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%corrections 6.4.02
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\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\ta{\tilde\alpha}
\def\tb{\tilde\beta}
\def\ba{\bar\alpha}
\def\bb{\bar\beta}
\def\PS{$ SU(4) \times $SU(2)$_L \times SU(2)_R$}
\begin{document}
\tighten

\title{SO(10) A LA  PATI-SALAM }
\author{Charanjit S. Aulakh${}^a$ and Aarti Girdhar}
\address{\it Dept. of Physics, Panjab University,
Chandigarh, INDIA, 160014\hfil\break
${}^a$  aulakh@pu.ac.in   }
\maketitle
\begin{abstract}
 
We present rules for rewriting SO(10) tensor and spinor invariants in 
terms of  invariants of its ``Pati-Salam'' maximal subgroup ($SU(4)\times 
SU(2)_L\times SU(2)_R $) supplemented by the discrete symmetry called
D parity. Explicit decompositions of quadratic and cubic invariants relevant
to GUT model building are presented and the role of D parity in organizing
the terms explained. We illustrate the application of our results by
obtaining the effective theory  below the scale where SO(10) is broken to
the Pati-Salam group(times D parity) in the R-parity preserving Susy SO(10)
GUT developed previously \cite{abmrs01}. Our results will facilitate analysis
of   symmetry breaking chains of SO(10) GUTS - since the matching
conditions across mass thresholds between $M_X$ and $M_W$ are easily
derived once the SO(10) couplings are rewritten in terms of Left-Right symmetric unitary labels.
\end{abstract}





\section {Introduction}

The virtues of    SO(10)
 supersymmetric GUTs{\cite{aulmoh,so10refs,leem,sato,abmrs,abmrs01}}
 are now widely appreciated although there are two contending points of view
regarding the type of 
Higgs fields that should be used (specifically whether large tensor
representations like the
${\overline{126}}$ may be legitimately employed in view of their strong effect on the 
SO(10) beta function above the GUT scale and the difficulty of obtaining
them from string theory).  SO(10) has the cardinal virtue
of exactly accomodating
the 16 chiral fermions of the Standard model (the right chiral neutrino 
has a legitimate claim to inclusion  since neutrino masses are now an 
inalienable part of particle phenomenology). 
Thus the seesaw mechanism \cite{seesaw,mohsen} finds a natural home in SO(10).
Moreover SO(10) provides an appealing rationale for the  parity   breaking
manifestation of the
Standard model by linking it to the natural embedding of Left-Right symmetry 
in $SO (10)$ via its Pati-Salam(PS)\cite{ps74} maximal subgroup $ G_{PS}=SU(4)\times 
SU(2)_L\times SU(2)_R $ ( More precisely $G_{PS} \times D$,where D is the
so called D parity\cite{Dparity1,Dparity2}).  In supersymmetric and other models
which employ a ``renormalizable see-saw mechansim'' (of either Type I or Type II
\cite{mohsen}) (i.e based on   B-L even Higgs multiplets) , 
R/M-parity becomes a part of the gauge symmetry and survives symmetry 
breaking down to the MSSM \cite{abs,ams,amrs,amrs99,abmrs01}.

 In previous work\cite{aulmoh,ams,amrs,abmrs,abmrs01}
it was shown that in supersymmetric theories the restricted form of the 
superpotential can leave  Renormalization Group(RG) significant multiplets with only 
intermediate  or even light masses . Thus a proper RG analysis of 
Susy GUTs should make use of the actual mass spectrum of the model 
in
question rather than the spectrum conjectured on the basis of the 
survival principle. To implement this program it is necessary to formulate 
the matching conditions for the couplings of the various mass multiplets 
at 
successive symmetry breaking and mass thresholds of the theory. Since 
the low energy theory is based upon a unitary
gauge group whereas the ultimate determinant of coupling constant 
relations is the overlying SO(10) gauge symmetry
it is necessary to write  the SO(10) invariants in 
terms of properly normalized fields carrying the unitary maximal subgroup 
labels. 
The maximal subgroups of SO(10) are $SU(5)\times U(1)$ and the 
Pati-Salam Group $SU(4)\times SU(2)_L\times SU(2)_R $ which is isomorphic to 
the
 $SO(6)\times SO(4) $   subgroup of $SO(10)$.  Very recently 
\cite{nathraza} the explicit forms of the Spin(10) invariants  of 
representations (with dimensions upto 210) were given in terms
of $SU(5)\times U(1)$ labels using the so called oscillator 
basis\cite{wilzee,mohsak} to effect the conversion. This rewriting,besides 
suffering from a certain lack of transparency (due precisely to the LR 
asymmetric nature of the embedding
of $SU(5)\times U(1)$),~is quite inappropriate for LR symmetric breaking 
chains. Thus it is necessary to obtain the invariants in terms of the 
PS subgroup separately. Moreover our results may be easily reassembled into SU(5) X U(1) invariants
and can serve as an alternative derivation and cross check.

Furthermore, a Discrete symmetry closely related to Parity , namely the the so called D-parity,
is important and useful in studying the possible symmetry breaking chains in SO(10) GUTs{\cite{Dparity2,abmrs01,geneal}.
D-Parity proves invaluable  for organizing  and cross checking the relative signs in our Left-Right symmetric expressions.
We have developed explicit rules for the action of D-parity on all fields according to their (SO(10) tensor or spinor)
origin and their PS labels.
 
Although the necessary basic tools for the translation have long existed (in somewhat implicit
form) in the work of Wilczek and Zee \cite{wilzee}  no explicit results
were available.
Thus the obstacle needed to be decisively surpassed by
carrying out the translations in a systematic and explicit manner 
rather than remaining content with the piece-meal  and incomplete approach
that had largely served the purposes of GUT RG analysis as long as the
survival principle could be invoked without excessive qualms.
Motivated by the need to carry out the proper RG analysis including Yukawa
couplings  and for estimating threshold effects we have attempted to fill
the long standing lacuna and provided rules for the translation to LR
symmetric unitary subgroup labels.
We indicate some applications of our results to the coupling matching 
problem but leave the actual RG calculation for a separate publication.

In Section II we introduce our notation and the embedding of 
$SO(6)\times SO(4)$ in $SO(10)$
and define D parity on tensor representations.We then show how to translate
tensor invariants of $SO(6)$ to $SU(4)$,
$SO(4)$  to $ SU(2)_L\times SU(2)_R$. In Section III we implement these 
rules on some tensor invariants to illustrate 
the procedures for translating from $SO(10)$ to $G_{PS}$. 
% However since an 
% exhaustive listing of invariants is both exhausting to produce and counterproductive 
% as regards actual utility for users of these techniques we have instead provided
%an appendix where we collect useful SO(6) and SO(4) contractions translated to unitary form.
 In Section IV, V  we perform the same tasks once spinor 
representations are included. In Section VI we apply our
 results to the Susy SO(10) GUT of \cite{abmrs01} to derive the effective
 theory for  RG analysis. We conclude with some remarks on future 
directions.


\section{SO(10) $\longrightarrow$ SO(6) $\times$ SO(4) $\sim$  ${\bf{G}}_{\bf{PS} }$}



The PS subgroup $G_{PS}\equiv SU(4) \times $ SU(2)$_L \times SU(2)_R$
of $SO(10)$ is actually isomorphic to the obvious maximal subgroup
$SO(6) \times SO(4)  \subset  SO(10)$. The essential components
 of the analysis are thus explicit translation between SO(6) and  
SU(4) on the one hand and SO(4) and SU(2)$_L \times  $SU(2)$_R$ on the 
other. Properly speaking we should speak of Spin(10) and  its subgroups,
however since this only means SO(10) with spinor representations included
we shall not abide by such niceties.
 Our notations and conventions follow those of \cite{wilzee} wherever 
possible.  A crucial difference with \cite{wilzee} concerning the 
explicit form of the charge conjugation  matrices for spinor 
representations of orthogonal groups will however emerge in the  section
on spinors.
Wherever feasible we repeat definitions so that the presentation is 
self contained.
 
We have adopted the rule that any submultiplet of an SO(10) field is 
always denoted by the {\it{same}} symbol as its parent field , its 
identity being 
established by the indices it carries or by supplementary indices , if 
necessary. Our notation for indices is as follows : The indices of the 
vector representaion of SO(10) (sometimes also SO(2N))
 are denoted by $i,j =1..10 (2N)$. The {\it {real}}
vector index of the upper left block embedding (i.e the embedding 
specified by the breakup of the vector multiplet $10=6 + 4$) of SO(6) in 
SO(10) are denoted $a,b=1,2..6$ and of the lower right block embedding of
SO(4) $\in$ SO(10) by  ${\tilde{\alpha},\tilde\beta= 7,8,9,10}$. These
indices are complexified  via a Unitary transformation and denoted by 
$\hat{a},\hat{b}=\hat{1},\hat{2},\hat{3},\hat{4},\hat{5},\hat{6} \equiv 
\bar{1},{\bar{1}^*,\bar{2},\bar{2}^*,\bar{3},\bar{3}^*}$
where $\hat{1}\equiv \bar{1}, \hat{2}\equiv \bar{1}^* $ etc.  Similarly we  
denote the complexified versions of  ${\tilde{\alpha},\tilde\beta}$  
by $\hat{\alpha},\hat{\beta}= \hat{7},\hat{8},\hat{9},\widehat{10}$. The 
indices of the doublet of $SU(2)_L$($SU(2)_R$)
are denoted $\alpha,\beta=1,2$($\dot\alpha,\dot\beta=\dot{1},\dot{2})$. 
Finally the index of the fundamental 4-plet 
of $SU(4)$ is denoted by a (lower) $\mu,\nu = 1,2,3,4$ and its 
upper-left block $SU(3)$ subgroup indices are 
$\bar\mu,\bar\nu = 1,2,3$ , while the corresponding indices on the $4^{*}$ are 
carried as superscripts.

\subsection  {\ SO(6) $ \longleftrightarrow  $\ SU(4)}
{{\bf{\underline{Vector/Antisymmetric}}}
\vspace{.3 cm}

The 6 dimensional vector  representation of SO(6) denoted by $V_a
 (a=1,2,...6)$ transforms  as

\begin{equation}
V'_a=(exp {i \over 2}\omega ^{cd}J_{cd})_{ab}V_b
\end{equation}

where the Hermitian generators $J_{cd}$ have the explicit form
\begin{equation}
(J_{cd})_{ef}=-i\delta_{c[e}\delta_{f]d}
\end{equation}
and thus satisfy the SO(6) algebra(square brackets denote antisymmetrization)
\begin{equation}
[J_{cd},J_{ef}]= i\delta_{e[c}J_{d]f}-i\delta_{f[c}J_{d]e} 
\end{equation}

It is useful to introduce complex indices
$\hat a, \hat b = {\hat 1} ...{\hat 6}  $ by the unitary change of 
basis
\begin{equation}
V_{\hat a}=U_{\hat a a}V_a \;, \quad U=U_2 \times I_3\;, \quad U_2={1 
\over\sqrt2}\left[
\matrix{1 & i\cr
        1 & -i\cr}
\right]
\end{equation}
so that  $V_{a}W_{a}=V_{\hat a}W_{\hat a^*}$. The decomposition of the 
fundamental 4-plet of  SU(4) w.r.t $SU(3)\times U(1)_{B-L}$ is 
$4=(3,1/3) \oplus(1,-1)$. The index for the 4 of SU(4) is denoted by 
$\mu=1,2,3,4$.
while $\bar \mu=1,2,3$ label its SU(3) subgroup.
%
In SU(4) labels,~the 6 of  SO(6) is the 2 index antisymmetric 
$V_{\mu\nu}$ and decomposes as $6=V_{\bar\mu}(3,-2/3) \oplus V_{{\bar\mu}^*
}(\bar 3, 2/3)$  and
we identify $V_{\bar \mu4}=V_\mu$,~$V_{\bar \mu\bar 
\nu}=\epsilon_{\bar \mu \bar \nu\bar \lambda}V_{\bar\lambda^*}$. In other
words , if one defines $V_{\mu\nu}=\Theta_{\mu\nu}^{\hat a} V_{\hat a}$ with
$\Theta_{{\bar \mu} 4}^{\hat a}=\delta^{\hat a}_{\bar \mu},
 \Theta_{{\bar \mu}{\bar \nu}}^{\hat a} =
\epsilon_{\bar \mu \bar \nu \bar \lambda}\delta^{\hat a}_{\bar
\lambda^*}$,~then since $\Theta_{\mu\nu}^{\hat a} \Theta_{\lambda\sigma}^
{\hat a*} \equiv \epsilon_{\mu\nu\lambda\sigma}$ it follows that the
translation of SO(6) vector index contraction is (${\widetilde V}^{\mu\nu}
=(1/2)\epsilon^{\mu\nu\lambda\sigma} V_{\lambda \sigma})$  

 \be
V_aW_a={1 \over4}\epsilon^{\mu\nu\lambda\sigma}V_{\mu\nu}W_{\lambda\sigma}
\equiv {1 \over2}\widetilde V^{\mu\nu}W_{\mu\nu}\label{6vec}
\ee

Representations carrying vector indices $a,b  ...$ are then translated 
by
replacing by each vector index by a  antisymmetrized pair of 
SU(4) indices $\mu_1\nu_1, \mu_2\nu_2,.....$. For example  

\be
A_{ab} B_{ab} = 2^{-4} \epsilon^{\mu_1\mu_2\mu_3\mu_4} 
\epsilon^{\nu_1\nu_2\nu_3\nu_4} A_{{\mu_1\mu_2},{\nu_1\nu_2}} 
B_{{\mu_3\mu_4},{\nu_3\nu_4}}
\ee




{\bf{\underline{Antisymmetric/Adjoint}}}:
\vspace{.3 cm}\\
 The 15 dimensional
antisymmetric
representation $A_{ab}$ of SO(6) translates to the adjoint (15) $A{_{\nu}}
^{\mu}$ of SU(4):

\be
A{{_\nu}}^{\mu}=-{1 \over 
4}\epsilon^{\mu\lambda\sigma\rho}A_{\lambda\sigma,\rho\nu} \quad
A_{\mu\nu,\rho\sigma}=-\epsilon_{\lambda\mu\nu[\rho}A_{\sigma]}^{~~\lambda}
\ee
The parameters $\omega_{ab}$ of SO(6) are identified with those of SU(4)
$({\theta^A,A=1...15})$
%
\be
\omega_{ab} \rightarrow {\omega_{\mu}}^{\nu} = -i\theta^{A}{(
\lambda^A)_{\mu}}^{\nu}
\ee
Where $\lambda^A,A=1..15 $ are the Gellmann matrices for $SU(4)$.   
We define
\bea
{A_{\nu}}^{\mu}={-i \over {\sqrt2}}{\lambda^A}_{\nu}^{~~\mu}A^A~,\qquad
{\lambda^A}_{\nu}^{~~\mu} \equiv {\lambda^A}_{\nu\mu}
\eea

Note that tracelessness $A{{_\mu}^\mu}=0$ is ensured by antisymmetry of 
$A_{\mu\nu,\lambda\sigma}$ and symmetry of 
$\epsilon^{\mu\nu\lambda\sigma}$ under interchange of index pairs $\mu\nu$
and $\lambda\sigma$. The normalization relation
\bea
(A{{_\nu}^\mu},A{{_\sigma}^\lambda})&=& 
\delta^{\lambda}_{\mu}\delta^{\nu}_
{\sigma}-{1 \over 4}\delta^{\mu}_{\nu}\delta^{\lambda}_{\sigma} 
\nonumber\\&=&
{1 \over 2}((\lambda^A){{_\nu}^\mu})^*(\lambda^A){{_\sigma}^\lambda} 
\eea
follows if
 $A_{ab},A^A$ are of unit norm :
 \be
(A_{ab},A_{cd})=\delta_{a[c}\delta_{d]b}~~ ;~~(A^{A},B^{B})=\delta^{AB}
\ee


 We denote the trace over SO(6) vector indices a,b ... by ``Tr'' and 
over the SU(4) fundamental index $\mu\nu...$ by ``tr''.
 Then 
\bea
TrAB &=& A_{ab} B_{ba} = 2 A{{_\nu}^\mu} B{{_\mu}^\nu}=2 trAB 
 \nonumber\\
TrABC &=& trA [B,C]
\eea

A notable point is that the invariant 6 index totally antisymmetric  
tensor of SO(6) leads to a distinct $SU(4)$ invariant involving the anti-
commutator.

\be
\epsilon_{abcdef}A_{ab} B_{cd} C_{ef} = 2^3 trA {\{B,C\}} 
\ee


{\bf{\underline{Symmetric traceless (20)/4 index mixed}}}
\vspace{.3 cm}\\
The 20 dimensional symmetric traceless representation $S_{ab}$ of SO(6) 
which has normalization 

\be (S_{ab},S_{cd}) = \delta^a_{(c} \delta^b_{d)} - {1\over 3} 
\delta^{ab} \delta_{cd}
\ee

appropriate to a traceless field 
translates to  $S_{\mu\nu,\lambda\sigma}=S_{\lambda\sigma,\mu\nu}$ with 
the
additional constraint (corresponding to tracelessness on SO(6) vector 
indices) 
\be
{1\over 4}\epsilon^{\mu\nu\lambda\sigma}S_{\mu\nu,\lambda\sigma}
\equiv S_{aa}=0
\ee

The normalization condition translates to
\be
(S_{\mu\nu,\lambda\sigma},S_{\theta\delta,\epsilon\rho}) = 
\delta^{\mu}_{[\theta} \delta^{\nu}_{\delta]} \delta^{\lambda}_{[\epsilon} 
\delta^{\sigma}_{\rho]} -{1\over 3} \epsilon^{\mu\nu\lambda\sigma} 
\epsilon_{\theta\delta\epsilon\rho}
\ee





{\bf{\underline{3 Index Antisymmetric (Anti) Self Dual/Symmetric 2 index}}}
\vspace{.3cm}\\
The invariant tensor $\epsilon_{abcdef}$ of SO(6) allows
the separation of the 3 index totally antisymmetric 20-plet
$T_{abc}$ of SO(6) into self dual and anti-self dual pieces 
$T^\pm_{abc}=\pm
\widetilde T^\pm_{abc}$ where the SO(6) dual is defined as 
\be
\widetilde T_{abc}={i \over 3!}\epsilon_{abcdef}T_{def}\label{tT}
\ee
$T^{+}_{abc}(T^{-}_{abc})$ translate into the 2 index symmetric
$10 (T_{\mu\nu})~~(\overline{10}({\overline T}^{\mu\nu}))$ of SU(4) via
\bea
T_{\mu\nu}&=&{1 \over 12}T^{+}_{\mu\lambda,\nu\sigma,\gamma\delta}
\epsilon^{\lambda\sigma\gamma\delta}\label{sd}\\
\overline T^{\mu\nu}&=&{1 \over 24}T^-_{\kappa\lambda,\rho\sigma,\pi\theta}
\epsilon^{\mu\kappa\lambda\pi}\epsilon^{\nu\rho\sigma\theta}\label{asd}\\ 
T^{(+)}_{\mu\nu,\rho\theta,\gamma\delta}&=& 
T_{{_[\mu}{^[\rho}}\epsilon_{{\nu_]}
{\theta^]}\gamma\delta}\\
T^{(-)}_{\kappa\lambda,\theta\rho,\sigma\delta}&=& -{\overline T}^{\mu\nu}
\epsilon_{\mu\kappa\lambda_[\sigma}\epsilon_{\delta_]\nu\theta\rho}
\eea
%
Note that to preserve unit norm one should define
\be
T^\pm_{abc}={{{T_{abc}\pm\widetilde T_{abc}} \over\sqrt2}}\label{tp}
\ee
%
The normalization conditions that follow from unit norm for 
$T^\pm_{abc}$ :
\be
{(T^\pm_{abc},T^\pm_{a'b'c'})}={\delta^a_{[a'} \delta^b_{b'}
\delta^c_{c']}}\label{tn1}\\
\ee

are
\be
{(T_{\mu\nu},T_{\lambda\sigma})}=\delta^{\mu}_{(\lambda}\delta^{\nu}_{\sigma)}
={(\overline T^{\lambda\sigma},\overline T^{\mu\nu})}\label{tn2}
\ee

So that $T_{\mu\mu}$(no sum) has norm squared 2 while 
$T_{\mu\nu}(\mu\neq \nu)$ has norm one.\\
One has the useful identities\\
\be
T^{+}_{abc}T^{-}_{abc}=6~T_{\mu\nu}~\overline{T}^{\mu\nu}
\ee

% In the appendix we collect translations of a  number of useful SO(6) contraction identities
%for ready reference .



\subsection{$SO(4) \leftrightarrow SU(2)_{L} \times SU(2)_{R}$}
{{\bf{\underline{Vector/Bidoublet}}}
\vspace{.2 cm}\\
 We use early greek indices ${ \ta, \tb =7,8,9,10}$ for the
vector of SO(4) corresponding to ${i,j=7....10}$ of the 10-plet of 
SO(10).
The Hermitian generators of SO(4) have the usual SO(2N) vector 
representation
form
\be
{(J_{\tilde\alpha\tilde\beta})_{\tilde\gamma\tilde\delta}}=-i{\delta_
{\tilde\alpha[\tilde\gamma}\delta_{\tilde\delta]\tilde\beta}}
\ee

The group element is $R=exp {i \over 
2}\omega^{\tilde\alpha\tilde\beta}J_{\tilde
\alpha\tilde\beta}$.
The generators of SO(4)  separate neatly into self-dual and anti-self- dual
sets of 3, ${J_{\tilde
\alpha\tilde\beta}^{\pm}}={1 \over 2}{(J_{\tilde\alpha\tilde\beta} \pm 
{\tilde
J_{\tilde\alpha\tilde\beta})}}$.
Then if ${\check\alpha,\check\beta=1,2,3}$ the generators and parameters 
of the
$SU(2)_{\pm}$ subgroups of SO(4) are defined to be
\be
J_{\check\alpha}^{\pm}={1 \over 2}\epsilon_{\check\alpha\check\beta\check
\gamma}
J_{(\check\beta +6)(\check\gamma +6)}^{\pm} ~ ;\quad 
\omega_{\check\alpha}^{\pm}={1 \over 2}\epsilon_{{\check \alpha}{\check
\beta }{\check\gamma}}\omega_{{(\check\beta+6}) (\check\gamma +6) } \pm
\omega_{(\check\alpha+6)10}
\ee
%
The group elements are ${exp(i{\vec   \omega}_{\pm} \cdot \vec J^{\pm})}.$
The vector 4-plet of SO(4) is a bi-doublet $(2,2)$ w.r.t to 
${SU(2)_{-} \otimes SU(2)_{+}}$.
%
We denote the indices of the doublet of ${SU(2)_L=SU(2)_{-}}$ 
${(SU(2)_R=
SU(2)_{+})}$ by undotted early greek indices ${\alpha,\beta=1,2}$ 
{(dotted
early greek indices ${\dot\alpha,\dot\beta=1,2}$)}.\\
Then one has
\be
V_{\hat4^*}=-V_{1\dot1}={{(V_7-iV_8)} \over \sqrt2}=V_{2\dot2}^*  \quad ,
\quad V_{\hat5}=V_{1\dot2}=
{{(V_9+iV_{10})}\over \sqrt2}=V_{2\dot1}^*
\label{bidoub}
\ee
${SU(2)_L {(SU(2)_R)}}$ indices are raised and lowered  with 
${\epsilon^
{\alpha\beta},\epsilon_{\alpha\beta}}$ 
${(\epsilon^{\dot\alpha\dot\beta},
\epsilon_{\dot\alpha\dot\beta})}$ with 
${\epsilon^{12}=+\epsilon_{21}=1}$ etc.
%
The SO(4) vector index contraction translates as
\bea
V_{\tilde\alpha}W_{\tilde\alpha} &=& 
-{V_{\alpha\dot\alpha}W_{\beta\dot\beta}
\epsilon^{\alpha\beta}\epsilon^{\dot\alpha\dot\beta}} \nonumber\\
&=& -{V^{\alpha\dot\alpha}W_{\alpha\dot\alpha}}.\label{4vec}
\eea
%
Separating the 2 index antisymmmetric tensor 
$A_{\tilde\alpha\tilde\beta}$
into self-dual and anti-self-dual parts of unit norm
\be 
A^{(\pm)}_{\ta\tb}={1 \over \sqrt2}{(A_{\ta\tb} \pm\tilde{A}_{\ta\tb})}\label
{at}
\ee
%
One finds ${A^{-}{(A^{+})}}$ is ${{(3,1)}{((1,3))}}$ w.r.t 
${SU(2)_L{(SU(2)_R})}$ . In fact these triplets are just 
\bea
A_{\check\alpha}^{(\pm)}&=& {\pm} A^{(\pm)}_{{\check\alpha +6},10} \nonumber
\\&=& {1 \over 2}\epsilon
_{\check\alpha \check\beta \check\gamma }
A_{(\check\beta+6)(\check\gamma +6)}^{(\pm)}
\eea
%
Defining ${{A_\alpha}^\beta}=i 
A_{\check\alpha}^{(-)}{{(\sigma^{\check\alpha})_\alpha}^\beta}=i\vec{A}_{L} \cdot
(\vec\sigma)_{\alpha}^{~~\beta}$~~,
 ${{A_{\dot\alpha}}^{\dot\beta}}=i A_{\check\alpha}^{(+)}{(\sigma^{\check
 \alpha})
 _{\dot\alpha}}^{\dot\beta}=i\vec{A}^{R} \cdot (\vec\sigma)_{\dot\alpha}^
 {\dot\beta}$, where $\sigma^{\check\alpha}$ are the Pauli matrices,
one has\\
 \bea
 A_{\hat\alpha\hat\beta}^{(+)}& \rightarrow & A_{\alpha\dot\alpha\beta\dot
\beta}^{(+)}\equiv\epsilon_{\alpha\beta}A_{\dot\alpha\dot\beta}=\epsilon_
{\alpha\beta}A_{\dot\beta\dot\alpha}\label{ap}\\
 %
 A_{\hat\alpha\hat\beta}^{(-)}&\rightarrow&A_{\alpha\dot\alpha\beta\dot\beta}
^{(-)}\equiv\epsilon_{\dot\alpha\dot\beta}A_{\alpha\beta}=\epsilon_{\dot
\alpha\dot\beta}A_{\beta\alpha}\label{am}\\
 \nonumber\eea
 Where the index pairs $\alpha\dot\alpha$ correspond to the complex 
indices
 $\hat\alpha$ as  given in (\ref{bidoub}) above.\\
 Then one has for the contraction of two antisymmetric tensors
 \bea       
 A_{\ta\tb}{B}_{\ta\tb}&=&{1 \over 
2}{(A_{\ta\tb}^{(+)}{B}_{\ta\tb}^{(+)}+
 A_{\ta\tb}^{(-)}{B}_{\ta\tb}^{(-)})}\\
 &=& 2{(\vec{A}_L.\vec{B}_L+\vec{A}_R.\vec{B}_R)}
 \eea
Similarly one gets the useful identity 

\be
A^{(\pm)}_{\ta\tb}B^{(\pm)}_{\ta\tilde\gamma}C^{(\pm)}_{\tb\tilde\gamma}=
4 \vec{A}^{(\pm)} \cdot ({\vec{B}^{(\pm)} \times \vec{C}^{(\pm)}})
\ee


{\bf{\underline{Symmetric Traceless(9)/Bitriplet(3,3)}}}
\vspace{.3 cm}\\
 The two index symmetric traceless tensor $S_{\ta\tb}$ of SO(4)
 which has dimension
 9 becomes the ${(3,3)}$ {(symmetry follows from tracelessness)} :
 \be
S_{\hat\alpha\hat\beta}=S_{\alpha\dot\alpha,\beta\dot\beta} \equiv S_
{\alpha\beta,
\dot\alpha\dot\beta}=S_{\beta\alpha,\dot\alpha\dot\beta}=S_{\alpha\beta,\dot
 \beta\dot\alpha}
 \ee
 so that e.g.
 \be
S_{\ta\tb}S'_{\ta\tb}=S^{\alpha\beta,\dot\alpha\dot\beta}S'_{\alpha\beta,\dot
 \alpha\dot\beta}
 \ee
 and are normalized as
 \be
{(S_{\alpha\beta,\dot\alpha\dot\beta},S_{\alpha'\beta',\dot\alpha'\dot\beta'}
)}=\delta_{\alpha\alpha'}\delta_{\beta\beta'}\delta_{\dot\alpha\dot\alpha'}
\delta_{\dot\beta\dot\beta'}+\delta_{\alpha\beta'}\delta_{\beta\alpha'}\delta_{\dot  
\alpha\dot\beta'}\delta_{\dot\beta\dot\alpha'}-{1 \over 
2}\epsilon_{\alpha\beta}  
\epsilon_{\dot\alpha\dot\beta}\epsilon_{\alpha'\beta'}\epsilon_{\dot\alpha'
  \dot\beta'}
  \ee
 The above treatment covers the     
the tensor SO(6) and SO(4) tensor  representations encountered in 
dealing with SO(10) representations upto dimension 210.
 
 
\vspace{.2cm}
{\bf{\underline{SO(10) Tensors {\&} D-Parity}}}
\vspace{.3 cm}\\
The proceedure for the decomposition of SO(10) tensor invariants is now
clear. Splitting the summation over
each SO(10) index i,j= 1,..10 into
summation over ${SO(6),SO(4)}$ indices $(a,\alpha)$, one replaces each SO(6)
(SO(4)) index by $SU(4)(SU(2)_L \times SU(2)_R)$ index pair contractions
according to the basic rules (\ref{6vec}) and (\ref{4vec}). Using ((\ref{tp})
(\ref{sd})(\ref{asd}) and (\ref{at})(\ref{ap})(\ref{am})) to decompose
according to self-duality and reduce to PS indices.\\
An important and useful feature of the decomposition is that it permits the
transparent implementation of the Discrete symmetry called D-Parity \cite{Dparity1,Dparity2} defined as \\
\be
D=exp(-i{\pi}J_{23})exp(i{\pi}J_{67})
\ee
On vectors this corresponds to rotations by $\pi$ in the (23) and (67) planes.
Thus components$(V_2,V_3,V_6,V_7)$ of $V_i$ change sign and the rest do not.
In PS language this becomes
\be
V_{\mu\nu} \leftrightarrow  (-)^{\mu+\nu+1}\widetilde{V}^{\mu\nu} \quad ,
\quad V_{2\dot{2}} \leftrightarrow V_{1\dot{1}}
\ee
%
While $V_{1\dot{2}},V_{2\dot{1}}$ remain unchanged.
For the self-dual multiplets of SO(4) one finds that under D parity
\be
V^{(\pm)}_{1} \leftrightarrow V^{(\mp)}_{1} \quad ;\quad V^{(\pm)}_{2,3}
\leftrightarrow -V^{(\mp)}_{2,3}\\
\ee
If we denote $\bar{1}=2$ and $\bar{2}=1$ for dotted and undotted
indices then these rules are just
\bea
V_{\alpha\dot\beta} \leftrightarrow V_{\bar\beta\dot{\bar\alpha}} \quad
,\quad V^{(-)}_{\alpha\beta} \leftrightarrow -V^{(+)}_{\dot{\bar\beta}\dot
{\bar\alpha}}
\eea
Then it follows that $\vec{A}_{L} \cdot \vec{B}_{L} \leftrightarrow
\vec{A}_{R} \cdot \vec{B}_{R}.$\\
The adjoint ${A_{\nu}}^{\mu}$ derived from the antisymmetric 15 has
D-parity property
\be
{A_{\nu}}^{\mu} \leftrightarrow (-)^{\mu+\nu+1}{A_{\mu}}^{\nu}
\ee
While the symmetric 10-plets transform as
\be
T_{\mu\nu} \leftrightarrow \overline{T}^{\mu\nu}(-)^{\mu+\nu+1}
\ee



\section {SO(10) Tensor Quadratic \& Cubic Invariants}
Using our rules we present examples of decompositions of SO(10) invariants
to illustrate the application of our method. 

%As noted above, however the reader may find
%the generative rules in the Appendix more convenient and complete in practice.



{\bf{\underline{$45 \cdot 45$}}}

\bea
45(A_{ij})&=&(15,1,1)A_{ab}+(A^{(-)}_{\ta\tb}(1,3,1) \oplus A^{(+)}_{\ta\tb}
(1,1,3))+(6,2,2)A_{a\ta}\\
A \cdot B&=&A_{ij}B_{ij}=A_{ab}B_{ab}+A_{\ta\tb}B_{\ta\tb}+2A_{a\ta}
B_{a\ta}\nonumber\\
&=&-2{A_{\nu}}^{\mu}{B_{\mu}}^{\nu}-A_{~~\mu\nu}^{\alpha\dot\alpha}\tilde{B}^
{\mu\nu}_{~~\alpha\dot\alpha}+2(\vec{A}_{L}.\vec{A}_{L}+\vec{A}_{R}.\vec{A}
_{R})
\eea

{\bf{\underline{$54 \cdot 54$}}}

\bea
54(S_{ij})&=&(20,1,1,)\widehat{S}_{ab}+(1,3,3)\widehat{S}_{\ta\tb}+(1,1,1)S+
(6,2,2)S_{a\ta}\\
 S_{ij}R_{ij}&=&\widehat{S}_{ab}\widehat{R}_{ab}+\widehat{S}
_{\ta\tb}\widehat{R}_{\ta\tb}+2\widehat{S}_{a\ta}\widehat{R}_{a\ta}+2S R\\
%
  &=&{1 \over 4}S^{\mu\nu\,,\lambda\sigma}R_{\mu\nu\,,\lambda\sigma}+
S^{\alpha\beta,\dot\alpha\dot\beta}R_{\alpha\beta\dot\alpha\dot\beta}-
S^{\mu\nu,\alpha\dot\alpha}R_{\mu\nu\,,\alpha\dot\alpha}+2SR\\
\nonumber\eea
where
\bea
\widehat{S}_{ab}&=&{S}_{ab}-{\sqrt{2 \over 15}}\delta_{ab}S\\
\widehat{S}_{\ta\tb}&=&{S}_{\ta\tb}+{\sqrt{3 \over 10}}\delta_{\ta\tb}S\\
S&=&{\sqrt{5 \over 24}}S_{aa}
\eea

{\bf{\underline{$54 \cdot 54 \cdot 54$}}}

\bea
S_{ij}R_{jk}T_{ki}&=&{1 \over {{2}^{3}}}{S}^{\mu\nu\lambda\sigma}
{{R}_{\lambda\sigma}}^{\theta\delta}{{T}_{\theta\delta,\mu\nu}}\nonumber\\
&-&{S}^{\alpha\beta,\dot\alpha\dot\beta}{R}_{\beta\gamma,
\dot\beta\dot\gamma}{{T}^{\gamma}_{~\alpha}},^{\dot\gamma}_{~\dot\alpha}
\nonumber\\
%
&-&{\sqrt {2 \over 15}}S.R.T \nonumber\\
%
&+&{\sqrt {1 \over 120}}{\{S}{R}^{\mu\nu,\lambda\sigma}T_{\lambda\sigma,
\mu\nu}+R{S}^{\mu\nu,\lambda\sigma}T_{\lambda\sigma,\mu\nu}+T{S}
^{\mu\nu,\lambda\sigma}R_{\lambda\sigma,\mu\nu}\}\nonumber\\
%
&-&{ \sqrt {1 \over30}}{\{S{R}^{\mu\nu\alpha\dot\alpha}T_{\mu\nu\alpha\dot
\alpha}+R{T}^{\mu\nu\alpha\dot\alpha}S_{\mu\nu\alpha\dot\alpha}+T{S}^{\mu\nu
\alpha\dot\alpha}R_{\mu\nu\alpha\dot\alpha}\}}\nonumber\\
%
&+&{1 \over 2}{\{{S}^{\alpha\beta,\dot\alpha\dot\beta}{R_{\beta\dot
\beta}}^{\mu\nu}T_{\mu\nu\,\alpha\dot\alpha}+{{R}^{\alpha\beta,\dot
\alpha\dot\beta}}{T_{\beta\dot\beta}}^{\mu\nu}S_{\mu\nu\,\alpha\dot\alpha}+
{T}^{\alpha\beta,\dot\alpha\dot\beta}{S_{\beta\dot\beta}}^{\mu\nu}
R_{\mu\nu\,\alpha\dot\alpha}\}}\nonumber\\
%
&+&{\sqrt{3 \over 40}}{\{S{R}^{\mu\nu,\alpha\dot\alpha}T_{\mu\nu\alpha\dot
\alpha}+R{T}^{\mu\nu,\alpha\dot\alpha}S_{\mu\nu,\alpha\dot\alpha}+T{S}^{\mu\nu,
\alpha\dot\alpha}R_{\mu\nu,\alpha\dot\alpha}\}}\nonumber\\
%
&-&{\sqrt{3 \over 10}}{\{S{R}^{\alpha\beta,\dot\alpha\dot\beta}{T}
_{\alpha\beta,\dot\alpha\dot\beta}+R{T}^{\alpha\beta,\dot\alpha\dot\beta}
{S}_{\alpha\beta,\dot\alpha\dot\beta}+T{S}_{\alpha\beta,\dot\alpha
\dot\beta}{R}_{\alpha\beta,\dot\alpha\dot\beta}\}}
%
\eea


{\bf{\underline{$(45)^{2} \cdot 54 $}}}


\bea
A_{ij}A_{jk}S_{ki}&=&2({A_\lambda}^\mu{A_\sigma}^{\nu}{{S}_{\mu\nu}}
^{~\lambda\sigma})+{\sqrt{8 \over 15}}({A_\nu}^{\mu}{A_\mu}^{\nu})S
\nonumber\\
%
&+&{\sqrt{1 \over 30}}(A^{\mu\nu,\alpha\dot\alpha}A_{\mu\nu,
\alpha\dot\alpha})S+{1 \over 4}{A_{\mu\nu}}^{\alpha\dot\alpha}A_
{\lambda\sigma,\alpha\dot\alpha}{S}^{\mu\nu,\lambda\sigma}\nonumber\\
%
&+&2 A^{\mu\nu,\alpha\dot\alpha}S_{\alpha\dot\alpha,\lambda\mu}{A_{\nu}}
^{\lambda}+{\sqrt{1 \over 2}}A^{\mu\nu,\beta\dot\beta}
(\epsilon_{\beta\alpha}A_{\dot\beta\dot\alpha}+\epsilon_{\dot\beta\dot\alpha}
A_{\beta\alpha}){S^{\alpha\dot\alpha}}_{\mu\nu}\nonumber\\
%
&-&{\sqrt{3 \over 40}}S{A}^{\mu\nu,\alpha\dot\alpha}A_{\mu\nu\alpha\dot\alpha}
-{1 \over 2}{S}^{\alpha\beta,\dot\alpha\dot\beta}{A^{\mu\nu}}_{\alpha
\dot\alpha}A_{\mu\nu,\beta\dot\beta}\nonumber\\
%
&+&{\sqrt{6 \over 5}}S(\vec{A}_L.\vec{A}_L+\vec{A}_R.\vec{A}_R)-2A^{\dot
\alpha\dot\beta}A^{\alpha\beta}{S}_{\beta\alpha,\dot\beta\dot\alpha}
\eea


{\bf{\underline{$\overline{126} \cdot 126$}}}


\bea
{1 \over 5!}\Sigma^{(-)}_{i_1...i_5}\Sigma^{(+)}_{i_1...i_5}
&=&\widetilde\Sigma^{(-){\mu\nu}}\Sigma^{(+)}_{\mu\nu}+
2 \Sigma^{(-)~\mu{\alpha\dot\alpha}}_{\nu} \Sigma^{(+)\nu}_{~~~\mu\alpha \dot\alpha}
\nonumber \\
%
& &+{\vec\Sigma}^{(-)}_{{R}\mu \nu} \cdot {\vec{\Sigma}^{(+)\mu \nu}}_{R}+
\vec{\Sigma}_{{L}\mu \nu}^{(+)} \cdot {\vec\Sigma}^{(-)\mu\nu}_{L}
\eea
Here $\Sigma^{(+)}(\Sigma^{(-)})$ is the self-dual (antiself-dual) 5 index
totally antisymmetric representation and the dual is defined as (note the
minus sign)
\be
\widetilde\Sigma_{i_1...i_5}=-{i \over 5!}\epsilon_{i_1....i_{10}}\Sigma_
{i_6...i_{10}} ; \quad \widetilde\Sigma^{(\pm)}={\pm}\Sigma^{(\pm)}
\ee
$\Sigma^+$  has the decomposition
\bea
\Sigma^{+}(126)&=&{\Sigma_{\nu}^{(+)\mu}}_{\alpha\dot\alpha}(15,2,2)+\vec
\Sigma_{\mu\nu}^{(-)L}(10,3,1)\nonumber\\
%
&+&\vec\Sigma^{\mu\nu}_R(\overline{10},1,3) +\Sigma_{\mu\nu}(6,1,1)
\eea
while $\Sigma^-({\overline{126}})$ has the conjugate expansion.
 

{\bf{\underline{$45 \cdot 126 \cdot \overline{126}$}}}
\vspace{.1 cm}\\
An example of the non trivial action of D parity is given by the terms
containing the (15,1,1) in the invariant $45 \cdot 126 \cdot \overline{126}$.

\bea
{1 \over {2(4!)}}A_{a_{1}a_{2}}\Sigma^{(-)}_{a_{1}i_1..i_4}\Sigma^{(+)}_{a_{2}
i_1..i_4}&=&{a_{\nu}}^{\mu}(-\Sigma^{(-)\lambda~\alpha\dot\alpha}_{\mu}
{\Sigma^{(+)}_{\lambda}}^{\nu}_{~\alpha\dot\alpha}+\Sigma^{(-)\nu~\alpha\dot
\alpha}_{\lambda}{\Sigma^{(+)\lambda}_{\mu~~\alpha\dot\alpha}})\nonumber\\
%
& &+(\vec\Sigma^{(-)}_{\mu\nu{R}} \cdot {a^{\nu}}_{\sigma} \cdot \vec\Sigma^{(+)
\sigma\mu}_{(R)}-\vec\Sigma^{(+)}_{\mu\nu{L}} \cdot {a^{\nu}}_{\sigma} \cdot
\vec\Sigma^{(-)\mu\nu}_{(L)})\nonumber\\
%
& &+{a_{\nu}}^{\mu}\widetilde\Sigma^{(-)\nu\lambda}\Sigma^{(+)}_{\lambda\mu}
\eea
Note the relative minus sign in the 
$(15,1,1)_A (15,2,2)_{\pm}(15,2,2)_{\mp})$ \hfil\break
and $((10,3_{\pm})(\overline{10},3_{\pm})(15,1,1)_A)$ terms due to
the property ${a_{\nu}}^{\mu} \stackrel{D}{\rightarrow}  (-)^{\mu+\nu+1}{a_
{\mu}}^{\nu})$.
The terms containing $A_{\ta\tb}$ are also useful and are given by
%
\bea
{1 \over 4!} A_{\ta\tb}\Sigma_{\ta{i_1..i_4}}^{(-)}\overline\Sigma_
{\tb{i_1..i_4}}^{(+)}
&=&\sqrt{2}\{\vec{A}_{R} \cdot ({\vec\Sigma^{{R}(-)}_{\mu\nu} \times \vec
\Sigma^{(+)\mu\nu}_{R}})+\vec{A}_{L} \cdot ({\vec\Sigma^{\mu\nu(-)}_{L}
\times \vec\Sigma^{(+)L}_{\mu\nu}})\nonumber\\
%
&-&({A^{\dot\alpha\dot\beta}\Sigma^{(-)\mu\alpha}_{\nu~~~~\dot\alpha}\Sigma^
{(+)\nu}_{\mu~~~\alpha\dot\beta}+A^{\alpha\beta}\Sigma^{(-)\mu~~\dot\beta}
_{\nu~~~\alpha}\Sigma^{(+)\nu}_{\mu~~~\beta\dot\beta}})\}
\eea
%
The invariance under D parity of the second term follows at once from our
rules,
while  the first two terms are seen to be invariant by rewriting them with
spinor indices or by the supplementary rule
\be
\vec{A}_{R} \cdot ({\vec{B}_{R} \times \vec{C}_{R}}) \leftrightarrow
\vec{A}_{L} \cdot ({\vec{B}_{L} \times \vec{C}_{L}})
\ee



\section {Spinor Representations}
\subsection{Generalties of SO(2N) Spinors}
In the Wilzcek and Zee \cite{wilzee}  notation  the $\gamma$ matrices of the
Clifford algebra of SO(2N),~$\gamma{{_i}^{(N)}}$ are defined iteratively as
direct products of Pauli matrices.
\bea
\gamma_{i}^{(n+1)}&=&\gamma{{_i}^{(n)}}\otimes \tau_3 ,\quad 
n=1.....N-1\\
\gamma_{(2n+1)}^{(n+1)}&=&1\otimes \tau_1\\
\gamma_{(2n+2)}^{(n+1)}&=&1\otimes \tau_2
\eea
starting with $\gamma^{(1)}_{1}=\tau_1 ,\quad \gamma^{(1)}_{2}=\tau_2$
One also defines
\be
\gamma^{(N)}_{F}={(-i)^N}\prod_{i=1}^{2N}\gamma_{i}^{(N)} \equiv 
\bigotimes_{i=1}
^{N}{(\tau_3)_i}=\gamma_F^{(m)} \otimes \gamma_F^{(N-m)} ,~m=1,...N-1
\label{gF}
\ee
so that $\gamma_{F}^2=1 ,\quad 
\gamma_{F}\gamma_{i}=-\gamma_{i}\gamma_{F}$\\

The generators of SO(2N) in the spinor representation are defined as $(i
=j)$
\be
J_{ij}=-{\sigma_{ij} \over 2}=-{i \over 4}[\gamma_i,\gamma_j]
\ee
 A crucial point (where we disagree with equation (A19) of \cite{wilzee})
 is the form of the charge conjugation matrix C.
Equation A(19) of \cite{wilzee} appears to
contradict equation A(11) of the same paper since $(-)^n\neq(-)^{n(n+1)
\over 2}$ in general. Recall that $\psi^{T}C\chi$ is a SO(2N) singlet
when
\be
\sigma_{ij}^{T}C=-C\sigma_{ij}
\ee
Two obvious choices for C are
\be
C_1=\prod_{j=1}^{n}\gamma_{2j-1}~~~~,\quad 
C_2=i^n\prod_{j=1}^{n}\gamma_{2j}\\
\ee
then~~~~~~~
\be
C_1^{(n)T}={(-)}^{n(n-1) \over 2}C_1^{(n)}~~~~,\quad
C_2^{(n)T}={(-)}^{n(n+1) \over 2}C_2^{(n)}\label{cT}
\ee
and both obey $C\gamma_F={(-)^n}\gamma_F C$. Their explicit forms are 
easily obtained from 
\bea
C_1^{(1)}&=& \tau_1 \qquad C_2^{(1)}=i \tau_2\\
C_1^{(n)}&=& \tau_1\times C_2^{(n-1)}\\
C_2^{(n)}&=& i\tau_2\times C_1^{(n-1)}
\eea

In particular $C_2^{(2m+1)}=i\tau_2 \times {\bigotimes}_{i=1}^{m}(\tau_1\times 
i\tau_2)_{i} $ is clearly very different from eqn.A(19) of\cite{wilzee} and obeys 
their eqn. A(11) (our eqn(\ref{cT})) while their A(19) does not.

We shall define the SO(2N) charge conjugation matrix  to be $C_2^{(N)}$.

 The Clifford algebra of SO(2N)
acts on a $2^N$ dimensional space which is given the convenient basis
of eigenvectors ${|\epsilon=\pm 1>}$ of $\tau_3$:
\be
{|\epsilon_1,.......\epsilon_n>}={|\epsilon_1>}\otimes......\otimes{|\epsilon_n>}
\ee
In this basis $\gamma_F=\prod_{i=1}^{n}\epsilon_i$. So the  basis spinors 
of SO(2N) decompose
into odd and even subspaces w.r.t $\gamma_F$.
\be
2^n=2_{+}^{n-1}+2_{-}^{n-1}
\ee
The SO(2N) dual of an N index object is
\be
\tilde F_{i_1.....i_N}=-{i^N \over 
N!}\epsilon_{i_1......i_{2N}}F_{i_{N+1}...i_{2N}}
\ee
The identity
\be
\gamma_{i_1}......\gamma_{i_M}\gamma_F={(-i)^{N}(-)^{{M(M-1)}\over 
2} \over {(2N-M)!}}
\epsilon_{i_1......i_{2N}}\gamma_{i_{M+1}}.........\gamma_{i_{2N}}
\ee
is also frequently needed.\\



\subsection{SO(6) Spinors}


The $4(\psi_\mu)$ and $\bar 4(\widehat{\psi}^\mu)$ of SU(4) may be consistently
identified with the $4_{-},4_{+}$ chiral spinor multiplets of SO(6) by
identifying components
${\psi_\mu}$ of the 4 with the coefficients of the states 
$|\epsilon_1\epsilon_2\epsilon_3>_{-}$
in $4_{-}=|\psi>_{-}$ as
\be
|\psi>_{-}=\psi_{1}|-++> +~\psi_{2}|+-+> +~\psi_3|++-> +~\psi_4|--->
\ee
and also ${\widehat\psi^\mu}$
in the $4_{+}=|\widehat\psi>_{+}$ as
\be
|\widehat\psi>_{+}=-\widehat\psi^{1}|+--> +~\widehat\psi^{2}|-+-> -
~\widehat\psi^{3}|--+> +~\widehat\psi^{4}|+++>
\ee
The reason for
the extra minus signs is that then the charge conjugation matrix 
$C_2^{(3)}$ correctly combines the $4,\bar 4$ components in the $2^3$-plet 
spinors of SO(6) to make SU(4) singlets and covariants . For example 
(we take $\psi,\chi$ to be non-chiral
$8=4_{+}+4_{-}$ spinors to preserve generality) 

\be
\psi^{T} C_{2}^{(3)}\chi=\widehat\psi^{\mu}\chi_{\mu}+\psi_{\mu}\widehat
\chi^{\mu}
\ee
while 
\be
D_{abc}^{\pm}\equiv {1 \over 
3!}\psi_{\mp}^{T}C_2\gamma_{[a}\gamma_{b}\gamma_{c]}
\chi_{\mp}=\pm\widetilde{D}_{abc}^{\pm}
\ee
i.e
\bea
{(4_{-} \times 4_{-})_{self-dual}}\leftrightarrow 10~~ of~~ SU(4)\\
{(4_{+} \times 4_{+})_{anti.s.d}}\leftrightarrow {\overline {10}}~~ of 
~~SU(4)
\eea
which is consistent with the identification $4_-\sim 4, 4_+\sim {\bar 4}$ 
and the multiplication rules in SU(4).

%
By writing the possible expressions using  SU(4) covariance and fixing the
coefficients by taking explicit coefficients,
one can now show that the following identities hold :
 \be
\begin{array}{cl}
\psi^{T}{C}_{2}^{(3)}\chi&=\psi_{\mu}{\widehat\chi}^{\mu}+{\widehat\psi}^{\mu}
\chi_{\mu}=\psi.\widehat\chi+\widehat\psi.\chi\\
%
\psi^{T}{C}_{2}^{(3)}\gamma_{\mu\nu}\chi&=\sqrt{2}{(-\psi_{[\mu}\chi_{\nu]}+
{\widehat\psi}^{\lambda}{\widehat\chi}^{\sigma}\epsilon_{\mu\nu\lambda\sigma})}\\
%
\psi^{T}{C}_{2}^{(3)}\gamma_{\mu\nu}\gamma_{\lambda\sigma}\chi&=-2{\{\widehat\psi^
{\theta}\chi_{[\lambda}\epsilon_{\sigma]\mu\nu\theta}+\psi_{[\mu}\epsilon_{\nu]
\lambda\sigma\theta}\widehat\chi^{\theta}\}}\\
%
\psi^{T}{C}^{(3)}_{2}\gamma_{\mu\nu}\gamma_{\lambda\sigma}\gamma_{\theta\delta}
\chi&={(\sqrt{2})^3}{\{\psi_{[\mu}\epsilon_{\nu]\lambda\sigma[\theta}\chi
_{\delta]}+{\widehat\psi}^{\omega}\widehat\chi^{\rho}\epsilon_{\omega\mu\nu
[\theta}\epsilon
_{\delta]\rho\lambda\sigma}\}}
\end{array}\label{so6spin}
\ee

The square root factors arise because the antisymmetric pair labels for 
the gamma matrices correspond to complex indices ${\hat a}, {\hat b}$.
These then imply the useful identities
\bea
\psi^T C_{2}^{(3)}\gamma_a\chi {O}_{a}&=&
{\sqrt 2}\{ O_{\mu\nu}{\widehat \psi}^{\mu}\widehat\chi^{\nu}-{\tilde O}^{\mu\nu}
\psi_{\mu}\chi_{\nu}
\}\nonumber\\
{1 \over {2!}}\psi^{T}C_{2}^{(3)}\gamma_{[a}\gamma_{b]}\chi{O}_{ab}&=&
4 {O_{\nu}}^{\mu}{\{\psi_{\mu}\widehat\chi^{\nu}-\widehat\psi^{\nu}\chi_{\mu}
}\}\nonumber\\
%
{1 \over {3!}} C^{(3)}_{2}\gamma_{[a}\gamma_{b}\gamma_{c]}\chi{O}_{abc}&=&
12{\{\overline{O}^{\mu\nu}\psi_{\mu}\chi_\nu-O_{\mu\nu}\widehat\psi^{\mu}
\widehat\chi^{\nu}}\}
\label{so6spinO}
\eea



\subsection {SO(4) Spinors}


In the case of SO(4) the spinor representation is 4 dimensional and 
splits into $2_+\oplus 2_-.$  It is not hard to see that with the 
definitions adopted for the generators of $SU(2)_{\pm}$ the  chiral spinors 
$2_{\pm}$ may be identified with the
doublets ${\psi_{\alpha},\psi_{\dot\alpha}}$ of $SU(2)_{-}=SU(2)_L$ and
$SU(2)_{+}=SU(2)_R$ as 
\be
|2>_{-}=|\psi>_{-}=\psi_{1}|+-> +~\psi_{2}|-+> ,\quad
|2>_{+}=|\psi>_{+}=\psi_{\dot{1}}|++> -~\psi_{\dot{2}}|-+>
\ee
The following expressions for spinor covariants then follow
\be
\begin{array}{cl}
{\psi}^{T}C_{2}^{(2)}\chi&=\psi^{\dot\alpha}\chi_{\dot\alpha}-\psi^{\alpha}
\chi_{\alpha}\\
{\psi}^{T}C_{1}^{(2)}\chi&=\psi^{\dot\alpha}\chi_{\dot\alpha}+\psi^{\alpha}
\chi_{\alpha}\\
\psi^{T}{C}_{2}^{(2)}\gamma_{\alpha\dot\alpha}\chi&=\sqrt{2}{( \psi_{\dot
\alpha}\chi_{\alpha}-\psi_{\alpha} \chi_{\dot\alpha})}\\
%
\psi^{T}{C}_{1}^{(2)}\gamma_{\alpha\dot\alpha}\chi&=\sqrt{2}{( \psi_{\dot
\alpha}\chi_{\alpha}+\psi_{\alpha} \chi_{\dot\alpha})}\\
%
\psi^{T}{C}_{2}^{(2)}\gamma_{\alpha\dot\alpha}\gamma_{\beta\dot\beta}\chi&=
{\epsilon_{\dot\alpha\dot\beta}\psi_{(\alpha}\chi_{\beta)}-\epsilon_{\alpha
\beta} \psi_{(\dot\alpha} \chi_{\dot\beta)}+\epsilon_{\alpha\beta}
\epsilon_{\dot\alpha\dot\beta}{(\psi^{\gamma}\chi_{\gamma}- \psi^{\dot
\gamma} \chi_{\dot\gamma})}}\\
%
\psi^{T}{C}_{1}^{(2)}\gamma_{\alpha\dot\alpha}\gamma_{\beta\dot\beta}\chi&=
{-\epsilon_{\dot\alpha\dot\beta}\psi_{(\alpha}\chi_{\beta)}-\epsilon_{\alpha
\beta} \psi_{(\dot\alpha} \chi_{\dot\beta)}-\epsilon_{\alpha\beta}
\epsilon_{\dot\alpha\dot\beta}{(\psi^{\gamma}\chi_{\gamma}+ \psi^{\dot
\gamma} \chi_{\dot\gamma})}}
\end{array}\label{so4spin}
\ee
From these the useful identities
\be
\label{so4spinO}
{1 \over 2!}\psi^{T}{{C^{(2)}_{2}} \brace C^{(2)}_{1}}\gamma_{[\ta}\gamma
_{\tb]}\chi{O}_{\ta\tb}=-2\sqrt{2}\{O^{\dot\alpha\dot\beta}\psi_{\dot\alpha}
\chi_{\dot\beta} \mp O^{\alpha\beta}\psi_{\alpha}\chi_{\beta}\}
\ee
easily follow.


\subsection {SO(10) Spinors}

The spinor representation of SO(10) is $2^5$ dimensional and splits as
\bea
2^5&=&2^{4}_{+}+2^{4}_{-}=16_{+}+16_{-}\\
16_{+}&=&(4_{+},2_{+})+(4_{-},2_{-})=(\overline{4},1,2)+(4,2,1) \\
\overline{16}&=&16_{-}=(4_{+},2_{-})+(4_{-},2_{+})=(\overline{4},2,1)+
(4,1,2)
\eea
Where the first equality follows from eqn(\ref{gF}) and second from the SO(6)
to SU(4) and SO(4) to $SU(2)_{L} \times SU(2)_{R}$ translations: $4_-=4,2_+=2_{R}
,2_-=2_{L}$.
Thus we see that the SU(4) and $SU(2)_L \times SU(2)_R$ properties of the
submultiplets  within the $16,\overline{16}$ are strictly correlated.
Use of the SO(6) and SO(4) spinor covariant identities
allows fast construction of SO(10) spinor invariants.
For example ,                    

\be 
\psi^{T}C_{2}^{(5)} \gamma_{\mu\nu}^{(5)}\chi = 
\psi^{T}(C_2^{(3)} \times C_{1}^{(2)})(\gamma_{\mu\nu}^{(3)} \times \tau_3
\times \tau_3)\chi
=\psi^T(C_2^{(3)}\gamma_{\mu\nu}^{(3)}\times C_2^{(2)})\chi
\ee

Next one uses the 
identities (\ref{so6spin},\ref{so4spin}) in parallel , keeping in mind that
the dotted ($SU(2)_R $) spinors are always $\bar 4$-plets of SU(4) and the 
undotted ones are 4-plets. Thus in the case when $\psi,\chi$ are both 16-plets ,
one can immediately read off the result 

\be
\psi^{T}C_{2}^{(5)} \gamma_{\mu\nu}^{(5)}\chi =
\sqrt {2}{(\psi^{\alpha}_{[\mu}\chi_{{\nu]}\alpha}+
{\widehat\psi}^{\lambda{\dot\alpha}}{\widehat\chi}^{\sigma}_{\dot\alpha}
\epsilon_{\mu\nu\lambda\sigma})}
\ee
While when $\psi=16_+,\chi=16_-$ one obtains a conflict in the application of
 (\ref{so6spin},\ref{so4spin}) which is resolved by realizing that this covariant is then
identically zero by chirality. 


{\bf{\underline{D parity on spinors}}}
\vspace{.3 cm}\\
 D parity acts on the spinors of
SO(10) as
\bea
D_{spinor}&=&e^{({-i{\pi}J_{23}})}e^{({i{\pi}J_{67}})}=-\gamma_2\gamma_3\gamma_6
\gamma_7\nonumber\\
& &=(\bigotimes_{i=1}^{3}i\tau_2) \times (i\tau_2 \times 1_2)=D^{(3)} \times
 D^{(2)}
\eea
Thus the action of D factorizes. Under $D^{(3)}$
one interchanges SO(6) spinors of opposite chirality with sign changes such that  :
\bea
\widehat\psi^\mu \rightarrow (-)^{\mu}\psi_\mu \\
\psi_\mu \rightarrow (-)^{\mu+1}\widehat\psi^{\mu}
\eea
%
%
Similarly for $D^{(2)}=i\tau_2 \times 1$ one finds interchange           
%
\bea
{\psi}_{\alpha} \rightarrow -{\psi}_{\dot{\bar\alpha}}  ,\quad {\psi}_{\dot\alpha}
\rightarrow {\psi}_{\bar\alpha} \Rightarrow \psi^{\alpha} \rightarrow \psi
^{\dot{\bar\alpha}},\psi^{\dot\alpha} \rightarrow -\psi^{\bar\alpha}
\eea
Where by $\bar\alpha$ we mean $\bar{1}=2,\bar{2}=1.$
%
This implies the contraction of spinors $\psi_{\alpha},\chi_{\dot\alpha}$ with
a bidoublet $V_{\alpha\dot\alpha}=V_{\hat{a}}$ tranforms  as
\be
V^{\alpha\dot\beta}\psi_{\alpha}\chi_{\dot\beta} \rightarrow -V^{\bar\beta
\dot{\bar\alpha}}\psi_{\dot{\bar\alpha}}\chi_{\bar\beta}
\ee
Similarly with $SU(2)_{L} (SU(2)_{R})$ vectors one gets
\be
V_{(-)}^{\alpha\beta}\psi_{\alpha}\chi_{\beta} \leftrightarrow -V_{(+)}
^{\dot{\bar\beta}\dot{\bar\alpha}}\psi_{\dot{\bar\alpha}}\chi_{\dot{\bar
\beta}}
\ee
While
\bea
\psi^{\alpha}\chi_{\alpha} \leftrightarrow -\psi^{\dot\alpha}\chi_{\dot
\alpha}\\
\widehat\psi^{\mu}\chi_{\mu} \leftrightarrow -\psi_{\mu}\widehat\chi^{\mu}
\eea
%
%
{\bf{\underline{SO(10) Spinor-Tensor Invariants  }}}
\vspace{.3 cm}\\
We next give the explicit decomposition of  quadratic and cubic
SO(10) invariants involving a pair ($16,16$ or $16,\overline {16}$ ) of SO(10) 
spinors contracted with (the conjugate of) one of the tensors in their
Kronecker product
decomposition :

\bea
16\otimes 16 &=& 10 \oplus 120\oplus 126\\
16\otimes {\overline{16}} &=& 1 \oplus 45 \oplus 210
\eea
\\
Besides use of the spinor identities (\ref{so6spin},\ref{so4spin})
the remainder of the task is merely to decompose the SO(10) index 
contractions into PS irrep. index contractions , take account of 
self-duality where relevant, and maintain unit reference norm.
 \\
\\
\\
${\bf{\underline{16 \cdot 16 \cdot 10}}}$
\vspace{.1 cm}\\
The 10-plet has decomposition:
$10=H_{a}{(6,1,1)}+H_{\ta}{(1,2,2)}$ and one gets 
%
\be
\psi^{T}{C}_{2}^{(5)}\gamma_{i}^{(5)}\chi H_{i}=\sqrt{2}
{\{H_{\mu\nu}\widehat{\psi}^{\mu{\dot 
\alpha}}\widehat{\chi}^{\nu}_{\dot\alpha}+ 
\widetilde{H}^{\mu\nu}\psi_{\mu}^{\alpha}\chi_{\nu\alpha}-H^{\alpha\dot\alpha}
{(\widehat{\psi}^{\mu}
_{\dot\alpha}\chi_{\alpha\mu}+\psi_{\alpha\mu}\widehat{\chi}_{\dot\alpha}^{\mu})}}\}\\
\ee
Note how D parity is maintained by the interplay between the SO(6) and SO(4)
sectors.
\vspace{.4 cm}
\\
${\bf{\underline{16 \cdot 16 \cdot 120}}}$
\\
Since
\bea
120&=&O_{abc}(10+\bar 10,1,1)+O_{ab \ta}(15,2,2)+O_{a\ta\tb}((6,1,3)+
(6,3,1))+ O_{\ta\tb\tilde\gamma}(1,2,2)\nonumber\\
%
&=&O^{(s)}_{\mu\nu}({10,1,1})+\overline{O}^{\mu\nu}_{(s)}(\overline{10},1,1)
+{O_{\nu}}^{\mu\alpha\dot\alpha}(15,2,2)\nonumber\\
%
&+&{O^{(a)}_{\mu\nu}}^{\dot\alpha
\dot\beta}(6,3,1)+{O^{(a)}_{\mu\nu}}^{\alpha\beta}(6,1,3)+O^{\alpha\dot
\alpha}(1,2,2)
\eea
%
%
one gets
\bea
{1 \over (3!)^2}\psi C^{(5)}_{2}\gamma_{[i}\gamma_{j}\gamma_{k]}\chi O_{ijk}&
=&
-2{(\bar{O}^{\mu\nu}_{(s)}\psi_{\mu}^{\alpha}\chi_{\nu\alpha}
+O_{\mu\nu}^{(s)}\widehat\psi^{\mu\dot\alpha}{\widehat\chi_{\dot\alpha}^{\nu}})}+
%
2\sqrt{2}{O^{\mu\alpha\dot\alpha}_{\nu}}
{{(\widehat\psi_{\dot\alpha}^{\nu}\chi_{\mu\alpha}-
{\psi_{\mu\alpha}\widehat\chi_{\dot\alpha}^\nu})}} \nonumber\\
%
& &-2({O_{\mu\nu}^{(a)}}^{\dot\alpha\dot\beta}
\widehat\psi^{\mu}_{\dot\alpha}\widehat\chi^{\nu}_{\dot\beta}+\widetilde{O}^
{\mu\nu\alpha\beta}_{(a)}\psi_{\mu\alpha}\chi_{\nu\beta})
%
+\sqrt{2}O^{\alpha
\dot\alpha}(+\psi_{\dot\alpha}^{\mu}\chi_{\mu\alpha}-\psi_{\mu\alpha}
\widehat\chi_{\dot\alpha}^{\mu})
\eea
%
Note $O^{\alpha\dot\alpha}$ is derived from $O_{\tilde\alpha}={1 \over
3!} \epsilon_{\ta\tb\tilde\gamma\tilde\delta}O_{\tb\tilde\gamma\tilde\delta}$
and so has opposite D parity to a vector $V_{\ta}$.

\vspace{.3 true cm}

${\bf{\underline{16 \cdot 16 \cdot \overline{126}}}}$
\vspace{.1cm}
\be
\overline{126}=O_{\mu\nu}^{(a)}(6,1,1)+{O_{\nu}^{~\mu}}_{\alpha\dot\alpha}
(15,2,2)+O_{\mu\nu\dot\alpha\dot\beta}(10,1,3)+{\overline{O}^{\mu\nu}}_{\alpha\beta}
({\overline {10}},3,1)
\ee
\bea
{1 \over (5!)^2}\psi^{T}C_{2}^{(5)}\gamma_{[i_{1}}.....\gamma_{i_{5}]}\chi
{\Sigma}^{(-)}_{i_{1}...{i_{5}}}&=&2\sqrt {2}{(\widetilde{O}^{\mu\nu}_{(a)}\psi_{\mu}^
{\alpha}\chi_{\nu\alpha}-O_{\mu\nu}^{(a)}\widehat \psi^{\mu\dot\alpha}{\widehat
 \chi}_{\dot\alpha}^{\nu})}-{4 \sqrt {2}}O^{\mu\alpha\dot\alpha}_{\nu}
 {(\widehat\psi_{\dot\alpha}^{\nu}\chi_{\alpha\mu}+\psi_{\mu\alpha}\widehat
\chi_{\dot\alpha}^{\nu})}\nonumber\\
%
& &+4(O_{\mu\nu}^{\dot\alpha\dot\beta}\widehat\psi_{\dot\alpha}^{\mu}
\widehat\chi_{\dot\beta}^{\nu}+\overline{O}^{\mu\nu\alpha\beta}\psi_{\mu\alpha}
\chi_{\nu\beta}) 
\eea
%
%
Here $(O_{\mu\nu}^{(a)}) \leftrightarrow (-)^{\mu+\nu}\widetilde{O}^{\mu\nu}
_{(a)} ,~~
{O^{\nu}}_{\mu} \leftrightarrow (-)^{\mu+\nu}{O_{\nu}}^{\mu}$ have reversed D
parity due to the dualization involved in their definition.
Henceforth we shall say a
representation is D-Axial if due to dualization it has an extra (-) in its D
transformation relative to that expected from its tensor structure.
\vspace{.2cm}\\
${\bf{\underline{16 \cdot \overline{16} \cdot 1}}}$
\bea
16(\psi)=(4,2,1)\psi_{\mu\alpha}+(\bar{4},1,2)\hat\psi^{\mu}_{\dot\alpha}\\
16(\overline\chi)=(\bar{4},2,1)\hat\chi^{\mu}_{\alpha}+(4,1,2)\chi_{\mu\dot\alpha}
\eea
%
\be
\psi^T{C}_{2}^{(5)}\overline\chi=\widehat\psi^{\mu\dot\alpha}\chi_{\mu\dot
\alpha}+\psi_{\mu\alpha}\widehat\chi^{\mu\alpha}=-\chi^{T}C_{2}^{(5)}\psi
\ee
\\
\\


${\bf{\underline{16 \cdot \overline{16} \cdot 45}}}$
\bea
45&=&{O_{\nu}}^{\mu}(15,1,1)+O_{\mu\nu\alpha\dot\alpha}(6,2,2)+O_{\alpha
\beta}(1,3,1)+O_{\dot\alpha\dot\beta}(1,1,3)\\
%
{1 \over (2!)^2}\psi^{T}{C}_{2}^{(5)}\gamma_{[i}\gamma_{j]}{\bar\chi}{O_{ij}}
%
&=&2O_{\kappa}^{~\mu}(\psi_{\mu}^{\alpha}\widehat\chi_{\alpha}^{\kappa}-
\widehat\psi^{\kappa\dot\alpha}\chi_{\mu\dot\alpha})-
\sqrt2{(O^{\dot\alpha
\dot\beta}\widehat\psi^{\mu\dot\alpha}\chi_{\dot\beta\mu}+O^{\alpha\beta}
\psi_{\mu\alpha}\widehat\chi^{\mu}_{\beta}})\nonumber \\
%
& &- {(\widetilde{O}^{\mu\nu\alpha\dot\alpha}\psi_{\mu\alpha}\chi_{\nu\dot
\alpha}+O_{\mu\nu}^{\alpha\dot\alpha}\widehat\psi^{\mu}_{\dot
\alpha}\widehat\chi_{\alpha}^{\nu})}
\eea


${\bf{\underline{16 \cdot \overline{16} \cdot 210}}}$:
\bea
210&=&{O_{\nu}}^{\delta}(15,1,1)+O_{\mu\nu\alpha\dot\alpha}(10,2,2) +
{\overline{O}^{\mu\nu}}_{\alpha\dot\alpha}(\overline{10},2,2)\nonumber\\
%
& &+O^{\nu}_{\mu\alpha\beta}(15,3,1)+
O^{\nu}_{\theta\dot\alpha\dot\beta}(15,1,3)+O(1,1,1)
\eea


\bea
{1 \over (4!)^2}\psi^{T}{C}_{2}^{(5)}\gamma_{[i_{1}}....\gamma_{i_{4]}}{\bar\chi}{O}_
{i_{1}...i_{4}}
%
&=&2i{O}^{\sigma}_{\delta}(\widehat\psi^{\delta\dot\alpha}\chi_
{\sigma\dot\alpha}+\psi_{\sigma\alpha}\widehat\chi^{\delta}_
{\alpha})\nonumber\\
%
& &+2 \sqrt{2}(\overline{O}^{\mu\nu\alpha\dot\alpha}\psi_{\mu\alpha}\chi_
{\nu\dot\alpha}+{O
_{\mu\nu}}^{\alpha\dot\alpha}\widehat\psi^{\mu}_{\dot\alpha}\widehat\chi^
{\nu}_{\alpha})\nonumber\\
%
& &+2 \sqrt{2}{\{  {{O_{\delta}}^{\lambda}}^{\dot\alpha\dot\beta}
\widehat\psi^{\delta\dot\alpha}\chi_{\lambda\dot\beta} 
{-O_{\delta}}^{\mu}}^{\alpha\beta}\psi_{\mu\alpha}{\widehat
\chi^{\delta}}_{\beta}   
\}\nonumber\\
%
& &+2{\{\widetilde{O}^{\mu\nu\alpha\dot\alpha}\psi_{\mu\alpha}\chi_{\nu\dot
\alpha}+{O_{\mu\nu}}^{\alpha\dot\alpha}
\widehat\psi^{\mu}_{\dot\alpha}{\widehat\chi^{\nu}}_{\alpha}\}}\nonumber\\
%
& &+O(\psi_{\mu}^{\alpha}{\widehat\chi^{\mu}}_{\alpha}
-\widehat\psi^{\mu\dot\alpha}\chi_{\mu\dot\alpha})
\eea
%
$O^{~\mu}_{\nu}$ , O are both D-Axial, while
%
\be
D(O_{\mu\nu}^{~\alpha\dot\beta})=(-)^{\mu+\nu+1}\overline{O}^{\mu\nu\bar\beta
\dot{\bar\alpha}}
\ee
%
Note that to obtain the results when the complex conjugate $16^*$ 
is used instead of $\overline{16}$
one need only replace
\be
\widehat\chi^{\mu\alpha} \rightarrow \chi^*_{\mu\alpha}  ,\quad
{\chi_{\mu}}^{\dot\alpha} \rightarrow ({\widehat\chi^{\mu}}_{\dot\alpha})^*
\ee
When calculating quartic invariants formed by contractions of SO(10) tensor covariants 
formed using 16,${\overline{16}}$ multiplets (which often arise in model 
building with non-renormalizable superpotentials\cite{so10refs})
 one need only  
apply the identities (\ref{so6spin},\ref{so4spin}) 
after decomposing the SO(10) vector indices while treating one of the covariants as an 
operator with appropriate PS indices. After using
 (\ref{so6spinO},\ref{so4spinO}) one uses (\ref{so6spin},\ref{so4spin}) again to expand
the dummy operator. 




\section {Effective Field Theory of an SO(10) GUT}

To illustrate the application of our results we  obtain the renormalizable supersymmetric 
Pati-Salam and D-parity invariant effective theory below the first symmetry breaking
threshold  in a previously developed SO(10) GUT\cite{abmrs01}. For clarity we consider
only one family$\Psi (16)$ and a single electro weak breaking 10-plet (H). 
  The Higgs multiplets  used to break SO(10) down to the SM gauge 
group are A(45), S(54),$\Delta (126),{\bar{\Delta}}(\overline{126})$ The superpotential is
%
\bea
W&=&{m_{s} \over 2}Tr{S}^{2}+{\lambda_{s} \over 3}Tr{S}^{3}+{m_{A} \over 2}
Tr{A}^{2}+{\lambda}Tr{A}^{2}S\nonumber\\
%
&+&{m_{\Sigma} \over 5!}\Delta\overline\Delta+
{\eta_{s} \over 4!}\Delta^{2}S+{\bar\eta_{s} \over 4!}{\bar\Delta}^{2}S+
{\eta_{A} \over 4!}A\Delta\overline\Delta\nonumber\\
%
& &+h\Psi^T C_2^{(5)} \gamma_{i}\Psi H_{i}+{f\over {5!}}\Psi\gamma_{[i_1}..
...\gamma_{i_5]}\Psi\bar\Delta_{i_1...i_5}\nonumber\\
%
&+&M_{H}H_{i}H_{i}+\eta_{H}S_{ij}H_{i}H_{j}
\eea

Note that our definition of the SO(10) dual {\it{differs by a sign}} from that in 
ref.\cite{abmrs01} :$\widetilde \Delta=-i\epsilon \Delta$ 
so that in terms of the fields $\Sigma,{\overline \Sigma}$ used in \cite{abmrs01} 
our 126 is $\Sigma^{+}=
\overline\Sigma=\Delta$ and $\overline{126}$ is $\Sigma^-=\Sigma=\overline
\Delta.$
The mass parameter $m_S>>m_A, m_{\Sigma}$ and the vev of the $(1,1,1)_S$ 
is determined to be $2 m_s/{\lambda_S}$. A fine tuning between parameters is
used to keep {\it{either}} the $(15,1,1)_A$ {\it{or}} the $(1,3,1)_A + (1,1,3)_A $
light at the first stage of symmetry breaking. The vev of $(1,1,1)_S$ gives mass 
to the $SO(10)/G_{PS}$ part of the gauge multiplet and all of the 54
is heavy. In the $126,{\overline{126}}$ only the decuplets remain light since they do not
couple to the vev of the 54 and have a smaller mass term. D-parity is not broken 
since the diagonal elements of S are D-even.  A renormalizable supersymmetric
effective Pati-Salam theory suitable for a simple RG analysis which employs a step
function approximation to the mass thresholds is 
 obtained by simply deleting heavy fields without 
a vev (including the auxiliary fields $D_{a\tilde\alpha}$ of the coset part
$SO(10)/G_{PS}$ of the gauge multiplet) and setting $(1,1,1)_S $ equal to its vev.
This lagrangian can be written in terms of unitary labels and allows one to
follow the couplings from the GUT scale downwards without confusion. The two
possible chains \cite{abmrs01} are implemented by the choice of triplets or
quindeciplet in the A(45) to be kept light.
Then the renormalizable effective super potential is 

\bea
W_{eff}&=&(m_{A}+2\lambda{s})A_{\nu}^{~\mu}A_{\mu}^{~\nu}\nonumber\\
&+&2\eta_{A}({{\overrightarrow {\overline \Delta^R}}}_{\mu\nu}
 \cdot {\overrightarrow{\Delta^{R\sigma\mu}}}-
{\overrightarrow{\Delta^L}}_{\mu\nu} \cdot {\overrightarrow{{\overline\Delta^{L\sigma\mu}}}})
A_{\sigma}^{~\nu} \nonumber\\
&+&{2\eta_A \over {\sqrt 2}}\{{\overrightarrow A}^R \cdot
 ({\overrightarrow{\overline\Delta^R}}_{\mu\nu} \times {\overrightarrow{\Delta^{R\mu\nu}}})
+ {\overrightarrow {A^L}} \cdot ({\overrightarrow {\Delta^L}_{\mu\nu}} \times
{\overrightarrow{{\bar\Delta}^{L\mu\nu}}})\}\nonumber\\
&+&4i{f}({\overrightarrow{\overline \Delta^R}}_{\mu\nu} \cdot \widehat\psi^{\mu}
i\sigma_2{\overrightarrow{\sigma}}\widehat\psi^{\nu}+
{\overrightarrow{{\bar\Delta}^{L\mu\nu}}} \cdot \psi_{\mu}i\sigma_2
{\overrightarrow{\sigma}}\psi_{\nu})\nonumber\\
&+&({3 \over 2}\lambda_{H}s  - M_{H}) H_{\alpha\dot\alpha}H^{\alpha\dot\alpha} 
\nonumber\\
&-&{\sqrt2}h H^{\alpha\dot\alpha}(\hat\psi^{\mu}_{\dot\alpha}\psi^{}
_{\mu\alpha}+\psi_{\mu\alpha}\hat\psi^{\mu}_{\dot\alpha})\nonumber\\
&+&m_{\Delta}({\overrightarrow{\overline\Delta^R}}_{\mu\nu} \cdot
{\overrightarrow {\Delta^{R\mu\nu}}}+
{\overrightarrow{\overline\Delta^L_{\mu\nu}}} \cdot
{\overrightarrow{\Delta^{L\mu\nu}}})
\eea
%
Notice that the diagonal elements of  of $A_{\nu}^\mu$  are D-parity {\it{odd}}
so that the spectrum becomes asymmetric when A gets a vev breaking $SU(4)$ down to 
$SU(3)\times U(1)_{B-L}$. As discussed in \cite{geneal} this model actually
embeds a variant MSLRM \cite{abs,ams,amrs} with a Parity Odd singlet.

In addition one may wish to study type II seesaw mass effects and Baryon violation.
This can be done by integrating out heavy fields such as those in S , the gauge
multiplet and the $126,{\overline {126}}$ using their mass terms and their couplings
to light fields to obtain higher dimension operators describing Baryon number violation ,
Type II see-saw effects etc. This procedure is particularly transparent and clear in
Unitary labels. Details will be reported elsewhere.
 
The translation of the GUT covariant derivatives may be seen from e.g
\bea
\overline{\psi}({{\partial}}+{i \over 2}{{{A}}}^{kl}g_{u}J_{kl})\psi&=&
%
\overline{\psi}_{\mu\alpha}{{\partial}}\psi_{\alpha}+\overline{\widehat\psi^{\mu}
_{\dot\alpha}}{\partial}\widehat\psi_{~~\dot\alpha}^{\mu}\nonumber\\
%
& &+{ig_u \sqrt2}\{\overline\psi_{\kappa\alpha}{{A}}^{A}({\lambda^{A}
\over 2})_{\kappa}^{\mu}\psi_{\mu}+\overline{\widehat\psi^{\mu}}_{~~\dot\alpha}{ A}^{A}
({({-\lambda^{A} \over 2})_{\mu}^{\kappa}})^*\widehat\psi^{\kappa}_{~~\dot\alpha}
\nonumber\\
%
& &+\overline{{\widehat\psi}^{\mu}_{\dot\beta}}({{\vec{A_R}} \cdot \vec\sigma \over 2})
_{\dot\beta}^{~~\dot\gamma}\widehat\psi^{\mu}_{\dot\gamma}
+\overline{\psi_{\mu\beta}}({\vec{A_L} \cdot \vec\sigma \over 2})
_{\beta}^{~~\gamma}\widehat\psi_{\mu\gamma}\}\nonumber\\
& &+{g_{u} \over 2}(\overline{\widehat\psi^{\nu}_{\dot\alpha}}\widetilde{{A}}
^{\mu\nu\alpha}_{~~\dot\alpha}\psi_{\mu\alpha}+\overline\psi_{\nu\alpha}
{{A}}_{\mu\nu\alpha}^{~~\dot\alpha}\widehat\psi^{\mu}_{~~\dot\alpha})
\label{covder}
\eea

Where we have retained terms corresponding to the heavy coset gauge fields for
use in deriving effectve lagrangians for B violation etc. 
We see that Pati-Salam coupling constants emerge as $g_{4}=g_{2}={g_u
\sqrt2}$. The GUT generators $T^{A},\vec{T}_{R},\vec{T}_{L}$ 
are each normalized to 2 on the 16-plet and have $\sqrt2{g_u}$ as their associated
coupling. Similarly the mass terms for the coset bosons 
 are \be
{1 \over 2}<((D\widetilde{S})_{ij})^{2}>=-{25 \over 4}{g_u}^{2}s^{2}\widetilde
{A}^{\mu\nu,\alpha\dot\alpha}A_{\mu\nu\alpha\dot\alpha}
\ee

 Integrating out these gauge fields using this 
mass term and the couplings to matter fields given in (\ref{covder}) above
gives dimension 6 operator relevant to B volation. 

Finally the vector representation covariant derivative behaves as 
\bea
\overline{V}_{i}(\partial+{i \over 2}g_u{{A}}^{kl}J_{kl})_{ij}V_{j}&=&
%
{1 \over 2}\overline{V}_{\mu\nu}\partial{V}_{\mu\nu}+{i \over 2}{g_u
\sqrt2}\overline{V}_{\mu\nu}{{A}}^{A}({\lambda^A \over 2})_{[\mu}^{~~\sigma}V_
{\nu]~\sigma}\nonumber\\
%
& &+ig_u\sqrt{2}{\overline{V}_{\alpha\dot\alpha}}({\vec{A}_{L}\cdot({\vec\sigma
 \over 2})_{\alpha}^{~~\beta}V_{\beta\dot\alpha}+\vec{A}_{R}\cdot({\vec\sigma \over
2})_{\dot\alpha}^{~~\dot\beta}V_{\alpha\dot\beta}})
\eea
This can easily be adapted to decompose the kinetic terms of any of the
tensor representations. 


\section{Discussion}                                                   
In this paper we have carried out the tedious calculations required to 
provide a tool kit for ready translation of any SO(10) invariant one is likely
to encounter in the course of SO(10) GUT model building into a convenient form where
the fields carry left-right symmetric unitary group labels. In addition we have obtained a very explicit 
description of the action of D parity on all fields. This  allows one to follow the operation
of D-parity, which implements Left-Right symmetry i.e parity, in such theories.
This translation is necessary in order to carry out RG analysis based on 
calculated mass spectra and will also be useful to obtain more accurate 
estimates of threshold uncertainties . We presented an explicit example of the use 
of these tools by giving the effective Susy Pati-Salam theory obtained in a particular 
SO(10) GUT model \cite{abmrs01}.
A systematic study of related theories along the lines of the program
outlined in \cite{geneal} using the tools developed here will be 
presented elsewhere. We hope that our techniques and results will be
found useful by other practitioners of the unwieldy and -so far- somewhat
obscure art of SO(10) GUT building, if only due to the 
simple minded and perhaps objectionably explicit
 approach we have taken to the analysis 
of this niggling group theoretical problem. Our rules may also be applied in other 
contexts where one encounters these groups. For example in 10 dimensional field theories where the Lorentz
group is $SO(1,9)$ and a translation to $SU(4)$ labels instead of $SO(6)$ labels for the compactified sector
may prove more convenient, specially for spinorial indices.

\vspace{.1 true cm}

{\bf{Acknowledgements}}
 \vspace{.1 true cm}

I am grateful to G. Senjanovic for discussions and encouragement and for 
hospitality at ICTP,Trieste where this work was initiated. This work is supported by the Department
of Science and Technology, Government of India under project No.SP/S2/K-07/99.
%\section{Appendix}
% In this section we have collected useful $SO(6)\leftrightarrow SU(4)$ 
% $SO(4)\leftrightarrow SU(2)\times SU(2)$ identities for the convenience of the reader
% while translating invariants of his choice using our methods.
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\end{document}
   



