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\hyphenation{ma-ni-fold La-gran-gi-ans di-men-si-o-nal
-di-men-si-o-nal La-gran-gi-an Ha-mil-to-ni-an multi-symplec-tic}

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{\parindent=0pt


{\large\bf Space-time BRST symmetries}
\bigskip


{\sc D.Bashkirov and G.Sardanashvily}

{\sl Department of Theoretical Physics, Moscow State University,
Russia}

\bigskip

{\small {\bf Abstract} Bearing in mind BV quantization of gauge
gravitation theory, we extend general covariant transformations to
the BRST ones. }

 }

\section{Introduction}


Let us consider a Lagrangian system on a smooth fiber bundle $Y\to
X$. It is called a gauge system if its Lagrangian $L$ admits a set
of symmetries depending on parameter functions $\xi^r$ and their
derivatives. BRST transformations come from gauge transformations
by replacement of gauge parameters with odd ghosts $c^r$.
Moreover, one completes these transformations with the terms
acting on ghosts such that the total ones become nilpotent.

In the case of general covariant transformations, parameter
functions are vector fields on $X$. We introduce the corresponding
ghosts and construct BRST transformations in the case of metric
and metric-affine gravitation theories.

Let $J^rY$, $r=1,\ldots$, be finite order jet manifolds of
sections of $Y\to X$. In the sequel, the index $r=0$ stands for
$Y$. Given bundle coordinates $(x^\la,y^i)$ on $Y$, jet manifolds
$J^rY$ are endowed with the adapted coordinates
$(x^\la,y^i,y^i_\La)$, where $\La=(\la_k...\la_1)$,
$k=1,\ldots,r$, is a symmetric multi-index. We use the notation
$\la+\La=(\la\la_k...\la_1)$ and
\mar{5.177}\beq
d_\la = \dr_\la + \op\sum_{0\leq|\La|} y^i_{\la+\La}\dr_i^\La,
\qquad d_\La=d_{\la_r}\circ\cdots\circ d_{\la_1}. \label{5.177}
\eeq
In order to describe gauge transformations depending on
parameters, let us consider Lagrangian formalism on the bundle
product
\mar{0681}\beq
E=Y\op \times_X V, \label{0681}
\eeq
where $V\to X$ is a vector bundle whose sections are gauge
parameter functions \cite{noether}. Let $V\to X$ be coordinated by
$(x^\la,\xi^r)$. Then gauge transformations are represented by a
differential operator
\mar{0509}\beq
\up= \op\sum_{0\leq|\La|\leq
m}\up^{i,\La}_r(x^\la,y^i_\Si)\xi^r_\La \dr_i \label{0509}
\eeq
on $E$ (\ref{0681}) which is linear on $V$ and takes its values
into the vertical tangent bundle $VY$ of $Y\to X$. By means of a
replacement of even gauge parameters $\xi^r$ and their jets
$\xi^r_\La$ with the odd ghosts $c^r$ and their jets $c^r_\La$,
the operator (\ref{0509}) defines a graded derivation
\mar{0680}\beq
\up= \op\sum_{0\leq|\La|\leq m}\up^{i,\La}_r(x^\la,y^i_\Si)c^r_\La
\dr_i \label{0680}
\eeq
of the algebra of the original even fields and odd ghosts. Its
extension
\mar{0684}\beq
\up= \op\sum_{0\leq|\La|\leq m}\up^{i,\La}_r c^r_\La \dr_i +
u^r\dr_r \label{0684}
\eeq
to ghosts is called the BRST transformation if it is nilpotent.
Such an extension exists if the original gauge transformations
form an algebra \cite{algebr}.

Note that, if gauge transformations are reducible, $(0\leq
k)$-stage ghosts are introduced \cite{brst}. Gauge theories in
question here are irreducible.

In the case of gravitation theory, a fiber bundle $Y\to X$ belongs
to the category of natural bundles, and it admits the canonical
lift of any vector field on $X$. One thinks of such a lift as
being an infinitesimal generator of general covariant
transformations of $Y$. In this case, the vector bundle $V\to X$
possesses the composite fibration $V\to TX\to X$, where $TX$ is
the tangent bundle of $X$.

Here, we consider the following three gauge theories: (i) the
gauge model of principal connections on a principal bundle (the
gauge symmetry (\ref{0660}), the BRST symmetry (\ref{gr20})), (ii)
this gauge model in the presence of a metric gravity (the gauge
symmetry (\ref{gr11}), the BRST symmetry (\ref{gr21})) and (iii)
in the presence of a metric-affine gravity (the gauge symmetry
(\ref{gr15}), the BRST symmetry (\ref{gr22})).


\section{Gauge systems on fiber bundles}

Lagrangian formalism on a fiber bundle $Y\to X$ is phrased in
terms of the following graded differential algebra (henceforth
GDA).

With the inverse system of jet manifolds
\mar{5.10}\beq
X\op\longleftarrow^\pi Y\op\longleftarrow^{\pi^1_0} J^1Y
\longleftarrow \cdots J^{r-1}Y \op\longleftarrow^{\pi^r_{r-1}}
J^rY\longleftarrow\cdots, \label{5.10}
\eeq
one has the direct system
\mar{5.7}\beq
\cO^*X\op\longrightarrow^{\pi^*} \cO^*Y
\op\longrightarrow^{\pi^1_0{}^*} \cO_1^*Y \ar\cdots \cO^*_{r-1}Y
\op\longrightarrow^{\pi^r_{r-1}{}^*}
 \cO_r^*Y \longrightarrow\cdots \label{5.7}
\eeq
of GDAs $\cO_r^*Y$ of exterior forms on jet manifolds $J^rY$ with
respect to the pull-back monomorphisms $\pi^r_{r-1}{}^*$. Its
direct limit
 $\cO_\infty^*Y$ is a GDA
consisting of all exterior forms on finite order jet manifolds
modulo the pull-back identification.

The projective limit $(J^\infty Y, \pi^\infty_r:J^\infty Y\to
J^rY)$ of the inverse system (\ref{5.10}) is a Fr\'echet manifold.
A bundle atlas $\{(U_Y;x^\la,y^i)\}$ of $Y\to X$ yields the
coordinate atlas
\mar{jet1}\beq
\{((\pi^\infty_0)^{-1}(U_Y); x^\la, y^i_\La)\}, \qquad
{y'}^i_{\la+\La}=\frac{\dr x^\m}{\dr x'^\la}d_\m y'^i_\La, \qquad
0\leq|\La|, \label{jet1}
\eeq
of $J^\infty Y$, where $d_\m$ are the total derivatives
(\ref{5.177}).  Then $\cO^*_\infty Y$ can be written in a
coordinate form where the horizontal one-forms $\{dx^\la\}$ and
the contact one-forms $\{\th^i_\La=dy^i_\La
-y^i_{\la+\La}dx^\la\}$ are local generating elements of the
$\cO^0_\infty Y$-algebra $\cO^*_\infty Y$. There is the canonical
decomposition $\cO^*_\infty Y=\oplus\cO^{k,m}_\infty Y$ of
$\cO^*_\infty Y$ into $\cO^0_\infty Y$-modules $\cO^{k,m}_\infty
Y$ of $k$-contact and $m$-horizontal forms together with the
corresponding projectors $h_k:\cO^*_\infty Y\to \cO^{k,*}_\infty
Y$ and $h^m:\cO^*_\infty Y\to \cO^{*,m}_\infty Y$. Accordingly,
the exterior differential on $\cO_\infty^* Y$ is split into the
sum $d=d_H+d_V$ of the nilpotent total and vertical differentials
\be
d_H(\f)= dx^\la\w d_\la(\f), \qquad d_V(\f)=\th^i_\La \w
\dr^\La_i\f, \qquad \f\in\cO^*_\infty Y.
\ee

Any finite order Lagrangian
\mar{0512}\beq
L=\cL\om:J^rY\to \op\w^nT^*X, \qquad \om=dx^1\w\cdots\w dx^n,
\qquad n=\di X, \label{0512}
\eeq
is an element of $\cO^{0,n}_\infty Y$, while
\mar{0513}\beq
\dl L=\cE_i\th^i\w\om=\op\sum_{0\leq|\La|}
(-1)^{|\La|}d_\La(\dr^\La_i \cL)\th^i\w\om\in \cO^{1,n}_\infty Y
\label{0513}
\eeq
is its Euler--Lagrange operator taking values into the vector
bundle
\mar{0548}\beq
T^*Y\op\w_Y(\op\w^n T^*X)= V^*Y\op\ot_Y\op\w^n T^*X. \label{0548}
\eeq

A Lagrangian system on a fiber bundle $Y\to X$ is said to be a
gauge theory if its Lagrangian $L$ admits a family of variational
symmetries parameterized by elements of a vector bundle $V\to X$
and its jet manifolds as follows.

Let $\gd\cO^0_\infty Y$ be the $\cO^0_\infty Y$-module of
derivations of the $\Bbb R$-ring $\cO^0_\infty Y$. Any $\vt\in
\gd\cO^0_\infty Y$ yields the graded derivation (the interior
product) $\vt\rfloor\f$ of the GDA $\cO^*_\infty Y$ given by the
relations
\be
\vt\rfloor df=\vt(f), \qquad  \vt\rfloor(\f\w\si)=(\vt\rfloor
\f)\w\si +(-1)^{|\f|}\f\w(\vt\rfloor\si), \quad f\in \cO^0_\infty
Y, \quad \f,\si\in \cO^*_\infty Y,
\ee
and its derivation (the Lie derivative)
\mar{0515}\beq
\bL_\vt\f=\vt\rfloor d\f+ d(\vt\rfloor\f), \qquad
\bL_\vt(\f\w\f')=\bL_\vt(\f)\w\f' +\f\w\bL_\vt(\f'), \qquad
\f,\f'\in \cO^*_\infty Y. \label{0515}
\eeq
Relative to an atlas (\ref{jet1}), a derivation
$\vt\in\gd\cO^0_\infty$ reads
\mar{g3}\beq
\vt=\vt^\la \dr_\la + \vt^i\dr_i + \op\sum_{|\La|>0}\vt^i_\La
\dr^\La_i, \label{g3}
\eeq
where the tuple of derivations $\{\dr_\la,\dr^\La_i\}$ is defined
as the dual of that of the exterior forms $\{dx^\la, dy^i_\La\}$
with respect to the interior product $\rfloor$ \cite{cmp}. Note
that the tuple of derivations $\{\dr^\La_i\}$ is the dual of the
basis $\{\th^i_\La\}$ of contact forms.

A derivation $\vt$ is called contact if the Lie derivative
$\bL_\vt$ (\ref{0515}) preserves the contact ideal of the GDA
$\cO^*_\infty Y$ generated by contact forms. A derivation $\up$
(\ref{g3}) is contact iff
\mar{g4}\beq
\vt^i_\La=d_\La(\vt^i-y^i_\m\vt^\m)+y^i_{\m+\La}\vt^\m, \qquad
0<|\La|. \label{g4}
\eeq
Any contact derivation admits the horizontal splitting
\mar{g5}\beq
\vt=\vt_H +\vt_V=\vt^\la d_\la + (\up^i\dr_i + \op\sum_{0<|\La|}
d_\La \up^i\dr_i^\La), \qquad \up^i= \vt^i-y^i_\m\vt^\m.
\label{g5}
\eeq
Its vertical part $\vt_V$  is completely determined by the first
summand
\mar{0641}\beq
\up=\up^i(x^\la,y^i_\La)\dr_i, \qquad 0\leq |\La|\leq k.
\label{0641}
\eeq
This is a section of the pull-back $VY\op\times_Y J^kY\to J^kY$,
i.e., a $k$-order $VY$-valued differential operator on $Y$. One
calls $\up$ (\ref{0641}) a generalized vector field on $Y$. Any
vertical contact derivation $\vt$ satisfies the relations
\mar{g232}\beq
\vt\rfloor d_H\f=-d_H(\vt\rfloor\f), \qquad
\bL_\vt(d_H\f)=d_H(\bL_\vt\f), \qquad \f\in\cO^*_\infty Y.
\label{g232}
\eeq

One can show that the Lie derivative of a Lagrangian $L$
(\ref{0512}) along a contact derivation $\vt$ (\ref{g5}) fulfills
the first variational formula
\mar{g8'}\beq
\bL_\vt L= \up\rfloor\dl L +d_H(h_0(\vt\rfloor\Xi_L)) +\cL d_V
(\vt_H\rfloor\om), \label{g8'}
\eeq
where $\Xi_L$ is a Lepagean equivalent of $L$ \cite{cmp}. A
contact derivation $\vt$ (\ref{g5}) is called  variational if the
Lie derivative (\ref{g8'}) is $d_H$-exact, i.e., $\bL_\vt
L=d_H\si$, $\si\in \cO^{0,n-1}_\infty$. A glance at the expression
(\ref{g8'}) shows that: (i) $\vt$ (\ref{g5}) is variational only
if it is projected onto $X$; (ii) $\vt$ is variational iff its
vertical part $\vt_V$ is well; (iii) it is variational iff
$\up\rfloor\dl L$ is $d_H$-exact. Therefore, we can restrict our
consideration to vertical contact derivations $\vt=\vt_V$. A
generalized vector field $\up$ (\ref{0641}) is called a
variational symmetry of a Lagrangian $L$ if it generates a
variational contact derivation.

Turn now to the notion of a gauge symmetry \cite{noether}. Let us
consider the bundle product $E$ (\ref{0681}) coordinated by
$(x^\la,y^i,\xi^r)$. Given a Lagrangian $L$ on $Y$, let us
consider its pull-back, say again $L$, onto $E$. Let $\vt_E$ be a
contact derivation of the $\Bbb R$-ring $\cO^0_\infty E$, whose
restriction
\mar{0508}\beq
\vt=\vt_E|_{\cO^0_\infty Y}=
\op\sum_{0\leq|\La|}d_\La\up^i\dr_i^\La \label{0508}
\eeq
to $\cO^0_\infty Y\subset \cO^0_\infty E$ is linear in coordinates
$\xi^r_\Xi$. It is determined by a generalized vector field
$\up_E$ on $E$ whose projection
\be
\up:J^kE\ar^{\up_E} VE\to E\op\times_Y VY
\ee
is a linear $VY$-valued differential operator $\up$ (\ref{0509})
on $E$. Let $\vt_E$ be a variational symmetry of a Lagrangian $L$
on $E$, i.e.,
\mar{0552}\beq
\up_E\rfloor \dl L=\up\rfloor \dl L=d_H\si. \label{0552}
\eeq
Then one says that $\up$ (\ref{0509}) is a gauge symmetry of a
Lagrangian $L$.

In accordance with Noether's second theorem \cite{noether}, if a
Lagrangian $L$ (\ref{0512}) admits a gauge symmetry $\up$
(\ref{0509}),  its Euler--Lagrange operator (\ref{0513}) obeys the
Noether identity
\mar{0550}\beq
[\op\sum_{0\leq|\La|\leq m} \Delta^{i,\La}_r d_\La \cE_i]
\xi^r\om=0, \label{0550}
\eeq
where
\mar{0511}\beq
\Delta^{i,\La}_r =\op\sum_{0\leq|\Si|\leq
m-|\La|}(-1)^{|\Si+\La|}C^{|\Si|}_{|\Si+\La|} d_\Si
\up^{i,\Si+\La}_r. \label{0511}
\eeq

For instance, if a gauge symmetry
\mar{0656}\beq
\up=(\up_r^i\xi^r +\up^{i,\m}_r\xi^r_\m)\dr_i \label{0656}
\eeq
is of first jet order in parameters,  the corresponding Noether
identity (\ref{0550}) reads
\mar{0657,8}\ben
&& \Delta^i_r=\up^i_r -d_\m \up^{i,\m}_r,\qquad \Delta^{i,\m}_r=-
\up^{i,\m}_r,\label{0657}\\
&& [\up^i_r\cE_i - d_\m(\up^{i,\m}_r\cE_i)]\xi^r\om=0.
\label{0658}
\een
Gauge models considered below are of this type.

\section{Space-time gauge symmetries}

We consider gravitation theory in the absence of spinor fields.
Its gauge symmetries are general covariant transformations (e.g.,
\cite{jack,fat94,book}). As was mentioned above, gravitation
theory is formulated on natural bundles which admit the canonical
lift of any vector field on $X$. One thinks of such a lift as
being an infinitesimal generator of general covariant
transformations. Natural bundles are exemplified by tensor
bundles, the bundles of world metrics and world connections. One
also considers non-vertical automorphisms of a principal bundle
$P\to X$  and their extensions to the gauge-natural prolongations
of $P$ and the associated  natural-gauge bundles
\cite{fat,fat04,pal}.

We here address the gauge theory of principal connections on a
principal bundle $P\to X$ with a structure Lie group $G$. These
connections are represented by sections of the quotient
\mar{0654}\beq
C=J^1P/G\to X,\label{0654}
\eeq
called the bundle of principal connections \cite{book}. This is an
affine bundle coordinated by $(x^\la, a^r_\la)$ such that, given a
section $A$ of $C\to X$, its components $A^r_\la=a^r_\la\circ A$
are coefficients of the familiar local connection form (i.e.,
gauge potentials). We consider the GDA $\cO^*_\infty C$.

Infinitesimal generators of one-parameter groups of automorphisms
of a principal bundle $P$ are $G$-invariant projectable vector
fields on $P\to X$. They are associated to sections of the vector
bundle $T_GP=TP/G\to X$. This bundle is endowed with the
coordinates $(x^\la,\tau^\la=\dot x^\la,\xi^r)$ with respect to
the fiber bases $\{\dr_\la, e_r\}$ for $T_GP$, where $\{e_r\}$ is
the basis for the right Lie algebra $\ccG$ of $G$ such that
$[e_p,e_q]=c^r_{pq}e_r.$ If
\mar{0652}\beq
 u=u^\la\dr_\la
+u^r e_r, \qquad v=v^\la\dr_\la +v^r e_r, \label{0652}
\eeq
are sections of $T_GP\to X$, their bracket reads
\mar{0651}\beq
[u,v]=(u^\m\dr_\m v^\la -v^\m\dr_\m u^\la)\dr_\la +(u^\la\dr_\la
v^r - v^\la\dr_\la u^r +c^r_{pq}u^pv^q)e_r. \label{0651}
\eeq
Any section $u$ of the vector bundle $T_GP\to X$ yields the vector
field
\mar{0653}\beq
u_C=u^\la\dr_\la +(c^r_{pq}a^p_\la u^q +\dr_\la u^r -a^r_\m\dr_\la
u^\m)\dr^\la_r \label{0653}
\eeq
on the bundle of principal connections $C$ (\ref{0654})
\cite{book}.

Let us consider a subbundle $V_GP=VP/G\to X$ of the vector bundle
$T_GX$ coordinated by $(x^\la, \xi^r)$. Its sections $u=u^re_r$
are infinitesimal generators of vertical automorphisms of $P$.
There is the exact sequence of vector bundles
\be
0\to V_GP \ar T_GP\to TX \to 0.
\ee
Its pull-back onto $C$ admits the canonical splitting which takes
the coordinate form
\mar{gr9}\beq
\tau^\la\dr_\la +\xi^r e_r = \tau^\la(\dr_\la +a^r_\la e_r) +
(\xi^r - \tau^\la a^r_\la)e_r. \label{gr9}
\eeq

Let us consider the bundle product
\mar{0659}\beq
E=C\op\times_X T_GP, \label{0659}
\eeq
coordinated by $(x^\la, a^r_\la, \tau^\la, \xi^r)$. It can be
provided with the generalized vector field
\mar{0660}\beq
\up_E= \up=(c^r_{pq}a^p_\la \xi^q + \xi^r_\la
-a^r_\m\tau^\m_\la-\tau^\m a_{\m\la}^r)\dr^\la_r, \label{0660}
\eeq
taking the form
\mar{gr8}\beq
\up=(c^r_{pq}a^p_\la \xi'^q + \xi'^r_\la +
\tau^\m\cF_{\la\m}^r)\dr^\la_r, \qquad \xi'^r=\xi^r -\tau^\la
a^r_\la, \label{gr8}
\eeq
due to the splitting (\ref{gr9}). For instance, this is a gauge
symmetry of the global Chern--Simons Lagrangian in gauge theory on
a principal bundle with a structure semi-simple Lie group $G$ over
a three-dimensional base $X$ \cite{chern}. Given a section $B$ of
$C\to X$ (i.e., a background gauge potential), this Lagrangian
reads
\mar{r50}\ben
&& L= [\frac12a^G_{mn} \ve^{\al\bt\g}a^m_\al(\cF^n_{\bt\g}
-\frac13 c^n_{pq}a^p_\bt a^q_\g)  -\frac12a^G_{mn}
\ve^{\al\bt\g}B^m_\al(F(B)^n_{\bt\g} \label{r50}\\
&& \qquad -\frac13 c^n_{pq}B^p_\bt B^q_\g) -d_\al(a^G_{mn}
\ve^{\al\bt\g}a^m_\bt B^n_\g)]d^3x, \nonumber\\
&& F(B)^r_{\la\m}=\dr_\la B^r_\m-\dr_\m B^r_\la +c^r_{pq}B^p_\la
B^q_\m, \qquad \cF^r_{\la\m}=a^r_{\la\m}-a^r_{\m\la}
+c^r_{pq}a^p_\la a^q_\m, \nonumber
\een
where $a^G$ is the Killing form. Its first term is the well-known
local Chern--Simons Lagrangian, the second one is a density on
$X$, and the Lie derivative of the third term is $d_H$-exact
because of the relations (\ref{g232}). The corresponding Noether
identities (\ref{0658}) read
\mar{gr1,2}\ben
&& c^r_{pq}a^p_\la\cE_r^\la - d_\la(\cE_q^\la)=0,
\label{gr1}\\
&& -a^r_{\m\la}\cE^\la_r +d_\la(a^r_\m\cE^\la_r)=0. \label{gr2}
\een
The first one is the well-known Noether identity corresponding to
the vertical gauge symmetry
\mar{0662}\beq
\up=(c^r_{pq}a^p_\la \xi^q + \xi^r_\la)\dr^\la_r. \label{0662}
\eeq
The second Noether identity (\ref{gr2}) is brought into the form
\be
-a^r_\m[c^r_{pq}a^p_\la\cE_r^\la - d_\la(\cE_q^\la)]
+\cF^r_{\la\m}\cE^\la_r=0,
\ee
i.e., it is equivalent to the Noether identity
\mar{gr6}\beq
\cF^r_{\la\m}\cE^\la_r=0, \label{gr6}
\eeq
which also comes from the splitting (\ref{gr9}) of the generalized
vector field $\up$.

In the case of Chern--Simons Lagrangian, the Noether identity
(\ref{gr6}) is trivial since its coefficients $\cF^r_{\la\m}$
vanish on the kernel of the Euler--Lagrange operator
\be
\dl L=a^G_{rn}\ve^{\la\m\nu}\cF^n_{\m\nu}\th_\la^r\w\om
\ee
of the Chern--Simons Lagrangian (\ref{r50}).

In order to obtain a gauge symmetry of the Yang--Mills Lagrangian,
one should complete the generalized vector field (\ref{0660}) with
the term acting on a world metric.

Let  $LX$ the fiber bundle of linear frames in the tangent bundle
$TX$ of $X$. It is a principal bundle with the structure group
$GL(n,\Bbb R)$, $n=\di X$, which is reduced to its maximal compact
subgroup $O(n)$. Global sections of the quotient bundle
$\Si=LX/O(n)$ are Riemannian metrics on $X$. If $X$ obeys the
well-known topological conditions, pseudo-Riemannian metrics on
$X$ are similarly described. Being an open subbundle of the tensor
bundle $\op\vee^2 TX$, the bundle $\Si$ is provided with bundle
coordinates $\si^{\m\nu}$. It admits the canonical lift
\mar{gr3}\beq
u_\Si=u^\la\dr_\la +(\si^{\nu\bt}\dr_\nu u^\al
+\si^{\al\nu}\dr_\nu u^\bt)\frac{\dr}{\dr \si^{\al\bt}}
\label{gr3}
\eeq
of any vector field $u=u^\la\dr_\la$ on $X$.

In order to describe the gauge theory of principal connections in
the presence of a dynamic metric field, let us consider the bundle
product
\mar{gr10}\beq
E=C\op\times_X \Si\op\times_X T_GP, \label{gr10}
\eeq
coordinated by $(x^\la,a^r_\la,\si^{\al\bt}, \tau^\la, \xi^r)$. It
can be provided with the generalized vector field
\mar{gr11}\beq
\up=(c^r_{pq}a^p_\la \xi^q + \xi^r_\la -a^r_\m\tau^\m_\la-\tau^\m
a_{\m\la}^r)\dr^\la_r + (\si^{\nu\bt}\tau_\nu^\al +\si^{\al\nu}
\tau_\nu^\bt-\tau^\la\si_\la^{\al\bt})\frac{\dr}{\dr
\si^{\al\bt}}. \label{gr11}
\eeq
This is a gauge symmetry of the sum $L=L_{\rm YM} + L_g$ of the
Yang--Mills Lagrangian $L_{\rm YM}(\cF^r_{\al\bt}, \si^{\m\nu})$
and a Lagrangian $L_g$ of a metric field. The corresponding
Noether identities read
\mar{gr12,3}\ben
&& c^r_{pq}a^p_\la\cE_r^\la - d_\la(\cE_q^\la)=0,
\label{gr12}\\
&& -a^r_{\m\la}\cE^\la_r +d_\la(a^r_\m\cE^\la_r)
-\si_\m^{\al\bt}\cE_{\al\bt} - 2d_\nu (\si^{\nu\bt}
\cE_{\mu\bt})=0. \label{gr13}
\een
The first one is the Noether identity (\ref{gr1}), while the
second identity is brought into the form
\be
-a^r_{\m\la}\cE^\la_r +d_\la(a^r_\m\cE^\la_r) - 2\nabla_\nu
(\si^{\nu\bt} \cE_{\mu\bt})=0,
\ee
where $\nabla_\nu$ are covariant derivatives with respect to the
Levi--Civita connection
\be
K=dx^\la\ot (\dr_\la + K_\la{}^\m{}_\nu \dot x^\nu\dot\dr_\mu),
\qquad K_\la{}^\m{}_\nu =-\frac12\si^{\nu\bt}(\si_{\la\bt\m} +
\si_{\m\bt\la} -\si_{\bt\la\m}).
\ee

In metric-affine gravitation theory, dynamic variables are a world
metric and a world connection. World connections are principal
connections on the frame bundle $LX$, and they are represented by
sections of the quotient fiber bundle
\mar{gr14}\beq
C_K=J^1LX/GL(n,\Bbb R). \label{gr14}
\eeq
This fiber bundle is provided with bundle coordinates
$(x^\la,k_\la{}^\nu{}_\al)$ such that, for any section $K$ of
$C_K\to X$, its coordinates $k_\la{}^\nu{}_\al\circ
K=K_\la{}^\nu{}_\al$ are coefficient of the linear connection
\be
K=dx^\la\ot (\dr_\la + K_\la{}^\m{}_\nu \dot x^\nu\dot\dr_\mu)
\ee
on $TX$. The bundle of world connections (\ref{gr14}) admits the
canonical lift
\be
u_K=u^\m\dr_\m +[\dr_\nu u^\al k_\m{}^\nu{}_\bt -\dr_\bt u^\nu
k_\m{}^\al{}_\nu -\dr_\mu u^\nu k_\nu{}^\al{}_\bt
+\dr_{\m\bt}u^\al]\frac{\dr}{\dr k_\mu{}^\al{}_\bt}.
\ee

In order to describe the gauge theory of principal connections in
the presence of metric-affine gravity, let us consider the bundle
product
\mar{gr16}\beq
E=C\op\times_X \Si\op\times_X C_K\op\times_X T_GP, \label{gr16}
\eeq
coordinated by $(x^\la,a^r_\la,\si^{\al\bt}, k_\la{}^\nu{}_\al,
\tau^\la, \xi^r, )$. It can be provided with the generalized
vector field
\mar{gr15}\ben
&& \up=(c^r_{pq}a^p_\la \xi^q + \xi^r_\la
-a^r_\m\tau^\m_\la-\tau^\m a_{\m\la}^r)\dr^\la_r +
(\si^{\nu\bt}\dr_\nu \tau^\al +\si^{\al\nu}\dr_\nu
\tau^\bt-\tau^\la\si_\la^{\al\bt})\frac{\dr}{\dr \si^{\al\bt}}
\label{gr15}\\
&& \qquad (\tau_\nu^\al k_\m{}^\nu{}_\bt -\tau_\bt^\nu
k_\m{}^\al{}_\nu -\tau_\mu^\nu k_\nu{}^\al{}_\bt
+\tau_{\m\bt}^\al-\tau^\la k_{\la\mu}{}^\al{}_\bt)\frac{\dr}{\dr
k_\mu{}^\al{}_\bt}.  \nonumber
\een
This is a gauge symmetry of the sum $L=L_{\rm YM} + L_{\rm MA}$ of
the Yang--Mills Lagrangian $L_{\rm YM}(\cF^r_{\al\bt},
\si^{\m\nu})$ and a metric-affine Lagrangian $L_{\rm MA}$.

\section{BRST symmetries}

In order to introduce BRST symmetries, let us consider Lagrangian
systems of even and odd variables. We describe odd variables and
their jets on a smooth manifold $X$ as generating elements of the
structure ring of a graded manifold whose body is $X$
\cite{cmp,ijmp}. This definition reproduces the heuristic notion
of jets of ghosts in the field-antifield BRST theory
\cite{barn,bran01}.

Recall that any graded manifold $(\gA,X)$ with a body $X$ is
isomorphic to the one whose structure sheaf $\gA_Q$ is formed by
germs of sections of the exterior product
\mar{g80}\beq
\w Q^*=\Bbb R\op\oplus_X Q^*\op\oplus_X\op\w^2
Q^*\op\oplus_X\cdots, \label{g80}
\eeq
where $Q^*$ is the dual of some real vector bundle $Q\to X$ of
fiber dimension $m$. In field models, a vector bundle $Q$ is
usually given from the beginning. Therefore, we consider graded
manifolds $(X,\gA_Q)$ where the above mentioned isomorphism holds,
and call $(X,\gA_Q)$ the simple graded manifold constructed from
$Q$. The structure ring $\cA_Q$ of sections of $\gA_Q$ consists of
sections of the exterior bundle (\ref{g80}) called graded
functions. Let $\{c^a\}$ be the fiber basis for $Q^*\to X$,
together with transition functions $c'^a=\rho^a_bc^b$. It is
called the local basis for the graded manifold $(X,\gA_Q)$. With
respect to this basis, graded functions read
\be
f=\op\sum_{k=0}^m \frac1{k!}f_{a_1\ldots a_k}c^{a_1}\cdots
c^{a_k},
\ee
where $f_{a_1\cdots a_k}$ are local smooth real functions on $X$.

Given a graded manifold $(X,\gA_Q)$, let $\gd\cA_Q$ be the
$\cA_Q$-module of $\Bbb Z_2$-graded derivations of the $\Bbb
Z_2$-graded ring of $\cA_Q$, i.e.,
\be
u(ff')=u(f)f'+(-1)^{[u][f]}fu (f'), u\in\gd\cA_Q, \qquad f,f'\in
\cA_Q,
\ee
where $[.]$ denotes the Grassmann parity. Its elements are called
$\Bbb Z_2$-graded (or, simply, graded) vector fields on
$(X,\gA_Q)$. Due to the canonical splitting $VQ= Q\times Q$, the
vertical tangent bundle $VQ\to Q$ of $Q\to X$ can be provided with
the fiber bases $\{\dr_a\}$ which is the dual of $\{c^a\}$. Then a
graded vector field takes the local form $u= u^\la\dr_\la +
u^a\dr_a$, where $u^\la, u^a$ are local graded functions. It acts
on $\cA_Q$ by the rule
\mar{cmp50'}\beq
u(f_{a\ldots b}c^a\cdots c^b)=u^\la\dr_\la(f_{a\ldots b})c^a\cdots
c^b +u^d f_{a\ldots b}\dr_d\rfloor (c^a\cdots c^b). \label{cmp50'}
\eeq
This rule implies the corresponding transformation law
\be
u'^\la =u^\la, \qquad u'^a=\rho^a_ju^j +
u^\la\dr_\la(\rho^a_j)c^j.
\ee
Then one can show that graded vector fields on a simple graded
manifold can be represented by sections of the vector bundle
$\cV_Q\to X$, locally isomorphic to $\w Q^*\ot_X(Q\oplus_X TX)$.

Accordingly, graded exterior forms on the graded manifold
$(X,\gA_Q)$ are introduced as sections of the exterior bundle
$\op\w\cV^*_Q$, where $\cV^*_Q\to X$ is the $\w Q^*$-dual of
$\cV_Q$. Relative to the dual local bases $\{dx^\la\}$ for $T^*X$
and $\{dc^b\}$ for $Q^*$, graded one-forms read
\be
\f=\f_\la dx^\la + \f_adc^a,\qquad \f'_a=\rho^{-1}{}_a^b\f_b,
\qquad \f'_\la=\f_\la +\rho^{-1}{}_a^b\dr_\la(\rho^a_j)\f_bc^j.
\ee
The duality morphism is given by the interior product
\be
u\rfloor \f=u^\la\f_\la + (-1)^{[\f_a]}u^a\f_a.
\ee
Graded exterior forms constitute the bigraded differential algebra
(henceforth BGDA) $\cC^*_Q$ with respect to the bigraded exterior
product $\w$ and the exterior differential $d$.

Since the jet bundle $J^rQ\to X$ of a vector bundle $Q\to X$ is a
vector bundle, let us consider the simple graded manifold
$(X,\gA_{J^rQ})$ constructed from $J^rQ\to X$. Its local basis is
$\{x^\la,c^a_\La\}$, $0\leq |\La|\leq r$, together with the
transition functions
\mar{+471}\beq
c'^a_{\la +\La}=d_\la(\rho^a_j c^j_\La), \qquad d_\la=\dr_\la +
\op\sum_{|\La|<r}c^a_{\la+\La} \dr_a^\La, \label{+471}
\eeq
where $\dr_a^\La$ are the duals of $c^a_\La$. Let $\cC^*_{J^rQ}$
be the BGDA of graded exterior forms on the graded manifold
$(X,\gA_{J^rQ})$. A linear bundle morphism $\pi^r_{r-1}:J^rQ \to
J^{r-1}Q$ yields the corresponding monomorphism of BGDAs
$\cC^*_{J^{r-1}Q}\to \cC^*_{J^rQ}$. Hence, there is the direct
system of BGDAs
\mar{g205}\beq
\cC^*_Q\ar^{\pi^{1*}_0} \cC^*_{J^1Q}\cdots
\ar^{\pi^r_{r-1}{}^*}\cC^*_{J^rQ}\ar\cdots. \label{g205}
\eeq
Its direct limit $\cC^*_\infty Q$ consists of graded exterior
forms on graded manifolds $(X,\gA_{J^rQ})$, $r\in\Bbb N$, modulo
the pull-back identification, and it inherits the BGDA operations
intertwined by the monomorphisms $\pi^r_{r-1}{}^*$. It is a
$C^\infty(X)$-algebra locally generated by the elements $(1,
c^a_\La, dx^\la,\th^a_\La=dc^a_\La -c^a_{\la +\La}dx^\la)$,
$0\leq|\La|$.

 In order to regard even and odd dynamic variables on the
same footing,  let $Y\to X$ be hereafter an affine bundle, and let
$\cP^*_\infty Y\subset \cO^*_\infty Y$ be the
$C^\infty(X)$-subalgebra of exterior forms whose coefficients are
polynomial in the fiber coordinates $y^i_\La$ on jet bundles $J^r
Y\to X$. Let us consider the product
\mar{0670}\beq
\cS^*_\infty=\cC_\infty^*Q\w\cP^*_\infty Y \label{0670}
\eeq
of graded algebras $\cC_\infty^*Q$ and $\cP^*_\infty Y$ over their
common graded subalgebra $\cO^*X$ of exterior forms on $X$
\cite{cmp}. It consists of the elements
\be
\op\sum_i \psi_i\ot\f_i, \qquad \op\sum_i \f_i\ot\psi_i, \qquad
\psi\in \cC^*_\infty Q, \qquad \f\in \cP^*_\infty Y,
\ee
modulo the commutation relations
\mar{0442}\ben
&&\psi\ot\f=(-1)^{|\psi||\f|}\f\ot\psi, \qquad
\psi\in \cC^*_\infty Q, \qquad \f\in \cP^*_\infty Y, \label{0442}\\
&& (\psi\w\si)\ot\f=\psi\ot(\si\w\f), \qquad \si\in \cO^*X.
\nonumber
\een
They are  endowed with the total form degree $|\psi|+|\f|$ and the
total Grassmann parity $[\psi]$. Their multiplication
\mar{0440}\beq
(\psi\ot\f)\w(\psi'\ot\f'):=(-1)^{|\psi'||\f|}(\psi\w\psi')\ot
(\f\w\f'). \label{0440}
\eeq
obeys the relation
\be
\vf\w\vf' =(-1)^{|\vf||\vf'| +[\vf][\vf']}\vf'\w \vf, \qquad
\vf,\vf'\in \cS^*_\infty,
\ee
and makes $\cS^*_\infty$ (\ref{0670}) into a bigraded $C^\infty
(X)$-algebra.  For instance, elements of the ring $S^0_\infty$ are
polynomials of $c^a_\La$ and $y^i_\La$ with coefficients in
$C^\infty(X)$.

The algebra $\cS^*_\infty$ is provided with the exterior
differential
\mar{0441}\beq
d(\psi\ot\f):=(d_\cC\psi)\ot\f +(-1)^{|\psi|}\psi\ot(d_\cP\f),
\qquad \psi\in \cC^*_\infty, \qquad \f\in \cP^*_\infty,
\label{0441}
\eeq
where $d_\cC$ and $d_\cP$ are exterior differentials on the
differential algebras $\cC^*_\infty Q$ and $\cP^*_\infty Y$,
respectively. It obeys the relations
\be
 d(\vf\w\vf')= d\vf\w\vf' +(-1)^{|\vf|}\vf\w d\vf', \qquad
\vf,\vf'\in \cS^*_\infty,
\ee
and makes $\cS^*_\infty$ into a BGDA. Hereafter, let the
collective symbols $s^A_\La$ stand both for even and odd
generating elements $c^a_\La$, $y^i_\La$ of the $C^\infty(X)$-ring
$\cS^0_\infty$. Then the BGDA $\cS^*_\infty$ is locally generated
by $(1,s^A_\La, dx^\la, \th^A_\La=ds^A_\La -s^A_{\la+\La}dx^\la)$,
$|\La|\geq 0$. We agree to call elements of $\cS^*_\infty$ the
graded exterior forms on $X$.

Similarly to $\cO^*_\infty Y$, the BGDA $\cS^*_\infty$ is
decomposed into $\cS^0_\infty$-modules $\cS^{k,r}_\infty$ of
$k$-contact and $r$-horizontal graded forms together with the
corresponding projections $h_k$ and $h^r$. Accordingly, the
exterior differential $d$ (\ref{0441}) on $\cS^*_\infty$ is split
into the sum $d=d_H+d_V$ of the total and vertical differentials
\be
d_H(\f)=dx^\la\w d_\la(\f), \qquad d_V(\f)=\th^A_\La\w\dr^\La_A
\f, \qquad \f\in \cS^*_\infty.
\ee
One can think of the elements
\be
L=\cL\om\in \cS^{0,n}_\infty, \qquad \dl (L)= \op\sum_{|\La|\geq
0}
 (-1)^{|\La|}\th^A\w d_\La (\dr^\La_A L)\in \cS^{0,n}_\infty
\ee
as being a graded Lagrangian and its Euler--Lagrange operator,
respectively.

A graded derivation  $\vt\in\gd \cS^0_\infty$ of the $\Bbb R$-ring
$\cS^0_\infty$ is said to be contact if the Lie derivative
$\bL_\vt$ preserves the ideal of contact graded forms of the BGDA
$\cS^*_\infty$. With respect to the local basis $(x^\la,s^A_\La,
dx^\la,\th^A_\La)$ for the BGDA $\cS^*_\infty$, any contact graded
derivation takes the form
\mar{g105}\beq
\vt=\vt_H+\vt_V=\vt^\la d_\la + (\vt^A\dr_A +\op\sum_{|\La|>0}
d_\La\vt^A\dr_A^\La), \label{g105}
\eeq
where $\vt^\la$, $\vt^A$ are local graded functions \cite{cmp}.
The interior product $\vt\rfloor\f$ and the Lie derivative
$\bL_\vt\f$, $\f\in\cS^*_\infty$, are defined by the formulae
\be
&& \vt\rfloor \f=\vt^\la\f_\la + (-1)^{[\f_A]}\vt^A\f_A, \qquad
\f\in \cS^1_\infty,\\
&& \vt\rfloor(\f\w\si)=(\vt\rfloor \f)\w\si
+(-1)^{|\f|+[\f][\vt]}\f\w(\vt\rfloor\si), \qquad \f,\si\in
\cS^*_\infty, \\
&& \bL_\vt\f=\vt\rfloor d\f+ d(\vt\rfloor\f), \qquad
\bL_\vt(\f\w\si)=\bL_\vt(\f)\w\si
+(-1)^{[\vt][\f]}\f\w\bL_\vt(\si).
\ee


The Lie derivative $\bL_\vt L$ of a Lagrangian $L$ along a contact
graded derivation $\vt$ (\ref{g105}) fulfills the first
variational formula
\mar{g107}\beq
\bL_\vt L= \vt_V\rfloor\dl L +d_H(h_0(\vt\rfloor \Xi_L)) + d_V
(\vt_H\rfloor\om)\cL, \label{g107}
\eeq
where $\Xi_L=\Xi+L$ is a Lepagean equivalent of a graded
Lagrangian $L$ \cite{cmp}.

A contact graded derivation $\vt$ is said to be variational if the
Lie derivative (\ref{g107}) is $d_H$-exact. A glance at the
expression (\ref{g107}) shows that: (i) a contact graded
derivation $\vt$ is variational only if it is projected onto $X$,
and (ii) $\vt$ is variational iff its vertical part $\vt_V$ is
well. Therefore, we restrict our consideration to vertical contact
graded derivations
\mar{0672}\beq
\vt=\op\sum_{0\leq|\La|} d_\La\up^A\dr_A^\La. \label{0672}
\eeq
Such a derivation is completely defined by its first summand
\mar{0673}\beq
\up=\up^A(x^\la,s^A_\La)\dr_A, \qquad 0\leq|\La|\leq k,
\label{0673}
\eeq
which is also a graded derivation of $\cS^0_\infty$. It is called
the generalized graded vector field. A glance at the first
variational formula (\ref{g107}) shows that $\vt$ (\ref{0672}) is
variational iff $\up\rfloor \dl L$ is $d_H$-exact.

A vertical contact graded derivation $\vt$ (\ref{0672}) is said to
be nilpotent if
\mar{g133}\beq
\bL_\up(\bL_\up\f)= \op\sum_{|\Si|\geq 0,|\La|\geq 0 }
(\up^B_\Si\dr^\Si_B(\up^A_\La)\dr^\La_A +
(-1)^{[s^B][\up^A]}\up^B_\Si\up^A_\La\dr^\Si_B \dr^\La_A)\f=0
\label{g133}
\eeq
for any horizontal graded form $\f\in S^{0,*}_\infty$ or,
equivalently, $(\vt\circ\vt)(f)=0$ for any graded function $f\in
\cS^0_\infty$. One can show that $\vt$ is nilpotent only if it is
odd and iff the equality
\mar{0688}\beq
\vt(\up^A)=\op\sum_{|\Si|\geq 0} \up^B_\Si\dr^\Si_B(\up^A)=0
\label{0688}
\eeq
holds for all $\up^A$ \cite{cmp}.

Return now to the original gauge system on a fiber bundle $Y$ with
a Lagrangian $L$ (\ref{0512}) and a gauge symmetry $\up$
(\ref{0509}). For the sake of simplicity, $Y\to X$ is assumed to
be affine. Let us consider the BGDA
$\cS^*_\infty[V;Y]=\cC^*_\infty V\w\cP^*_\infty Y$ locally
generated by $(1,c^r_\La, dx^\la, y^i_\La, \th^r_\La, \th^i_\La)$.
Let $L\in \cO^{0,n}_\infty Y$ be a polynomial in $y^i_\La$, $0\leq
|L|$. Then it is a graded Lagrangian $L\in \cP^{0,n}_\infty
Y\subset \cS^{0,n}_\infty[V;Y]$ in $\cS^*_\infty[V;Y]$. Its gauge
symmetry $\up$ (\ref{0509}) gives rise to the generalized vector
field $\up_E=\up$ on $E$, and the latter defines the generalized
graded vector field $\up$ (\ref{0673}) by the formula
(\ref{0680}). It is easily justified that the contact graded
derivation $\vt$ (\ref{0672}) generated by $\up$ (\ref{0680}) is
variational for $L$. It is odd, but need not be nilpotent.
However, one can try to find a nilpotent contact graded derivation
(\ref{0672}) generated by some generalized graded vector field
(\ref{0684}) which coincides with $\vt$ on $\cP^*_\infty Y$. Then
$\up$ (\ref{0684}) is  called a BRST symmetry.

In this case, the nilpotency conditions (\ref{0688}) read
\mar{0690,1}\ben
&& \op\sum_\Si d_\Si(\op\sum_\Xi\up^{i,\Xi}_rc^r_\Xi)
\op\sum_\La\dr^\Si_i (\up^{j,\La}_s)c^s_\La +\op\sum_\La d_\La
(u^r)\up^{j,\La}_r
=0, \label{0690}\\
&& \op\sum_\La(\op\sum_\Xi d_\La(\up^{i,\Xi}_r c^r_\Xi)\dr^\La_i
+d_\La (u^r)\dr_r^\La)u^q=0\label{0691}
\een
for all indices $j$ and $q$. They are equations for graded
functions $u^r\in\cS^0_\infty[V;Y]$. Since these functions are
polynomials
\mar{0693}\beq
u^r=u_{(0)}^r + \op\sum_\G u_{(1)p}^{r,\G} c^p_\G +
\op\sum_{\G_1,\G_2} u_{(2)p_1p_2}^{r,\G_1\G_2}
c^{p_1}_{\G_1}c^{p_2}_{\G_2} +\cdots \label{0693}
\eeq
in $c^s_\La$, the equations (\ref{0690}) -- (\ref{0691}) take the
form
\mar{0694}\ben
&& \op\sum_\Si d_\Si(\op\sum_\Xi\up^{i,\Xi}_rc^r_\Xi)
\op\sum_\La\dr^\Si_i (\up^{j,\La}_s)c^s_\La +\op\sum_\La d_\La
(u_{(2)}^r)\up^{j,\La}_r
=0, \label{0694a}\\
&& \op\sum_\La d_\La (u_{(k\neq 2)}^r)\up^{j,\La}_r =0,
\label{0694b}\\
&& \op\sum_\La\op\sum_\Xi d_\La(\up^{i,\Xi}_r c^r_\Xi)\dr^\La_i
u_{(k-1)}^q +\op\sum_{m+n-1=k}d_\La (u_{(m)}^r)\dr_r^\La u_{(n)}^q
=0. \label{0694c}
\een
One can think of the equalities (\ref{0694a}) and (\ref{0694c})
 as being the generalized commutation relations and
generalized Jacobi identities of original gauge transformations,
respectively \cite{algebr}.



\section{Space-time BRST symmetries}

Let us consider gauge symmetries (\ref{0660}), (\ref{gr11}) and
(\ref{gr15}). Following the procedure in Section 4, we replace
parameters $\xi^r$ and $\tau^\la$ with the odd ghosts $c^r$ and
$c^\la$, respectively, and obtain the generalized graded vector
fields
\mar{gr20-22}\ben
&& \up= (c^r_{pq}a^p_\la c^q + c^r_\la -a^r_\m c^\m_\la-c^\m
a_{\m\la}^r)\dr^\la_r +(-\frac12c^r_{pq}c^pc^q -c^\m c^r_\m)\dr_r
+ c^\la_\m c^\m\dr_\la, \label{gr20}\\
&& \up= (c^r_{pq}a^p_\la c^q + c^r_\la -a^r_\m c^\m_\la-c^\m
a_{\m\la}^r)\dr^\la_r  + (\si^{\nu\bt} c_\nu^\al +\si^{\al\nu}
c_\nu^\bt-c^\la\si_\la^{\al\bt})\frac{\dr}{\dr \si^{\al\bt}}
+ \label{gr21}\\
&& \qquad (-\frac12c^r_{pq}c^pc^q -c^\m c^r_\m)\dr_r + c^\la_\m
c^\m\dr_\la, \nonumber\\
&& \up=(c^r_{pq}a^p_\la c^q + c^r_\la -a^r_\m c^\m_\la-c^\m
a_{\m\la}^r)\dr^\la_r + (\si^{\nu\bt} c_\nu^\al +\si^{\al\nu}
c_\nu^\bt-c^\la\si_\la^{\al\bt})\frac{\dr}{\dr \si^{\al\bt}}+
\label{gr22}\\
&& \qquad (c_\nu^\al k_\m{}^\nu{}_\bt -c_\bt^\nu k_\m{}^\al{}_\nu
-c_\mu^\nu k_\nu{}^\al{}_\bt +c_{\m\bt}^\al-c^\la
k_{\la\mu}{}^\al{}_\bt)\frac{\dr}{\dr k_\mu{}^\al{}_\bt}
+(-\frac12c^r_{pq}c^pc^q -c^\m c^r_\m)\dr_r + c^\la_\m
c^\m\dr_\la. \nonumber
\een
The vertical contact graded derivations (\ref{0672}) generated by
these generalized graded vector fields are nilpotent, i.e., these
generalized graded vector fields are BRST symmetries.

It should be noted that all the BRST symmetries (\ref{gr20}) --
(\ref{gr22}) possesses the same ghost term
\be
(-\frac12c^r_{pq}c^pc^q -c^\m c^r_\m)\dr_r + c^\la_\m c^\m\dr_\la.
\ee
It is not surprised because this term corresponds to the bracket
of the vector fields (\ref{0652}).


\begin{thebibliography}{ddd}

\bibitem{barn} G.Barnich, F.Brandt  and M.Henneaux, Local
BRST cohomology in gauge theories, {\it Phys. Rep.} {\bf 338}, 439
(2000).

\bibitem{noether} D.Bashkirov, G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Noether's second theorem in a general setting.
Reducible gauge theories, {\it E-print arXiv}: math.DG/0411070.

\bibitem{brst} D.Bashkirov, G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Noether's second theorem for BRST symmetries,
{\it E-print arXiv}: math-ph/0412034.


\bibitem{bran01} F.Brandt, Jet coordinates for local BRST
cohomology, {\it Lett. Math. Phys.} {\bf 55}, 149 (2001).

\bibitem{jack} Dongsu Bak, D.Cangemi and R.Jackiw, Energy-momentum
conservation in gravity theories, {\it Phys. Rev. D} {\bf 49}
(1994) 5173-5181.

\bibitem{fat94} L.Fatibene, M.Ferraris and M.Francaviglia, N\"other
formalism for conserved quantities in classical gauge field
theories, {\it J. Math. Phys.} {\bf 35} (1994) 1644-1657

\bibitem{fat} L.Fatibene and M.Francaviglia, {\it Natural and
Gauge Natural Formalism for Classical Field Theories. A Geometric
Perspective Including Spinors and Gauge Theories} (Kluwer,
Dordrecht, 2003).

\bibitem{fat04} L.Fatibene, M.Ferraris and M.Francaviglia, On the
gauge natural structure of modern physics, {\it Int. J. Geom.
Methods Mod. Phys.} {\bf 1} (2004) 443-466.


\bibitem{book} G.Giachetta, L.Mangiarotti and G.Sardanashvily,
{\it New Lagrangian and Hamiltonian Methods in Field Theory}
(World Scientific, Singapore, 1997).

\bibitem{chern} G.Giachetta, L.Mangiarotti and G.Sardanashvily, Noether
conservation laws in higher-dimensional Chern-Simons theory, {\it
Mod. Phys. Lett. A} {\bf 18}, 2645 (2003); {\it E-print arXiv:}
math-ph/031067.

\bibitem{cmp} G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Lagrangian supersymmetries depending on
derivatives. Global analysis and cohomology, {\it Commun. Math.
Phys.} (accepted); {\it E-print arXiv}: hep-th/0407185.

\bibitem{pal} M.Palese and E.Winterroth, Global generalized
Bianchi identities for invariant variational problems on
gauge-natural bundles, {\it E-print arXiv:} math-ph/0311003.

\bibitem{ijmp} G.Sardanashvily, SUSY-extended field theory,
{\it Int. J. Mod. Phys. A} {\bf 15}, 3095 (2000).

\bibitem{algebr} G.Sardanashvily, On algebras of gauge
transformations in a general setting, {\it E-print arXiv}:
math.QA/0411635.

\end{thebibliography}

\end{document}




\bibitem{fulp02} R.Fulp, T.Lada and J.Stasheff, Sh-Lie
algebras induced by gauge transformations, {\it Comm. Math. Phys.}
{\bf 231}, 25 (2002); {\it E-print arXiv}: math.QA/0012106.


\bibitem{gom} J.Gomis, J.Par\'\i s, J. and S.Samuel,
Antibracket, antifields and gauge theory quantization, {\it Phys.
Rep} {\bf 295}, 1 (1995).





\bibitem{book00} L.Mangiarotti and G.Sardanashvily, {\it
Connections in Classical and Quantum Field Theory} (World
Scientific, Singapore, 2000).

%13
\bibitem{fat} L.Fatibene, M.Ferraris, M.Francaviglia and  R.McLenaghan,
 Generalized symmetries in mechanics and
field theories, {\it J. Math.. Phys.} {\bf 43}, 3147 (2002).

%14

hep-th0412232.extracted_bib
\begin{thebibliography}{ddd}

\bibitem{barn} G.Barnich, F.Brandt  and M.Henneaux, Local
BRST cohomology in gauge theories, {\it Phys. Rep.} {\bf 338}, 439
(2000).

\bibitem{noether} D.Bashkirov, G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Noether's second theorem in a general setting.
Reducible gauge theories, {\it E-print arXiv}: math.DG/0411070.

\bibitem{brst} D.Bashkirov, G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Noether's second theorem for BRST symmetries,
{\it E-print arXiv}: math-ph/0412034.


\bibitem{bran01} F.Brandt, Jet coordinates for local BRST
cohomology, {\it Lett. Math. Phys.} {\bf 55}, 149 (2001).

\bibitem{jack} Dongsu Bak, D.Cangemi and R.Jackiw, Energy-momentum
conservation in gravity theories, {\it Phys. Rev. D} {\bf 49}
(1994) 5173-5181.

\bibitem{fat94} L.Fatibene, M.Ferraris and M.Francaviglia, N\"other
formalism for conserved quantities in classical gauge field
theories, {\it J. Math. Phys.} {\bf 35} (1994) 1644-1657

\bibitem{fat} L.Fatibene and M.Francaviglia, {\it Natural and
Gauge Natural Formalism for Classical Field Theories. A Geometric
Perspective Including Spinors and Gauge Theories} (Kluwer,
Dordrecht, 2003).

\bibitem{fat04} L.Fatibene, M.Ferraris and M.Francaviglia, On the
gauge natural structure of modern physics, {\it Int. J. Geom.
Methods Mod. Phys.} {\bf 1} (2004) 443-466.


\bibitem{book} G.Giachetta, L.Mangiarotti and G.Sardanashvily,
{\it New Lagrangian and Hamiltonian Methods in Field Theory}
(World Scientific, Singapore, 1997).

\bibitem{chern} G.Giachetta, L.Mangiarotti and G.Sardanashvily, Noether
conservation laws in higher-dimensional Chern-Simons theory, {\it
Mod. Phys. Lett. A} {\bf 18}, 2645 (2003); {\it E-print arXiv:}
math-ph/031067.

\bibitem{cmp} G.Giachetta, L.Mangiarotti and
G.Sardanashvily, Lagrangian supersymmetries depending on
derivatives. Global analysis and cohomology, {\it Commun. Math.
Phys.} (accepted); {\it E-print arXiv}: hep-th/0407185.

\bibitem{pal} M.Palese and E.Winterroth, Global generalized
Bianchi identities for invariant variational problems on
gauge-natural bundles, {\it E-print arXiv:} math-ph/0311003.

\bibitem{ijmp} G.Sardanashvily, SUSY-extended field theory,
{\it Int. J. Mod. Phys. A} {\bf 15}, 3095 (2000).

\bibitem{algebr} G.Sardanashvily, On algebras of gauge
transformations in a general setting, {\it E-print arXiv}:
math.QA/0411635.

\end{thebibliography}