%Paper: hep-th/9409062
%From: "Clifford Johnson" <cvj@puhep1.Princeton.EDU>
%Date: Sun, 11 Sep 94 20:05:21 -0400
%Date (revised): Sun, 18 Sep 94 16:30:15 -0400
%Date (revised): Sun, 5 Mar 95 19:31:29 -0500

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\vskip2pt\rightline{hep-th/9409062}
\vskip2pt\rightline{12th September 1994}
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\vskip-1.0cm
\title{\Bigtitlerm Heterotic Coset Models}
\vskip-1.0cm \author{Clifford V.
Johnson\myfoot{$^\dagger$}{\sl email: cvj@puhep1.princeton.edu}}{}{}
\vskip1.0cm
\addressline{\sl Joseph Henry Laboratories}
\addressline{\sl Jadwin Hall}
\addressline{\sl Princeton University}
\addressline{\sl Princeton NJ 08544}
\addressline{\sl U S A}

\abstract

\bigskip


A description is given of how to construct $(0,2)$ supersymmetric
conformal field theories as coset
models. These models
 may be used as non--trivial backgrounds for Heterotic
String Theory. They are realised as a combination  of an anomalously
gauged  Wess--Zumino--Witten model, right--moving supersymmetric
fermions, and left--moving current algebra fermions. Requiring the
sum of the gauge anomalies from the bosonic and fermionic sectors to cancel
yields the final model.  Applications discussed include exact models of
extremal  four--dimensional
charged black holes and Taub--NUT solutions of string theory.
These coset models may also be used to construct  important families of
$(0,2)$ supersymmetric Heterotic String compactifications. The Kazama--Suzuki
models are the left--right symmetric  special case of these  models.
\vskip-1.5cm
%\draft
\date{(To appear in {\sl Mod. Phys. Lett.} 1995)}

\section{Introduction and Motivation}
The  aim here is to show  how to construct non--trivial conformal field
theories with $(0,2)$ supersymmetry. The motivation is clear: It is well
known that in order to obtain the desired $N=1$ spacetime supersymmetry in
heterotic string theory, the minimum requirement is world
sheet $N=2$
supersymmetry. There are many well established results
for the  highly studied $(2,2)$
conformal field theories and many  of the fascinating facts
about their moduli spaces  (e.g. Mirror Symmetry) have been
uncovered to date. However, these models are
in a sense over--specialised examples
of the generic $(0,2)$ supersymmetric conformal field theories which
heterotic string theory demands. In this sense, the task of studying the
moduli space of heterotic string vacua has only just begun. A search for
many $(0,2)$ models and understanding of their moduli has to begin in
earnest. This letter will describe the construction of  isolated points in
this moduli space. We first describe the general case and then end with some
examples and a brief discussion.

\section{(2,2) cosets: Kazama--Suzuki Models}

Coset models were first invented by Bardakci and Halpern\ref\bardhal{K.
Bardakci and M. B. Halpern, Phys. Rev. {\bf  D3} (1971) 2493\semi M. B.
Halpern, Phys. Rev. {\bf D4} (1971) 2398.} and later generalised by Goddard,
Kent and Olive\ref\gko{P. Goddard and D. Olive, Nucl. Phys. {\bf B257} (1985)
226\semi P
Goddard, A Kent and D Olive, Phys Lett {\bf B152} (1985) 88\semi
P Goddard, A Kent and D Olive, Commun Math Phys {\bf 103} (1986) 105.}\ as
algebraic realisations of new conformal systems, `$G/H$' based upon
affine Lie algebras (a special case of Kac--Moody algebras\ref\km{V. G. Kac,
Funct. Anal. App. {\bf 1} (1967) 328\semi R. V. Moody, Bull. Amer. Math. Soc.
{\bf 73} (1967) 217.}\ref\kmmore{V. G. Kac,  {\sl `Infinite-dimensional Lie
Algebras---An Introduction'}, Birkh\"auser, Basel 1983, 2nd Edition Cambridge
University Press, Cambridge 1985.}) for a group $G$ and a subgroup $H$. The
$N=1$
supersymmetric extension was worked out soon after and is based upon
analogous constructions using affine Lie superalgebras\ref\supergko{V. G. Kac
and T. Todorov, Commun. Math. Phys. {\bf 103} (1986) 105.}. (For a review see
ref.\ref\goddardolive{P. Goddard and D. Olive, Int. Jour. Mod. Phys. {\bf A1}
(1986) 303.}.)

When the space $G/H$ is K\"ahler, it was shown by Kazama and
Suzuki\ref\ks{Y. Kazama and H. Suzuki, Nucl. Phys. {\bf B321} (1989) 232\semi
Y. Kazama and H. Suzuki, Phys. Lett. {\bf B216} (1989) 112.}\
using an algebraic construction that  $N=1$  is promoted
to an $N=2$ supersymmetry. This realises a large family of $(2,2)$
models, of which the $N=2$ minimal models (used for example by Gepner in
his construction of non--trivial heterotic string vacua\ref\gepner{D. Gepner,
Nucl. Phys. {\bf B290} (1987) 10\semi
D. Gepner, Nucl. Phys. {\bf B296} (1988) 757.}) are the simplest
case (they are realised as $SU(2)/U(1)$). The most straightforward examples
are the `hermitian symmetric
spaces'.


For  many reasons it is advantageous to have a Lagrangian definition of a
conformal field theory which realises the algebraic structures as a field
theory. It is very often a powerful supplement to the algebraic description.
The {\sl gauged Wess--Zumino--Witten model} is the appropriate device to
use.

\section{(2,2) Cosets as Gauged Wess--Zumino--Witten Models}

An action for a conformal field theory with all of the
algebraic structures of the Kazama--Suzuki models is:
\def\DG{{g^{-1}\d g}}\def\DBG{{\db gg^{-1}}}
\eqn\SGWZW{\eqalign{I^{(2,2)}& =I_{WZW}(g)+I(g,A)+I_F(\Psi_L,\Psi_R,A)=\cr
&-{k\over4\pi}\int_{\Sigma} d^2z\,\, \Tr[g^{-1}\d g\cdot g^{-1}\db
g]\cr &-{i\over12\pi}\int_B d^3\sigma\,\,
\epsilon^{ijk}\Tr[g^{-1}\partial_ig\cdot g^{-1}\partial_jg\cdot g^{-1}
\partial_kg]\cr
&+{k\over2\pi}\int_\Sigma d^2z\,\,\Tr[\Az\DG-\Azb\DBG+\Azb g^{-1}\Az
g-\Az\Azb]\cr
&+{i\over4\pi}\int_\Sigma d^2z\,\,\Tr[\Psi_+{\cal D}_{\bar
z}\Psi_++\Psi_-{\cal D}_z \Psi_-]
}}
where the two dimensional surface $\Sigma=\partial B$ has complex coordinates
$(z,{\bar z})$ and
$$
\eqalign{g\in& G;\,A^a\in{\rm Lie}H;\cr
\Psi_\pm\in&{\rm Lie}G-{\rm Lie}H,\,\,\,{\cal D}_a\equiv \partial_a+
[A_a,\,\,\,]}
$$
and we have gauged the group invariance $$\eqalign{g\to&hgh^{-1},\cr
\Psi_\pm\to&h\Psi_\pm h^{-1}\cr
A\to&hdh^{-1}+hAh^{-1}\cr
{\rm where}\,\,\,h(z,\zb)\in& H.}$$
This action has an $N=1$ supersymmetry:
\eqn\Neqone{\eqalign{
&\delta g=i\epsilon_-g\Psi_++i\epsilon_+\Psi_-g\cr
&\delta \Psi_+=\epsilon_-(1-\Pi_0)\cdot(g^{-1}{\cal D}_zg-i\Psi_+\Psi_+)\cr
&\delta \Psi_-=\epsilon_+(1-\Pi_0)\cdot({\cal D}_{\bar
z}gg^{-1}+i\Psi_-\Psi_-)\cr
}}
where $\Pi_0$ is the orthogonal projection of Lie$G$ onto Lie$H$.

Now, just as in the algebraic construction of Kazama and Suzuki, an $N=2$
supersymmetry arises from this $N=1$ when the space $G/H$ is K\"ahler. We
will not dwell on this further here, save to note that this action was first
studied in this context  by Witten\ref\ed{E. Witten, Nucl. Phys. {\bf B371}
(1992) 191.}\
and Nakatsu\ref\Nakatsu{S. Nakatsu, Prog. Theor. Phys. {\bf 87} (1992) 795.}.
Witten used this action (after twisting) to
do explicit calculations in certain topological field theories.  The
explicit $N=2$ transformations are written down
in ref.\ref\mans{M. Henningson, Nucl. Phys. {\bf B413} (1994) 73\semi
M. Henningson, Institute for Advanced Study preprint IASSNS--HEP--94/13,
hep-th/9402122.}\ for example and there Henningson uses the  models to study
 important properties of the Kazama--Suzuki models which are more
easily accessible via field theoretic methods. This includes  an
investigation of mirror symmetry for the Kazama--Suzuki models and a
calculation of the elliptic genus for the $N=2$ minimal models.

Note by the way that the bosonic and fermionic sectors in \SGWZW\ are
separately
consistent models. In particular, the bosonic sector of the gauged \WZW\
 (WZW) model is of course a consistent model realising the  bosonic
cosets\ref
\gauging{See for example D. Karabali, Q.--H. Park, H. Schnitzer and
Z. Yang, Phys. Lett. {\bf B216} (1989) 307.}\
and the action for the  chiral fermions, when written in this `coset'
basis, is just a simple minimal coupling to the gauge fields\ref\rohm{R.
Rohm, Phys. Rev. {\bf D32} (1985) 2849.}.
The chiral anomalies which potentially arise from this coupling exactly
cancel due to the identical nature of the left and right fermion couplings.
The anomalies contribute with opposite sign (due to opposite chirality)
 but equal magnitude.

\section{(0,2) Cosets: Potential Problems and a Solution}

\noindent
(1) To get a $(0,2)$ conformal field theory, we need to remove the left
$N=2$.
Simply deleting or changing the couplings of the left moving fermions to
the gauge fields would certainly do this for us, without spoiling the
right--moving $N=2$. The only problem is that this procedure is bound to
produce anomalies. The right--movers' chiral anomaly will either have
nothing to cancel against (if we deleted the left--movers), or will not
completely cancel (if we changed the couplings of the left--movers to
spoil the third symmetry in \Neqone).


\noindent
(2)
For many other reasons (as will be illustrated later), it would also be nice
to gauge other symmetries of the WZW model. To get a consistent model,
one has to gauge a restricted class of embeddings of
subgroups of the full $G_L\times
G_R$ symmetry which exists for the basic WZW. These are called
`anomaly--free' subgroups, mainly because one of the first uses of this
type of model (in higher dimensional gauge theories) was to study the
structure of anomalies\ref\anomalyref{See for example the book {\sl
`Current Algebra and Anomalies'}, S. B. Treiman, E. Witten, R. Jackiw and B.
Zumino, World Scientific, Singapore 1986.}\ by deliberately
studying anomalous subgroups, and then letting the Wess--Zumino term
produce {\sl classically} the familiar quantum gauge anomalies.
Since Witten's paper on the use of the Wess--Zumino term to define a
conformally invariant sigma model in two  dimensions---the \WZW\
model---most of
the efforts involving them in 2D, including their gauged versions, have
made sure that there are no anomalies. This is simply because the model
would not correctly reproduce the coset algebra---it would not  be
conformally invariant, in general.


Given the language  just used to describe the problems we would like to
solve, it is clear that a solution presents itself in the form of {\sl
cancelling the anomalies against one another.} If we arrange things
correctly, this will work\foot{Indeed, the idea of achieving quantum
consistency of chiral fermions by cancelling their anomalies against
anomalies generated by Wess--Zumino terms goes back to Faddeev and
Shatashvili, where it was studied both in four
dimensions\ref\ludwigsamson{L. D. Faddeev, Phys. Lett. {\bf B145} (1984)
81.\semi L. D. Faddeev and S. L. Shatashvili, Theor. Math. Phys. {\bf 60}
(1985) 770\semi L. D. Faddeev and S. L. Shatashvili, Phys. Lett. {\bf B167}
(1986) 225. } and two dimensions\ref\samson{S. L. Shatashvili, Theor. Math.
Phys. {\bf 71} (1987) 366.}.  The author is grateful to Samson Shatashvili for
pointing this out. See also ref.\ref\halliday{I. G. Halliday, E. Rabinovici, A.
Schwimmer and M. Chanowitz, Nucl. Phys. {\bf 268} (1986) 413.}.}.
The next section describes just how to do
this.

\section{Anomalies}

There are anomalies arising from three sectors now. The classical anomaly
from the WZW  and the chiral anomalies at one--loop from each chirality
of fermion. We will discuss each in turn.

\bigskip

\noindent
{\bf The WZW anomalies.}

\bigskip

In general gauging the following  symmetry of the WZW model
$$\eqalign{&g\to h_Lgh_R^{-1}\cr
{\rm for}\,\,\,&(h_L,h_R)\in(H_L,H_R)\subset(G_L,G_R)}$$ is anomalous. This
simply means that one cannot write down an extension of the
WZW model which promotes this symmetry to a local invariance: There will
always be terms which spoil gauge invariance. (This is because of the
Wess--Zumino term; the `metric' term may be simply minimally coupled.)

Knowing that we will get an anomaly, let us choose to write {\sl some}
gauge extension such that under gauge transformations the `anomalous'
piece does not depend upon the group element
$g$. This results in the anomalous piece taking
the form of the standard 2D chiral anomaly. The  {\sl unique} action
is\ref\edagain{E. Witten, Commun. Math. Phys. {\bf 144} (1992) 191.}:
\eqn\extend{\eqalign{I^{G_k}_{GWZW}(g,A)&=-{k\over4\pi}\int_\Sigma\,\,
d^2z \,\,\Tr[g^{-1}{\cal D}_zg\cdot g^{-1}{\cal D}_{\bar z}g]\cr
&-{ik\over12\pi}\int_B d^3\sigma\,\,
\epsilon^{ijk}\Tr[g^{-1}\partial_ig\cdot g^{-1}\partial_jg\cdot g^{-1}
\partial_kg]\cr
&-{k\over4\pi}\int_\Sigma A^a\wedge\Tr[t_{a,L}\cdot
dgg^{-1}+t_{a,R}g^{-1}dg]\cr
&-{k\over8\pi}\int_\Sigma A^a\wedge A^b\Tr[t_{a,R}g^{-1}t_{b,L}g-
t_{b,R}g^{-1}t_{a,L}g].}}
Under the infinitesimal variation
$$\eqalign{g\to&g+ \epsilon^Lg-g\epsilon^R\cr
A^{R(L)}\to& A^{R(L)}-dA^{R(L)}-[A^{R(L)},\epsilon^{R(L)}]\cr
}$$
where
$$
\eqalign{ \epsilon^{R(L)}\equiv&\epsilon^a t_{a,R(L)}\cr
A^{R(L)}\equiv&A^a t_{a,R(L)}\cr
Dg\equiv&dg+A^Lg-gA^R,}
$$
the variation is
\eqn\vary{
\eqalign{
\delta I(g,A)&={k\over4\pi}\Tr[t_{a,R}\cdot t_{b,R}-t_{a,L}\cdot
t_{b,L}]\int_\Sigma d^2z \epsilon^{(a)}F^{(b)}_{z\bar z}\cr
{\rm where}\,\,\,&t_{a,L(R)}\in {\rm Lie}H_{L(R)}.
}}
Notice in particular that for the popular diagonal gaugings of WZW models
this variation is zero and the action reduces to the familiar one.

\bigskip
\vfill\eject

\noindent
{\bf The right movers}

\bigskip

As mentioned before, it is sufficient to minimally couple the coset fermions
to the gauge fields:
\eqn\IFR{\eqalign{I_F^R(\Psi_R,A)=&{k\over4\pi}\int_\Sigma\,\,
i\Tr[\Psi_R{\cal D}_{\bar z}\Psi_R]\cr
{\rm where}\,\,{\cal D}_{\bar z}\Psi_R&=\db\Psi_R+\sum_a
\Azb^a[t_{a,R},\Psi_R],\,\,\Psi_R\in{\rm Lie}G-{\rm Lie}H.}}

There are $D=$dim$G-$dim$H$ fermions $\psi^i_R$ in $\Psi_R$, all coupled
with charges derived from the generators $t_{a,R}$.
The chiral anomalies appear at one loop and  are\foot{Here and for the
remainder of the letter, it is implicit that a consistent regularisation scheme
has been chosen for calculation of the fermion anomalies, and such that the
normalisation of the anomalies is chosen to be of this simple
form.}:
\eqn\Ranomalies{{1\over4\pi}\Tr[t_{a,R}\cdot t_{b,R}]\int_\Sigma d^2z
\epsilon^{(a)}F^{(b)}_{z\bar z}.}
(Note here  the absence of $k$, which plays the role of $1/\hbar$. This
 really is a one loop effect.)

\bigskip

\noindent
{\bf The left movers}

\bigskip

Let us couple into the model some left movers. Let us add
$D=$dim$G-$dim$H$ of
them (a good choice, as we will see later) with arbitrary couplings. To
be precise, arrange them into a fundamental vector
$\Lambda_L=\{\lambda^i_L\}$ of the
group \def\SOD{SO({\rm dim }G-{\rm dim}H)}
$SO(D)_L$ which acts on them as a global symmetry, and minimally couple
them to the
$H_L$ subgroup with generators $Q_{a,L}$ in this fundamental representation:
\eqn\IFL{I_F^L(\lambda^i_L,A)={k\over4\pi}\int_\Sigma\,\,
i\Lambda^T_L(\d+\sum_a\Az^aQ_{a,L})\Lambda_L.}

Their chiral anomalies
appear at one loop and are:
\eqn\Lanomalies{-{1\over4\pi}{\widetilde{\Tr}}[Q_{a,L}\cdot Q_{b,L}]\int_\Sigma
d^2z \epsilon^{(a)}F^{(b)}_{z\bar z}.}
(Here $\widetilde{\Tr}$ is the
trace in the fundamental representation of $SO(D)$.
Note again the absence of $k$. Also note the minus sign relative to
\Ranomalies, due to the opposite chirality.)

So if we add together the three actions \extend,\IFR\ and \IFL, we get
a gauge invariant  model if we ensure
that all of the  anomalies (classical and
quantum) cancel: \eqn\cancelone{k\Tr[t_{a,R}\cdot
t_{b,R}-t_{a,L}\cdot t_{b,L}]+\Tr[t_{a,R}\cdot
t_{b,R}]-{\widetilde{\Tr}}[Q_{a}\cdot Q_{b}] =0.}

Our model has $(0,2)$ supersymmetry as advertised (because we have not
touched the right--moving sector), and is conformally invariant.

Well, our model is gauge invariant when we take into account the
one--loop effects, but we still have not written a {\sl classically}
gauge invariant action. This means that we cannot truly carry out
procedures like path--integral quantisation, etc. We have not quite
achieved our goal of  a Lagrangian realisation of a $(0,2)$ conformal
field theory.

The answer is to {\sl bosonize} the fermions. The bosonic action
equivalent to $I_R^F+I_L^F$ is {\sl classically} anomalous. It is a
theory of $D/2$ real bosons with the same anomalies
as above.

\section{Bosonisation}

In the specific examples to be mentioned later, the bosonisation was worked
out `by hand' and was for abelian cases.  After a little thought,
however, it is clear  once one realises that  a
classically anomalous bosonic theory equivalent to an anomalous
fermionic theory is to be found, it might be that  the bosonic theory is
something like another anomalously gauged WZW. This can be seen as follows.

Note that before gauging there are $D$ free fermions on the left and right.
They therefore carry a global $SO(D)_L\times SO(D)_R$ symmetry. Witten
showed in ref.\ref\ed{E. Witten, Commun Math Phys {\bf 92} (1984) 455}\ that
this system of free fermions
is equivalent to a \WZW\ model based on $SO(D)$ at level 1. Considering
what we saw about WZW anomalies in section 5 it is clear that the
classically anomalous bosonic theory equivalent to the fermionic theory
is just this $SO(D)$ WZW gauged anomalously with different embeddings of
$H$ in $SO(D)$ on the left and on the right:
\def\tg{{\tilde g}}
\def\th{{\tilde h}}
$$
\eqalign{{\tilde g}\to& {\tilde h}_L{\tilde g}{\tilde h}_R^{-1}\cr
{\rm for}\,\,\,{\tilde g}\in& SO(D)\,\,\,{\rm and}\cr
(\th_L,\th_R)\in&(H_L,H_R)\subset(SO(D)_L,SO(D)_R)
}
$$
Let the $(H_L,H_R)$ be generated by $(Q_{a,L},Q_{a,R})$. Choose the
$Q_{a,R}$ such that when acting on the $\psi_R^i$'s in the fundamental
representation of $SO(D)$ they are equivalent to the $t_{a,R}$ acting on
the $\psi^i_R$ in the coset fermion $\Psi_R\in{\rm Lie}G-{\rm Lie}H$.
This will ensure that the right moving fermions are correctly coupled and
preserve the (now hidden) $N=2$ on the right.

Then the bosonic action equivalent to the interacting fermions is just an
action of the form \extend\ (with level 1), which yields the classical
anomalies:
$$
{1\over 4\pi}{\widetilde{\Tr}}
[Q_{a,R}\cdot Q_{b,R}-Q_{a,L}\cdot Q_{b,L}]
\int_\Sigma d^2z \,\,\epsilon^{(b)}F_{z\bar z}^{(a)}.
$$
So cancelling this against the anomaly of the $G/H$ bosonic model
(and recalling from the above paragraph that
${\widetilde{\Tr}}[Q_{a,R}\cdot Q_{b,R}]=\Tr[t_{a,R}\cdot t_{b,R}]$), we
recover \cancelone\ as the condition for a consistent model.

\section{(0,2) Cosets as Gauged \WZW\ Models}

So finally we can write a classically gauge invariant analogue of
\SGWZW\ which realises a $(0,2)$ conformal field theory as a gauge
invariant action written as the sum of two gauged \WZW\ models which are
separately anomalous:
\eqn\final{I^{(0,2)}=I_{GWZW}^{G_k}(g,A)+I_{GWZW}^{SO(D)_1}(\tg,A),}
where $ D={\rm dim}G-{\rm dim}H$.

The heterotic coset is realised as:
$\left[G_k\times SO(D)_1\right]/ H$ with the gauged symmetry:
$$
\eqalign{g\to&h_Lgh_R^{-1}\cr
\tg\to&\th_L\tg\th_R^{-1}\cr
\hbox{\rm subject to}\,\,\,k\Tr[t_{a,R}\cdot t_{b,R}-t_{a,L}\cdot t_{b,L}]+&
\Tr[t_{a,R}\cdot t_{b,R}]-{\widetilde{\Tr}}[Q_{a,L}\cdot Q_{b,L}]=0.}
$$
Note that $h_R$ and $\th_R$ are chosen so as to recover right--moving
supersymmetry  in the fermion picture.

Note that in \final\ the gauge extensions to each WZW (written using
\extend) are generally not  gauge
invariant, but together they are because of the anomaly equation above.
In the special case of $h_L=h_R$ and $\th_L=\th_R$, they are each
separately gauge invariant extensions, the anomaly equation is trivially
satisfied, and we recover the $(2,2)$ case, the Kazama--Suzuki models.
{\sl In this sense, the $(2,2) $ models can now be seen as a special case
of a more general class of $(0,2)$ models.}


\section{Some examples.}
These ideas were originally used by the author to study some particular
cases\ref\paperone{C. V. Johnson, Phys. Rev. {\bf D50} (1994) 4032,
hep-th/9403192}.
The prototype model for this construction is the `monopole theory' of
Giddings, Polchinski and Strominger\ref\gps{S. Giddings, J. Polchinski and A.
Strominger, Phys. Rev. {\bf D48} (1993) 5784. hep-th/9405083} (GPS). It
is a conformal field theory of a heterotic string in a Dirac monopole
background of charge $Q$ on a two--sphere of radius $Q$.
GPS described it as an asymmetric orbifold
of $SU(2)$. In ref.\paperone, when described as a heterotic coset, it is based
upon an $SU(2)$
 WZW with
the $U(1)$ subgroup of the right $SU(2)$ gauged. Adding supersymmetric
right  movers and left movers of charge $Q$ gives an anomaly equation
$k=2(Q^2-1)$. Bosonising the fermions it is possible to correctly
determine the quadratic terms in the gauge fields which
turns out to only depend upon $Q$. After integrating
out the gauge fields (valid for large $Q$), and correctly re--fermionising the
action, the
heterotic sigma model describing the above system is recovered.
This is described in detail in ref.\paperone.
As pointed out by GPS, the tensor product of this model with a supersymmetric
 $SL(2,\rline)/U(1)$
2D black hole coset\ref\bh{E. Witten,
Phys. Rev. {\bf D44} (1991) 314.}\ yields
a 4D solution which is the extremal
limit of the magnetically charged dilaton black hole of Gibbons, Maeda
 and Garfinkle, Horowitz and Strominger\ref\QblackHole{G. W. Gibbons and K.
Maeda, Nucl. Phys. {\bf B298} (1988) 741\semi
D. Garfinkle, G. T. Horowitz and  A. Strominger, Phys. Rev. {\bf D43} (1991)
3140, erratum Phys. Rev. {\bf D45} (1992) 3888.}.

Notice that in the construction just described  for the monopole
theory, one cannot have a charge $Q=0$ solution, as then the anomaly
equation would not be satisfied. After a little thought, it is clear that
there is a quick way out of this problem: simply gauge $g\to hg$ instead
and  keep everything else the same. Then the sign of the WZW anomaly
changes and the condition $k=2(1-Q^2)$ should now be satisfied. Now it is
possible to get a $Q=0$ solution. (In constructing their neutral
solution in their paper, Giddings, Polchinski and Strominger arrive
at this simple modification in an equivalent way. This is indeed the same
solution).
Now naively, the interpretation of the model  would be as a heterotic string on
a neutral two--sphere background. However, it is easy to see that this is
wrong. The problem of incorrectly identifying
the two--sphere as the background manifold for small $Q$ has its roots in the
fact that the final form of the metric for the model  is obtained by
integrating out the constraining 2D gauge fields, a process which is well
defined only for large $Q$. which is equivalent to small $\alpha^\prime$, or
large $k$. Here, the neutral solution has  $k=2$, and no sensible metric
interpretation may be made of the target space via perturbation theory, as all
length scales (in units of $\alpha^\prime$) contribute equally to the
$\beta$--function equations.

In the light of the work of GPS, the most obvious application of this
construction  was to find
more general 4D solutions which were dyons
(i.e. with both magnetic and electric
charge). Applying this construction to general gaugings of $SL(2,\rline)$
was carried  out in ref.\paperone, generalising Witten's conformal field theory
of a 2D black hole\bh\ by  yielding the conformal field theory of  the known
2D charged black hole heterotic string solutions of McGuigan,
Nappi and Yost\ref\charged{M. D. McGuigan, C. R. Nappi and S. A. Yost, Nucl.
Phys. {\bf B375} (1992) 421. hep-th/9111038.}. Then 4D dyonic solutions were
defined by tensor product with the
GPS theory. At about the same time, Lowe and Strominger wrote a
paper\ref\lowe{D. A. Lowe and A. Strominger, Phys. Rev. Lett. {\bf 73} (1994)
1468, hep-th/9403186.}\
about 4D dyons which were defined by tensoring the GPS theory with an
asymmetric orbifold of $SL(2,\rline)$. This asymmetric orbifold may be
realised an  $SL(2,\rline)$ heterotic coset.


Instead of tensor products of these 2D theories,
it is possible to obtain 4D dyon solutions which
are not tensor products, by gauging
 (for example) a $U(1)\times U(1)$ subgroup of $SL(2,\rline)\times SU(2)$
embedded non--trivially such that the action of the $U(1)$'s was shared
among the two parent groups. In  a 4D
dyon
with a non--trivial mixing of the angular and radial coordinates was obtained
in ref.\paperone. This solution is the extremal limit of
a dyonic, axionic analogue of the Taub--NUT solution of General Relativity, as
confirmed by Myers and the author in ref.\ref\papertwo{C. V. Johnson and R. C.
Myers, Phys. Rev, {\bf D50} (1994) 6512, hep-th/9406069.}\ and also by
Kallosh, Kastor Ortin and Torma
\ref\kallosh{R. Kallosh, D.
Kastor, T. Ortin and T. Torma, Phys. Rev, {\bf D50} (1994) 6374,
hep-th/9406059.}.
It would have been difficult to construct
such a non--trivial solution
as a conformal field theory
without the use of the heterotic coset
technique.

\section{Future directions}

There are a {\sl huge} number of avenues opened by allowing such freedom to
gauge any subgroup of the WZW model's symmetries, obtaining consistency by
adding heterotic fermions. One general point is that it allows one to
consider leaving  important WZW symmetries untouched, which in turn
leaves certain spacetime symmetries intact. For example,
in the simple GPS monopole
model (and its neutral cousin)
in the last section, leaving the $SU(2)_L$ (or $SU(2)_R$) action untouched
meant that a  spacetime spherically symmetric system was obtained
from an $SU(2)$ WZW.
This type of freedom will certainly lead to many more interesting
heterotic string backgrounds. The search for more 4D cosmological
heterotic  string
backgrounds seems a promising area to apply this technique to.

It is worth noting here that the presence of the right moving supersymmetry is
of course not neccessary for the consistency of these models. It is present in
this paper only because the of the context of superstrings. Indeed, one may
simply relax the requirement of the right $N=2$ and give the right moving
fermions arbitrary couplings also. As long as the anomalies cancel, this will
still produce  consistent conformal field theories, with an asymmetric
combination of left and right characters for the partition functions.
The  values of the fermion
couplings which correspond to $(0,2)$, $(2,0)$ and $(2,2)$ conformal field
theories would then be regarded as special points in the moduli space of
non--supersymmetric  backgrounds. The $(0,2)$ monopole theory described in the
previous section was
first described by GPS as a such a special point in a family of
non--supersymmetric models.

Of great interest is the problem of calculating the spectrum and partition
function for these models. This will be of course a highly non--trivial
combination of right $N=2$ characters and general $N=0$ characters. It is
a hard problem to discover the heterotic modular invariant combinations
algebraically of course (see e.g. ref.\ref\gannon{See for example T.
Gannon, Nucl. Phys. {\bf B402} (1993) 729. hep-th/9209042. }), but there are
promising signs that their Lagrangian
description using  a gauged WZW with fermions,
as described here may provide some guidance.
Work is in progress on this and related matters with Berglund, Kachru and
Zaugg\ref\paperthree{P. Berglund, C. V. Johnson, S. Kachru and P. Zaugg,
{\sl `Heterotic Coset Models II: The spectrum of $(0,2)$ conformal field
theories'}, in
preparation.}.

The problem of starting to map out the  moduli space of $(0,2)$ models
can be
attacked successfully by studying the marginal perturbations of these
models. This is of course much easier when there exists a Lagrangian
description of the type constructed here. Such marginal perturbations
would help to find the geometrical interpretation of the nieghbourhoods of
these models, in the case of their use as string compactifications.

Marginal perturbations
would also represent interesting geometrical freedom in some 4D
solutions, where they correspond to such
 processes as widening the throat of some of the extremal solutions of
the type mentioned in the last section, connecting onto the
asymptotically flat 4D exterior solution\gps.

There are of course many more questions which need to be
answered about the moduli
space of  $(0,2)$ conformal field theories.
Hopefully  this construction may go some way to help to answer them.


%\vfill\eject

\vskip2cm
\noindent
{\bf Acknowledgements}
\bigskip


I am grateful for the hospitality of the International
Centre For Theoretical Physics at Trieste during my stay there in July 1994,
 where many of the
details of this letter were worked out and presented as a talk in the Summer
School and Workshop on High Energy Physics and Cosmology.

I would also like once again to
thank Ed Witten for originally pointing out to me well over a year ago
that the GPS monopole theory might be described by a construction
of the type described in this talk.

This work was supported by an EPSRC (UK) NATO Fellowship.


\listrefs
 \bye




