ is a Poisson bracket and
for odd $n$ $\om^{ij}$ is an antibracket. But only for $n=1,2$ do these
brackets have the conventional ghost number
prescriptions.\footnote{\,$(\;,\;)_1$ is the conventional Poisson
bracket.} The boundary conditions must be consistent with \r{141}.

When $\om^{ij}(X)$ satisfies the above properties and when the boundary
conditions are
consistent with \r{141} then the action
\be
&&\Sigma[X,X^*]=\int d^nu d^n\tau \left( X^*_iDX^i(-1)^{\ve_i+n}-S\right)
\e{592}
satisfies the master equation $(\Sigma, \Sigma)=0$.
 However, for this case the original models can only be written down after
we have
given a more explicit form for
$\om^{ij}(X)$. The only exception is for $n=2$ and $gh(X^i)=0$ in which case,
 according to the rules \r{37}, we
directly obtain
\be
A=\!\int\!\!\Big(x^*_i\wedge
dx^i-\half x^*_j\wedge x^*_i\om^{ij}(x)\Big)\,,
\e{65}
 where $x^*_i$ and
$x^i$ are one-form and zero-form fields respectively. (The Grassmann parities
are $\ve(x^*_i)=\ve(x^i)\equiv\ve_i$~.) This is just the
well-known Poisson sigma 
model\,\cite{Ikeda,SS} for which Cattaneo and Felder also gave the
superfield master action\,\cite{CF} given here for $\ve(x^i)=0\,$. One can
easily check that the boundary conditions\,\cite{CF} are consistent
with \r{141}.

In Ref.\,\protect\citebk{Sigma} generalized Poisson sigma models for $n\!=\!2$
were constructed by means of the algorithm presented here. This construction
can also be generalized to arbitrary $n$. We consider then the following
expression for $S$: 
\be
S=\half X^*_jX^*_i\om^{ij}(X)(-1)^{\ve_j+n}+\Lambda^*_{\al}\theta^{\al}(X)\,,
\e{651}
where we have introduced new superfield pairs
$\Lambda^*_{\al}$,
$\Lambda^{\al}$ with the properties
\be
\ve(\Lambda^*_{\al})\!\!\!\!&=&\!\!\!\!
\ve_{\al}+n\,,\hspace{1.2cm}
\ve(\Lambda^{\al})=\ve_{\al}+1\,,\qquad\qquad
\ve_{\al}\equiv \ve(\theta^{\al})\,,\nn\\
gh(\Lambda^*_{\al})\!\!\!\!&=&\!\!\!\!
n-gh(\theta^{\al})\,,\quad
gh(\Lambda^{\al})=gh(\theta^{\al})-1\,.
\e{6511}
 Apart from \r{59}, the local master equation $(S,S)_n=0$ requires now that
\be
&&\theta^{\al}(X)\ldif_j\om^{ji}(X)=0\,,
\e{652}
which means that $\om^{ji}(X)$  can be interpreted as a Dirac bracket of
the $n-1$
type, \ie a bracket $(\cdot,\cdot)_{n-1}$ satisfying
$(\cdot,\theta^{\al}(X))_{n-1}=0$.

The expression \r{651} is
not the general expression of $S$ in terms of these fields. We can also add
the terms
\be
(-1)^{\ve_j+n+(\ve_i+n)(\ve_k+n)}\,{1\over6}
\!\!\!\!\!\!&&\!\!\!\!\!\!
X^*_iX^*_jX^*_k\om^{kji}_{\al}(X)\La^{\al}\nn\\&&\!\!\!\!+\,
(-1)^{\ve_i+n}X^*_i\La^*_{\al}\om^{\al i}_{\beta}(X)\La^{\beta}+\ldots\,,
\e{653}
where \vspace{-3mm}
\be
\ve(\om^{kji}_{\al}(X))
\!\!\!\!&=&\!\!\!\!
\ve_i+\ve_j+\ve_k+\ve_{\al}\,,\nn\\
\ve(\om^{\al i}_{\beta}(X))
\!\!\!\!&=&\!\!\!\!
\ve_i+\ve_{\al}+\ve_{\beta}+n\,,\\
gh(\om^{kji}_{\al}(X))
\!\!\!\!&=&\!\!\!\!
gh(X^i)+gh(X^j)+gh(X^k)-gh(\theta^{\al})+4-2n\,,\nn\\
gh(\om^{\al i}_{\beta}(X))
\!\!\!\!&=&\!\!\!\!
gh(X^i)+gh(\theta^{\al})-gh(\theta^{\beta})+2-n\,,\nn
\e{661}
and where $\om^{kji}_{\al}$ is totally antisymmetric in $i$, $j$, $k$ with
Grassmann parity \hfill
$\ve_{ijk}=(\ve_i+n)(\ve_j+n)+(\ve_j+n)(\ve_k+n)+(\ve_k+n)(\ve_i+n)$. 
In this case the master equations yield a weak form of the Jacobi
identities \r{59}
and the degeneracy condition \r{652}, which means that $\om^{ij}(X)$ now is
a weak Dirac bracket of the $n\!-\!1$ type. For all these forms of $S$ the
master action is given by
\be
\!\!\!\!\Sigma[X,X^*\!,\Lambda,\Lambda^*]\!=\!\!\int\!\!d^nud^n\tau
\Big(X^*_iDX^i(-1)^{\ve_i+n}\!+\Lambda^*_{\al}
D\Lambda^{\al}(-1)^{\ve_{\al}+n+1}\!-S\Big).
\e{6611}

We can also add further superfield pairs $\Xi^{\al_k}$, $\Xi^*_{\al_k}$,
$k=1,\ldots,L$, which enter $S$ to leading order in the form 
\[
\sum_{k=0}^{L-1}\Xi^*_{\al_{k+1}}Z^{\al_{k+1}}_{\al_k}(X)\,\Xi^{\al_k}\,,\qquad
(k\geq0\,,\quad\al_0\equiv\al\,,\quad\Xi^{\al_0}\equiv\Lambda^{\al})\,,
\]
where
\be
\ve(Z^{\al_{k+1}}_{\al_k})\!\!\!\!&=&\!\!\!\!\ve_{\al_{k+1}}+\ve_{\al_k}\,,
\nn\\
\ve(\Xi^{\al_k})\!\!\!\!&\equiv&\!\!\!\!\ve_{\al_k}+k+1\,,\\
gh(Z^{\al_{k+1}}_{\al_k})
\!\!\!\!&=&\!\!\!\!
gh(\Xi^{\al_{k+1}})-gh(\Xi^{\al_k})+1\,.\nn
\e{662}
Here the master action is
\be
%\hspace{-1.2cm}
&&
%\hspace{-1.2cm}
\Sigma\,\big[X,X^*,\Xi,\Xi^*\big]=\\
%\hspace{-4mm}
&=&\hspace{-4mm}
%\!=\!\!
\int\!\!d^nud^n\tau
%\!\!\!\!\!&&\!\!\!\!\!
%\times\nn\\
\Big(X^*_iDX^i(-1)^{\ve_i+n}\!+\sum_{k=0}^{L}\Xi^*_{\al_{k}}
D\,\Xi^{\al_k}(-1)^{\ve_{\al_k}+k+1+n}\!-S\Big)\,,\nn
\e{6621}
and the master equations tell us that $\theta^{\al}$ is reducible to stage
$L$, which means $Z^{\al_1}_{\al}\theta^{\al}=0\,,\,
Z^{\al_3}_{\al_2}Z^{\al_2}_{\al_1}\approx 0\,,\,\ldots,\,
Z^{\al_L}_{\al_{L-1}}Z^{\al_{L-1}}_{\al_{L-2}}\approx 0$\,.

 The original model
and its properties of all these modified forms of $S$ can only be written
down when we
have given a more explicit form for the ghost dependent functions. However,
for $n=2$
and
$gh(X^i)=0$
 we
 obtain,
 according to the rules
\r{37}, for all the above cases the original action,
\be
A\!=\!\!\int\!\!\Big(x^*_i\wedge dx^i-\half\,x^*_j\wedge x^*_i\om^{ij}(x)-
\la_{\al}\theta^{\al}(x)\Big)\,,
\e{66}\\[-3mm]
where $\la_{\al}$ is a Lagrange multiplier $2$-form field.

Still more generalized forms of Poisson sigma models might be possible to
derive
if we allow $S$ to have explicit $\tau$-dependence which is possible
according to the
generalization 3 in Sec.\,3.


\section*{Acknowledgements}
\addcontentsline{toc}{section}{\numberline{}Acknowledgements}
I.A.B. would like to thank Lars Brink for
his very warm hospitality at the
Department of Theoretical Physics, Chalmers
and G\"oteborg University.
 The work of I.A.B. is supported by the grants 99-01-00980 and
99-02-17916 from Russian foundation for basic researches and by the
President grant
00-15-96566 for supporting leading scientific schools. This work is partially
supported by the grant INTAS 00-00262.

%\appendix
\section*{Appendix A: Further applications}
\addcontentsline{toc}{section}{\numberline{}Appendix A: Further applications}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
%\section{Further applications}
\subsection*{Models in $n=1$}
\addcontentsline{toc}{subsection}{\numberline{A.1}Models in $n=1$}
Consider first superfield pairs $X^i$ and $X^*_i$ with
ghost number zero and arbitrary Grassmann parities $\ve(X^*_i)=\ve(X^i)$.
If these are the only superfields  then $S=0$,  since
$S$ must have ghost number one. The original model is trivial then and is
of the form
$A=\int x^*_idx^i$. If we also have the superfield pairs
$\La^*_{\al}$ and $\La^{\al}$ with ghost numbers one and minus one
respectively, then
$S$ has the general form ($\ve(\La^*_{\al})=\ve(\Lambda^{\al})=\ve_{\al}+1$,
$\ve_{\al}\equiv\ve(\theta^{\al})$)
\vspace{-1mm}
\be
S=\La^*_{\al}\theta^{\al}(X,
X^*)+\half\Lambda^*_{\al}\Lambda^*_{\beta}U^{\beta\al}_{\ga}(X,X^*)
\Lambda^{\ga}(-1)^{\ve_{\al}}+\ldots\ ,
\e{60}\\[-4mm]
\pagebreak[3]where the dotted terms are determined by the condition
$(S,S)_1=0$. Notice that $S$ is odd and that $(\;,\;)_1$ is a conventional
Poisson bracket. In fact,
\be
(S,S)_1\!=0\quad\Rightarrow\quad(\theta^{\al}\!,\theta^{\beta})_1\!=
U^{\al\beta}_{\ga}\theta^{\ga}\,.
\e{61}
Thus, the master action 
\be
\Sigma\big[X,X^*,\Lambda,\Lambda^*\big]\!=\!\!\int\!\!du\,d\tau
\left(X^*_iDX^i(-1)^{\ve_i+1}\!+\Lambda^*_{\al}D\Lambda^{\al}
(-1)^{\ve_{\al}}\!-S\right),
\e{611}
where $S$ is defined by \r{60},  satisfies the master
equations provided
the constraint variables $\theta^{\al}$ are in involutions. We can also add
further
pairs
$\Xi^*_{\al_k}$, $\Xi^{\al_k}$ with ghost numbers $k>1$ and $-k$. The
resulting
$S$ satisfying $(S,S)_1=0$ will then imply that the constraint variables
$\theta^{\al}$ are reducible (linearly dependent) up to a certain stage.
Thus, $S$
may attain the general form of a BFV-BRST charge for a constraint theory
where the
constraints are in arbitrary involutions on the phase space spanned by the
canonical
coordinates
$X^i$ and
$X^*_i$.
$\La^*_{\al}$ are ghosts and $\Xi^*_{\al_k}$ ghost for ghosts.
The corresponding BRST
charge for the original model is the $\tau=0$ component of the $S(u,\tau)$
in \r{60}.
The boundary condition
\r{141} requires the conservation of this charge, \ie $S(u_2,0)-S(u_1,0)=0$
where
$u_1$ and
$u_2$ are the limits of the $u$-integration in $\Sigma$.
 The original action is obtained from the corresponding
master action using the rules \r{37}. We find for all the above cases the
original
action
\be
A=\!\!\int\!\!\left(x^*_idx^i-\la_{\al}\theta^{\al}(x, x^*)\right),
\e{62}
where $\la_{\al}$ is a Lagrange multiplier one-form field. This action  is
defined on the phase space where $x^*_i$ and $x^i$ are canonical conjugate
variables ($x^*_i$ are conjugate momenta to $x^i$).
 It is gauge invariant
under the gauge transformations
\be
&&\del x^i=(x^i, \theta^{\al})_1\beta_{\al},\quad \del
x^*_i=(x^*_i, \theta^{\al})_1\beta_{\al},\nn\\&&
\del\la_{\al}=d\beta_{\al}(-1)^{\ve_{\al}}-
\beta_{\mu}\la_{\nu}U^{\nu\mu}_{\al}(-1)^{\ve_{\mu}},
\e{63}
where $\beta^{\al}$ are the gauge parameters.
 The theory \r{62} is an arbitrary constraint theory with zero
Hamiltonian, which implies that the theory is reparametrization invariant.
In fact,
any one-dimensional Hamiltonian model can be cast into this form, since 
any such
theory can be cast into a reparametrization invariant form. However, a nonzero
Hamiltonian
$H(X, X^*)$ is possible to introduce if we  allow $S$ to have an explicit
$\tau$-dependence  (generalization 3 in Sec.\,3, see 
also Ref.\,\protect\citebk{GD}). Condition \r{15}  requires then  $H(X, X^*)$
to have zero local Poisson bracket with the $S$ in \r{60}.
We can also choose fields which are  arbitrary symplectic coordinates if
we make use of the general master actions following from \r{47} and \r{491}.
These results  agree with  those obtained in \mbox{Ref.\,\protect\citebk{GD}}.

\subsection*{The Chern-Simon model and generalizations}
\addcontentsline{toc}{subsection}{\numberline{A.2}The Chern-Simon
 model and generalizations}
In $n=3$ the natural choice of  superfield pairs are  $X^i$ and $X^*_j$ both
with  ghost number one. Since $S$ must have ghost number three, $S$ must be
trilinear
in
$X^i$ and $X^*_j$. This case is simpler to analyze if we let the superfield
$Z^I$ represent both $X^i$ and $X^*_j$.
Then, $Z^I$ are Darboux coordinates on a symplectic manifold.
The local Poisson bracket is then
\be
(Z^I, Z^K)_3=E^{IK}\,,
\e{70}
where $E^{IK}$ is constant. (Notice that the bracket $(\;,\;)_3$ carries an
extra ghost number $-2$ which implies $gh(E^{IK})=0$ for $gh(Z^I)=1$). The
general form of $S$ is in this case
\vspace{-3mm}
\be
S={1\over 6}\,C_{IJK}Z^IZ^JZ^K\,.
\e{71}
If we let the fields $Z^I$ be odd fields corresponding to original fields
which are even one-form fields, then
 $C_{IJK}$ in \r{71} are even real constants. Notice also that $E^{IK}$ in
\r{70} is
symmetric for odd fields. In this case the master action with the Lagrangian
density \r{47} becomes
\be
\Sigma[Z]=\half\!\int\!\!d^3u\,d^3\tau\Big(E_{IK}Z^KDZ^I\!-
{1\over 3}\,C_{IJK}Z^IZ^JZ^K\Big)\,.
\e{72}
The master equation $(\Sigma, \Sigma)=0$ allows us to interpret
$C_{IJK}$ as
structure coefficients of a Lie group, since $C_{IJK}$ is totally antisymmetric
from \r{71} and since
$(S,S)_3=0$ requires them to satisfy the Jacobi identities. $E^{IK}$ acts
as a group metric. According to the rules \r{37}, the original model is
\be
A=\half\int\!E_{IK}z^I\wedge\Big(dz^K\!-
{1\over 3}\,E^{KL}{C_{LMN}}\,{z^M\!\!\wedge\!z^N}\Big)\,,
\e{73}
which is a non-abelian Chern-Simon model, $z^I$ being even, one-form fields.
The Chern-Simon model was also treated in Ref.\,\protect\citebk{AKSZ}.


We can also introduce superfield pairs with ghost numbers two and zero
respectively, and  pairs with ghost numbers three and minus one. These
superfields allow for many more terms in $S$ which are consistent with
the ghost number prescription. It seems likely that the local master
equation \r{15} will allow for new nontrivial solutions, in which
case one will find generalized Chern-Simon models. All fields must  satisfy
boundary conditions which are consistent with
\r{141}.

For $n=4$ we can obtain something similar to the Chern-Simon model if we
choose superfield pairs  $X^i$ and $X^*_i$ with ghost number one and two,
respectively. It is natural then to let $X^i$ be odd and
$X^*_i$ be even since the
corresponding original one- and two-form fields, $x^i$ and $x^*_i$,
then are
even according to the rule \r{37}.
The general form of
$S$ is then
\be
S=\half\,C^{ij}X^*_iX^*_j+\half\,C^{i}_{\;jk}\,X^*_iX^jX^k+
{1\over 24}\,C_{ijkl}X^iX^jX^kX^l\,,
\e{80}
where the coefficients are even real constants. In this case the condition
$(S,S)_4=0$ yields the following three conditions:
\be
C^{i}_{\;jk}\,C^{kl}X^*_iX^*_lX^j
\!\!\!\!&=&\!\!\!\!0\,,\label{85}\\
\Big(C^{i}_{\;jm}\,C^{m}_{\;\;kl}+{1\over 3}\,C_{jklm}C^{mi}\Big)
X^*_iX^jX^kX^l\!\!\!\!&=&\!\!\!\!0\,,\label{83}\\
C_{ijkn}C^{n}_{\;\;lm}X^iX^jX^kX^lX^m\!\!\!\!&=&\!\!\!\!0\,.
\e{81}
We notice the following special solutions:\\\\
i) If $C^{ij}=C_{ijkl}=0$ then $C^{i}_{\;\;jk}$ can be interpreted as
structure coefficients of a Lie group since $(S,S)_4=0$ requires the Jacobi
identities due to (\ref{83}).\\ \\
ii) If only $C_{ijkl}=0$, then the same interpretation of $C^{i}_{\;\;jk}$ is
possible. However, in this case $(S,S)_4=0$ requires not only the Jacobi
identities for $C^{i}_{\;\;jk}$ but also the condition \vspace{-3mm}
\be
C^{i}_{\;jl}\,C^{lk}+C^{k}_{\;jl}\,C^{li}=0\,,
\e{811}
from (\ref{85}).
If $C^{ij}$ is an invertible matrix, then $C_{ij}$ exists satisfying
$C^{ij}C_{jk}=\del^i_k$ and \r{811} can be rewritten as \vspace{-1mm}
\be
C_{ijk}+C_{kji}=0\,,\qquad C_{ijk}\equiv C_{il}C^{l}_{\;jk}\,.
\e{821}
Thus, the symmetric matrix $C^{ij}$ can be interpreted as a group metric and
$C_{ijk}$ is totally antisymmetric as required by a semi-simple Lie
group. This
corresponds to what we had in the Chern-Simon model in three dimensions.
According to the rules \r{37}, the original model is
\be
A=\!\int\!\!\Big(x^*_i\!\!\wedge\!dx^i-\half\,C^{ij}x^*_i\!\!\wedge\!x^*_j-
\half\,C^{i}_{\;jk}\,x^*_i\!\!\wedge\!x^j\!\!\wedge\!x^k\Big)\,,
\e{822}
where $x^i$ and $x^*_i$ are even one- and two-form fields, respectively. (They
are the ghost number zero coefficients of $X^i$ and $X^*_i$.) This action
is gauge
invariant under the transformations
\be
\del x^i
\!\!\!\!&=&\!\!\!\!d\beta^i+C^{ij}\ga_j+C^{i}_{\;jk}\,\beta^jx^k\,,\nn\\
\del x^*_i\!\!\!\!&=&\!\!\!\!d\ga_i+C^{k}_{\;ij}\,x^*_k\beta^j+C^{k}_{\;ij}
\,\ga_k\!\wedge x^j,
\e{823}
\pagebreak[3]where the gauge parameters $\beta^i$ and $\ga_i$ are zero-
and one-forms, respectively.

Further, we can also introduce superfield pairs here, with ghost numbers
three and zero respectively and pairs with ghost numbers four and minus one,
etc, which will allow for new terms in $S$. The local master equation \r{15}
should then allow for new nontrivial solutions in which case one  obtains
generalized models. Again, all fields must satisfy boundary conditions
which are consistent with
\r{141}.


From the kinetic terms in the superfield Lagrangians \r{2} it is clear that
we get a
BF-theory in any dimension $n$, and  in any dimension we can also have a
Lagrangian
multiplier which is an $n$-form field in the original actions. However, in
higher and
higher dimensions there are more and more different $k$-form fields
allowed. The most
general structure will therefore be more and more  complex in higher and
higher dimensions.

%\appendix
\section*{Appendix B: Superfields in terms of component fields}
\addcontentsline{toc}{section}{\numberline{}Appendix B:
Superfields in terms of component fields}
%\section{Superfields in terms of component fields}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0} 
A decomposition of the superfields in
component fields satisfying \r{20} can be obtained by means of the
following recursive formula. We start with zero components $\Phi^A_0$
and $\Phi^*_{0B}$ satisfying
\be
&&(\Phi^A_0(u), \Phi^*_{0B}(u'))=\del^A_B\del^{n}(u-u')\,.
\e{900}
We write then the superfields as follows
\be
\Phi^A(u,\tau^1,\ldots,\tau^{n-1}, \tau^n)\!
=\!\Phi^A_0(u,\tau^1,\ldots,\tau^{n-1})\!+\!
\tau^n\Phi^A_1(u,\tau^1,\ldots,\tau^{n-1})\,,
\e{901}\vspace{-7mm}
\be
\hspace{-1.4cm}
\Phi^*_A(u,\tau^1,\ldots,\tau^{n-1}, \tau^n)=\hspace{6.5cm}\nn\\[1mm]
\hspace{-2.5mm}=\Big((-1)^{n-1}\Phi^*_{1A}(u,\tau^1,\ldots,\tau^{n-1})-
\Phi^*_{0A}(u,\tau^1,\ldots,\tau^{n-1})\tau^n\Big)(-1)^{\ve_A}\,,
\e{902}\vspace{-5mm}
where
\be
\left(\Phi^A(u,\tau^1,\ldots,\tau^{n-1}),
\Phi^*_B(u',\tau^{\prime 1},\ldots,\tau^{\prime n-1})\right)\hspace{2cm}\nn
\\[1mm]
\hspace{4cm}
=\del^{n}(u-u')\del^{n-1}(\tau-\tau')\del^A_B(-1)^{(n-1)\ve_A}\,.
\e{903}
Then the $n$-parametric superfields \r{901} and \r{902} satisfy by
construction the relations \r{903} with the formal replacement
$n\rightarrow n+1$. One can easily check that the following expressions
satisfy these recursion formulas:
\be
\Phi^A(u,\tau)\!\!\!\!&=&\!\!\!\!\Phi^{A}_0(u)+\tau^a\Phi^{A}_a(u)+
\half\,\tau^{a_1}\tau^{a_2}\Phi^{A}_{a_1a_2}(u)+\cdots\nn\\
\cdots\!\!\!\!&+&\!\!\!\!{1\over k!}\,\tau^{a_1}\cdots\,\tau^{a_k}
\Phi^A_{a_1\cdots a_k}(u)+\cdots\nn\\[1mm]
\cdots\!\!\!\!&+&\!\!\!\!{1\over n!}\,\tau^{a_1}\cdots\,\tau^{a_n}
\Phi^A_{a_1\cdots a_n}(u)\,,\\
\Phi^*_A(u,\tau)\!\!\!\!&=&\!\!\!\!\biggl({1\over n!}\,{\Phi^*_A}^{a_1\cdots 
a_{n}}(u)-{1\over (n-1)!}\,{\Phi^*_A}^{a_1\cdots
a_{n-1}}(u)\,\tau^{a_{n}} \nn\\
\!\!\!\!&+&\!\!\!\!\half\,{1\over(n-2)!}\,{\Phi^*_A}^{a_1\cdots
a_{n-2}}(u)\,\tau^{a_{n-1}}\tau^{a_{n}}+\cdots\nn\\
\cdots\!\!\!\!&+&\!\!\!\!(-1)^k{1\over k!(n-k)!}\,{\Phi^*_A}^{a_1\cdots
a_{n-k}}(u)\,\tau^{a_{n-k+1}}\cdots\tau^{a_n}+\cdots\nn\\
\cdots\!\!\!\!&+&\!\!\!\!(-1)^n{1\over n!}\,{\Phi^*_A}^0(u)\,\tau^{a_1}
\cdots\tau^{a_n}\biggr)(-1)^{n\ve_A}\ve_{a_1\cdots a_n}\,,
\e{904}
where $\ve_{a_1a_2\cdots
a_{n}}$ is totally antisymmetric such that $\ve_{12\cdots
{n}}=1$. We also define\\$\ve_{1}\equiv 1$, $\ve_0\equiv 1$. 
Notice that $\Phi^A_{a_1\cdots a_k}$ and $\Phi_A^{*a_1\cdots a_k}$ 
are totally antisymmetric, and that we have
\be
(\Phi^A_{a_1\cdots a_k}(u), {\Phi_B^*}^{a_1\cdots a_k}(u'))=
\del^A_B\,\del^n(u-u')\,.
\e{905} 

%\newpage
\vspace{1cm}
\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}


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\end{document}


