Non-Commutative GUTs, Standard Model and $C,P,T$ properties from Seiberg-Witten map
Paolo Aschieri
\noindent Noncommutative generalizations of Yang-Mills theories using Seiberg-Witten map are in general not unique. We study these ambiguities and see that $SO(10)$ GUT, at first order in the noncommutativity parameter $\th$, is unique and therefore is a truly unified theory, while $SU(5)$ is not. We then present the noncommutative Standard Model compatible with $SO(10)$ GUT. We next study the reality, hermiticity and $C,P,T$ properties of the Seiberg-Witten map and of these noncommutative actions at all orders in $\theta$. This allows to compare the Standard Model discussed in \cite{SM} with the present GUT inspired one.
PR1.tex

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\def\QEDp{{{\rm{QED}}_+}}
\def\QEDm{{{\rm{QED}}_-}}
\def\SW{{{\mbox {\small {SW}}}}}
\def\P*{{\Psi^*}}
\def\PL*{{\Psi_{\!L}^{*}}}
\def\PR*{{\Psi_{\!R}^{*}}}
\def\p*{{\psi^*}}
\def\lh{{\widehat \Lambda}}
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\def\stella[#1,#2]{[#1\stackrel{{\displaystyle\star}}{,}#2]}
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\begin{document}

\title*{Non-Commutative GUTs, Standard Model and 
$C,P,T$ properties from Seiberg-Witten map}
\titlerunning{NCGUTs, NCSM and C,P,T}
\author{Paolo Aschieri}
\institute{Dipartimento di Scienze e Tecnologie
Avanzate, Universit\'a del Piemonte\\ Orientale, and INFN, Piazza
Ambrosoli 5, I-15100,  Alessandria, Italy\\
Max-Planck-Institut f\"{u}r Physik F\"{o}hringer Ring 6,
D-80805 M\"{u}nchen\\
Sektion Physik,
Universit\"{a}t M\"{u}nchen Theresienstra\ss e 37, D-80333
M\"{u}nchen
\texttt{aschieri@theorie.physik.uni-muenchen.de}}

\maketitle

\begin{abstract}
\noindent
Noncommutative generalizations of Yang-Mills theories using
Seiberg-Witten map are in general not unique. We study these
ambiguities and see that 
$SO(10)$ GUT, at first order in the noncommutativity 
parameter $\th$, is unique and therefore is a truly 
unified theory, while $SU(5)$ is
not. We then present the noncommutative Standard Model
compatible with $SO(10)$ GUT.
We next study the reality, hermiticity and $C,P,T$ properties of 
the Seiberg-Witten map and of these noncommutative actions at all
orders in $\theta$.
This allows to compare the Standard Model discussed in \cite{SM} with the 
present GUT inspired one.
\end{abstract}

% ----------------------------------------------------------

\section{Introduction}

There are different examples of noncommutative theories,
we here concentrate on the case where noncommutativity
is described by a constant parameter $\theta^{\mu\nu}$.
The commutation relations among the coordinates read
$\stella[x^\mu,x^\nu]\equiv x^\mu\*x^\nu-x^\nu\* x^\mu=i\th^{\mu\nu}$,
where the star product between functions $f,g$ 
is given by $f\*g=f {_{\,}}e^{{i\over 2}{\th}^{\mu\nu}
{\stackrel{\leftarrow}{\partial}_{\!\mu}}{\stackrel{\rightarrow}
{\partial}_{\!\nu}}} {_{\,}}g_{_{_{_{_{}}}}}$.
We do not claim that spacetime has exactly this noncommutativity, 
rather we are interested in investigating a mathematically
sound gauge theory based on this easiest noncommutative structure.
General aspects of this noncommutative theory will then probably 
be in common with more refined choices of $\th$. In particular the 
choice $\theta^{\mu\nu}=$ constant breakes the Lorentz group in a 
spontaneous 
way; in a bigger theory we would like to consider $\th^{\mu\nu}$ 
(or the related $B$ field)  dynamical and not frozen 
to a constant value, thus recovering Lorentz covariance. 
One can also consider gauge theories with $\th$ nondynamical
but frozen to a particular nonconstant value, 
linear in the coordinates, such that one has a (kappa) deformed 
Poincar\'e symmetry, see \cite{Wess}.

Using Seiberg-Witten map \cite{SW}, 
that relates commutative 
gauge fields to noncommutative ones in such a way that commutative 
gauge transformations are mapped in NC gauge transformations, one can
construct NC gauge theories with arbirary gauge groups 
\cite{Madore, Jurco1}.
%see also B.~Jur\v {c}o contibution to the proceedings. 
These theories are invariant under both commutative and noncommutative 
gauge transformations. Along these lines noncommutative 
generalizations of the 
standard model and GUT theories have been studied 
\cite{SM,A}.  
The SW map and the $\*$ product 
allow us to expand these noncommutative actions order by order in $\th$ 
and to express them in terms of ordinary commutative fields
so that one can then study the physics properties of these
$\th$-expanded commutative actions, see for ex. \cite{astron}.
% see also P.~Schupp  contribution to the proceedings.

It turns out that given a commutative YM theory, 
SW map and commutative/noncommutative gauge invariance 
are in general not enough in order to single out a unique 
noncommutative generalization of the
original YM theory. One can follow different criteria in order to
select a specific noncommutative generalization. We here focus on a
classical analysis, in particular imposing the constraint that 
the noncommutative generalization of the Standard Model should 
be compatible with noncommutative GUT theories. 
Another issue would be to single out a noncommutative SM or GUT that 
is well behaved at the quantum level. 
We refer to the problems relative to 
renormalization, see for ex. \cite{Wulkenhaar2}. On the other hand 
chiral gauge anomalies are absent in these models
\cite{Brandt}.
% see also F.~Brandt contribution to the proceedings.

\sk
In this talk, following \cite{A}, we present a general study of the ambiguities
that 
appear when constructing NCYM theories. We then see that at first order in 
$\th$ there is no ambiguity in $SO(10)$ NCYM theory. In particular 
no triple gauge bosons coupling  of the kind  $\th FFF$
 is present.
We further study the noncommutative SM compatible with $SO(10)$.

We next study the reality, hermiticity, 
charge conjugation, parity and time reversal properties of 
the SW map and of $\th$-expanded NCYM theories. 
This constraints the possible freedom in the choice of a
``good'' SW map.
In \cite{Sheikh-Jabbari:2000vi} the $C,P,T$ properties of NCQED
were studied assuming the usual $C,P$ and $T$ transformations
also for noncommutative fields. 
We here  show that the usual
$C,P,T$ transformation on commutative spinors and nonabelian
gauge potentials imply, via SW map, the same $C,P,T$
transformations for the noncommutative spinors and gauge potentials.
We also see that $CPT$ is always a  symmetry of noncommutative
actions. In \cite{Chai} $CPT$ is studied more axiomatically.

The reality property of the SW map is then used to 
analyze the difference between the SM in \cite{SM} and the GUT inspired 
SM proposed here. 
It is a basic one, and can be studied also in a QED model. 
While in \cite{SM}, and in general in the literature,
left and right handed components of a noncommutative spinor field are 
built with the same SW map, we here use and advocate a different
choice: if noncommutative left handed fermions are built with the 
$+\th$ SW map then their right handed companions should be built with 
the $-\th$ SW map; this implies that both noncommutative $\psi_L$ and
$\psi^{\scriptscriptstyle C}_{~L}\equiv
-i\sigma_2{_{_{\,}}}{\psi_{_{\!}R}}^{^{_{_{_{\scriptstyle *}}}}}$
are built with the $+\th$ SW map.
In other words, with this choice, noncommutativity does not
distinguish 
between a left handed fermion and a left handed antifermion, but 
does distinguish between fermions with different chirality.
This appears to be the only choice compatible with GUT theories.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Seiberg-Witten map and NC particle models}
Consider an ordinary ``commutative'' YM action with  gauge group $G$, 
and one fermion multiplet,
$\,\int d^4x \,{-1\,\over 2 g^2}^{\,} 
\Tr(F_{\mu\nu}F^{\mu\nu})  + 
\overline{\Psi}  i
{\fmslashmio D} \Psi \,.$
This action is gauge invariant under 
$
\delta\Psi=i\rho_\Psi(\Lambda)\Psi\label{psi}
$
where $\rho_{\Psi}$ is the representation of $G$  determined by $\Psi$.
Following \cite{Jurco1}
the noncommutative generalization of 
this action is given by 
\beq
\Sh=\int d^4x \,{-1\,\over 2 g^2}^{\,} 
T^{\!}r(\Fh_{\mu\nu}\*\Fh^{\mu\nu})  + 
\overline{\widehat \Psi} \star i
\widehat{\fmslashmio D} \widehat \Psi \label{Action1multiplet}
\eeq
where the noncommutative field strength $\Fh$ is defined by
$
\widehat F_{\mu\nu}  = \pa_\mu\widehat A_\nu 
- \pa_\nu\widehat A_\mu  -i\s[\widehat A_\mu,\widehat A_\nu]\,.
$
The covariant derivative is given by
\beq
\widehat D_\mu \widehat\Psi = \pa_\mu \widehat\Psi 
- i \rho_\Psi(\widehat A_\mu)\star\widehat\Psi\,.
\label{covder}
\eeq
The action (\ref{Action1multiplet}) is invariant under the 
noncommutative gauge transformations 
\beq
\hat\delta\widehat\Psi = i \rho_\Psi(\widehat\Lambda) \star
\widehat\Psi ~~~,~
~~\hat\delta \widehat A_\mu = \pa_\mu\widehat\Lambda 
+ i\s[\widehat\Lambda,\widehat A_\mu]~.
\eeq

The fields $\Ah$, $\Psih$ and $\lh$ are functions of the commutative 
fields $A,\Psi, \Lambda$ and the noncommutativity parameter $\th$
via the SW map \cite{SW}. At first order in $\th$ we have
\begin{eqnarray}
\widehat A_\xi[A,\th] & = & A_\xi 
+ \frac{1}{4} \theta^{\mu\nu}\{A_\nu,\pa_\mu A_\xi\} + 
\frac{1}{4} \theta^{\mu\nu}\{F_{\mu\xi},A_\nu\} +  \mathcal{O}(\theta^2)
\label{SWA}\\
\widehat\Lambda[\Lambda, A, \th]  &=&  \Lambda 
+ \frac{1}{4} \theta^{\mu\nu}\{\pa_\mu \Lambda , A_\nu\}+ \mathcal{O}(\theta^2)\label{SWLambda}~\\
\widehat \Psi[\Psi,A,\th] & = & \Psi 
+ \frac{1}{2} \theta^{\mu\nu}\rho_\Psi(A_\nu)\pa_\mu\Psi
+\frac{i}{8}\theta^{\mu\nu}[\rho_\Psi(A_\mu), \rho_\Psi(A_\nu)] \Psi + \mathcal{O}(\theta^2)
 \label{SWPsi}
\end{eqnarray}
In terms of the commutative fields 
the action (\ref{Action1multiplet}) is also invariant
under the ordinary gauge transformation 
$\delta A_\mu = \pa_\mu\Lambda  + i[\Lambda, A_\mu]$,
$\delta\Psi=i\rho_\Psi(\Lambda)\Psi$.
\sk
In (\ref{Action1multiplet}) the information on the gauge group $G$ 
is through the dependence of the noncommutative fields on the 
commutative ones. The commutative
gauge potential $A$ and gauge parameter $\Lambda$ are valued in the
$G$  Lie algebra, $A=A^aT^a, \L =\L^aT^a$; and from 
(\ref{SWA}), (\ref{SWLambda})
it follows that $\Ah$ and $\Lambdah$ 
are valued in the universal enveloping 
algebra of the $G$ Lie algebra. 
However, due to the SW map, the degrees of freedom of $\Ah$ are the same as that of $A$.
Similarly to $\Ah$, also $\Fh$ is valued in the universal enveloping 
algebra of $G$.
Now expression (\ref{Action1multiplet}) is ambiguous because 
in  $T^{\!}r(\Fh_{\mu\nu}\*\Fh^{\mu\nu})$ we have not specified 
the representation $\rho(T^a)$. 
We can render explicit the ambiguity in (\ref{Action1multiplet}) 
by writing
\beq
{1\over g^2}T^{\!}r(\Fh_{\mu\nu}\*\Fh^{\mu\nu})=
\sum_\rho c_\rho\Tr(\rho(\Fh_{\mu\nu})\*\rho(\Fh^{\mu\nu}))
\label{ambiguity}
\eeq
where the sum is extended over all unitary irreducible and
inequivalent representations $\rho$ of $G$. 
The real coefficients $c_\rho$ parametrize the ambiguity in 
(\ref{ambiguity}). They are constrained by 
requiring that in the commutative limit, 
$\th\rightarrow 0$, (\ref{ambiguity}) becomes the correctly normalized
commutative gauge kinetic term.

The ambiguity (\ref{ambiguity}) in the action (\ref{Action1multiplet})
can also be studied by expanding (\ref{ambiguity}) in terms of the
commutative fields $\Psi,A,F$.  At first order in $\th$ we have
\bea
\widehat{S}_{gauge}&=&
-\frac{1}{4g^2}\int \! d^4x 
\sum_{a=1}^{{\mbox{{\tiny{dim }}}}{\!G}}
F^a_{\mu \nu}F^{a\,\mu \nu}_{\!}\nonumber\\
& &\,+\,(_{\,}{\mbox{$\sum_\rho$} c_\rho D_\rho^{abc}}) \,\,\,
{\theta^{\mu \nu}\over 4}\!\int \!d^4x^{\,\,\,} {1\over 4}F^a_{\mu \nu}
F^b_{\rho \sigma} F^{c\,\rho \sigma}\, -\,F^a_{\mu \rho} F^b_{\nu \sigma}
F^{c\,\rho\sigma}~~~~~~~~~
\label{traccia}
\eea
where 
\beq
{1\over 2}D_\rho^{abc}\equiv\Tr(\rho(T^a)\{\rho(T^b),\rho(T^c)\})
= {\cal A(\rho)}\Tr(t^a\{t^b,t^c\})\equiv
{1\over 2}{\cal A(\rho)}d^{abc}\label{Dterm}~.
\eeq
Here $t^a$ denotes the fundamental representation, and we are
using that the completely symmetric $D_\rho^{abc}$ tensor in the
representation $\rho$ is proportional to the $d^{abc}$ one defined
by the fundamental representation. In particular 
for all simple Lie groups, except $SU(N)$ with $N\geq 3$, 
we have $D_\rho^{abc}=0$
for any representation $\rho$.
Thus from  (\ref{traccia}) we see that at first order in $\th$  
the ambiguity (\ref{ambiguity}) is present just for  $SU(N)$ Lie
groups.
% and it is equivalent to the choice
%of the real number $\sum_\rho c_\rho{\cal A(\rho)}$. 
\sk
Among the possible representations that one can choose in 
(\ref{ambiguity}) there are two natural ones. 
The fermion representation and the adjoint representation. 
The adjoint representation is particularly appealing if we just have a pure
gauge action, then, since only the structure constants appear in the
commutative gauge kinetic term $\sum_a F_{\mu\nu}^aF^{a\,\mu\nu}$, 
a possible choice is indeed to consider only the
adjoint representation. This is a minimal choice in the sense that in
this case only structure constants enter (\ref{ambiguity}).
(It can be shown \cite{A} that in this case the gauge action is even
in $\th$).
If we also have matter fields then from (\ref{covder})
we see that we must consider the particle representation  
$\rho_\Psi$ given by the multiplet $\Psi$ (and inherited by $\Psih$). 
In (\ref{ambiguity}) one could then make the minimal choice
of selecting just the $\rho_\Psi$ representation.

Along the lines of the above NCYM theories framework we now examine
the $SO(10)$, the $SU(5)$  and the Standard Model noncommutative 
gauge theories.
\sk

\noindent{\bf Noncommutative \mbox{\boldmath ${ SO(10)}~$}}
We consider only one fermion generation: the $16$-dimensional spinor representation of
$SO(10)$ usually denoted $16^+$ (no relevant new effects appear
considering all three families).
We write the left handed multiplet as 
\beq
\Psi_L^+=(u^i,d^i\,,\,-u^{{\scriptscriptstyle C}}_i\,,\,
d^{{\scriptscriptstyle   C}}_i\,,\,\nu,e^-\,,\,e^+\,,\,
-\nu^{\scriptscriptstyle  C})_L
\label{ml}
\eeq
where $i$ is the $SU(3)$ color index and 
$\nu^{\scriptscriptstyle  C}_{~L}=
-i\sigma_2{_{_{\,}}}{\nu_{_{\!}R}}^{^{_{_{_{\scriptstyle *}}}}}$
is the charge conjugate of the neutrino particle $\nu_R$ 
(not present in the Standard Model).
The gauge and fermion sector of noncommutative $SO(10)$ is then simply
obtained by replacing  $\Psih$ with $\Psih^+_L$ in (\ref{Action1multiplet}).
Notice that no linear term in $\th$, i.e. no cubic term in $F$ can appear. 
This is so because $SO(10)$ is anomaly free: $D_\rho^{abc}=0$ forall $\rho$.
In other words, at first order in $\th$, noncommutative $SO(10)$ gauge
theory is unique. 
\sk
\noindent{\bf Noncommutative \mbox{\boldmath ${ SU(5)}~$}}
 The fermionic sector of $SU(5)$ has the
${\psi^{\scriptscriptstyle C}}_L$ multiplet 
that transforms in the $\overline{5}$ of $SU(5)$ and the 
$\chi_L$ multiplet that
transforms according to the $10$ of $SU(5)$. 
In this case we expect that the adjoint, the ${\overline 5}$
and the $10$ representations enter in (\ref{ambiguity}). 
In principle one can consider the coefficients
$c_{\overline 5}\not=c_{10}$, i.e. while the 
$({\psi^{\scriptscriptstyle C}}_L,\chi_L)$ 
fermion rep.
is $\overline 5\oplus 10$, in (\ref{ambiguity}) the weights
$c_\rho$ of the  ${\overline 5}$ and the $10$ can possibly 
be not the same.
It turns out that only if $c_{\overline 5}\not=c_{10}$ then  
$\sum_\rho c_\rho D_\rho^{abc}\not=0$ in (\ref{traccia}). 
We see that, already at first order in the noncommutativity 
parameter $\th$,  noncommutative $SU(5)$ gauge theory 
is not uniquely determined by the gauge coupling constant $g$, 
but also by the value of $\sum_\rho c_\rho D_\rho^{abc}$.
Thus $SU(5)$ is \emph{not} a truly unified
theory in a noncommutative setting.
It is tempting to set $c_{\overline 5}=c_{10}$ so that 
exactly the fermion representation ${\overline 5}\oplus 10$ enters 
(\ref{traccia}). We then have $\sum_\rho c_\rho D_\rho^{abc}=0$,
(however this relation is not protected by symmetries).

\sk 
\noindent {\bf (GUT inspired) Noncommutative Standard Model $~$}
One proceeds similarly for the SM gauge group. The full ambiguity of
the gauge kinetic term is given in \cite{A}.
About the fermion kinetic term, the fermion vector  
${\widehat \Psi_L}$ is constructed from 
$\Psi_L=(u^i,d^i\,,\,-u^{\scriptscriptstyle C}_i\,,
\,d^{\scriptscriptstyle C}_i\,,\,\nu,e^-\,,\,e^+)_L$.
The  covariant derivative is as in (\ref{covder}), 
with $\Psi\rightarrow \Psi_L$ and with $A_\mu=A_\mu^A\Tcal^A$,
where $\{\Tcal^A\}=\{ Y,T^a_L,T^\ell_S\}$ are the generators of 
$U(1)\otimes SU(2)\otimes SU(3)$.
The fermion kinetic term is then as in  (\ref{Action1multiplet}).
This Standard Model is built using only left handed fermions and
antifermions. We call it GUT inspired because its noncommutative
structure can be embedded in $SO(10)$ GUT. Indeed $\Psi_L$ and 
$\Psi_L^+$  differ just by the extra neutrino 
$\nu^{\scriptscriptstyle  C}_{~L}=
-i\sigma_2{_{_{\,}}}{\nu_{_{\!}R}}^{^{_{_{_{\scriptstyle *}}}}}$;
moreover under an infinitesimal gauge transformation $\lh$,
all fermions in $\Psi_L$ transform with $\lh$ on the left.
This GUT inspired Standard Model differs from the one considered 
in \cite{SM}; indeed here we started from the chiral vector
$\Psi_L$, while there the vector $\Psi'=(u^i_L,d^i_L\,,\,u^i_R\,d^i_R\,,\,\nu_L,e^-_L\,,\,e^-_R)$
is considered. In the commutative case $\int \overline{\Psi'} 
{\fmslashmio D} \Psi'=\int \overline{\Psi_L}  
{\fmslashmio D} \Psi_L$ but in the noncommutative case (see later)
%Subection 4.1 (see also (\ref{er}) and the last lines of
%the next subsection),
this is no more true:  $\int \overline{\widehat{\Psi'\,}} \*
{\widehat{\fmslashmio D}} \widehat{\Psi'\,}\not=
\int \overline{\widehat{\Psi_{\!L}}} \*
{\widehat{\fmslashmio D}}_{_{\,}} \widehat{\Psi_{\!L}}\,$;
if we change  $\th$ into $-\th$  in the right handed sector of 
$\int \overline{\widehat{\Psi'\,}} \*
{\widehat{\fmslashmio D}} \widehat{\Psi'\,}$, 
then the two expressions coincide.

Finally it is a natural choice to consider in the SM gauge kinetic term
only the adjoint rep. and the fermion rep.,
we then have  that at first order in $\theta$ there are no
modifications to the SM gauge kinetic term.
This is so because the fermion rep. is anomaly free:
$D_{\rho_{{_{\rm{fermion}}}}}^{A^{\,_{\!\!}}A'\!A''}=0$ 
and because for $U(1)$ the adjoint rep. is trivial.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sk
\noindent {\bf Higgs Sectors$~$}
While the noncommutative Higgs kinetic and potential
terms are given by 
\beq
%\int d^4x \,\bigg(\,
(\widehat D_\mu \widehat \phi)^\dagger
\star \widehat D^\mu \widehat \phi           
\,+\, \mu^2 {\widehat{\phi}}^{\,\dagger} \star  \widehat \phi - \lambda\,
\widehat{\phi}^{\,\dagger} \star  \widehat{\phi}
\star
{\widehat \phi}^{\,\dagger} \star {\widehat \phi}\,   ~,
%\bigg)
\eeq
a noncommutative version of the SM and GUT Yukawa
terms is not straighforward and requires the introduction of the
hybrid Seiberg-Witten 
maps  ${\widehat{{\quad}}}^{^{_{_{_{\!\!H}}}}}\!$ 
on fermions.
A typical noncommutative Yukawa term then reads
\beq
\widehat{\phi}^{{{\,}}\dagger} \*
\widehat{L_L}^{^{_{_{_{\!\!H}}}}}\*
\widehat{e_R^*}+~herm. ~con_{\!}j.
\label{hsw}
\eeq
where 
$L_L^{}=\left({^{^{^{\!}}}}{_{_{_{\!\!}}}}\right.
{}^{^{\mbox{\sma{$\nu^{}_L $}}}}_{_{\mbox{\sma{$e^{}_L $}}}}
\left.{^{^{^{\!}}}}{_{_{_{\!\!\!}}}}\right)\,$.  
Under an infinitesimal $U(1)\otimes SU(2)\otimes SU(3)$ 
gauge transformation
$\Lambda$, 
${\widehat{{L_L^{}}}}^{^{_{_{_{\!\!H}}}}}\!$ transforms as
$
\delta\,{\widehat{{L_L^{}}}}^{^{_{_{_{\!\!H}}}}}\!=
i\rho_\phi(\lh)\*{\widehat{{L_L^{}}}}^{^{_{_{_{\!\!H}}}}}\!
-i{\widehat{{L_L^{}}}}^{^{_{_{_{\!\!H}}}}}\!\*
\rho_{e_R^*}(\lh)
$.
We see that in the hybrid SW map $\lh$ appears both on the
left and on the right of the fermions, moreover
the representation of $\lh$
is inherited from the Higgs and fermions that sandwich 
${\widehat{{L_L^{}}}}^{^{_{_{_{\!\!H}}}}}\!$.
The Yukawa term (\ref{hsw}) is thus  invariant under
noncommutative gauge transformations. Of course in the $\th\rightarrow
0$ limit we recover the usual gauge transformation for the leptons.
An explicit formula for the hybrid SW map at first order in $\th$ is in 
\cite{SM, A}.
The Yukawa terms (\ref{hsw}) differ from those studied in \cite{SM}.
There the hybrid SW map is considered on $\phi$, in particular there
${\widehat{\phi}}^{^{^{{{H}}}}}\!$ is not invariant under $SU(3)$
gauge transformations (and this
implies that in \cite{SM} gluons couple directly to the Higgs field).

The Higgs sector in the $SO(10)$ and $SU(5)$ models can be constructed
with similar techniques \cite{A}.




\sk
\section{Hermiticity and reality properties of SW map}
From (\ref{SWA}) we see that if $A$ is hermitian
then $ \Ah$ is also  hermitian.
Actually, to all orders in $\th$, $\Ah$ and $\lh$ 
can be chosen hermitian if $A$ and $\Lambda$ are hermitian. 
Otherwise stated,
SW map can be chosen to be compatible with hermitian conjugation.
Compatibility of SW map with complex conjugation reads,
\beq
\widehat{\P*}={\widehat{\Psi}}^{{^{\,\mbox{\scriptsize{$*$}}}}}
\label{conj}
\eeq
where 
$
\Psih= \SW [\Psi,\rho_\Psi(A), \th]
%\widehat{\U}\equiv \SW [\U,\rho_\U(A), \th]~~~,~~~~~~
%\widehat{\P*}\equiv \SW [\P*,\rho_\P*(A), - \th]
%\label{30}
$
denotes the SW map of $\Psi$ constructed with the representation 
$\rho_\Psi$ of the potential $A$, and the SW map of the complex
conjugate spinor $\P*$ is defined by 
\beq
\widehat{\P*}\equiv \SW [\P*,\rho_\P*(A), - \th]
\label{30}
\eeq
where $\rho_\P*$ is the representation conjugate to 
$\rho_\Psi$\footnote{ 
Given the group element $g=e^{i\Lambda}=e^{i\Lambda^aT^a}$ we have 
$\rho_{\Psi^*}(g)\equiv\overline{\rho_\Psi(g)}$ and, 
since 
$\L^a$,  $A^a$ are real, we have
$\rho_{\P*}(\L)=-\overline{\rho_\Psi(\L)}
{}~,~~\rho_\P*(A)=-\overline{\rho_\Psi(A)}\;
$.}. Notice that in (\ref{30}) the noncommutativity parameter $\theta$
appears with opposite sign w.r.t. the $\theta$ in the SW map of
$\Psi$.
Similarly to (\ref{conj}) we have 
$\widehat{\rho_\P* (A)}=-\overline{{\widehat{\rho_\Psi(A)\,}}}$ and
$\widehat{\rho_\P* (\L)}=-\overline{{\widehat{\rho_\Psi(\L)\,}}}$
where 
$\widehat{\rho_\P* (A)}\equiv\Ah[\rho_\P*(A),-\th]$ and 
$\widehat{\rho_\P* (\L)}\equiv\lh[\rho_\P*(A),\rho_\P*(\L),-\th]$.
{}The proof of (\ref{conj}) relies on showing that the SW differential
equations \cite{SW} (obtained by requiring
that gauge equivalence classes of  the  gauge
theory with noncommutativity 
$\th+\delta\th$, correspond to 
gauge equivalence classes of the gauge
theory with noncommutativity $\th$)
are themselves compatible with complex
conjugation \cite{A}. One proceeds similarly for 
the case of hermitian conjugation.
\sk
\noindent{\bf Noncommutativity and chirality$~$} We can now discuss a further ambiguity of noncommutative gauge theories, and
resolve it by requiring compatibility with grand unified theories.
{}For simplicity we consider noncommutative QED.
% however 
%the analysis immediately applies to general nonabelian gauge theories
%with fermionic matter.
Let $\psi$ be a $4$-component Dirac spinor, and
decompose it into its Weil spinors $\psi_L$ and $\psi_R$.
Their charge conjugate spinors are
$\psi^{\;C}_{\!L}=\psi^C_{~L}=-i\sigma_2\psi_R^{\,*}$
and  $\psi^{\;C}_{\!R}=\psi^C_{~R}=i\sigma_2\psi_L^{\,*}$.
Consider 
the  noncommutative left-handed spinor
$\widehat{\psi_L}=\SW [\psi_L,\rho_{\psi_L}(A),\th]~,$
we then have the $\mbox{\boldmath ${\pm}$}\th$ choice 
\beq
\widehat{\psi_R}=\SW [\psi_R,\rho_{\psi_R}(A),\mbox{\boldmath ${\pm}$}
\th]~
\label{choice}
\eeq
for the right handed one.
In the literature the choice $+\th$ is usually
considered so that for the $4$-component Dirac spinor $\psi$ we can
write  $\widehat{\psi}=\SW [\psi,A,\th]$, 
$\delta\widehat{\psi}=i\widehat{\Lambda}\*\widehat{\psi}$.
We here advocate the opposite choice ($-\th$) in (\ref{choice}).
Indeed from  (\ref{conj}) we have that 
$\widehat{\psi_{\!L}^{\;C}}=-i\sigma_2\,{\widehat{\psi_R}}^{\,*}$ and therefore
\beq
\widehat{\psi_R}=\SW [\psi_R,\rho_{\psi_R^{}}(A),-\th]~~~
\Longleftrightarrow~~~
\widehat{\psi_{\!L}^{\;C}}=\SW [{\psi_{\!L}^{\;C}},
\rho_{\psi_{\!L}^{\;C}}(A),+\th]
\label{equiv}
\eeq
so that with the $-\th$ choice in (\ref{choice}), 
both left handed fermions $\widehat{\psi_L}$,
$\widehat{{\psi^{\;C}_{\!L}}}$
are associated with $\th$ while the right handed ones $\widehat{\psi_R}$, 
$\widehat{{\psi^{\;C}_{\!R}}}$ are associated with $-\th$.
In GUT theories we have multiplets of definite chirality
and therefore this is the natural choice to
consider in this setting.

These observations allow us to compare  
QED$_+$ with  $\QEDm$, the two different QED theories obtained with the two
different $\pm\th$ choices (\ref{choice}). This difference 
immediately extends to the fermion kinetic terms 
of nonabelian gauge theories and allows us to compare the 
NCSM discussed in
\cite{SM} with the present GUT compatible one. We have (up to gauge kinetic terms)
$$
S_\QEDp \!
%=\int \overline{\widehat{\psi_{}}}\* i
%\widehat{\fmslashmio D} \widehat \psi_{}
=\int {\widehat{\psi_{\!L}}}^\dagger\* i
\widehat{\fmslashmio D}_{\,} \widehat \psi_{\!L}
\,+\,{\widehat{\psi_{\!R}}}^\dagger\* i
\widehat{\fmslashmio D}_{\,} \widehat \psi_{\!R}
~~~,~~~
S_\QEDm\! =\int {\widehat{\psi_{\!L}}}^\dagger\* i
\widehat{\fmslashmio D}_{\,} \widehat \psi_{\!L}
\,+\,{\widehat{\psi^{\;C}_{\!L}}}^\dagger\* i
\widehat{\fmslashmio D}_{\,} \widehat{\psi^{\;C}_{\!L}}\:~~
$$
where the GUT inspired $\QEDm$ is obtained 
using the left handed spinor
 \(\left({^{^{^{\!}}}}{_{_{_{\!\!}}}}\right.
{}^{^{\mbox{\sma{$_{\psi_{\!L}^{} }$}}}}_{{\mbox{\sma{$^{\psi^C_{\!L} }$}}}}
\left.{^{^{^{\!}}}}{_{_{_{\!\!\!\!}}}}\right)\) 
so that $\widehat{\psi^C_{\!L}}
=\SW[\psi^C_{\!L},\rho_{\psi^C_{\!L}}(A),\th]$. 
Now from 
$\widehat{\psi_{\!L}^{\;C}}=-i\sigma_2\,{\widehat{\psi_R}}^{\,*}$
and from $\sigma$ matrix algebra we have
$\,\int {\widehat{\psi^{\;C}_{\!L}}}^\dagger\* i
\widehat{\fmslashmio D}_{\,} \widehat{\psi^{\;C}_{\!L}}=
\int {\widehat{\psi_{\!R}}}^{\!{{
\mbox{${_{^{{{o{^{_{^{_{\!\!}}}}}p}}}}}$}}}}
{}{^{\mbox{${}^\dagger$}}}\m* i
\widehat{\fmslashmio D}^{\!{{
\mbox{${_{^{{{^{\:{_{\!\!\!}}}}{o{^{_{^{_{\!\!}}}}}p}}}}}$}}}}
_{\,} \widehat{\psi_{\!R}}^{\!{{
\mbox{${_{^{{{o{^{_{^{_{\!\!}}}}}p}}}}}$}}}}
$,
where we have emphasized  that we are using the $-\th$ convention
in the SW map by writing 
${~}\widehat{^{}}^{\!{{\mbox{${_{^{{\,\;{o{^{_{^{_{\!\!}}}}}p}}}}}$}}}}$ 
instead of ${~}\widehat{^{}}{~\,}$. 
We conclude that in order to obtain QED$_-$ from QED$_+$ we just need to 
change $\th$ into $-\th$ in the right handed fermion sector of
QED$_+$.


\section{$C,P,T$ properties of SW map and  of NCYM actions}
Using  compatibility of SW map with complex conjugation and the
 tensorial properties of SW map (i.e. that
 SW map preserves the space-time index)
one can study the properties of SW map with respect to the $C$, $P$
 and $T$ operations. In particular these same expressions as in the
 commutative case holds:
\beq
\widehat{\Psi_{\!L}}^{\:T}={-i\sigma_1\sigma_3
\widehat{\Psi_{\!L}}}~~~,~~~
{\widehat{\Psi_{\!R}}^{\:T}}
={-i\sigma_1\sigma_3\widehat{\Psi_{\!R}}}
~~~,~~~
\Ah_{\,\mu}^{\textstyle{^{\;T}}}=(\Ah_0,-\Ah_i) 
\label{LT}
\eeq
{\vskip -2.5em}
\beq
{\widehat{\Psi_{\!L}}^{\:C\!P}}=i\sigma_2{\widehat{\Psi_{\!L}}}^{*}
      ~~~,~~~
{\widehat{\Psi_{\!R}}^{\:C\!P}}=-i\sigma_2{\widehat{\Psi_{\!R}}}^{*} 
~~~,~~~\Ah_{\,\mu}^{\textstyle{^{\;C\!P}}}
=(-\overline{\Ah_0},\overline{\Ah_i}) 
\label{LCP}
\eeq
where the action of the $P$ and $C$ operators on spinors is given by 
$$
\widehat{\Psi_{\!L}}^{\,P}=
 \SW[\Psi_{\!L}^{\,P},\rho_{\Psi_{_{\!}L}}^{}(A^P),\th^P,\partial^P,i]~~~,~~~
\widehat{\Psi_{\!L}}^{\,C}=
\SW[\Psi_{\!L}^{\,C},{(\rho_{\Psi_{_{\!}L}^{}}(A))^C},
\th^C,\partial,i]~,$$
while the time inversion is given by 
$
{\widehat{\Psi_{\!L}}^{\:T}}
=\SW[\Psi_{\!L}^{{^{^{\,}}}T},
\rho_{\Psi_{\!L}^{{^{^{\,}}}T}}(A^T),
\th^T,\partial^T,-i]\,
$.
In these expressions we have written explicitly the dependence on the 
partial derivatives, and
the imaginary unit $i$ in the last slot marks that the coefficients in
the SW map are in general complex coefficients. 
The $-i$ in the last expression means that we are considering the
complex conjugates of the coefficients in the SW map, this is so
because $T$ is antilinear and multiplicative. 
Relations (\ref{LT}) and (\ref{LCP}) hold provided that $\theta^{\mu\nu}$
transforms under $C,P,T$  as a $U(1)$ field strenght $F_{\mu\nu}$.
If we choose $+\theta$ in (\ref{choice}) then parity and charge
conjugation sepatately assume the same expression as 
in the commutative case.
\sk
Now we discuss the transformations properties
of NCYM actions under $C,P$ and $T$. With the $+\th$
choice (\ref{choice}) we have that NCYM actions are 
invariant under $C,P$ and $T$ iff in the commutative limit they are 
invariant. On the other hand, with the $-\th$ choice
NCYM actions are invariant under $C_{\!}P$ and $T$ iff in the 
commutative limit they are invariant.
For the fermion kinetic term these statements are
a straighforward consequence of $\int {\widehat{\Psi_{\!L}}}^\dagger\*
{\fmslashmio\, \partial}{\widehat{\Psi_{\!L}}}
=\int {\widehat{\Psi_{\!L}}}^\dagger
{\fmslashmio \,\partial}{\widehat{\Psi_{\!L}}}$.
Since $\Fh$ transforms like $F$ under $C_{\!}P$ and $T$, and in the
$+\th$ case also under $C$ and $P$ separately,
the $C,P$,$T$ properties of the gauge kinetic term 
$\int T\!r(\Fh\*\Fh)=\int T\!r(\Fh\Fh)$ easily follow.
Inspection of the fermion gauge bosons interaction term
leads also to the same conclusion. 


We have studied the $C,P$ and $T$ symmetry properties of NCYM 
actions where $\th$ transforms under $C,P$ and $T$ 
as a field strenght.
Viceversa, {\sl if  we keep $\th$ fixed} under $C,P$ and $T$ 
transformations, we in general have that NCYM theories 
{\sl break}  $C,P$ and $T$ symmetries. 
%Notice however that in the $-\th$ case, if we keep  $\th$ fixed
%under $C$, then the SW map is well behaved under $C$, and $C$ is a symmetry 
%of a NCYM action if it is a symmetry of the corresponding commutative one
%and the gauge kinetic term $\int T^{\!}r(\Fh\Fh)$ is even in $\th$.

Finally a $U(1)$ 
field strenght is invariant under the combined $CPT$ transformation,
 and therefore $\th$ does not change. 
This implies that $CPT$ is always a  
symmetry of NCYM actions.

\sk
\noindent{\bf Acknowledgments$~$}
%\noindent 
It is a pleasure to thank the organizers for the nice and 
stimulating
atmosphere at the conference and its efficient organization.

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%%CITATION = HEP-TH 0006246;%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{Jurco1}
%\cite{Jurco:2001rq}
%\bibitem{Jurco:2001rq}
B.~Jurco, L.~Moller, S.~Schraml, P.~Schupp and J.~Wess,
%``Construction of non-Abelian gauge theories on noncommutative spaces,''
Eur.\ Phys.\ J.\ C {\bf 21}, 383 (2001)
[hep-th/0104153].
%%CITATION = HEP-TH 0104153;%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibitem{SM}
X.~Calmet, B.~Jurco, P.~Schupp, J.~Wess and M.~Wohlgenannt,
%``The standard model on non-commutative space-time,''
Eur.\ Phys.\ J.\ C {\bf 23} (2002) 363
[arXiv:hep-ph/0111115].
%%CITATION = HEP-PH 0111115;%%

%\cite{Aschieri:2002mc}
\bibitem{A}
P.~Aschieri, B.~Jurco, P.~Schupp and J.~Wess,
%``Non-commutative GUTs, standard model and C, P, T,''
Nucl.\ Phys.\ B {\bf 651} (2003) 45
[arXiv:hep-th/0205214].
%%CITATION = HEP-TH 0205214;%%




%\cite{Schupp:2002up}
\bibitem{astron}
P.~Schupp, J.~Trampetic, J.~Wess and G.~Raffelt,
``The photon neutrino interaction in non-commutative gauge field
theory  and astrophysical bounds,''
arXiv:hep-ph/0212292.
%%CITATION = HEP-PH 0212292;%%

%\cite{Buric:2004ms}
\bibitem{Wulkenhaar2}
M.~Buric and V.~Radovanovic,
%``Non-renormalizability of noncommutative SU(2) gauge theory,''
JHEP {\bf 0402} (2004) 040
[arXiv:hep-th/0401103].
%%CITATION = HEP-TH 0401103;%%

%\cite{Brandt:2003fx}
\bibitem{Brandt}
F.~Brandt, C.~P.~Martin and F.~R.~Ruiz,
%``Anomaly freedom in Seiberg-Witten noncommutative gauge theories,''
JHEP {\bf 0307} (2003) 068
[arXiv:hep-th/0307292].
%%CITATION = HEP-TH 0307292;%%




%\cite{Sheikh-Jabbari:2000vi}
\bibitem{Sheikh-Jabbari:2000vi}
M.~M.~Sheikh-Jabbari,
%``Discrete symmetries (C,P,T) in noncommutative field theories,''
Phys.\ Rev.\ Lett.\  {\bf 84} (2000) 5265
[arXiv:hep-th/0001167].
%%CITATION = HEP-TH 0001167;%%



%\cite{Chaichian:2002vw}
\bibitem{Chai}
M.~Chaichian, K.~Nishijima and A.~Tureanu,
%``Spin-statistics and CPT theorems in noncommutative field theory,''
Phys.\ Lett.\ B {\bf 568}, 146 (2003)
[arXiv:hep-th/0209008]\\
%%CITATION = HEP-TH 0209008;%%
%\cite{Alvarez-Gaume:2003mb}
%\bibitem{Alvarez-Gaume:2003mb}
L.~Alvarez-Gaume and M.~A.~Vazquez-Mozo,
%``General properties of noncommutative field theories,''
Nucl.\ Phys.\ B {\bf 668} (2003) 293
[arXiv:hep-th/0305093]\\
%%CITATION = HEP-TH 0305093;%%
%\cite{Mahajan:2003vs}
%\bibitem{Mahajan:2003vs}
N.~Mahajan,
%``PCT theorem in field theory on noncommutative space,''
Phys.\ Lett.\ B {\bf 569} (2003) 85
[arXiv:hep-th/0305105].
%%CITATION = HEP-TH 0305105;%%


\end{thebibliography}