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\def\yarim{{{1}\over{2}}}
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\def\ft{\phi^{\dag}}
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\def\ds{\displaystyle}
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\def\kok{\sqrt{2}}
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\begin{center}
{\Large{\bf N=2 Super Yang Mills Action}}\\
{\Large{\bf as a BRST Exact Term,}}\\
{\Large{\bf Topological Yang Mills and Instantons}}
%
\vskip1.5em
K. \"{U}lker\\\vskip3em
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{\sl Feza G\"{u}rsey Institute,}\\
{\sl \c{C}engelk\"{o}y, 81220, \.{I}stanbul, Turkey}\\
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\end{center}
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%++++++++++++++++++++++++++++++++++++++ABSTRACT
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\begin{abstract}

By defining an extended BRST operator that includes the chiral part of N=2 global supersymmetry, it is shown that the
full N=2 off-shell Super Yang Mills Action can be represented as an exact BRST term. The action written in this form
suggests that the fields of the Topological Yang Mills theory can be defined in terms of composite fields of
supersymmetry ghosts and N=2 fields in a natural way. Topological Yang Mills theory is obtained from the ordinary
Euclidean N=2 SYM directly as field redefinitions without using twisting procedure. With the help of these results,
relation between the recent instanton calculations in N=2 Super Yang Mills  and Topological Yang Mills theories is also
discussed.

\end{abstract}
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{\small{e-mail: kulker@gursey.gov.tr}}
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%++++++++++++++++++++++++++++++++++++++++++ INTRODUCTION
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\section{Introduction}

N=2 Super Yang Mills Theory (SYM) Theory has been extensively studied these last years after the work of Seiberg and
Witten \cite{sw} where a self consistent non-perturbative effective action was calculated
by using certain ans\"{a}tze dictated by physical intuitions. This solution is unique \cite{fmrss}.

After this seminal paper \cite{sw} one of the main area of research in N=2 SYM is to calculate directly the
multi-instanton
contributions to the holomorphic prepotential and consequently to check the correctness of the results of \cite{sw}.
These multi-instanton contributions are calculated in a pioneering work \cite{dkm} for one and two instantons by using
a semi-classical expansion around {\it approximate} saddle points of the action \cite{dkm} \footnote{One instanton
contribution is also calculated in \cite{finnel, ito}.}. The results are found to be in agreement with the one of
\cite{sw}. (For a self-contained review see Ref. \cite{dkhm}.) However a natural question was posed by several
researchers: how it could be possible that an approximate approach gives an exact result and what was the mechanism
behind \cite{fb}?

In order to get answer to the above questions, in Ref.\cite{bftt2} the instanton calculus is performed in the framework
of Topological Yang Mills (TYM) theory and it is found that the results are the same with that of {\it traditional}
instanton calculus \cite{dkm}. As noted in \cite{bftt2}, the underlying fact of this result is, the action of TYM
theory, which can be obtained as a twist of the ordinary N=2 SYM \cite{witten},
can be written as a BRST-exact term \cite{bs} and that the functional integration over the antifields of the topological
theory gives the same field configurations of the constrained instantons without any approximation.

Therefore an interesting question (at least for the present author) is whether the action of ordinary N=2 SYM can be
written as a BRST exact term or not? The answer, as we will show, is affirmative.

To approach this goal we first note that when the cohomological structure of SYM actions are studied, it is seen that
these actions can be written as chiral (or antichiral) multiple supervariations of lower dimensional field polynomials
\cite{ben1, ben2}\footnote{For a similar approach of constructing N=1 globally and locally supersymmetric actions and
also for the discussion of anomalies, see \cite{bra2}.}. Secondly, the extension of BRST formalism (also called BV or
field-antifield formalism \cite{bv}) to include global supersymmetry (SUSY) is well known \cite{w,mag,mpw}
\footnote{Such an extension of BRST transformations that includes rigid symmetries is first introduced in \cite{bon}.
The problem of how to extend the BRST formalism to include arbitrary global symmetries can be found in Ref.
\cite{bra1}.}. Following these lines  our strategy is to define a nilpotent extended BRST operator on the field space
that only contains the chiral part of the N=2 SUSY as a global supersymmetry. It is then straightforward to find an
expression as a {\it gauge fermion} of which the full off-shell N=2 action is its BRST variation.

As it will be discussed in this note, the gauge fermion is proportional to the inverse power of the chiral SUSY ghost.
Consequently, this means that to write the action as a BRST exact term, one has to give up analyticity in the global
SUSY ghost. In other words, action belongs to the trivial cohomology of BRST operator if this operator is defined on the
space of field polynomials that are not necessarily analytic in global SUSY ghosts. A similar situation occurs in the
twisted SYM. As noted in Ref.\cite{sorlec,sor1,sor2}, twist of N=2 SYM becomes a topological theory if the analyticity
 requirement of scalar SUSY ghost is given up.

Therefore, the fact that N=2 SYM action can be written as a BRST exact term, suggests a natural variable redefinition
between the ordinary Euclidean N=2 SYM and TYM. Indeed, as it will be shown explicitly, the fields of Witten's
topological theory \cite{witten} can be found as composite fields of SUSY ghosts and the original fields. The dimensions
and the ghost numbers of the topological fields defined in this way, appear {\it per se} SUSY ghosts. In other words,
the topological theory is really obtained as a variable redefinition, i.e. without twisting, in a pedestrian way.

On the other hand,  when the the field strength is taken to be (anti)self-dual with the corresponding restrictions due
to supersymmetry \cite{zumino}, it is possible to define a modified extended BRST operator that respects these
restrictions. Consequently, it is also possible to derive an action as an exact term from a convenient gauge fermion
by using such a modified operator. It is then seen that the supersymmetric instanton equations can be obtained
without using any approximation from this action functional. Not surprisingly after performing the above mentioned
variable redefinition one gets exactly the Baulieu-Singer formulation of TYM \cite{bs} .

\section{Extended BRST  transformations\\ and N=2 SYM action as an exact term}

N=2 SYM theory is a rather old and well studied theory. In this work we will study the off-shell formulation of the
theory given in \cite{gsw,soh} by using the conventions of \cite{wb}. The action of the theory is given in Minkowskian
space as
\bea
S_{N=2} &=& \frac{1}{g^2}Tr \int d^4 x (-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}  -i\la^i D \!\!\!\! / \labar_i  + \f D_{\mu}
D^{\mu} \f^{\dag}\nonumber\\
&&-\frac{i\kok}{2}(\la_i [\la^i , \ft ] +\labar^i [\labar_i ,\f ]) -\yarim [\f ,\ft]^2 + \yarim\vD . \vD)
\eea
where the gauge field $A_\mu  $ and the scalar fields $\f\, ,\,\ft $ are singlets, the Weyl spinors $\la_{i\a} \,\,
\labar^{i}_{\adot}$ are doublets and the auxiliary field $\vD $ is a triplet under the $SU(2)_R$ symmetry group. These
fields are members of N=2 vector multiplet $V=(A_\mu\, ,\,\f \, ,\, \ft \, ,\, \la_{i\a} \, ,\,
\labar^{i}_{\adot} \, ,\, \vD )$ \cite{gsw,soh}. The  $SU(2)_R$ indices of the spinors are raised and lowered due to
\bea
\la^i = \se^{ij} \la_j \quad&,&\quad \la_i = \la^j \se_{ji}
\\
\labar_i = \se_{ij} \labar^j \quad&,&\quad \labar^i = \labar_j \se^{ji}
\eea
where the antisymmetric tensor $\se^{ij}$ is given as\footnote{Note that in our convention the $\se^{ij}$ is different
then the one, $\e_{\a\b}$,  used for spinor indices},
$$
\se_{12} = \se^{12} = -\se_{21} = - \se^{21}=1.
$$

The extension of the BRST transformations with the global N=2 SUSY and translation symmetry is first given in
Ref.\cite{mag}:
\be
s=s_0 + \x^i Q_i + \xbar_i \bar{Q}^i -i \eta^{\mu}\del_{\mu}
\ee
where $s_0$ is the ordinary BRST transformations, $Q_i \, ,\, \bar{Q}^i $ are chiral and antichiral parts of N=2 SUSY
transformations and $\x^{i\a} \, ,\, \xbar_{i\adot} $ and $\eta _{\mu}$ are the constant commuting chiral,
antichiral SUSY ghosts and constant imaginary anticommuting translation ghost respectively.

The extended BRST transformations read on the elements of the N=2 vector multiplet as\footnote{Here, $\vt$'s are Pauli
spin matrices},
\bea
sA_{\mu}&=& D_{\mu}c +\x_i \s_{\mu}\labar^i + \xbar^i \sbar_{\mu}\la_i -i \eta^{\nu}\del_{\nu}A_{\mu}\\
s\la_i &=& i\{c,\la_i \} - i\s^{\mu\nu}\x_i F_{\mu\nu} +\x_i [\f ,\ft ] \nonumber\\
 && \qquad\qquad\qquad\qquad  -\kok \s^{\mu}\xbar_i D_{\mu} \f + \vt_i ^j \x_j.\vD -i \eta^{\mu}\del_{\mu} \la_i\\
s\labar^i &=& i \{ c,\labar^i \} - i \sbar^{\mu\nu} \xbar^i F_{\mu\nu} -\xbar^i [\f ,\ft ]\nonumber\\
 && \qquad\qquad\qquad\qquad  -\kok\sbar^{\mu}\x^i D_{\mu} \ft -\xbar^j \vt_i ^j . \vD -i \eta^{\mu}\del_{\mu}\labar^i
\\
s\f &=& i[c,\f ]-i\kok \x_i\la^i -i \eta^{\mu}\del_{\mu}\f\\
s\ft &=& i[c,\ft ]-i\kok \xbar^i \labar_i -i \eta^{\mu}\del_{\mu}\ft\\
s\vD &=& i[c, \vD] + i\vt_i ^j(\x_j \Ds\labar^i -\xbar^i\bar{\Ds}\la_j \nonumber\\
 && \qquad\qquad\qquad\qquad +\kok\x^i [\la_j ,\ft] -\kok\xbar_j
[\labar^i,\f]) -i \eta^{\mu}\del_{\mu}\vD
\eea
In order to get a nilpotent $s$,
\be
s^2 = 0
\ee
 Fadeev-Popov ghost field and the global ghosts are asked to transform as
\bea
sc &=& \frac{i}{2}\{ c,c\}-2i\x_i\s^{\mu}\xbar^i A_{\mu} -\kok\x_i\x^i \ft -\kok\xbar^i\xbar_i \f -i
\eta^{\mu}\del_{\mu}c \\
s\eta_{\mu}  &=& 2\x^i\s_{\mu}\xbar_i  \\
s\x_i &=& s\xbar_i = 0
\eea
Note that with the help of extra terms in $sc$, the characteristic complication that SUSY algebra is modified by
field-dependent gauge transformations is solved, whereas the closure on translations disappeared due to inclusion of
translation ghosts. We summarize the dimension, ghost number and the R-charges of the fields and ghosts in Table 1.
%+++++++++++++++++++++++++++++++++++++++++++ table
%
\begin{table}[hbt]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|}
\hline
&$A_\mu$&$\f$&$\ft$&$\la^i$&$\labar_i$&$\vD$&$\,\,c\,\,$&$\x^i$&$\xbar_i$&$\eta_{\mu}$\\
\hline
$Dim$&1&1&1&3/2&3/2&2&0&-1/2&-1/2&-1 \\
\hline
$Gh$&0&0&0&0&0&0&1&1&1&1 \\
\hline
$R$&0&-2&2&-1&1&0&0&-1&1&0\\
\hline
$GP$&0&0&0&1&1&0&1&0&0&1 \\
\hline
\end{tabular}
\caption[t1]{Dimensions   $d$, Grassmann parity $GP$, ghost number $Gh$
  and R-weights.}
\end{table}
%
%


From the definition of $s$ it is still possible to derive another nilpotent operator by using a suitable filtration of
global ghosts. We choose this filtration to be
\be
\mathcal{N} = \xbar_{i\adot} \frac{\d}{\d \xbar_{i\adot}} + \eta_{\mu} \frac{\d}{\d \eta_{\mu}} \quad ;\quad s =
\sum{s^{(n)}} \quad ,\quad [\mathcal{N},
s^{(n)}]=ns^{(n)},
\ee
so that the zeroth order in the above expansion is a operator that includes ordinary BRST and chiral SUSY on the space
of the fields of the N=2 vector multiplet
\be
\bs:= s^{(0)} = s_0 + \x^i Q_i
\ee
that is also nilpotent,
\be
\bs ^2 =0.
\ee
The $\bs$ transformation of the fields are now given as
\bea
\bs A_{\mu}&=& D_{\mu}c +\x_i \s_{\mu}\labar^i \\
\bs \la_i &=& i\{c,\la_i \} - i\s^{\mu\nu}\x_i F_{\mu\nu} +\x_i [\f ,\ft ] + \vt_i ^j \x_j.\vD \\
\bs \labar^i &=& i \{ c,\labar^i \}  -\kok \sbar^{\mu}\x^i D_{\mu} \ft \\
\bs \f &=& i[c,\f ]-i\kok \x_i\la^i \\
\bs \ft &=& i[c,\ft ]\\
\bs \vD &=& i[c, \vD] + i\vt_i ^j(\x_j \Ds\labar^i  + \kok\x^i [\la_j ,\ft]) \\
\bs c &=& \frac{i}{2}\{ c,c\} -\kok\x_i\x^i \ft\\
\bs \eta_{\mu}  &=& \bs \x_i = \bs \xbar_i = 0
\eea

It is known from cohomological arguments that N=2 SYM action can be written as pure multiple chiral SUSY transformation
of lower dimensional field monomial $Tr\f ^2$ \cite{ben1} in component formalism of SUSY. Since $\bs$ contains chiral
SUSY and the action is gauge invariant, we assume that the action can be written also as a $\bs$ exact
term of a gauge invariant field polynomial which is independent of Fadeev-Popov ghost fields. In other words, we assume
that the action can be chosen to be a trivial element of equivariant cohomology\footnote{See for instance
Ref.s\cite{sorlec,sor1,sor2} and the references therein.} of $\bs$
\be
I=\bs \Psi.
\ee

It is clear that $\Psi$, the so called gauge fermion in BV formalism, has negative ghost number, $Gh(\Psi )=-1$.
Therefore a further assumption is the gauge fermion has the following form:
\be
\Psi=\frac{1}{\x _k \x^k} \x^i \psi_i
\ee
where $\psi_i ^\a $ is a dimension $7/2$  fermion that is made from the fields of the N=2 vector multiplet. The most
general such gauge fermion that is covariant in its Lorentz, spinor and $SU(2)_R$ indices is
\be
\Psi=\frac{1}{\x _k \x^k}Tr \int d^4 x \{ (k_1 \x ^i  \la_i  [\f ,\ft ] + k_2 \x^i \vt _i ^j \la_j . \vD+ k_3 \x^i
\s^{\mu\nu}\la_i F_{\mu\nu} + k_4 \f \x^i \Ds \labar_i ) \}
\ee
Here $k$'s are constants. In order that the $\bs$ variation of $\Psi$ to be free of chiral ghosts after some algebra it
is seen that constants $k$ should be fixed and as a result
\bea
\bs\Psi &=& \bs \frac{1}{\x _k \x^k} Tr \int d^4 x \{ (\yarim \x ^i  \la_i  [\f ,\ft ] -\yarim \x^i \vt _i ^j \la_j .
\vD\nonumber\\
&& \qquad\qquad\qquad\qquad  -\frac{i}{2} \x^i \s^{\mu\nu}\la_i F_{\mu\nu} +\frac{\kok}{2} \f \x^i \Ds \labar_i ) \}\\
&=&Tr \int d^4 x (-\frac{1}{4} F_{\mu\nu} F^{\mu\nu}
-\frac{i}{8}\e^{\mu\nu\la\rho}F_{\mu\nu}F_{\la\rho} \nonumber\\
&&\qquad \qquad\qquad\qquad -i\la^i D \!\!\!\! / \labar_i  + \f D_{\mu} D^{\mu} \f^{\dag}\nonumber\\
&&-\frac{i\kok}{2}(\la_i [\la^i , \ft ] +\labar^i [\labar_i ,\f ]) -\yarim [\f ,\ft]^2 + \yarim\vD . \vD)
\eea
is found. This expression is exactly the action of N=2 SYM theory with a topological term
$\e^{\mu\nu\la\rho}F_{\mu\nu}F_{\la\rho} $ ,
\be
I_{N=2} = \frac{1}{4\pi} Im [\Upsilon \, (\bs\Psi)] \quad,\quad \Upsilon = \frac{i4\pi}{g^2}+\frac{\theta}{2\pi}.
\ee

However, note that, in above algebra auxiliary field $\vD$ plays an important role. In other words, to write
the action as a BRST exact term, one has to use the off-shell formulation of N=2 supersymmetry.

\section{Euclidean N=2 SYM and\\TYM as a variable redefinition }

As it is well known TYM theory can be obtained by twisting N=2 SYM theory in euclidean space \cite{witten}. In
summary twisting procedure is simply identifying the $SU(2)_R$ index $i$ with the spinor index of one
chirality\footnote{In $\mathbb{R}^4$ the symmetry group of N=2 SYM is $SU(2)_L \otimes SU(2)_R \otimes SU(2)_R \otimes
U(1)_R $. The twist $( i\equiv\a )$ consists of replacing the rotation group $SU(2)_L \otimes SU(2)_R $ with $SU(2)_L '
\otimes SU(2)_R$ where $SU(2)_L '$ is the diagonal sum of $SU(2)_L \otimes SU(2)_R $. For a detailed analysis of
topological theories see for instance \cite{top}.}  i.e. $\a$  and $R$-charges of the fields with ghost number. It is
then possible to show that TYM action can be written as a BRST-exact term up to some field redefinitions \cite{bs}.
Therefore, it is natural to look for an analogy between the results of previous section and TYM theory.

However, before formulating the results of previous section in Euclidean space, we find it useful to clarify our
approach to Euclidean N=2 SYM. First of all, obviously in Euclidean space the chiral and antichiral spinors are not
related with each other. Nevertheless, it is still possible to find consistent reality conditions for the spinors of
extended supersymmetry \cite{bvn}. We will take these reality conditions in our conventions as,
\be
(\la_i ^\a )^\dag = i \e_{\a\b} \se^{ij} \la_j ^\b \quad ,\quad (\labar^{i\adot})^\dag =i \e_{\adot\bdot}
\se_{ij}\labar^{j\bdot}.
\ee
On the other hand, since the spinors of different chirality are independent from each and since  supersymmetry is
manifest, it is clear that one should also consider the complex scalar field $\f$ and its hermitean conjugate $\ft$
defined in Minkowskian space as independent fields from each other in Euclidean space \footnote{Indeed, this
somehow unusual treatmet of ours appears naturally if one defines a continuous Wick rotations to Euclidean space
\cite{vnw}:  pseudoscalar field $B$ where $\f =A+iB$, goes over into $iB_E$ in Euclidean space .

However, we will not explicitly represent the scalar field as $\f_E=A_E + B_E$ and $\ft _E=A_E - B_E$ in Euclidean
space.}. For this reason, to avoid confusion we will denote Euclidean scalar fields as  $M:=\f_E$,  $N:=\ft_E$ from now
on.

Following above remarks, we perform a Wick rotation to formulate the results of previous section in Euclidean
space i.e. Minkowskian vector quantities $v^\mu=(v^0,\vec{v})$, $\mu=0,1,2,3$ become Euclidean ones $v_\mu
=(\vec{v},iv^0)$, $\mu=1,2,3,4$ and Euclidean sigma matrices are taken as $e_{\mu\a\adot}=(i\vt,1)$ and
${\bar{e}_{\mu}^{\adot,\a}=(-i\vt,1)}$. We will also take the gauge field anti-hermitean rather then hermitean in order
to follow the instanton literature\footnote{We use the  Euclidean conventions of
Ref.\cite{dkhm}.}.

In Euclidean space, $\bs$-transformations of the fields are now given as
\bea
\bs A_{\mu}&=&  D_{\mu}c - \x_i e_{\mu}\labar^i \\
\bs \la_i &=& - \{c,\la_i \} - e_{\mu\nu}\x_i F_{\mu\nu} +\x_i [M ,N ] + \vt_i ^j \x_j.\vD \\
\bs \labar^i &=& - \{ c,\labar^i \}  +i \kok \bar{e}_{\mu}\x^i D_{\mu} N \\
\bs M &=& - [c, M ] + i\kok \x^i\la_i \\
\bs N &=& - [c, N ]\\
\bs \vD &=& - [c, \vD] + \vt_i ^j(\x_j e_\mu D_\mu \labar^i  + i \kok\x^i [\la_j ,N]) \\
\bs c &=& - \frac{1}{2}\{ c,c\} + i\kok\x_i\x^i N
\eea
and  the gauge fermion (28) as
\be
\Psi_E = \frac{1}{\x _k \x^k} Tr \int d^4 x \{ (\yarim \x ^i  \la_i  [M ,N ] -\yarim \x^i \vt _i ^j \la_j .
\vD - \frac{1}{2} \x^i e_{\mu\nu}\la_i  F_{\mu\nu} - \frac{i\kok}{2} M \x^i e_\mu D_\mu \labar_i ) \}.
\ee

The supersymmetric Euclidean action that is constructed by Zumino \cite{zumino}  can now be written as the
$\bs$-variation of $\Psi_E$,
\bea
I_E&=&\bs\Psi_E\nonumber\\
&=&Tr \int d^4 x (\frac{1}{4} F_{\mu\nu} F_{\mu\nu}
+\frac{1}{8}\e_{\mu\nu\la\rho}F_{\mu\nu}F_{\la\rho} - \la^i D \!\!\!\! / \labar_i  + M D_{\mu} D_{\mu}
N\nonumber\\
&&-\frac{i\kok}{2}(\la_i [\la^i , N ] +\labar^i [\labar_i , M ]) -\yarim [M,N ]^2 + \yarim\vD . \vD)
\eea
up to the topological term $\e_{\mu\nu\la\rho}F_{\mu\nu}F_{\la\rho}$ and the auxiliary term $\yarim\vD . \vD$ since in
our
approach the inclusion of the auxiliary fields i.e. off-shell formulation is mandatory. Note also that the action (41)
is  hermitean with respect to reality conditions\footnote{The scalar fields are hermitean, i.e. $(M)^\dag =M$,
$(N)^\dag=N$} (32).

We are now ready to look for a relation between the above expressions and topological theory. For this
purpose, since after twisting physical nature of some fields are changed it is useful summarize the dimensions and
ghost numbers of the topological fields $(A_{\mu},\, \Phi, \, \bar{\Phi},  \,
\psi_{\mu}, \, \eta, \,  \X_{\mu\nu}, B_{\mu\nu})$ in Table2.
%+++++++++++++++++++++++++++++++++++++++++++ table
\begin{table}[hbt]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline
&$A_\mu$&$\Phi$&$\bar{\Phi}$&$\psi_{\mu}$&$\eta$&$\X_{\mu\nu}$&$B_{\mu\nu}$&$\,\,c\,\,$\\
\hline
$Dim$&1&0&2&1&2&2&2&0 \\
\hline
$Gh$&0&2&-2&1&-1&-1&0&1 \\
\hline
$GP$&0&0&0&1&1&1&0&1 \\
\hline
\end{tabular}
\caption[t1]{Dimensions   $d$, Grassmann parity $GP$, ghost number $Gh$}
\end{table}
%
%

When the $\bs$-transformations of the fields (33-39) are compared with the BRST transformations defined in the
topological theory which contains the Witten's scalar SUSY, and when also the
gauge fixing fermion $\Psi_E$ given in (40) is studied it is easy to read off the following relations that have the
correct dimensionality and ghost number\footnote{We have chosen the coefficients in the definitions
of the topological fields in order to get the conventions of \cite{top}.}:
\be
A_\mu = A_\mu
\ee
\be
\psi_\mu = - \x_i e_\mu \labar^i
\ee
\be
\Phi=i\kok \x_i\x^i N \quad ,\quad \bar{\Phi}= \frac{i}{\kok\x_i \x^i} M
\ee
\be
\eta = \frac{1}{\x _k \x^k} \x_i\la^i \quad , \quad \X_{\mu\nu}= \frac{-2}{\x _k \x^k} \x^i e_{\mu\nu} \la_i
\ee
\be
B_{\mu\nu}= \frac{-2}{\x _k \x^k} \x^i e_{\mu\nu}\vt _i ^j \x_j . \vD
\ee
It is straightforward to show that when the above variable redefinitions are plugged in the ordinary Euclidean
action (41) and in the transformations (33-39), the action and corresponding BRST transformations that are found, is
exactly the ones of Topological Yang Mills theory. In other words, as mentioned by several authors (but not shown
explicitly to our best knowledge), TYM theory in flat Euclidean space can be obtained as variable redefinitions from
the ordinary N=2 SYM theory. As it is obvious from the above definitions of the topological fields, the ghost numbers
assigned to the fields in the twisted version, appears here naturally due to the composite structure of the topological
fields in terms of global ghosts $\x_i$ and the original fields i.e. with respect to the power of $\x_i$'s in the
definitions.

To be clear, (and hoping not to be too tedious) we demonstrate the above points explicitly. By using the field
redefinitions (33-39), $\bs$-transformations can be rewritten as\footnote{Here we note that the derivation of the above
results depends crucially on the commuting nature of the global ghost
$\x_i$. For example it is easy to verify that $\bs$-transformation of $\la_i$ (34) decomposes into (52,53) by using 
$\x^i
e_{ \mu\nu}\x_i=0$ and $\x^i\vt_i ^j\x_j =0$.}
\bea
\bs A_{\mu}&=& D_{\mu}c + \Psi_{\mu} \\
\bs \psi_\mu  &=& - \{ c, \Psi_\mu \}  -  D_{\mu} \Phi \\
\bs \Phi &=& - [c,\Phi ]\\
\bs c &=& - \frac{1}{2}\{ c,c\} +\Phi
\eea
and
\bea
\bs \bar{\Phi} &=& - [c,\bar{\Phi} ] + \eta \\
\bs \eta &=& - \{c, \eta \} + [\Phi ,\bar{\Phi} ]\\
\bs \X_{\mu\nu} &=& -[c,\X_{\mu\nu} ] + F_{\mu\nu}^+ + \B_{\mu\nu}\\
\bs B_{\mu\nu} &=& - [c, B_{\mu\nu} ] + [\Phi , \X_{\mu\nu} ] - (D_\mu \psi_\nu -D_\nu \psi_\mu)^+
\eea
where $F_{\mu\nu}^+ = F_{\mu\nu}+\frac{1}{2}\e_{\mu\nu\la\rho}F_{\mu\nu}$ is the self-dual part of the field strength
$F_{\mu\nu}$. If one decomposes $\bs$ on the fields $(A_{\mu},\, \Phi, \,\bar{\Phi},  \,\psi_{\mu}, \,\eta, \,
\X_{\mu\nu})$ as
\be
\bs=s_o + \d
\ee
one can see that the $c$ independent part  $\d$ is exactly  the same with the scalar supersymmetry introduced by Witten
\cite{witten} with the auxiliary field $B_{\mu\nu}$.

In terms of these topological fields that are found as a variable redefinition the gauge fermion (40) now reads as
\be
\Psi_{top} = Tr \int d^4 x \{ (-\yarim \eta  [\Phi ,\bar{\Phi} ] + \frac{1}{8}\B_{\mu\nu}  F_{\mu\nu}^+
-\frac{1}{8}\X_{\mu\nu}B_{\mu\nu} + \bar{\Phi} D_\mu \psi_\mu ) \}.
\ee
and the corresponding action is
\bea
I_{top} &=& \bs \Psi_{top}\\
&=&Tr\int d^4 x (\frac{1}{8} F_{\mu\nu}^+ F_{\mu\nu}^+ + \eta D_\mu \psi_\mu -\frac{1}{4} \X_{\mu\nu}(D_\mu \psi_\nu -
D_\nu \psi_\mu)^+ -\bar{\Phi}D^2 \Phi\nonumber\\
 &-&\! \yarim \Phi \{\eta ,\eta \} -\frac{1}{8} \Phi \{ \X_{\mu\nu}, \X_{\mu\nu}\} + \bar{\Phi} \{\psi_\mu ,
\psi_\mu  \} -\yarim [\Phi ,\bar{\Phi}]^2 -\frac{1}{8} B_{\mu\nu}B_{\mu\nu} )
\eea
which is the Topological Yang Mills action that is found by twisting the original N=2 SYM \cite{witten, top}
with an auxiliary field term\footnote{We remark once more that the auxiliary field is crucial in order to write the
action as an exact term.
Indeed, the reason why the action could not be written as an exact term in \cite{witten} is that the twisted theory was
obtained from  the on-shell SYM. Note that, since $\Psi_{top}$ given in (56) is gauge invariant we have
$I_{top}=\bs\Psi_{top}=\d\Psi _{top}$.}.

\section{Instantons and \\Baulieu-Singer approach to TYM}

As it is well known, instantons are finite action solutions of the Euclidean field theories. Aiming to incorporate the
instantons into supersymmetric theories Zumino have constructed a supersymmetric field theory directly in the Euclidean
space \cite{zumino}. This theory has N=2 supersymmety with a hermitean action\footnote{Recently, this theory has  been
derived by defining a continuous Wick rotation \cite{vnw} and also by using dimensional reduction via time direction
from six dimensional N=1SYM\cite{bvn}.}.  It is then possible to show that when one takes for instance an anti self-dual
field strength, $$F_{\mu\nu}^+ = F_{\mu\nu}+\frac{1}{2}\e_{\mu\nu\la\rho}F_{\mu\nu} = 0 $$
with the following restrictions $$ M=\la_i=0 $$ the equations of motion from (41) take a simple form \cite{zumino},
\bea
F_{\mu\nu}^+ &=& F_{\mu\nu}+\frac{1}{2}\e_{\mu\nu\la\rho}F_{\mu\nu} = 0\\
D^2 N &=& \frac{i\kok}{2}\{ \labar^i , \labar_i \} \\
e_\mu D_\mu \labar^i &=& 0
\eea
that are covariant under the supersymmetry transformations found by applying the above restrictions \cite{zumino}.

The equations (59-61) are also the ones whose solutions are used as approximate solutions of the saddle point
equations in  the context of constraint instanton method \cite{dkm,dkhm}. On the other hand, similar equations are
obtained in TYM without any approximation \cite{bftt2} from an action functional that can be written as a BRST
transformation of a  gauge fermion given by Baulieu-Singer \cite{bs}. Both of the approaches to the instanton
calculations give the same result \cite{bftt2}.

Therefore, since Euclidean N=2 SYM action can be written as a BRST exact term (41) and  Wittens TYM \cite{witten} can be
obtained by using simple field redefinitions (42-46), we look for another analogy between the above instanton equations
and the Baulieu-Singer approach to TYM.

The first step towards for this purpose is to define a restricted $\ts$ ,
\be
\ts = \bs|_{F_{\mu\nu}^+ = \Ds \labar^i =M = \la_i = 0}
\ee
such that,
\bea
\ts A_{\mu}&=&  D_{\mu}c - \x_i e_{\mu}\labar^i \\
\ts \labar^i &=& - \{ c,\labar^i \}  +i \kok \bar{e}_{\mu}\x^i D_{\mu} N \\
\ts N &=& - [c, N ]\\
\ts c &=& - \frac{1}{2}\{ c,c\} + i\kok\x_i\x^i N
\eea
and
\bea
\ts M &=&  i\kok \x^i\la_i \\
\ts \la_i &=&  \vt_i ^j \x_j.\vD \\
\ts \vD &=& 0
\eea
The reason why we do not set $\la_i = \vD =0$ in equations (67,68) is that the pairs $(M,\x^i\la_i)$ and $(\x^i
\vt_i^j\la_j,\vD)$ behaves like the trivial pairs that are introduced in order to fix the gauge
symmetry in BRST formalism, i.e. like $(\cbar,b)$ such that $s_0 \cbar = b \, ,\, s_0 b=0$. It is straightforward to
derive that $\ts$ is also nilpotent $$\ts^2 = 0$$ and after the field definition (42-46) $\ts$-transformations is
exactly that of Baulieu-Singer \cite{bs, top}.

The gauge fermion compatible with the restrictions of Zumino \cite{zumino} is chosen to be,
\be
\Psi_{inst.} = \frac{1}{\x _k \x^k} Tr \int d^4 x \{  - \frac{\a}{2} \x^i \vt _i ^j \la_j . \vD  - \frac{1}{2} \x^i
e_{\mu\nu}\la_i F_{\mu\nu}^+  + \frac{i\kok}{2}  \x^i e_\mu  \labar_i D_\mu M ) \}.
\ee
Note that this gauge fermion is different then the one given for Euclidean case (40). The reason becomes apparent when
the corresponding action is driven,
\bea
I_{inst.}^{(\a)}&=&\bs\Psi_E\nonumber\\
&=&Tr \int d^4 x (-\frac{\a}{8}B_{\mu\nu}B_{\mu\nu} +\frac{1}{4}B_{\mu\nu}  F_{\mu\nu}^+  - \la^i e_\mu D_\mu \labar_i
\nonumber\\&&\qquad\qquad\qquad + M (D_{\mu} D_{\mu}N -\frac{i\kok}{2}\{\labar^i ,\labar_i \}) \nonumber\\
&& +  \frac{1}{\x _k \x^k} (- \frac{1}{2}\x^i e_{\mu\nu}\la_i [c,F_{\mu\nu}^+] + \frac{i\kok}{2} M \{ c ,\x^i e_\mu
D_\mu
\labar_i \}) \,\,) \nonumber\\
&& +  \frac{1}{\x _k \x^k} Tr\int d^4 x \del_\mu( \ts \frac{i\kok}{2}M \x^i e_\mu  \labar_i )
\eea
where we have used the definition of $B_{\mu\nu}$ in order to have notational simplification. First of all, the gauge
fermion (70) and the above action is exactly the ones given in Baulieu-Singer approach \cite{bs} and it is
straightforward to rewrite the above given $\ts$-transformations, gauge fermion and the action by using the
field redefinitions given in (42-46) in order to get the results of \cite{bs}. However, if the above relations are
considered on their own, to be able to derive the instanton equations exactly from the action functional without having
any dependence on the constant ghosts, the coefficients of the Euclidean $\Psi_E$ has to be modified. For instance since
with the restrictions of \cite{zumino} the $\x_i $ dependence of the term $Tr\ts \x^i\la_i [M,N]$ cannot be eliminated
the coefficient of this term has to be chosen to vanish. The coefficient of $Tr \ts \x^i \vt _i ^j \la_j . \vD$ can be
left arbitrary since after performing the Gaussian integration over the auxiliary field $B_{\mu\nu}$ the action is
\bea
I_{inst.}^{(\a)}&=&\bs\Psi_E\nonumber\\
&=&Tr \int d^4 x (\frac{1}{8\a}F_{\mu\nu}^+  F_{\mu\nu}^+  - \la^i e_\mu D_\mu \labar_i
+ M (D_{\mu} D_{\mu}N -\frac{i\kok}{2}\{\labar^i ,\labar_i \}) \nonumber\\
&& +  \frac{1}{\x _k \x^k} (- \frac{1}{2}\x^i e_{\mu\nu}\la_i [c,F_{\mu\nu}^+ ] + \frac{i\kok}{2} M \{ c ,\x^i e_\mu
D_\mu \labar_i \}) \,\,) \nonumber\\
&& +  \frac{1}{\x _k \x^k} Tr\int d^4 x \del_\mu( \ts \frac{i\kok}{2}M \x^i e_\mu  \labar_i )
\eea
and Fadeev-Popov ghost field, $c$, independent part of the action is also SUSY ghost $\x_i$ free. The
form of the last term in $\Psi_{inst}$ is inspired from Ref.\cite{bftt2} in order to get a surface contribution, since
 $$ Tr\ts \x^i e_\mu  \labar_i D_\mu M  = Tr \del^\mu \ts \x^i e_\mu  \labar_i  M - Tr  \ts M \x^i e_\mu D_\mu  \labar_i
$$ if the scalar field has non-trivial boundary conditions.

On the other hand, the free parameter $\a$ can be thought as a gauge parameter, since
it is so in Baulieu-Singer approach \cite{bs}. Therefore by choosing directly $\a=0$, the action (71) takes the form,
\bea
I_{inst.}&=&\bs\Psi_E\nonumber\\
&=&Tr \int d^4 x (\frac{1}{4}B_{\mu\nu}  F_{\mu\nu}^+  - \la^i e_\mu D_\mu \labar_i
+ M (D_{\mu} D_{\mu}N -\frac{i\kok}{2}\{\labar^i ,\labar_i \}) \nonumber\\
&& +  \frac{1}{\x _k \x^k} (- \frac{1}{2}\x^i e_{\mu\nu}\la_i [c,F_{\mu\nu}^+ ]+ \frac{i\kok}{2} M \{ c ,\x^i e_\mu
D_\mu
\labar_i \}) \,\,) \nonumber\\
&& +  \frac{1}{\x _k \x^k} Tr\int d^4 x \del_\mu( \ts \frac{i\kok}{2}M \x^i e_\mu  \labar_i )
\eea
Note that by performing a functional integration over the fields $\la_i \, ,M$ and $\B_{\mu\nu}$, the configurations of
the constraint instanton method (59-61) are obtained without using any approximation procedure.

\section{Conclusion and discussion}

We have shown how to write the off-shell N=2 SYM action as a BRST exact term by using an nilpotent extended BRST
operator that includes chiral supersymmetry transformations. The corresponding gauge fixing fermion is not analtic in
global SUSY ghosts $\x_i$. In other words the action belongs to trivial cohomology of this extended BRST operator which
acts on the field polynomials that are not necessarily analytic in these parameters.

Then, we have seen that the Topological Yang Mills theory can be obtained as a simple variable redefinition from the
Euclidean N=2 SYM. Due to this variable change, the fields of TYM are found to be composite fields of N=2 vector
multiplet and the chiral supersymmetry ghosts. The ghost number of the topological fields, that are assigned by hand
when it is formulated by twisting, appears in our approach naturally according to this composite structure.

In order to be clear, we should stress that above results do not mean that N=2 SYM is a topological theory. It has its
own physical degrees of freedom. The topological theory can be obtained only if the analyticity requirement of the
SUSY ghosts is relaxed. As it is pointed out in Ref.\cite{sorlec,sor1,sor2} when perturbative calculations are
considered one should obviously require analyticity of the constant ghosts $\x_i$ since they are parameters of the
theory and the twist of the N=2 theory can be interpreted as a topological theory only if the analyticity is lost in
SUSY ghosts.

On the other hand, when the restrictions on the fields in order to get supersymmetric instanton
configurations given in \cite{zumino} are used, we show that with the help of a  restricted BRST operator $\ts$ an
action can be written as an exact term. After using the given variable redefinition, it is seen that this formulation is
 exactly  TYM theory in the approach of Baulieu-Singer \cite{bs}. Moreover, the instanton equations, that are used in
the {\it traditional} instanton computations can be found from this action without using any approximation. It is known
that the Witten's action \cite{witten} , and hence the Euclidean N=2 action, can be obtained from the one given in
Baulieu-Singer approach by a continuous deformation of the gauge fixing \cite{bs}. Indeed, the instanton calculations
performed in Baulieu-Singer approach of TYM \cite{bftt2} gives exactly the same result with the one performed in
Euclidean space in the framework of constraint instantons \cite{dkm, dkhm}, that is approximate. Therefore, since both
theories are shown to be equivalent by simple variable redefinitions, it seems unnecessary to consider instanton
equations as approximate saddle points. As in the topological case they are exact.

As a concluding remark, we note that Seiberg-Witten's prepotential is composed of a perturbative and a non-perturbative
parts:
\bea
\sf(\varphi)&=&\sf_{pert}(\varphi)+\sf_{inst}(\varphi)\nonumber\\
&=&\frac{i}{2\pi}\varphi^2ln\frac{2\varphi^2}{\Lambda^2} \, - \frac{i}{\pi}\sum^{\infty}_{n=1} \sf_n (
\frac{\Lambda}{\varphi} ) ^{4n} \varphi^2 \nonumber
\eea
where $\varphi $ is the N=2 chiral superfield. A simple rescaling, $\varphi = \rho \varphi $ , shows that the
perturbative part of the prepotential is analytic in the parameter $\rho$ where as the non-perturbative part is not. Our
results confirm this observation: instantons localizes in the topological sector of the theory where the functional
space of field polynomials is not necessarily analytic in the constant global ghosts $\x_i $.

\newpage
{\it Acknowledgments

I am indebted to R. Flume for an initiation to the subject. \\
I also gratefully acknowledge the  numerous enlightening discussions with \"{O}. F. Day\i{} and M.
Horta\c{c}su.}




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\addcontentsline{toc}{section}{References}
\newcommand{\anp}[3]{{\sl Ann.~Phys.} {\bf #1} (19#2) #3}
\newcommand{\app}[3]{{\sl Acta~Phys.~Pol.} {\bf #1} (19#2) #3}
\newcommand{\cmp}[3]{{\sl Commun. Math. Phys.} {\bf #1} (19#2) #3}
\newcommand{\cpc}[3]{{\sl Comp. Phys. Commun.} {\bf #1} (19#2) #3}
\newcommand{\cqg}[3]{{\sl Class. Quant. Grav.} {\bf #1} (19#2) #3}
\newcommand{\ijmpA}[3]{{\sl Int. J. Mod. Phys.} {\bf A#1} (19#2) #3}
\newcommand{\jmp}[3]{{\sl J. Math. Phys.} {\bf #1} (19#2) #3}
\newcommand{\JHEP}[3]{{\sl JHEP} {\bf #1} (20#2) #3}
\newcommand{\JPA}[3]{{\sl J. Phys. A} {\bf #1} (20#2) #3}
\newcommand{\mpl}[3]{{\sl Mod. Phys. Lett.} {\bf #1} (19#2) #3}
\newcommand{\mpla}[3]{{\sl Mod. Phys. Lett.A} {\bf #1} (20#2) #3}
\newcommand{\npB}[3]{{\sl Nucl. Phys.} {\bf B #1} (19#2) #3}
\newcommand{\np}[3]{{\sl Nucl. Phys.} {\bf B #1} (20#2) #3}
\newcommand{\nphbps}[3]{{\sl Nucl. Phys.} {\bf B} {\it (Proc. Suppl.)}
{\bf #1} (19#2) #3}
\newcommand{\plB}[3]{{\sl Phys. Lett.} {\bf B #1} (19#2) #3}
\newcommand{\plb}[3]{{\sl Phys. Lett.} {\bf B #1} (20#2) #3}
\newcommand{\prD}[3]{{\sl Phys. Rev.} {\bf D #1} (19#2) #3}
\newcommand{\prd}[3]{{\sl Phys. Rev.} {\bf D #1} (20#2) #3}
\newcommand{\prl}[3]{{\sl Phys. Rev. Lett.} {\bf #1} (19#2) #3}
\newcommand{\prep}[3]{{\sl Phys. Rep.} {\bf #1} (19#2) #3}
\newcommand{\pre}[3]{{\sl Phys. Rep.} {\bf #1} (20#2) #3}
\newcommand{\pl}[3]{{\sl Phys. Lett.} {\bf #1} (19#2) #3}
\newcommand{\ptp}[3]{{\sl Prog. Theo. Phys.} {\bf #1} (19#2) #3}
\newcommand{\sptp}[3]{{\sl Suppl. Prog. Theo. Phys.} {\bf #1} (19#2) #3}
\newcommand{\sjnp}[3]{{\sl Sov. J. Nucl. Phys.} {\bf #1} (19#2) #3}
\newcommand{\zp}[3]{{\sl Z. Phys.} {\bf #1} (19#2) #3}
\newcommand{\vj}[4]{{\sl #1~}{\bf #2} (19#3) #4}
\newcommand{\ej}[3]{{\bf #1} (19#2) #3}
\newcommand{\vjs}[2]{{\sl #1~}{\bf #2}}

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\end{thebibliography}



\end{document}
