\documentclass[aps,secnumarabic,showpacs,nobibnotes,amsmath,amssymb,prd]{revtex4}
%\usepackage{graphicx}% Include figure files
%\usepackage{dcolumn}% Align table columns on decimal point
%\usepackage{bm}% bold math
%\newcomand{\be}{\begin{equation}}
%\newcomand{\ee}{\end{equation}}
\begin{document}
\title{\large \bf Supersymmetric Standard Model from String Theory}
\medskip
%\preprint{}
%\draft
\author{B. B. Deo }
\affiliation{ Physics Department, Utkal University, Bhubaneswar-751004, India.}
\begin{abstract}
N=1, D=4 Superstring possessing $SO(6)\otimes SO(5)$ gauge symmetry is constructed 
from the open bosonic string in twenty six dimension. Without breaking supersymmetry, 
the gauge symmetry of the model descends to the supersymmetric standard model of 
the electroweak scale in four flat dimension.

\end{abstract}
\pacs{11.25-w,11.30Pb,12.60.Jv}
\maketitle
The model begins from the Nambu-Goto~\cite{ng} open bosonic string in the world sheet 
($\sigma ,\tau$) which makes sense in 26-dimension. Following Mandelstams's~\cite{m}, 
proof of equivalence between one boson to two fermionic modes, one can rewrite the action 
as the sum of the four bosonic coordinates $X^{\mu}$ of SO(3,1) and forty four fermions
which are scalars in the world sheet having symmetry SO(44). To have an action with 
SO(3,1) invariance and noting that Majorana Lorentz fermions are in bosonic 
representation of SO(3,1), the forty four fermions are grouped into eleven Lorentz vectors.
The action, so obtained, is not supersymmetric. The eleven vectors have to be further 
divided into two species; $\psi^{\mu,j}$, j=1,2,..6 and $\phi^{\mu, k}$,$ k$=7,8..11. 
For the group of six, the the positive and negative parts are 
$\psi^{\mu,j }=\psi^{(+)\mu,j} +\psi^{(-)\mu,j}$  whereas for the group of five, 
allowing the freedom of phase of creation operators for Majorana fermions
,$\phi^{\mu,k} = \phi^{(+)\mu,k}-\phi^{(-)\mu,k}$. The action is now
\begin{equation}
S= -\frac{1}{2\pi}\int d^2\sigma\left [ \partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}
-i\; \bar{\psi}^{\mu,j}\;\rho^{\alpha}\;\partial_{\alpha}\;  \psi_{\mu,j}
+ i\; \bar{\phi}^{\mu,k}\;\rho^{\alpha}\;\partial_{\alpha}\;  \phi_{\mu,k}\right ]
\label{a}
\end{equation}
The upper indices i,j refer to a row and lowers to a column and
\begin{eqnarray}
\rho^0 =
\left (
\begin{array}{cc}
0 & -i\\
i & 0\\
\end{array}
\right )
\end{eqnarray}
and
\begin{eqnarray}
\rho^1 =
\left (
\begin{array}{cc}
0 & i\\
i & 0\\
\end{array}
\right )
\end{eqnarray}

Dropping indices,
\begin{equation}
\bar{\psi}=\psi^{\dag}\rho^o,\;\;\;\;\;\;\;\bar{\phi}=\phi^{\dag}\rho^o
\end{equation}
Besides SO(3,1), the action (\ref{a}) is invariant under $SO(6)\otimes SO(5)$. It is also 
invariant under the transformation
\begin{eqnarray}
\delta X^{\mu} =\bar{\epsilon}(e^j\psi^{\mu}_j - e^k\phi^{\mu}_k),\\
\delta\psi^{\mu,j}= - ie^j\rho^{\alpha}\partial_{\alpha}X^{\mu}\epsilon\\
\delta\phi^{\mu,k}= ie^k\rho^{\alpha}\partial_{\alpha}X^{\mu}\epsilon.
\end{eqnarray}
$\epsilon$ is a constant anticommuting spinor. $e^j\;\;,e^k$ are unit component 
of c-number row vectors with $e^j\;e_l = \delta^j_l$, $e^je_j$=6, $e^ke_k$=5.
The commutors of two supersymmetric transformations, lead to a translation with 
the coefficient $a^{\alpha}= 2\;i\;\bar{\epsilon}^1\rho^{\alpha}\epsilon_2$
provided
 \begin{equation}
\psi_j^{\mu} = e_{ j}\Psi^{\mu} ,\;\;\;\;\;\;\;\phi_k^{\mu} = e_{ k}\Psi^{\mu}\label{ae}
\end{equation}
and
\begin{equation}
\Psi^{\mu} = e^j\psi^{\mu}_j - e^k\phi^{\mu}_k\label{be}
\end{equation}
It is easy to verify that
\begin{eqnarray}
\delta X^{\mu}=\bar{\epsilon}\Psi^{\mu}, \;\;\;\;\;\;\;\;\; \delta \Psi^{\mu}=-i\;
\epsilon\;\rho^{\alpha}\;\partial_{\alpha}\;X^{\mu}\label{b}
\end{eqnarray}
and 
\begin{equation}
[\delta_1 ,\delta_2]X^{\mu } = a^{\alpha}\partial_{\alpha}X^{\mu },\;\;\;\;\;\;
[\delta_1 ,\delta_2]\Psi^{\mu } = a^{\alpha}\partial_{\alpha}\Psi^{\mu }\label{c}
\end{equation}
Thus the action(\ref{a}) is supersymmetric. However there are eleven fermionic and 
two bosonic ghosts which have to be eliminated by an equal number of subsidiary 
conditions following from a gauge symmetry.
From equations (\ref{b}) and (\ref{c}), it follows that the superpartner of $X^{\mu}$
is $\Psi^{\mu}$. Introducing another supersymmetric pair, the Zweibein 
$e^{\alpha}(\sigma,\tau)$ and the gravitons $\chi_{\alpha}= \nabla_{\alpha}\epsilon$,
the local 2-d supersymmetric action first written down by Brink, Di Vecchia, Howe, Deser
and Zumino~\cite{br}
\begin{equation}
S= -\frac{1}{2\pi}\int d^2\sigma ~~e~~\left [ h^{\alpha\beta}\partial_{\alpha}X^{\mu }
\partial_{\beta}X_{\mu } -i\bar \Psi^{\mu}\rho^{\alpha}\partial_{\alpha}
\bar \Psi_{\mu}+ 2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}
\partial_{\beta}\chi^{\mu}+\frac{1}{2}
\bar{\Psi }^{\mu}\Psi_{\mu}\bar{\chi}_{\beta} \rho^{\beta}\rho^{\alpha}\chi_{\alpha}
\right ]\label{d}
\end{equation}
This action has several invariances. For detailed discussions, see reference~\cite{g}.
Varying the field and Zweibein, the vanishing of the Noether current $J^{\alpha}$ and 
the energy momentum tensor $T_{\alpha\beta}$ is derived.
\begin{equation}
J_{\alpha}= \frac{\pi}{2e}\frac{\delta S}{\delta \chi^{\alpha}}=\rho^{\beta}
\rho_{\alpha}\bar{\Psi}^{\mu}
\partial_{\beta}X_{\mu}=0
\end{equation}

\begin{equation}
T_{\alpha\beta}=\partial_{\alpha}X^{\mu }
\partial_{\beta}X_{\mu }- \frac{i}{2}\bar{\Psi}^{\mu}\rho_{(\alpha}\partial_{\beta )}
\Psi_{\mu}=0
\end{equation}
In a light cone basis, the vanishing of the light cone components are
\begin{equation}
J_{\pm}=\partial_{\pm}X_{\mu}\Psi^{\mu}_{\pm}=0
\end{equation}
and
\begin{equation}
T_{\pm\pm}=
\partial_{\pm}X^{\mu}\partial_{\pm}X_{\mu}+\frac{i}{ 2}\psi^{\mu j}_{\pm}\partial_{\pm}
 \psi_{\pm\mu,j }- \frac{i}{2}\phi_{\pm}^{\mu k}\partial_{\pm}\phi_{\pm\mu,k}\label{e}
\end{equation}
where
$\partial_{\pm}=\frac{1}{2}(\partial_{\tau} \pm\partial_{\sigma})$. Using the 
equation(\ref{ae}), the component constraints are
\begin{equation}
\partial_{\pm}X_{\mu}\psi_{\pm}^{\mu,j} = \partial_{\pm}X_{\mu}
e^j\Psi^{\mu}_{\pm }=0,~~~~~~~~j=1,2...6.\label{f}
\end{equation}
\begin{equation}
\partial_{\pm}X_{\mu}\phi_{\pm}^{\mu k} = \partial_{\pm}X_{\mu}e^k\Psi^{\mu}
_{\pm }=0,~~~~~~~~k=7,8,..11.\label{ga}
\end{equation}
Equations (\ref{e}), (\ref{f}) and (\ref{ga}) constitute 13 constraints and eliminates 
all the metric ghosts from the Fock space.

The action in equation (\ref{d}) is not space time supersymmetric. However, in the 
fermionic representation SO(3,1) fermions are Dirac spinor with four components $\alpha$. 
We construct Dirac spinor like equation (\ref{be}) as the sum of component spinor
\begin{equation}
\theta_{\alpha}=\sum^6_{j=1}e^j\theta_{j\alpha} -
\sum^{11}_{k=7}e^k\theta_{k\alpha}
\end{equation}
With the usual Dirac matrices $\Gamma^{\mu}$, since the identity
\begin{equation}
\Gamma_{\mu}\psi_{[1}\bar{\psi}_2\Gamma^{\mu}\psi_{3]} =0
\end{equation}
is satisfied due to the Fierz transformation in four dimension, the Green Schwarz action
~\cite{gs} for N=1 is
\begin{equation}
S=\frac{1}{2\pi}\int d^2\sigma \left ( \sqrt{g}g^{\alpha\beta}\Pi_{\alpha}\Pi_{\beta}
+2i\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\bar{\theta}\Gamma_{\mu}
\partial_{\beta}\theta\right )\label{l}
\end{equation}
where 
\begin{equation}
\Pi^{\mu}_{\alpha}=\partial_{\alpha}X^{\mu}- i\bar{\theta}\Gamma^{\mu}\partial_
{\alpha}\theta .
\end{equation}
This is the N=1 and D=4 superstring originating from the D=26 bosonic string. It is difficult 
to quantise this action covariantly. It is better to use NS-R~\cite{ns} formulation with G.S.O
projection~\cite{gl}. This has been done in reference~\cite{dm} where the explicit 
elimination of ghosts, modular invariance and a derivation of Einstein's field equation 
have been presented.

The principal success has been to achieve the actions contained in equation(\ref{a})
or equation(\ref{l}) which possess\\ $SO(6)\otimes SO(5)$ gauge supersymmetry besides the SO(3,1).
To descend to the standard model group\\ $SU_C(3)\otimes SU_L(2)\otimes U_Y(1)$, it is normally done by 
introducing Higgs which breaks gauge symmetry and supersymmetry. However, one uses the method
of symmetry breaking by using Wilson's lines, supersymmetry remains in tact but the 
gauge symmetry is broken. This Wilson loop is
\begin{equation}
U_{\gamma}= P\;exp(\oint_{\gamma} A_{\mu}\;dx^{\mu})
\end{equation}
P represents the ordering of each term with respect to the closed path $\gamma$. SO(6)=SU(4)
descends to $SU_C(3)\otimes U_{B-L}(1)$. This breaking can be accomplished by choosing one element of
$U_o$ of SU(4) such that
\begin{equation}
U_o^2=1
\end{equation}
This elements generates the permutation group $Z_2$. Thus
\begin{equation}
\frac{SO(6)}{Z_2}= SU_C(3)\otimes U_{B-L}(1)
\end{equation}
Without breaking supersymmetry. Similarly $SO(5)\rightarrow SO(3)\otimes SO(2)=SU(2)\otimes U(1)$. 
We have
\begin{equation}
\frac{SO(5)}{Z_2}=SU(2)\otimes U(1)
\end{equation}
Thus
\begin{equation}
\frac{SO(6) \otimes SO(5)}{Z_2 \otimes Z_2}=SU_C(3)\otimes U_{B-L}(1)\otimes U_R(1) \otimes SU_L(2)
\end{equation}
making identification with the usual low energy phenomenology. But this is not the 
standard model. We have an additional U(1). But there is an instance in $E_6$ where there 
is a reduction of rank by one. Following the same idea~\cite{g}, we take
\begin{equation}
U_{\gamma}=(\alpha_{\gamma}) \otimes \left(\begin{array}{ccc}\beta_{\gamma}&&\\&\beta_{\gamma}&\\&&
\beta^{-2}_{\gamma}\end{array}\right) \otimes \left( \begin{array}{ccc}\delta_{\gamma} &&\\&
\delta^{-1}_{\gamma}&\\&&
\end{array}\right) 
\end{equation} 
$\alpha_{\gamma}^3$ =1 such that $\alpha_{\gamma}$ is the cube root of unity. This structre
lowers the rank by one. We have
\begin{equation}
\frac{SO(6)\otimes SO(5)}{Z_3}=SU_C(3)\otimes SU_L(2)\otimes  U_Y(1)
\end{equation}
arriving at the supersymmetric standard model.

We can elaberately discuss ~\cite{spm} the $Z_3$ described by 
\begin{equation}
g(\theta_1, \theta_2,\theta_3) =(\frac{2\pi}{3}-2\theta_1,\frac{2\pi}{3}+\theta_2,
\frac{2\pi}{3}+\theta_3)
\end{equation}
For the Wilson loop, the angle integrals for $\theta_1 =\frac{2\pi}{9}$ to $\frac{2\pi}
{3} - \frac{4\pi}{9} = \frac{2\pi}{9}$, the loop integral vanishes. $\theta_2 =0$ to
$2\pi -\frac{2\pi}{3} =\frac{4\pi}{3}$ for the loop described the length parameter R and 
$\theta_2$, $\theta_3 =0$ to $2\pi -\frac{2\pi}{3}= \frac{4\pi}{3}$ for the remaining 
loop. We take the polar components of the gauge fields as non zero constants as given 
below.\\
\[ g A_{\theta_2}^{15} = \vartheta_{15}\] for SO(6) = SO(4) 
for which the diagonal generator $t_{15}$ breaks the symmetry and
\[ g'A_{\theta_3}^{10} =\vartheta'_{10} \] for SO(5), the diagonal generator being $t'_{10}$
. The generators of both SO(6) and SO(5)  are $4\times 4$ matrices. We can write the $Z_3$
group as
\begin{eqnarray}
T = T_{\theta_1}T_{\theta_2}T_{\theta_3}~~~~~~~~~~~~~and~~~~~~~~~~T_{\theta_1} = 1.
\end{eqnarray}
This leaves the unbroken symmetry $SU(3)\times SU(2)$ untouched.
\begin{equation}
T_{\theta_2}= exp\left( i~t_{15}\int_o^{\frac{4\pi}{3}}\vartheta_{15} R 
d\theta_2\right )
\end{equation}
$T_{\theta_2} \neq 1$ breaks the SU(4) symmetry.
\begin{equation}
T_{\theta_3}= exp\left( i~t'_{10}\int_o^{\frac{4\pi}{3}}\vartheta'_{10} R 
d\theta_3 \right )
\end{equation}
$T_{\theta_3} \neq 1$ breaks the SO(5) symmetry.
But
\begin{equation}
T_{\theta_2} T_{\theta_3} =exp \left(
  i\int_o^{\frac{4\pi}{3}}(\vartheta'_{10}t'_{10} +
\vartheta_{15}t_{15}) R d\theta \right )
\end{equation}

Since~~ $\vartheta_{15}$ and $\vartheta'_{10}$~~ are arbitrary constants,
we can choose in such a way that ~~
$t_{15}\vartheta_{15} + t'_{10}\vartheta'_{10}$=0.~~~ Thus T= U(1) and equation (29)
is obtained reducing the rank by one.
 
Thus we have made the successful attempt in constructing a N=1, D=4 superstring with 
$SO(6)\otimes SO(5)$ which, with the help of Wilson lines, descend to the SUSY standard model. 
Since the $Z_3$ has three generators, we get three generations of the standard model because 
$SO(6)\otimes SO(5) \rightarrow Z_3\otimes SU_C(3)\otimes SU_L(2)\otimes U_{Y}$.

I have profitted from a discussion with Prof L.Maharana which is thankfully acknowledged.
 

\begin{thebibliography}{99}
\bibitem{ng} Y. Nambu, {\it Lectures at Copenhagen Symposium(1970)}, T. Goto, 
Prog. Theo. Phys. {\bf 46}(1971) 1560.
\bibitem{m} S. Mandelstam, Phys. Rev. {\bf D11}(1975) 3026
\bibitem{br} L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. {\bf 65B}(1976) 471;
S. Deser and B. Zumino, Phys. Lett. {\bf 65B}(1976) 369
\bibitem{g} M.B. Green, J. H. Schwarz and E. Witten, {\it Supersymmetry Theory} Vol-I
and Vol-II, Cambridge University Press, Cambridge, England(1987).
\bibitem{gs} M. Green and J. H. Schwarz, Phys. Lett. {\bf 136B}(1984) 367, Nucl. Phys. 
{\bf B198}(1982)252,441
\bibitem{ns} A. Neveu and J. H. Schwarz, Nucl. Phys. {\bf B31}(1971) 86; P. Ramond, Phys.Rev.
{\bf D3}(1971)2415
\bibitem{gl} F. Gliozzi, J. Scherk and D. Olive, Phys. Lett.{\bf 65B}(1976)282
\bibitem{dm} B. B. Deo, `A New Type of Superstring in four Dimension', hep-th/0211223,
Physics Letters B to be published\\
B. B. Deo and L. Maharana, ` Derivation of Einstein Equation from a new type of 
superstring in four dimension' hep-th/0212004
\bibitem{spm} S. P. Misra, {\it Introduction to Supersymmetry and Supergravity},
Wiley Eastern limited(1992)
\end{thebibliography}
\end{document}

