\documentstyle[12pt]{article}
\textheight 220mm
\textwidth 170mm
\renewcommand{\baselinestretch}{1.5}
\hoffset=1.5cm
\voffset=2cm
%\oddsidemargin -5 mm
\topmargin -10 mm
\newcommand{\NP}[1]{Nucl.Phys.\ {\bf #1}\ }
\newcommand{\PL}[1]{Phys.Lett.\ {\bf #1}\ }
\newcommand{\CMP}[1]{Comm.Math.Phys.\ {\bf #1}\ }
\newcommand{\PRL}[1]{Phys.Rev.Lett.\ {\bf #1}\ }
\begin{document}
\newpage\pagenumbering{arabic}
89 - 2\\
January 1989
\vspace{15mm}
\begin{center}
{\huge Standard Model Like String Theories From Covariant Lattices}
\\[1cm]{\large Bengt E. W. Nilsson\\[4mm] Patrick Roberts\\[5mm]
Per Salomonson}\\[4mm]{\sl Institute of Theoretical Physics\\
CTH/GU\\S-412 96 G\"{o}teborg\\SWEDEN\\Bitnet\\ TFEBN@SECTHF51\\
TFEPS@SECTHF51}\\
\end{center}
\vspace{1cm}
\begin{abstract}
We argue that the covariant lattice approach to four dimensional strings
may be used in the search for a string theory containing the standard
model of low energy phenomenology. The argument employs the recent
observation that large numbers of theories are connected by means of
continuous parameters and that often lower rank models are related to
maximal rank (i.e. 22) ones. Many examples of lattices with various
properties are constructed demonstrating the simplicity and flexibility
of this approach. These include models with a gauge group containing
$SU(3)\times SU(2)\times U(1)$ and three generations.
\end{abstract}
\newpage
\section{Introduction}
During the last few years there has appeared a variety of approaches to
four dimensional strings \cite{CHSW}-\cite{LNS}. Although some of these are
quite different
in spirit, the sets of theories they lead to overlap to a large extent
\cite{D,N}. It is an important task to analyze the structure of the space of
all conformal field theories relevant to four dimensional strings. For
practical purposes, however, it might be even more urgent to understand
how the various approaches differ and to what extent they produce inequivalent
and unrelated string theories. Plotting in a common graph the sets of
currently known theories produced by the different methods, a rather
complicated picture emerges \cite{D,N}. One might hope that as we gather
further
information this picture will simplify due to among other things an
increase of the overlaps.
One could, in fact, obtain a substantial increase of the overlaps simply
by including in each set all theories connected to it via parameters
that can be varied continuously. Some values of these parameters might
even give rise to theories that are not yet possible to formulate in the
exact terms of conformal field theory. Instead these theories are given
as field theory perturbation expansions in the continuous parameters. Examples
of such a phenomenon were recently discussed by
Font et al \cite{FINQb} in the context of $Z_3$ orbifolds with Wilson
lines. (In the case of Calabi-Yau vacua recent results indicate that even
vacua with different topology might be related via continuous parameters
\cite{CGH}.)
The approach that is affected the most by such a reinterpretation is
certainly the covariant lattice approach \cite{LLS,LNS,LSW}. This approach
has suffered
from the fact that it contains, in its present formulation, only theories
with rank 22 gauge groups. In view of the above discussion, however, it
seems possible
to break these to groups of lower rank by changing the
values of certain continuous parameters. These correspond to the
expectation values of spacetime fields which define flat
directions in the potential of the theory. The basic assumption here, of
course, is that the overlap between the orbifold and the
covariant lattice approaches is almost complete as far as rank 22 models are
concerned. The studies conducted in ref.\cite{SW} give some relevant
information which support the above assumption.
The covariant lattice approach has many virtues, and with the main negative
feature circumvented, this approach could become a powerful
tool in the search for a realistic string theory. The virtues we have in
mind are, for instance, its simplicity due to the fact that the whole string
is described in a uniform way (namely in terms of a single lattice), and the
very direct way in which modular invariance is implemented. More
importantly, there is also a very close connection between properties of
the lattice and features of the low energy field theory obtained from the
string theory. Many examples of such connections are given in refs.
\cite{LLS,LNS,LSW}.
Two features of particular relevance are spacetime supersymmetry and
chirality. It turns out that N=4, 2, or 1 supersymmetry
is realized in spacetime if the lattice extends the $D_5$ spacetime group
(see Section 2) to
$E_8$, $E_7$, or $E_6$, respectively, whereas
the appearance of chirality is dictated by the form of the supercurrent.
This in turn is connected to properties of the lattice. We will
return to this question below but mention here that what is required of
the lattice is that certain vectors of length squared 3 appear, while in the
same sector of the lattice length squared 1 vectors must not appear.
Naturally,
it is highly desirable to uncover the origin in the lattice of as many
properties of the low energy field theory as possible. The importance of
such considerations was recently emphasized by the authors
of refs.\cite{BDFM,BD}.
\section{Shifting Covariant Lattices}
Before turning to explicit examples of covariant lattices, we describe
briefly the method of construction used in this paper. This method has
already been discussed extensively in the literature \cite{LT,LS,SW} where
the reader could find a more comprehensive account of the procedure. Very
schematically it could be described by the following two steps.
\begin{enumerate}
\item In the even lattice version of the covariant lattice
approach \cite{LLS} the theory is defined by
choosing an even self-dual lattice of the form $\Gamma_{22;14}$, i.e., with
signature $(+)^{22}(-)^{14}$. The lattice is required to contain the
$D_5$ lattice on the right hand side and, furthermore, there must exist vectors
on the lattice from which one can obtain a supercurrent.
\item From this lattice one may derive another even self-dual lattice, and
thereby another string theory, by introducing a shift vector $\vec{\delta}$.
The new lattice is constructed by first defining a sublattice
\begin{equation}
\Gamma_0 = \{\vec{v}\in \Gamma_{22;14} : \vec{\delta}\cdot\vec{v}\in
{\bf Z}\}
\end{equation}
and then forming
\begin{equation}
\Gamma_{22;14}^{'} = \bigcup_{k=0}^{n-1}(\Gamma_0 + k\vec{\delta})
\end{equation}
The new lattice thus constructed is even self-dual provided
\begin{equation}
n\,\vec{\delta}\in \Gamma_{22;14},\hspace{1cm} \vec{\delta}^2 \in 2{\bf Z}
\end{equation}
\end{enumerate}
Here n is an integer and is called the order of the shift. Of course, the
requirements imposed on the original lattice in the first step must also
be fulfilled by the new lattice. Note, however, that it is possible to
choose a shift vector such that the supercurrent is projected out in
eq.(1) as long as the shifted sectors produce another viable supercurrent
\cite{LT,SW}.
In the construction of new lattices, it is natural to use the language of
conjugacy classes. For instance the even self-dual lattice
$\Gamma_{E_8}=(0)\bigcup (s)$ using $D_8$ conjugacy classes, $\Gamma_{E_8}
=(0)\bigcup (3)\bigcup (6)$ using $A_8$ conjugacy classes or in terms of
the group $A_2\times {\cal U}(\sqrt{12})\times A_1\times {\cal U}(\sqrt{18})\times
{\cal U}(2)\times {\cal U}(\sqrt{18})\times {\cal U}(\sqrt{18})$, one would
need 1296 combinations of
the various conjugacy classes of these groups. (For the
number 1296 see ref.\cite{LS}.) Here ${\cal U}(r)$ ,$ r^2\in
2{\bf Z}_+$,\cite{LS} denotes
an abelian group with "root"-lattice $\{m\cdot r : m\in {\bf Z}\}$
and conjugacy classes $(k) =\{ \frac{k}{r}:k=0,1,2...,r^2-1\}$. In the case
of $D_n$ groups (0), (v), (s), and (c) refer to adjoint, vector, spinor, and
cospinor conjugacy classes, respectively while for $A_n$ (m) denotes the
conjugacy class of the m'th basic representation. If now the shift vectors
are also written in this language, the above construction will describe an
infinite lattice in terms of a finite number of combinations of conjugacy
classes.
To get a flavour of how this is done in practice we now consider some
examples. These will also demonstrate the simplicity as well as the
flexibility of this approach. In the first examples we start from the even
self-dual lattice $\Gamma_{22;14}=(E_8\times E_8\times (A_2)^3;(A_2)^3
\times E_8)$. This lattice satisfies the requirements stated above since
$D_5$ is a subgroup of $E_8$, and the supercurrent is constructed using the following
vectors of length squared 1 and 3: $\{(\vec{\alpha},\vec{0},\vec{0},1,0,0),
(\vec{0},\vec{\alpha},\vec{0},0,1,0), (\vec{0},\vec{0},\vec{\alpha},0,0,1)
\}$ where
$\vec{\alpha}=\vec{0}, \vec{\alpha}_1, \vec{\alpha}_2$, or $-(\vec{\alpha}_1
+ \vec{\alpha}_2)$ ($\vec{\alpha}_1$ and $\vec{\alpha}_2$ are the simple
roots of $A_2$). The first three entries in these vectors refer to the
$(A_2)^3$ factor on the right hand side of the full lattice, and the last
three entries are the directions in the right hand side $E_8$ not in
$D_5$.
The most general form of the supercurrent is \cite{LNS}
\begin{equation}
T_F = \sum_{\vec{l}} \vec{B}(\vec{l})\partial\vec{X}e^{i\vec{l}\cdot \vec{X}}
+ \sum_{\vec{t}}A(t)e^{i\vec{t}\cdot\vec{X}}
\end{equation}
where $\vec{l}^2$=1, $\vec{B}\cdot\vec{l}=0$ and $\vec{t}^2=3$. With
the vectors just given we see that both terms in $T_F$ appear and as a
consequence the low energy field theory cannot be chiral (it is really
the first term in the supercurrent that prevents this).
The d=4 low energy theory obtained this way is in fact an $E_8\times E_8
\times (A_2)^3$ gauge theory with N=4 supersymmetry and corresponds to a
standard torus compactification of the d=10 theory. We will now consider two
different shift vectors and discuss some of their implications for the
structure of the four-dimentional low energy theory.
The first shift vector is
\begin{equation}
\vec{\alpha} \in ((1),(4),(0),(4),(0),(4),(0),(2),(0);(2),(0),(2),(0),
(2),(0),(2),(0))
\end{equation}
where $(k)$ are conjugacy classes of, respectively, $A_8$, ${\cal U}(6)$, $D_7$,
${\cal U}(\sqrt{18})$, ${\cal U}(\sqrt{6})$, ${\cal U}(\sqrt{18})$, ${\cal U}(\sqrt{6})$,
${\cal U}(\sqrt{18})$, ${\cal U}(\sqrt{6})$ in the 22 left moving
dimensions, and
${\cal U}(\sqrt{18})$, ${\cal U}(\sqrt{6})$,${\cal U}(\sqrt{18})$, ${\cal U}(\sqrt{6})$,${\cal U}(\sqrt{18})$,
${\cal U}(\sqrt{6})$, $A_2$, $E_6$ in the 14 right moving dimensions.
This model has gauge group $SU(9)\times SO(14)\times (U(1))^7$ and the
massless sector is chiral with the following states: in the unshifted
sector, apart from the N =1 supergravity and N =1 Yang-Mills
supermultiplets,
\begin{eqnarray}
3((6),(0),(0),(0)^6),\hspace{3mm} 3((0),(-6),(v),(0)^6),\hspace{3mm}
3((0),(3),(c),(0)^6),\\
3((0),(0),(0),\alpha_i,(0)^4),\hspace{3mm} 3((0),(0),(0),(0)^2,
\alpha_i,(0)^2),\hspace{3mm}3((0),(0),(0),(0)^4,\alpha_i)
\end{eqnarray}
and in the shifted sector
\begin{eqnarray}
((1),(4),(0),\frac{1}{3}\alpha_i,\frac{1}{3}\alpha_j,\frac{1}{3}\alpha_k)
\end{eqnarray}
where i,j,k =1,3,5 and $\alpha_i$ are $SU(3)$ roots, explicitly:
\begin{eqnarray}
\frac{1}{3}\alpha_1 = (2),(0)\nonumber\\
\frac{1}{3}\alpha_3 = (-1),(1)\nonumber\\
\frac{1}{3}\alpha_5 = (-1),(-1)
\end{eqnarray}
In all cases the massless sector corresponds to the shortest vectors in each
conjugacy class. This is another reason why the conjugacy class notation
is convenient. This example has also been
discussed in \cite{FINQb} as an example of a $Z_3$ orbifold.
As a second example, we will discuss an asymmetric shift.
By noting that the length squares of ((4),(0),(4),(0),(2),(0))
$\in 2{\bf Z}$ (where the conjugacy classes refer to
$({\cal U}(\sqrt{18}) \times {\cal U}(\sqrt{6}))^{3})$ we may actually
remove these from the previous shift vector and thus obtain
\begin{equation}
\vec{\alpha}^{'} \in ((1),(4),(0),((0),(0))^{3};((2),(0))^{3},(2),(0))
\end{equation}
This modified shift vector gives rise to the gauge group
$SU(9)\times {\cal U}(6) \times SO(14) \times (SU(3))^{3}$ and drastically
changes the abundance of chiral multiplets in the shifted sectors.
In fact, the unshifted sector remains the same, but the shifted sectors
now contain (with multiplicity ${\bf one}$):
\begin{eqnarray}
((1),(34),(v),(0)^3),\hspace{3mm} ((7),(4),(0),(0)^3),\nonumber\\
((1),(4),(0),\underline{(1),(0),(0)}),\hspace{3mm}
((1),(4),(0),\underline{(2),(0),(0)})
\end{eqnarray}
where in the six internal dimensions, the $(SU(3))^3$ conjugacy classes
are given and underlined entries refer to all possible permutations.
One notices that the $SU(9)$ anomaly appearing in the unshifted sector
is cancelled by the states of the shifted sectors, but this cancellation
is due to entirely different $SU(9)$ representations as compared to the
previous example.
The supercurrent is the same in these two examples and has the form
\begin{equation}
T_F=\frac{1}{\sqrt{3}}\sum_{\vec{t}}(e^{i\vec{t}\cdot\vec{X}}+
e^{-i\vec{t}\cdot\vec{X}})
\end{equation}
where the sum runs over the nine vectors of length square 3 given above
(when the unshifted lattice was introduced) and
hence $T_F$ has the correct form for the spectrum to be chiral. This
supercurrent has been discussed by various authors \cite{KLT,LNS} and
appears in the
classification of supercurrents presented in ref.\cite{SW}.
Our third example is similar to the previous ones except that we now
compactify
on an $(E_6)_L \times (E_6)_R$ lattice with conjugacy classes
((0);(0)), ((1);(1)), and ((2);(2)). To describe this lattice we note
that $E_6$ has a maximal subgroup $SU(3)\times SU(3)\times SU(3)$, and we
may therefore describe it in terms of an $(SU(3))^3_L\times (SU(3))^3_R$ lattice.
The $E_6$ conjugacy classes can be decomposed into $(SU(3))^{3}$
conjugacy classes as follows:
\begin{eqnarray}
(0) = ((0),(0),(0)) \bigcup ((1),(1),(1))\bigcup
((2),(2),(2)) \nonumber\\
(1) = ((0),(1),(2)) \bigcup ((2),(0),(1))\bigcup
((1),(2),(0)) \nonumber\\
(2) = ((2),(1),(0)) \bigcup ((0),(2),(1)) \bigcup((1),(0),(2))
\end{eqnarray}
We now choose our shift vector $\vec{\alpha}$ to belong to the conjugacy class
((0);(0),(0),(1)) of $(E_6)_{L}\times (SU(3)\times SU(3)\times SU(3))_{R}$.
This shift breaks $(E_6)_{R}$ to $(SU(3))^3$ and
admits a supercurrent of the form (8), but with the following choice for
the nine orthogonal vectors
\begin{eqnarray}
(\omega_i,\omega_i,\omega_i,1,0,0),\hspace{3mm}
(\omega_i,\omega_{i+1},\omega_{i+2},0,1,0),\hspace{3mm}
(\omega_i,\omega_{i-1},\omega_{i-2},0,0,1)
\end{eqnarray}
where $\omega_i$ are the weights of the (1) representation of $SU(3)$.
Using the same $E_8\times {E}_8^{'}\times (E_8)_{R}$ components of
$\vec{\alpha}$
as in the previous examples leads to the Yang-Mills gauge group
$SU(9)\times {\cal U}(6)\times SO(14)\times E_6$, three copies of
\begin{eqnarray}
((6),(0),(0),(0)), \hspace{3mm}
((0),(-6),(v),(0)), \hspace{3mm}
((0),(3),(c),(0))
\end{eqnarray}
massless chiral multiplets from the unshifted sector, and three copies of
\begin{eqnarray}
((7),(4),(0),(0)), \hspace{3mm}
((1),(-2),(v),(0))
\end{eqnarray}
from the shifted sector. The multiplicity 3 of states in the shifted sector
is due to the states being triplets under the last $SU(3)$ of the broken
$(E_6)_{R}$.
All of these examples have rather large gauge groups which have to be
broken to yield realistic models. Alternatively it may be possible to find
interesting models with small visible sectors and large hidden sectors. To
get an example of such a chiral model with a hidden $E_8\times E_8«$ we
start
from the lattice $(E_8\times E_8«\times E_6)_{L}\times (E_6\times E_8)_{R}$
and choose the shift vector $\vec{\alpha}$ to lie in the conjugacy class
$((0),(0),(0),(0),(1);(1),(1),(2),(1),(0))$ of $(E_8\times E_8«\times
(SU(3))^{3})_{L}
\times ((SU(3))^{3}\times SU(3)\times E_6)_{R}$. This produces an N =1
Yang-Mills theory with gauge group $E_8^{2}\times (SU(3))^{3}$ in which the
chiral matter multiplets belong to the $(SU(3))^{3}$ representations
3((1),(1),(1)) from the unshifted sector and 3((1),(2),(2)) $\oplus$
3((2),(1),(2)) $\oplus$ 3((2),(2),(1)) from the shifted sector.
\section{Two-Shift models}
Having found an easy way to construct models with a small number of
generations we now turn to the question of how to reduce the size of the
gauge
group.(Of course, here "size" refers to the dimension and not the rank). It
is clear that when choosing a particular shift vector given in terms of
conjugacy classes of certain groups, the gauge group is guaranteed to be at
least as big as these groups. However, trying to break the gauge group of the
original lattice to some preselected "small" subgroups one often discovers
accidental roots enhancing the gauge group to a bigger one.
By means of some models constructed from two shift vectors, we will discuss
when this can and
cannot happen. The case of three shifts is different and will be commented
upon
in section 4. At this point we should emphasize that the process of shifting
with several vectors, one after the other, is in general not equivalent to
shifting with
the sum of the vectors (unless the order of the shifts are relative primes).
We now strive at breaking the $SU(9)$ gauge group appearing in the models
given in the previous section to $SU(5)$ by introducing a second shift
vector $\vec{\beta}$. Once again we start from the lattice $\Gamma_{22;14}=
(E_8\times E_8«\times (A_2)^{3};(A_2)^{3}\times E_8)$. Using the more
familiar vector notation, the two shift vectors may be chosen as follows
\begin{eqnarray}
\vec{\alpha}=\frac{1}{3}(1,1,1,1,2,0,0,0,2,0^{5},0^{6};\vec{\alpha}_1,
\vec{\alpha}_1,\vec{\alpha}_1,1,1,-2,0^{5})\\
\vec{\beta}=\frac{1}{3}(0^7,2,0,1,1,0^5,\vec{\alpha}_1,\vec{\alpha}_1,
\vec{\alpha}_1;3\vec{\omega}_3,-3\vec{\omega}_3,0^2,0^8)
\end{eqnarray}
where $\vec{\alpha}_1=(\sqrt{2},0)$ and
$\,\vec{\omega}_3=(0,-\sqrt{\frac{2}{3}})$.
$\vec{\alpha}$ is actually a representative of the conjugacy class given
in eq.(6) and the form of $\vec{\beta}$ was chosen in analogy with the
first Wilson line in the third model of ref.\cite{IKNQ}. The structure of
$\vec{\beta}$ in the internal $\Gamma_{6;6}$ part is such
as to preserve the supercurrent and give $\vec{\alpha}\cdot\vec{\beta}=0$
mod(1).
By inspecting the effect of $\vec{\beta}$ in the $E_8\times E_8^{'}$
directions one finds that $SU(9)\times U(1)^{'}\times SO(14)^{'}$ is broken to
$SU(5)\times(SU(2))^{2}\times (U(1))^{2}\times SO(10)^{'}\times
SU(2)^{'}\times (U(1)^{'})^{2}$
(together with the $(U(1))^6$ from the internal directions). The chiral
($A_4$ non-singlet) matter
consists of the following ($A_4$, $D_2$, $A_1$, $D_5$) multiplets:
3((4),(v),(0),(0)) $\oplus$ 3((2),(0),(0),(0)) from the unshifted sector,
6((4),(0),(0),(0)) from the $\pm\vec{\beta}$ shifted sector, and
((3),(0),(0),(0)) $\oplus$ ((1),(0),(0),(v)) $\oplus$
6((1),(0),(0),(0)) from the
$\vec{\alpha}$ shifted sector. Consequently, in this
particular model the former sector is crucial for anomaly cancellation. This
observation is of some interest when comparing to orbifolds where the
corresponding sector do not occur at all.
The complete spectrum discloses, however, the presence of some seemingly
unexpected and in most situations unwanted states, namely states
corresponding to gauge bosons not present in the gauge groups given above.
In fact, one checks easily that the gauge group is enhanced to
$SU(5)\times SO(6)\times U(1)\times SO(10)^{'}\times SU(2)^{'}\times
(U(1)^{'})^{2}\times (SU(2))^{4}\times (U(1))^{2}$.
By scrutinizing the full spectrum one sees that these states come from the
sector shifted by $\vec{\beta}$ but not by $\vec{\alpha}$, i.e., the sector
that does not exist in the orbifold approach.
There is a way to make this sector less relavant in that it does not
contribute to the non-abelian anomaly. It does however still add gauge group enhancing
states, but in a slightly different fashion as compared to the previous
model. The specific model we have in mind here is given by
\begin{eqnarray}
\vec{\alpha}=\frac{1}{3}(1,1,1,1,2,0,0,0,2,0^7,\vec{\alpha}_1,\vec{\alpha}_1
,\vec{\alpha}_1;\vec{\alpha}_1,\vec{\alpha}_1,2\vec{\alpha}_1,1,1,-2,0^{5})\\
\vec{\beta}=\frac{1}{3}(0^7,2,0,1,1,0^5,\vec{0},\vec{0},\vec{0}
;\vec{0},\vec{0},3\vec{\omega}_3,0^8)
\end{eqnarray}
which gives rise to the same naive gauge group as before, but it is now
enhanced to $A_5\times D_3\times (A_1^{'})^2\times U(1)^{'}\times D_5^{'}$
Before discussing the reason behind the appearance of these gauge group
enhancing states we turn to the twice shifted $E_6$-models. That is
$\Gamma_{22;14}$ = ($E_8\times E_8^{'}\times E_6;E_6\times (E_8)_{R}$)
and we choose, corresponding to the first $(SU(3))^3$ model above,
\begin{eqnarray}
\vec{\alpha}=\frac{1}{3}(1,1,1,1,2,0,0,0,2,0^7,0^6;-3\vec{\omega}_3
,0^4,1,1,-2,0^5)\\
\vec{\beta}=\frac{1}{3}(0^7,2,0,1,1,0^5,0^6;\vec{\alpha}_1,\vec{\alpha}_1,
\vec{\alpha}_1,0^8)
\end{eqnarray}
Of course the naive gauge group coming from $E_8\times E_8^{'}$ is the same
as above, but in
this case there is $\em no$ enhancement of it. The reason is easily
understood.
In (14) and (15) the two sets of six internal entries on the right hand side
do not correspond to vectors present in this part of $\Gamma_{22;14}$.
Therefore there is no
vector in this part of $\Gamma_{22;14}$ that can be shifted back to the
origin giving
rise to the possibility of a new gauge boson. This is not the case for the
previous
($A_2)_L^3\times (A_2)_R^3$ models. It is also clear that states with only
$\vec{\beta}$-shifts cannot occur. The $E_6$ model similar to the
second $(SU(3))^3$ model is also easy to obtain, but we will not discuss
it here since it is part of a model appearing in the next section.
\section{3-shift Models}
The technique for generating three generation models with gauge group
$SU(3)\times SU(2)\times (U(1))^5$ in the observable sector outlined below has
many features in common with the orbifold construction with two Wilson lines
discussed in ref. \cite{IKNQ}. In
fact the models to be presented are very similar to the ones appearing
in this reference to the degree that we have simply taken over the components
of the two Wilson lines which make up the $(E_8\times E_8)_L$ part of our
second and third shift vectors. This is irrelevant, however, since the main
purpose of this paper is to demonstrate that standard model like theories
are indeed obtainable from covariant lattices. It might even be
that starting from rank 22 points in the space of conformal field theories
one can reach a larger class of models than if one had committed oneself
to a particular set of background field expectation values already from
the outset.
As we will see below, both $E_6$ and $(SU(3))^3$ lattices now tend to give
rise
to enhanced gauge groups and it is of course crucial to understand if and
how
these are eliminated or reduced when the vacuum expectation values of
scalar fields are varied along flat directions in the potential.
Adding the second Wilson line used in ref.\cite{IKNQ} we can construct the
following two candidate sets of three shift vectors:
\begin{enumerate}
\item
\begin{eqnarray}
\vec{\alpha} = \frac{1}{3}(1,1,1,1,2,0,0,0,2,0^7,0^6;
\vec{\alpha}_1,\vec{\alpha}_1,\vec{\alpha}_1,1,1,-2,0^5) \nonumber \\
\vec{\beta} = \frac{1}{3}(0^7,2,0,1,1,0^5,\vec{\alpha}_1,\vec{\alpha}_1,
\vec{\alpha}_1;3\vec{\omega}_3,-3\vec{\omega}_3,0,0^8) \nonumber \\
\vec{\gamma} = \frac{1}{3}(1,1,1,2,1,0,1,1,1,1,0^6,0^6;
0,3\vec{\omega}_3,-3\vec{\omega}_3,0^6)
\end{eqnarray}
\item
\begin{eqnarray}
\vec{\alpha} = \frac{1}{3}(1,1,1,1,2,0,0,0,2,0^7,
\vec{\alpha}_1,\vec{\alpha}_1,\vec{\alpha}_1;
\vec{\alpha}_1,\vec{\alpha}_1,2\vec{\alpha}_1,1,1,-2,0^5) \nonumber \\
\vec{\beta} = \frac{1}{3}(0^7,2,0,1,1,0^5,0^6;0,3\vec{\omega}_3,0,
0^8) \nonumber \\
\vec{\gamma} = \frac{1}{3}(1,1,1,2,1,0,1,1,1,1,0^6,0^6;
0,-3\vec{\omega}_3,3\vec{\omega}_3,0^6)
\end{eqnarray}
\end{enumerate}
These two cases differ only in how the internal sublattice $\Gamma_{6;6}$
is shifted but, as we will discuss below, this has drastic consequences
for the field content of the two resulting theories. For the naive gauge
group we find in both cases
$SU(3)\times SU(2) \times SO(10) \times (U(1))^{14}$.
However, this is enhanced by the appearance of extra gauge bosons, in the
first case, to
\begin{equation}
(SU(3))^3\times (SU(4))^2 \times SO(10) \times (U(1))^7
\end{equation}
and, in the second case, to
\begin{equation}
SU(3)\times SU(2) \times SO(10) \times (SU(4))^2\times (U(1))^8
\end{equation}
Here we notice that the enhancement affects the naive gauge group in the
two cases differently. Interestingly enough, the $SU(3)\times SU(2)$ that
we wish to identify with the same groups in the standard model is
enhanced to $(SU(3))^2$ in the first case, but is unaffected in the
second.
In terms of the naive gauge group $(A_2,A_1,D_5)$ the unshifted sector
contains, in both cases, three copies of the chiral supermultiplets
\begin{equation}
((1),(1),(0))\oplus ((2),(0),(0))\oplus ((0),(1),(0))
\oplus ((0),(0),(c))
\end{equation}
Case two has in addition to these states nine $(A_2,A_1,D_5)$
singlets.
The difference between the two cases becomes even more pronounced in the
shifted sectors. The two chiral spectra contain, in addition to
$(A_2,A_1,D_5)$ singlets,
\begin{enumerate}
\item
\begin{eqnarray}
((2),(1),(0))\oplus 13((1),(0),(0))\oplus
14((2),(0),(0))\oplus ((0),(0),(s)) \nonumber \\
\oplus ((1),(0),(v))\oplus ((2),(0),(v))\oplus 31((0),(1),(0))
\end{eqnarray}
\item
\begin{eqnarray}
12((1),(0),(0))\oplus 15((2),(0),(0))\oplus 36((0),(1),(0))
\end{eqnarray}
\end{enumerate}
Several comments are now in order. First, we notice that in the second case
the spectrum is very similar to the one obtained for
the symmetric $Z_3$ orbifold with two Wilson lines as discussed in
\cite{FINQc}. In fact, in the shifted sectors ${\bf with}$ an
$\vec{\alpha}$-shift
the two spectra are identical. The main difference occurs in the non-
$\vec{\alpha}$-shifted sectors which do not appear at all in the orbifold
version of this theory.
In our first case the spectrum is entirely different in the shifted sectors
where, among other things, one even finds states in the non-$\vec{\alpha}$-
shifted
part that are needed to cancel the non-abelian anomaly.
The spectrum is slightly altered if one compacifies on
$\Gamma_{6;6}=((E_6)_L;(E_6)_R)$ instead. Namely, if in this case one uses
the set of three shift vectors above, but with internal components
\begin{eqnarray}
(\vec{\omega}_1,\vec{0},\vec{0};
\vec{\omega}_1,\vec{\omega}_1,\vec{0})\\
(\vec{\alpha}_1,\vec{\alpha}_1,\vec{\alpha}_1;
\vec{\alpha}_1,\vec{\alpha}_1,2\vec{\alpha}_1)\\
(\vec{\alpha}_1,\vec{\alpha}_1,\vec{\alpha}_1;
\vec{0},\vec{0},\vec{0})
\end{eqnarray}
one discovers that the spectrum in the shifted sectors now is
\begin{eqnarray}
15((1),(0),(0))\oplus 18((2),(0),(0))\oplus 42((0),(1),(0))
\end{eqnarray}
Looking at the spectrum for this model reveals that the gauge group
enhancing states come from the sector shifted by $\vec{\gamma}$ alone.
The complete gauge group is $SU(4)\times SU(3)\times (SU(2))^4 \times
(U(1))^8 \times D_5$. Thus we discover that the standard model gauge group
$SU(3)\times SU(2)$ is enhanced.
Finally, we would like to comment upon the
Green-Schwarz mechanism $\cite{GS,LNS}$ which is at work in all our cases.
An analysis of the $U(1)$ anomalies, like the one carried out
in \cite{CKM}, is needed to identify the anomalous direction of $U(1)$
charges. In fact, we have checked explicitly how the Green-Schwarz
mechanism works in these models. There are three types of anomalous
triangle diagrams specified by their three external legs corresponding to
three $U(1)$'s, one $U(1)$ and two $SU(3)$'s, and one $U(1)$ together
with two gravitons. The last one is proportional to $c_i = \frac{1}{12}
\sum_n Q_i(n)$. Here $Q_i(n)$ is the i'th $U(1)$ charge of the n'th
chiral state normalized so that the total length squared (of both abelian
and nonabelian charges) of the vector $\vec{Q}(n)$ is two. The index i
runs over the fourteen $U(1)$ groups. The other two anomalous diagrams are
proportional to $\sum_n k(n)Q_i(n)$ and $\sum_n Q_i(n)Q_j(n)Q_k(n)$
respectively. Here $k(n)$ is the index of the $SU(3)$ representation
(zero or one in the present case).
By explicit calculation one finds that
\begin{center}
\begin{eqnarray}
\sum_n k(n)Q_i(n) = c_i\\
\sum_n Q_i(n)Q_j(n)Q_k(n) = c_i\delta_{jk}+ c_j\delta_{ki}+ c_k\delta_{ij}
\end{eqnarray}
\end{center}
This is the unique form required for the Green-Schwarz mechanism to work.
\section{Summary and Conclusion}
We have seen that using the covariant lattice approach it is quite easy
to construct theories that resemble the standard model. Using different
internal lattices we have also shown that small (but nevertheless crucial)
changes in the spectrum might be obtained. Concerning the gauge group
in these rank 22 string theories it seems to be a general feature that they
get enhanced to larger groups than would have been the case in the orbifold
approach. In at least one interesting situation however does
the enhancement not affect the $SU(3)\times SU(2)$ group of the
standard model namely in the second $(SU(3))^3$ model
above. In the corresponding $E_6$ model there is an enhancement of
$SU(3)\times SU(2)$ to $SU(4)\times SU(3)$.
Of some importance is the total number of massless states transforming
as (3,1) and (1,2) under $SU(3)\times SU(2)$, since the values of the
coupling constants depend on these numbers
through the renormalization group equations. As recently
discussed in \cite{CMa} this may cause difficulties for many phenomenological
string models. For the $\beta$ functions to behave in accordance with
the standard unification picture, these numbers
must assume rather small values. As it turns out neither of the two models
presented in \cite{CMb} and \cite{FINQc} seem to satisfy these constraints. One can
hope to remedy this by either adding new physics, i.e., new energy scales
below the Planck scale, or choosing other v.e.v.'s in existing models, or
perhaps we must simply find new string models which give more realistic
values for these numbers. We think that the methods presented here could
be very useful in this search.
\include{refCLA}
\end{document}