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%Date: 5 March  2002 Marinov Volume
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\markboth{\small \quad \em A.~Losev,  M.~Shifman and
A.~Vainshtein\hfill}{\hfill\small \em Single state supermultiplet\quad}

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\def\beq{\begin{eqnarray}}   \def\eeq{\end{eqnarray}}
\newcommand{\nhalf}{${\cal N}\!\!=\!1/2\,$}
\newcommand{\ntwo}{${\cal N}\!\!=\!2\;$}
\newcommand{\none}{${\cal N}\!\!=\!1\,$}
\newcommand{\nfour}{${\cal N}\!\!=\!4\,$}

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\begin{document}
\sloppy


\title{SINGLE STATE SUPERMULTIPLET IN 1+1 DIMENSIONS}

\author{A.~LOSEV}

\address{Institute of Theoretical and Experimental Physics,\\
B. Cheremushkinskaya 25, Moscow
117259, Russia}

\author{M.~SHIFMAN and A.~VAINSHTEIN}
\address{Theoretical Physics Institute, University of Minnesota, \\
116 Church St SE, Minneapolis, MN 55455, USA}


\maketitle\abstracts{
We consider multiplet shortening for BPS solitons in ${\cal N}\!\!=\!1\,$
two-dimensional models. Examples of the single-state multiplets
were established previously in ${\cal N}\!\!=\!1\,$ Landau-Ginzburg models. 
The shortening comes at a price of loosing the fermion parity $(-1)^F$ 
due to boundary effects. This implies the disappearance of the boson-fermion 
classification resulting in abnormal statistics. We discuss an appropriate 
index that counts such short multiplets. \\
\rule{4mm}{0mm} A broad class of hybrid models which extend the
Landau-Ginzburg models to include a nonflat metric on the target space
is considered. Our index turns out to be related to the index of the
Dirac operator on the soliton reduced moduli space (the moduli space
is reduced by factoring out the translational modulus). The index
vanishes in most cases implying the absence of shortening. In
particular, it vanishes when there are only two critical points on the
compact target space and the reduced moduli space has
nonvanishing dimension. \\
\rule{4mm}{0mm} We also generalize the anomaly in the central charge
to take into account the target space metric.
}

\vspace{0.6cm}

\tableofcontents

\newpage

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\section{Introduction}
\label{sec:intro}

The minimal supersymmetry, \none,  in two-dimensional field 
theories has two supercharges $Q_\alpha $  ($\alpha =1,2$) which form 
a Majorana spinor in 1+1 dimensions.\footnote{%
  ~By \none we mean what is often denoted in the literature as $(1,1)$
  SUSY, with two supercharges.  }  
The centrally extended superalgebra
contains a central charge ${\cal Z}$ which has a topological
meaning.\cite{WO} Solitons can be defined as states with nonvanishing
${\cal Z}$.  Critical, or BPS (Bogomol'nyi-Prasad-Sommerfield)
saturated solitons are such that their mass $M$ and the corresponding
central charge ${\cal Z}$ are rigidly related,
\begin{equation}
M-|{\cal Z}|=0\,.
\label{MZ}
\end{equation}
The BPS saturated solitons preserve 1/2 of the original SUSY: one of
two supercharges annihilates the soliton state, while the remaining
supergenerator acts nontrivially.  The irreducible representation of
the superalgebra in this situation is one-dimensional, i.e., the
supermultiplet consists of a single state.\cite{1}
$^{\mbox -}$\cite{LSV} The shortening of the supermultiplet protects the
relation (\ref{MZ}) against small variations of parameters and quantum
corrections.

If such supershort multiplet does exist (and is not accompanied by
another short multiplet), the fermion-boson classification, i.e.
$(-1)^F$, is lost.  In this paper we address this particular issue:
whether or not such single-state supermultiplets are dynamically
realized. We consider various models in the weak coupling regime using
the quasiclassical approach. In this approach the signature of the
problem is an odd number of the fermion zero modes. In the simplest
case we deal just with one fermion zero mode (one fermion modulus)
produced by the action of the remaining supercharge.

Let us elucidate in somewhat more detail the algebraic aspect.
The centrally extended \none superalgebra has the form
\beq
\{ Q_\alpha\,,  {\bar Q}_\beta \}=2\left(\gamma^\mu 
P_\mu +\gamma^5 {\cal Z}\right)_{\alpha\beta}\, ,
\qquad 
\left[Q_\alpha, P_\mu\right]=0\,,
\label{alg}
\eeq
where ${\bar Q}_\beta=Q_\alpha (\gamma^0)_{\alpha\beta}$ and
\begin{equation}
\gamma^0=\sigma_2\,,\qquad \gamma^1=i\sigma_3\,,\qquad
\gamma^5=i\gamma^0\gamma^1=-i\sigma_1
\label{gamma}
\end{equation}
 are purely imaginary two-by-two matrices.

The states for which ${\cal Z}\neq 0$ are called {\em solitons}.  The
sign of the central charge ${\cal Z}$ differentiates between {\em
  solitons}, ${\cal Z}>0$, and {\em antisolitons}, ${\cal Z}<0$.  This
definition refers to both, the ground states and excitations.  The
algebra (\ref{alg}) implies that $P_\mu P^\mu \geq {\cal Z}^2$.  The
solitons which saturate this lower bound, $P_\mu P^\mu = {\cal Z}^2$,
are called the {\em BPS states}. They are the ground states in the
soliton sector.  In the BPS soliton rest frame, where 
$P_\mu=\left\{ {\cal Z}, 0\right\}$, it is $Q_2$ that annihilates the soliton,
$Q_2|{\rm sol}\rangle=0$.  Correspondingly, in the soliton sector the
algebra reduces to
\begin{equation}
\left(Q_1\right)^2=2\,{\cal Z}\,,\qquad \left(Q_2\right)^2=0\,,
\end{equation}
so there is only one supercharge $Q_1$ which is realized nontrivially.
In the semiclassical soliton construction it is just this supercharge
that generates the fermion zero mode. The irreducible representation
of superalgebra consists of a single state (the representation is
one-dimensional), and the action of the supercharge reduces to
multiplication by a number,
\begin{equation}
Q_1 \,|\,{\rm sol}\rangle=\pm \,\sqrt{2 {\cal Z}} \, |\,{\rm sol}\rangle\,.
\end{equation}

The choice of a particular sign in the irreducible representation
breaks $Z_2$ symmetry which reverses the sign of the fermionic
operators, $Q_\alpha\to -Q_\alpha$.  Such unusual representation
implies that decomposition into bosons and fermions (the standard
$Z_2$ grading) is broken. There are surprising manifestations of this
phenomenon.  For instance, in analyzing statistics of such solitons
one finds\,\cite{Tsvelik}$^{\mbox -}$\cite{FS} that an effective
multiplicity (defined through the entropy per soliton in the ideal
soliton gas) is $\sqrt{2}$ instead of 1.

Note that the problem is not specific for supersymmetric models, it
appears always when the total number of the fermion zero modes is odd.
In a more general context, the problem exists even in quantum
mechanics, when the Clifford algebra has odd number of generators. In
fact, an example of such algebra is known since long: it is the
Grassmannian description of nonrelativistic spin 1/2 discovered by
Berezin and Marinov.\cite{BM} These authors introduced three Grassmann
variables, $\xi_k$ ($k=1,2,3$) which were quantized by
anticommutators,
\begin{equation}
\{\xi_k\,,\xi_l\}=\delta_{kl}\,.
\label{spin}
\end{equation}

The irreducible representations of the above algebra are
two-dimensional.  For instance, one can choose
$\xi_k=\sigma_k/\sqrt{2}$.  There exists a unitary nonequivalent
choice, $\xi_k=-\sigma_k/\sqrt{2}$. The Hamiltonian $H$ depends on
$\xi$ only through the spin operators, 
$S_i=-(i/2)\,\epsilon_{ikl}\,\xi_k\,\xi_l$, which are bilinear in $\xi_k$.
Therefore, the change of the sign of $\xi_k$ looks as a classical
$Z_2$ symmetry of the problem.

 At the quantum level, however, this symmetry is not realized:
there is no operator $G$ such that
\begin{equation}
G^2=1\,,\qquad G H G= H\,,\qquad G \xi_k G= -\xi_k\,.
\end{equation}
Alternatively, the breaking of $Z_2$ could be seen from the following.
The operator $i\xi_1 \xi_2 \xi_3$ commutes with all $\xi_i$ and its
square is equal to 1.  Therefore, it can be realized as $\pm 1$. The
choice of a particular sign breaks $Z_2$, the symmetry that
interchanges the signs. The absence of $Z_2$ implies the breakdown of
the fermion-boson classification: the operator trilinear in fermions
is a number.

Returning to supersymmetric field theories in two dimensions, let us
note that the issue of the multiplet shortening and the related loss
of the fermion parity, $(-1)^F$, has a long and sometimes confusing
history.  Probably, the first encounter with the problem occurred in
the context of integrable two-dimensional models, in particular, in
the supersymmetric sine-Gordon model,\cite{Tsvelik} and in the
tricritical Ising model, field-theoretical limit of which is the
Landau-Ginzburg model with a polynomial superpotential.\cite{Zam} Even
earlier the question was discussed\,\cite{WittGN} in the framework of
the Gross-Neveu model which at $N=3$ is equivalent to the
supersymmetric sine-Gordon.

A certain clash was apparent in these considerations: on the one hand, the
existence of  $(-1)^F$ was taken for granted,\footnote{%
  ~In a recent private discussion E.~Witten mentioned that he had
  changed his opinion as to the existence of $(-1)^F$ in the soliton
  sector.  } 
implying the absence of the single-state supermultiplets.
On the other hand, as was already mentioned, an abnormal multiplicity
factor $\sqrt{2}$ was discovered through an entropy calculation. This
unusual factor was recently discussed anew by Witten\,\cite{WitEd} and
Fendley and Saleur.\cite{FS}

The approach used in the above works was based on the exact
$S$-matrices and the thermodynamic Bethe {\em ansatz}. Close to this
is the approach based on massive deformations of conformal
theories.\cite{Ahn} Another line of development is based on the
quasiclassical analysis at weak coupling. The main goal was the
calculation of quantum corrections to supersymmetric soliton masses,
that would generalize the textbook calculations of
Ref.\,\protect\citebk{DHN} done in nonsupersymmetric models. In recent
years the interest to this issue was revived by P.~van Nieuwenhuizen
and collaborators.\cite{RN,NRS} These works were the starting point
for investigations of the MIT group\,\cite{Jaffe} and
ours.\cite{1,2,LSV}

Already in Ref.\,\protect\citebk{1} it was mentioned that the
irreducible representation of the superalgebra (\ref{alg}) consists of
a single state, i.e. the representation is one-dimensional. This issue
was not elaborated in detail in Ref.\,\protect\citebk{1} although the
assertion was in contradiction with the previous analysis of
Ref.\,\protect\citebk{NRS} where it was assumed that the global $Z_2$
symmetry (associated with $(-1)^F$) is maintained.

The fact that the irreducible one-dimensional multiplet is realized
without doubling was explicitly demonstrated in
Ref.\,\protect\citebk{2} where the consideration starts from the \ntwo
extended version of the model.  As was shown in
Ref.\,\protect\citebk{FI} in such \ntwo model the solitonic multiplet
is shortened (i.e. it is two- rather than four-dimensional).  A soft
breaking of \ntwo to \none was then introduced in
Ref.\,\protect\citebk{2}. When the breaking parameter reaches some
critical value, one of the two soliton states disappears from the
physical spectrum\,\cite{2} via the phenomenon of delocalization.
Delocalization means that fields are not localized near the soliton
center. The BPS state that remains localized is single.  The existence
of the supershort multiplets and the loss of $(-1)^F$ was emphasized
in the first version of the present paper;\,\cite{LSV2} in
Ref.\,\protect\citebk{GLN} the authors came to the same conclusion.

In our recent publication\,\cite{LSV} the weak coupling consideration
was generalized to a wide class of models including those with nonflat
target space.  The present text combines (and extends) the first
version of the paper in Ref.\,\protect\citebk{LSV2} with
\mbox{Ref.\,\protect\citebk{LSV}}.

Let us now summarize our main points:\\[1mm]
(1) 
We introduce a new index which counts supershort multiplets in \none
two-dimensional theories. Let us remind that the first SUSY index,
${\rm Tr}\, (-1)^F$, was introduced by Witten twenty years
ago\,\cite{WittenIndex} to count the number of supersymmetric vacua.
About ten years ago, Cecotti, Fendley, Intriligator and Vafa
introduced\,\cite{CFIV} another index, Tr$\,[F\,(-1)^F]$, counting the
number of short multiplets in \ntwo theories in two dimensions. No
index counting single-state multiplets in \none theories in two
dimensions was known. This is probably not
surprising,  since it was always assumed that $(-1)^F$ does exist.\\[1mm]
(2) 
We will show that the appropriate index is 
$\left\{{\rm Tr}\,Q_1\right\}^2\!/2{\cal Z}$ --- it vanishes for long 
multiplets and is equal to 1 for one-dimensional multiplets.
If the value of this index  does not vanish in the given \none theory, 
short multiplets {\em do exist} with necessity.\\[1mm]
(3) 
We consider a wide class of \none hybrid models which include, along
with a superpotential ${\cal W}(\phi)$, a nonflat metric
$g_{ab}(\phi)$ of the target space for the fields $\phi^a$. These are
hybrids between the sigma models and the Landau-Ginzburg models.  The
result depends on the soliton moduli space.  More exactly, what counts
is a {\em reduced} moduli space, namely the moduli space with the
translational modulus (corresponding to the motion of the center of
inertia) factored out, together with its fermionic superpartner.  The
index we introduce, $\left\{{\rm Tr}\,Q_1\right\}^2\!/2{\cal Z}$,
turns out to be a square of the index of the Dirac operator defined on
the reduced moduli space.\\[1mm]
(4) 
We find, with surprise, that the existence of the BPS solitons
belonging to one-dimensional supermultiplets is quite a rare occasion.
It happens only in the problems with a single modulus, translational.
The reduced moduli space is then trivial. Although the index of Dirac
operator on a generic compact space may be nonvanishing, the reduced
moduli spaces arising in the hybrid models have a special geometry
(similar to spherical) for which the index vanishes.  If the reduced
moduli space has dimension 1 or larger and is compact there are no
BPS solitons.\\[1mm]
(5) 
Another issue we address is the generalization of the anomaly in the
central charge found previously in the Landau-Ginzburg models,\cite{1}
where the target space metric is flat, to the hybrid models with a
nonflat metric. Our result for the anomaly in this case is a
straightforward extension of Ref.\,\protect\citebk{1} and can be
formulated as a substitution
\begin{equation}
{\cal W}(\phi) \longrightarrow \widetilde {\cal W}(\phi )={\cal
W}(\phi)+\frac{1}{4\pi}\,\nabla^a
\nabla_a {\cal W}(\phi)
\label{anoW}
\end{equation}
for the superpotential. The quantum anomaly is represented by 
the second term, with the covariant Laplacian on
the target space, $\nabla^a \nabla_a\equiv g^{ab}\nabla_a\nabla_b\,$.
The anomaly-corrected superpotential enters  into the energy-momentum tensor,
the supercharges and the central charge. In particular, the operator of the 
central charge  becomes
\begin{equation}
{\cal Z} = 
\widetilde{\cal W} (\phi (z\to \infty))-\widetilde{\cal W} (\phi (z\to 
-\infty))
\,.
\end{equation}

\vspace{0.2cm}

The paper is organized as follows.  In Sec.\,\ref{sec:algeb} we 
thoroughly discuss algebraic aspects. The relation between the multiplet 
shortening and the loss of $(-1)^F$ is explained.  We also make a remark 
on short multiplets in 2+1 dimensions (Sec.\,2.5). Section \ref{sec:index} 
treats Tr$\,Q_1$ as an index. In Sec.\,4 we begin a systematic consideration 
of various models. We start from a generic hybrid model combining 
the Ginzburg-Landau and sigma models, with \none supersymmetry. 
The target space ${\cal T}$ is an arbitrary Riemann manifold. Section
4.1 presents generalities. In Sec.\,4.2 we derive the quantum anomaly
in the central charge for the hybrid model of the general form.
Classification of the models is presented in Sec.\,4.3.  In Sec.\,5 we
treat particular examples with flat target space.  Section 6 is
devoted to a model in which both, the target space and the spatial
coordinate are circles.  Section 7 is devoted to nonflat target
spaces.  It is demonstrated that the index Tr$\,Q_1$ is related to the
index of the Dirac operator on the soliton moduli space.  The main
example here is $S^3$ as the target space where we find the index
Tr$\,Q_1$ to be vanishing.  Correspondingly, all supermultiplets are
long, there are no BPS solitons in this model.  Sections 8 summarizes
our findings and results.  Appendix presents a proof that in any
hybrid model the situation is similar: the reduced moduli space is
such that the index of the Dirac operator on it (and, hence,
Tr$\,Q_1$) vanishes provided that the reduced moduli space is a
compact manifold of positive dimension.

\section{Representations of superalgebra}
\label{sec:algeb}

\subsection{Automorphisms of superalgebra}
\label{auto}
Let us address the question: what extra symmetries are compatible with 
the centrally extended algebra  (\ref{alg})? It is clear that 
 the Lorentz boost can be included,
\begin{eqnarray}
&& Q_\alpha \to \left[\exp\left(-\frac{i}{2}\,\beta\,
\gamma^5\right)\right]_{\alpha\beta} Q_\beta\;,\qquad
\bar Q_\alpha \to \left[\exp\left(\frac{i}{2}\,\beta\,
\gamma^5\right)\right]_{\alpha\beta}\bar Q_\beta\;,
\nonumber\\[2mm]
&& P_\mu (\gamma^\mu )_{\alpha\beta}\to \left[\exp\left(-\frac{i}{2}\,\beta\,
\gamma^5\right)\right]_{\alpha\gamma}\,\left[\exp\left(\frac{i}{2}\,\beta\,
\gamma^5\right)\right]_{\beta\delta} P_\mu (\gamma^\mu
)_{\gamma \delta }\;, \nonumber\\[2mm]
&&{\cal Z}\to {\cal Z}\;,
\label{boost1}
\end{eqnarray}
where $\beta $ is a real parameter (in our convention $i \gamma^5$ is Hermitean).

If the supercharges $Q_\alpha$ were complex, as in \ntwo\!\!, an extra
symmetry would emerge --- in addition to the real $\beta $ one could
consider transformations with purely imaginary $\beta$. This symmetry
would express the conservation of the fermion charge $F$.  In \none
where the supercharges are real there is no fermion charge. What
survives, however, is the fermion parity $G=(-1)^F$. The action of $G$
(given by putting $\beta =2i\,\pi $) reduces to changing the sign for
the fermion operators leaving the boson operators intact,
\begin{equation}
G\,Q_\alpha\, G^{-1}= -Q_\alpha\,,\qquad G\,P_\mu\,  G^{-1}=P_\mu\,.
\end{equation}
The fermion parity $G$ realizes $Z_2$ symmetry associated with
changing the sign of the fermion fields. From this standpoint it seems
that this symmetry is guaranteed. In fact we will show in some
examples that in the soliton sector the very classification of states
as either bosonic or fermionic is broken.  In constructing
representations of superalgebra we would not necessarily assume that
the fermion parity $(-1)^F$ is the valid symmetry but the Lorentz
symmetry is certainly assumed.

\subsection{Beginning the construction}
\label{begin}
Now let us start the construction.  The Lorentz symmetry implies that
$P^\mu P_\mu$ is invariant, $P^\mu P_\mu=M^2$ where $M$ is the mass of
the state.  In what follows we will treat $M$ and ${\cal Z}$ as
$c$-numbers characterizing each given irreducible representation.  It
is convenient to choose the rest frame where $P_\mu=(M,0)$ and the
algebra (\ref{alg}) takes the form
\beq
(Q_1)^2 =M+{\cal Z}\,,\qquad (Q_2)^2 = M-{\cal Z}\,, \qquad \{ Q_1\,, Q_2\}
= 0\,.
\label{algone}
\eeq
Positive definiteness  leads to $$M^2\geq {\cal Z}^2\,.$$
The analysis bifurcates at this point: one should consider separately 
the non-BPS cases, $M^2> {\cal Z}^2$,  and the BPS case, $M^2={\cal Z}^2$.

Let us note that for solitons for which $M-{\cal Z}\ll {\cal Z}$ we
are in domain of nonrelativistic description (it is implied that
${\cal Z}>0$). The quantity which plays the role of the Hamiltonian in
the supersymmetric quantum mechanics of solitons is then $M-{\cal Z}$,
where $M$ should be understood as the operator $\sqrt{P^\mu P_\mu}\,$,
\begin{equation}
H_{\rm SQM}=M-{\cal Z}=(Q_2)^2\,.
\label{sqm}
\end{equation}
The relation $H_{\rm SQM}=(Q_2)^2$ is a subalgebra of the SUSY algebra 
which is stationary for the BPS states.\footnote{~The construction 
(\ref{sqm}) is called \nhalf\; quantum mechanics with ${\cal N}$ counting 
the number of {\em pairs} of supercharges that square to $H_{\rm SQM}$.}

In constructing $H_{\rm SQM}$ we separate dynamics of the center of
mass, passing to the rest frame.  The total spatial momentum $P_z$ is
conjugated to the center of mass coordinate, $Q_1$ is its
superpartner.  Both, $P_z$ and $Q_1$ commute with $H_{\rm SQM}$. Their
partnership becomes evident in the moduli dynamics example which will
be considered in Sec.\,\ref{sec:sigma}, zero modes generated by $P_z$
and $Q_1$ have the same dependence on the coordinate $z$.

\subsection{Non-BPS multiplets}
\label{sec:nonbps}

If $M^2\neq {\cal Z}^2$ Eq.\,({\ref{algone}}) represents  the Clifford 
algebra with two generators, its irreducible representation is 
two-dimensional. For instance, one can choose
\begin{equation}
Q_1=\sigma_1\sqrt{M+{\cal Z}}\,,\qquad Q_2=\sigma_2\sqrt{M-{\cal Z}}\,,
\label{two-d}
\end{equation}
where $\sigma_{1,2}$ are the Pauli matrices. 

There is an obvious automorphism: one can substitute $\sigma_1$, $\sigma_2$
by rotated matrices $\tilde \sigma_1=\sigma_1\cos\alpha +\sigma_2\sin \alpha $
and $\tilde \sigma_2=-\sigma_1\sin\alpha +\sigma_2\cos \alpha $.
It means that the two-dimensional representation at hand
admits  introduction of the fermion number $F=(1-\sigma_3 )/2 $. 
Generically, the fermion number operator $F$  can be expressed in terms 
of  the supersymmetry generators $Q_\alpha$.  First, let us  consider 
a bosonic operator
\begin{equation}
S=-\frac i 2 \,\left[Q_1\,, Q_2\right]=\frac 1 2\,
\,\bar Q\, Q\,,
\label{Rop}
\end{equation}
which is an element of the enveloping algebra.
The expression after the second equality sign is not bound to the rest frame.
This operator has the following features:
\begin{eqnarray}
&& S^2=P_\mu P^\mu-{\cal Z}^2\,,\quad \{S,Q_\alpha\}=0\,,\nonumber\\[2mm]
&& \left[S\,, Q_\alpha\right]=-2
\,\left(\gamma^\mu P_\mu +\gamma^5 {\cal Z}\right)_{\alpha\beta} Q_\beta
\nonumber\\[2mm]
&& \left[S\,, P_\mu\right]=0\,,\quad \left[S\,, {\cal Z}\right]=0\,.
\end{eqnarray}

In the representation (\ref{two-d})  the operator $S$ has the form,
\begin{equation}
S=\sqrt{M^2-{\cal Z}^2}\,\sigma_3\,.
\end{equation}
This expression shows that the operator $S$ can be introduced only for
non-BPS representations, $M^2\neq {\cal Z}^2$. Then 
$S/\sqrt{M^2-{\cal  Z}^2}$ is a generator of SO(2) rotations, associated 
with the fermion charge $F$,
\begin{equation}
\frac{S}{\sqrt{M^2-{\cal Z}^2}}=1- 2\,F \,,\qquad
F=\frac{1}{2}-\frac{\bar Q Q}{4\,\sqrt{M^2-{\cal Z}^2}}\,.
\label{fchar}
\end{equation}
The fact that $S$ is bilinear in $Q_\alpha$ results in $F^2=F$, i.e.
the operator $F$ acts as a projection operator. Its eigenvalues are 0
and 1 and it measures the number of fermions modulo two. It means that
the very same operator $S$ defines also the fermion parity $G$ of
state
\begin{equation}
G=(-1)^F =\frac{\bar Q Q}{2\,\sqrt{M^2-{\cal Z}^2}}\,.
\label{gpar}
\end{equation}

Let us emphasize that in the \none models there is no local current
associated with the fermion charge. Therefore, the fermion charge we
have introduced has no local representation.  A local current does
exists in the case of extended \ntwo ~supersymmetry. The corresponding
fermion charge is different from $F$ defined in Eq.\,(\ref{fchar}).  It
is known that the fermion charge defined by the local current is
noninteger for solitons\,\cite{FI} (the fractional fermion charge of
the soliton was discovered by Jackiw and Rebbi\,\cite{JR}). At the
same time the fermion charge (\ref{fchar}) is always integer.  In the
topologically trivial one-particle sector of \ntwo theories both
fermion charges coincide.

Note an analogy between the introduction of the operator $S$ above, in
constructing representations of the superalgebra, and the introduction
of the Pauli-Lubanski spin operator
$\Gamma^\mu=\epsilon^{\mu\nu\gamma\delta}M_{\nu\gamma}P_\delta$ for
the Poincar\'e group. In this case there is no local current too, and
the Pauli-Lubanski operator is not defined for massless particles,
$P_\sigma P^\sigma=0$.  Let us emphasize once more that $S$ vanishes
for the BPS states.


\subsection{BPS representations}
\label{sec:bps}

Now let us consider the special case $M^2= {\cal Z}^2$. Assuming ${\cal Z}$ 
to be positive we consider  the BPS soliton, $M= {\cal Z}$, for which 
the supercharge $Q_2$ is trivial, $Q_2=0$. 
Thus,  we are left with a single supercharge
$Q_1$ realized nontrivially. The algebra reduces to a single relation
\begin{equation}
(Q_1)^2=2\,{\cal Z}\,. 
\label{q2z}
\end{equation}
The irreducible representations of this algebra are one-dimensional, 
there are two such representations,
\begin{equation}
Q_1=\pm  \,\sqrt{2{\cal Z}}\,,
\label{twosigns}
\end{equation}
i.e., two types of solitons,
\begin{equation}
Q_1|\,{\rm sol_+}\,\rangle=\sqrt{2{\cal Z}}\,|\,{\rm sol_+}\,\rangle\,,\qquad
Q_1|\,{\rm sol_{\,-}}\,\rangle=-\sqrt{2{\cal Z}}\,|\,{\rm sol_{\,-}}\,\rangle\,.
\label{1dim}
\end{equation}
It is clear that these two representations are unitary nonequivalent.

The one-dimensional irreducible representation implies multiplet
shortening: the short BPS supermultiplet contains only one state while
non-BPS supermultiplets contain two. However, the possibility of such
supershort one-dimensional multiplets is usually discarded. It is for
a reason: while the fermion parity $(-1)^F$ is granted in any local
field theory based on fermionic and bosonic fields, it is not defined
in the one-dimensional irreducible representation. Indeed, if it were
defined, it would be $-1$ for $Q_1$, which is incompatible with any of
the equations (\ref{1dim}).  The only way to recover $(-1)^F$ is to
have a reducible representation containing both 
$|\,{\rm sol}_{+}\,\rangle$ and $|\,{\rm sol}_{\,-}\,\rangle$.  Then,
\begin{equation}
Q_1=\sigma _3\sqrt{2{\cal Z}}\,,\qquad
(-1)^F=\sigma_1\,. 
\label{pm}
\end{equation}

Does it mean that the one-state multiplet is not a possibility in the
local field theory? It was argued in Refs.\,\protect\citebk{1,2,GLN}
(and we are going to review this again in Sec.\,\ref{box}) that
solitons in certain models do realize such supershort multiplets
indeed defying $(-1)^F$.

Thus, for the BPS representations 
 we come to two scenarios:
\begin{itemize}
\item[{\bf (i)}]
Fermion parity $(-1)^F$ is broken and the irreducible representation is
realized. The supermultiplet is short, contains only one state.
\item[{\bf (ii)}]
Fermion parity $(-1)^F$ is not broken, the representation is reducible.
The multiplet of degenerate states is not short, containing bosonic and 
fermionic components.
\end{itemize}
The important point is that  only  short multiplets  of the BPS
states are protected against becoming non-BPS under small perturbations. 
It is clearly not the case in the scenario {\bf (ii)}. This leads us 
to introduction of a new index, $({\rm Tr}\, Q_1)^2$, which counts short 
multiplets with broken $(-1)^F$. This index will be carefully defined 
in Sec.\,\ref{sec:index}.

The discrete $Z_2$ symmetry $(-1)^F$ discussed above is nothing but
the change of sign of all fermion fields, $\psi\to -\psi$. This
symmetry is seemingly present in any theory with fermions. How this
symmetry can be lost in the soliton sector will be explained later.
Here we would like to mention the following.  Although the overall
sign of $Q_1$ on the irreducible representation is not observable, the
relative sign is.  For instance, there are two types of reducible
representations of dimension two: one is $\{+,-\}$ (see
Eq.\,(\ref{pm})), and another $\{+,+\}$ (equivalent to $\{-,-\}$).  Our
index $(\,{\rm Tr}\, Q_1)^2/2{\cal Z}$ discriminates between these two
cases --- it vanishes in the first case and equals to 4 in the second.
Another example of dimension three is the reducible representation
$\{+,+,-\}$. The index is 1 in this case, implying that the pair
$\{+,-\}$ can leave the BPS bound leaving a single BPS state.

\subsection{Massless supermultiplets in 2+1 dimensions}
\label{sec:2+1d}

The SUSY algebra (\ref{alg}) with the central charge ${\cal Z}$ we
have considered in 1+1 dimensions becomes identical to the
superalgebra in 2+1 dimensions (without central extension) provided
one identifies the central charge ${\cal Z}$ with the momentum $P_2$
in the extra spatial dimension. Indeed, after this identification the
algebra (\ref{alg}) can be rewritten as
\begin{equation}
\{ Q_\alpha\,,  {\bar Q}_\beta \}=2\left(\gamma^M\right)_{\alpha\beta} 
P_M \, ,\qquad (M=0,1,2) \, ,
\end{equation}
where $\gamma^2$ coincides with $\gamma^5$ from Eq.\,(\ref{gamma}).

The one-dimensional representation we have constructed for the BPS states,
$P_\mu P^\mu={\cal Z}^2$, in 1+1 dimensions, in 2+1 becomes  a representation 
$|P_1,P_2\rangle$ for the massless particle, $P_M P^M= 0$. Assume that we choose
a Lorentz frame where $P_M=(E, 0, E)$. In this frame the supercharges
are represented as
\begin{equation}
Q_1=\pm \sqrt{2E}\,,\qquad Q_2=0\,,
\end{equation}
cf.  Eq.\,(\ref{twosigns}).  Again, although irreducible
representations are one-dimensional, maintaining $(-1)^F$ makes the
representation two-dimensional and reducible, 
see e.g. Ref.\,\protect\citebk{BdW}. 

Is it possible to break $(-1)^F$ in 2+1D similarly to solitons in 1+1?
We are aware of no dynamical example of this type.\footnote{\, Of course, 
here we speak of spatially localized states, rather then of extended 
objects which can appear as an elevation from 1+1D.} For example, in the free
massless supersymmetric theory there are two states of the $|\pm\rangle$ type,
\begin{equation}
|P_1=0,P_2=E\rangle_\pm=\int \!{\rm d}^2 x\, {\rm e}^{i{\vec P}{\vec x}} \left(
\sqrt{2E}\,\phi \pm
\psi_1\right) |0\rangle \,,
\label{state}
\end{equation}
which are irreducible representations. Here $\phi$ and $\psi_\alpha$ 
are free field operators and we chose $P_1=0$. Together they form 
the reducible representation for which $(-1)^F$ is well defined.

Note that models where the breaking of $(-1)^F$ may occur, emerge for
domain wall junctions in 3+1 dimensions.\cite{junctions} In these
models the junctions effectively reduce to 1+1 dimensional objects.

\section{Tr\,{\em Q}$_{\,1}$ as index}
\label{sec:index}

In the context of supersymmetric theories ${\rm Tr}\,(-1)^F$ as an
index was introduced by Witten. The Witten index counts the difference
between the numbers of the bosonic and fermionic states of zero
energy, i.e. vacua which are annihilated by supercharges.  For all
supermultiplets with nonzero energy this difference vanishes.

In particular, it vanishes in the soliton sector in \ntwo two-dimensional 
theories. However, the BPS solitons are annihilated by a part of supercharges 
and form short multiplets.  This is counted\,\cite{Ntwo} by 
the Cecotti-Vafa-Fendley-Intriligator index  
${\rm Tr}\,F\,(-1)^F$.  The fermion number $F$ is well defined 
in \ntwo theories. 

In ${\cal N}=1$ theories  the fermion number $F$, and even the fermion parity
$(-1)^F$, are not defined for short multiplets.
Is there an  index in the \none soliton problems which would count the
supershort multiplets? We assert that 
$\left\{\,{\rm Tr}\,Q_1\right\}^2$ does the job. 
More exactly, the  definition of the index  is as follows
\begin{equation}
{\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)=\frac{1}{2\,{\cal Z}}\left\{\lim_{\beta \to
\infty}{\rm Tr}
\left[Q_1\,\exp(-\beta\,(Q_2)^2 )\right]\right\}^2\,.
\label{Ind}
\end{equation}
The exponential factor in Eq.\,(\ref{Ind}) is introduced for 
the UV regularization.  The necessity of taking the $\beta \to\infty$ 
limit is due to continuous spectrum, as explained in Ref.\,\protect\citebk{CFIV}.

This index vanishes for non-BPS multiplets for which the fermion parity $(-1)^F$ 
can be consistently defined. Equation (\ref{two-d}) demonstrates this explicitly.
For each irreducible BPS representation the index is unity,
$$
{\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)\,[\,{\rm irreducible~BPS}\,] = 1 \,.
$$
If a reducible representation contains a few irreducible BPS multiplets
the index may or may not vanish depending on the numbers of the $\{+\}$ and   
$\{-\}$ irreps. In the case of  the vanishing index one can introduce
$(-1)^F$, and small deformations or quantum corrections can destroy 
the BPS saturation. 

Note, that our index is not additive: it is not equal to the sum of
the indices of the irreducible representations.  An interesting
example of an \none reducible BPS representation is provided by
solitons in \ntwo models. The \ntwo BPS multiplet consists of two
\none multiplets of the opposite types, leading to the vanishing of
the index (\ref{Ind}).

The definition (\ref{Ind}) has a technical drawback --- it refers
to the soliton rest frame. It is simple to make it Lorentz invariant,
\begin{equation}
{\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)=\frac{1}{2\,{\cal Z}^2}\left(\,{\rm Tr}\,
\bar Q\right)\! \not \! P \left(\,{\rm Tr}\, Q\right)
\,,
\label{IndL}
\end{equation}
where the trace refers to the Hilbert space but not to the Lorentz indices
of the supercharges $Q_\alpha$ and $\bar Q_\alpha=Q_\beta
(\gamma^0)_{\beta\alpha}$. Here we have omitted the regularizing exponent.

\section{Theories with \none supersymmetry in 1+1 dimensions}
\label{GL}

In this section we consider a generic \none  field theory  in 1+1
dimensions, presenting a realization  of SUSY with two real supercharges
$Q_\alpha$ and the central charge, see Eq.\,(\ref{alg}). 
The two-dimensional space $x^\mu=(t,z)$ is flat. The time $t\in R$ while the
spatial coordinate $z$ lives either on the line $R$ (noncompact), or on the
circle $S^1$ (compact).  We will deal with
$n$  superfields 
\begin{equation}
\Phi^a=\phi^a+\bar\theta \psi^a +\frac{1}{2\,}\bar\theta \theta F^a\,,\qquad
(a=1,\ldots,n)\,.
\end{equation}
Each superfield contains a real  boson field $\phi^a$,
a two-component Majorana  spinor $\psi_\alpha^a$ ($\alpha=1,2$)
and an auxiliary field $F^a$.  The target space formed by $\phi ^a$ 
is an arbitrary Riemann manifold ${\cal T}$ endowed with the metric 
$g_{ab}(\phi )$.
Moreover, we introduce a superpotential ${\cal W}(\phi )$. Thus,
the generic model  is a hybrid between  the
$\sigma$-model and the Landau-Ginzburg theory.

The generic form of the Lagrangian is (for a review see
Ref.\,\protect\citebk{quantmat})
\begin{eqnarray}
{\cal L}&=&\!\frac{1}{2}\,g_{ab}\left[\,\partial_\mu\phi^a \,
\partial^\mu\phi^b + \bar\psi^a \,i\gamma^\mu {\cal D}_\mu \psi^b +F^a F^b\right]
+ \frac{1}{12} \, R_{abcd}\,(\bar\psi^a \psi^c)(\bar\psi^b \psi^d)\nonumber\\[1mm]
&+&\! F^a\partial_a {\cal W} - \frac1 2 \,(\nabla_a \partial_b {\cal W})\,\bar\psi^a
\psi^b
\,,
\label{sigmaL}
\end{eqnarray}
where  $\bar \psi=\psi^T \gamma^0$, and
$\Gamma^b_{c\,d}(\phi)$ and $R_{abcd}(\phi)$ are the Christoffel symbols
and the Riemann tensor, respectively. Furthermore,
${\cal W}(\phi)$ is the superpotential,  and
$\partial_a$ and
$\nabla_a$ denote usual and  covariant derivatives in the target space, e.g.,
\begin{equation}
{\cal W}_{,\,a}=\nabla_a {\cal W}=\partial _a{\cal W}\,;\qquad
\nabla_a {\cal W}_{,b}=\partial_a {\cal W}_{,b} -
\Gamma^c_{ab} \, {\cal W}_{, c}\;,
\label{covar}
\end{equation}
while the covariantized space-time derivative ${\cal D}_\mu$ is
\begin{equation}
{\cal D}_\mu \, \psi^b=\frac{\partial\, \psi^b}{\partial x^\mu} +\Gamma^b_{cd}\, 
\frac{\partial\, \phi^c}{\partial x^\mu}\, \psi^d\,.
\label{covd}
\end{equation}
The Lagrangian (\ref{sigmaL}) implies that 
the auxiliary field $F^a=-g^{ab}\partial_b{\cal W}$.

\subsection{SUSY, central charge and BPS saturation}
The \none \,supersymmetry of the model  is expressed by two supercharges,
\begin{equation}
Q_\alpha=\int {\rm d} z \,S_{\alpha}^0\,,\qquad
S^\mu=g_{ab}\left(\not \!\!\,\partial \,\phi^a-i \,F^a\right)
\gamma^\mu\,\psi^b \,,
\label{supercur}
\end{equation}
where $S^\mu$ is the conserved supercurrent.
These supercharges form the \none algebra (\ref{alg}) with the metric
independent central charge 
\begin{equation}
{\cal Z}=\int  {\rm d} z\, \partial_z \phi^a \,\partial_a {\cal W}\,.
\label{cch}
\end{equation}
The central charge does not vanish for classical solitons
interpolating between different vacua of the theory. These vacua
correspond to critical points of the superpotential ${\cal W}(\phi)$
at which $\partial_a{\cal W}=0$.  For the soliton interpolating
between critical points $\phi=A$ and $\phi=B$ the central charge is
equal to
\begin{equation}
{\cal Z}_0=\Delta{\cal W}={\cal W}(A)-{\cal W}(B)\,,
\label{claz}
\end{equation}
where we assume, by convention,  that  ${\cal Z}>0$. Certainly, the inverse
interpolation (antisoliton) with negative   ${\cal Z}$ also exists. 

For the BPS saturated soliton and antisoliton their masses  are equal to
$|{\cal Z}|$. The BPS soliton configuration $\phi_0$ satisfies the following 
equation:
\begin{equation}
\frac{ {\rm d} \phi^a_{0}}{{\rm d} z}= \, g^{ab}\,\partial_b \,{\cal
W}(\phi_{0})\,.
\label{AAone}
\end{equation}


\subsection{Ultraviolet aspects and quantum anomaly}
\label{sec:anomaly}

The expressions (\ref{supercur}), (\ref{cch})   and (\ref{claz}) above 
are obtained at the classical level. If the target space manifold ${\cal T}$ 
is flat, i.e., we deal with the
Landau-Ginzburg model, the theory is superrenormalizable: logarithmic
divergences appear only at one loop. For generic nonflat  ${\cal T}$
the theory is nonrenormalizable: as well known, loops will generate an
infinite series of new structures in the target space metric.  However, if 
${\cal T}$ is symmetric, the number of structures is finite, and the theory
is renormalizable: divergences can be absorbed into  a finite number 
of parameters. 

Already in the superrenormalizable Landau-Ginzburg models (where
$g_{ab}=\delta_{ab}$) loop corrections lead to a quantum anomaly.\cite{1} 
The anomaly occurs in the energy-momentum tensor, supercurrent and 
the central charge density. In fact, these anomalies form a supermultiplet. 
Moreover, the impact of the anomaly is local and universally expressed 
through the substitution
\begin{equation}
{\cal W}\longrightarrow \widetilde{\cal W}=
{\cal W}+\frac{1}{4\pi}\,\partial_a\partial_a{\cal W}\,.
\label{anoma}
\end{equation}

It is clear, then, that a similar anomaly must occur in the hybrid models 
as well. Below we present its form for the generic model. Note, first, that 
the distinction between the superrenormalizable Landau-Ginzburg
models and renormalizable hybrid models is irrelevant for the analysis
of the one loop anomaly. The form of the anomaly is severely limited
by the following considerations: (i) dimension and locality;
(ii) general covariance in the target space; (iii) the  flat metric limit.
The one-loop calculation presented in Ref.\,\protect\citebk{1} can be readily
extended to include the target space metric and leads to
\begin{equation}
{\cal W} \longrightarrow \widetilde{\cal W}=
{\cal W} +\frac{1}{4\pi}\,g^{ab}\,\nabla_a\nabla_b {\cal W} 
\,.
\label{hyban}
\end{equation}
This differs from the Landau-Ginzburg case only by covariantization
of the Laplacian, 
\begin{equation}
\delta^{ab}\partial_a \partial_b{\cal W} \to 
g^{ab}\,\nabla_a \nabla_b {\cal W}=g^{-1/2}\, \partial_a
\, g^{1/2}g^{ab}\partial_b\, {\cal W}\,.
\end{equation}
The corrected superpotential $\widetilde{\cal W}$ should be substituted 
into the expressions for  the energy-momentum tensor, supercurrent and 
the central charge.

In particular,  the central charge  ${\cal Z}$ becomes
\begin{equation}
{\cal Z}=\widetilde{\cal W}(A)-\widetilde{\cal W}(B)\,.  
\end{equation}
A novel feature compared with the flat target space is the occurrence of 
a metric dependence. Let us remind that at the loop level the metric
``runs''. It is this running metric that enters the anomaly (\ref{hyban}).
As a result, the anomaly 
which was one-loop in the flat target space\,\cite{1} becomes multi-loop.

It is worth singling out  interesting cases of superpotentials which are 
eigenfunctions of the covariant Laplacian,
\begin{equation}
g^{ab}\nabla_a\nabla_b {\cal W} = c{\cal W}\,.
\label{eigenlap}
\end{equation}
 The simplest example of this type
is provided by the sine-Gordon model, ${\cal W}=mv^2\sin(\phi /v)$ (see
Sec.\,\ref{box}).  The impact of the anomaly is the following
replacement of the classical central charge
${\cal Z}_0$:
$$
{\cal Z} ={\cal Z}_0\left[1 -\frac{1}{4\pi v^2}\right]\,.
$$
The equivalent  replacement $v^2 \to v^2-(1/4\pi )$ is well-known in the
framework of the CFT treatment of the integrable models.\cite{Ahn}

Another similar example is given by the $S^3$ model of Sec.\,\ref{sec:sigma}.
In this model the superpotential satisfies Eq.\,(\ref{eigenlap})  with $c=-3f$
where $f$ is the running coupling constant.\footnote{%
~At one-loop level,
$
1/f (\mu )=1/f_0-(1/\pi)
\, \ln(\mu_0/\mu)\,.
$
}
The anomaly shifts ${\cal Z}_0$,
\begin{equation}
{\cal Z} = {\cal Z}_0 \left( 1-\frac{3f}{4\pi}
\right)\,.
\label{shifz}
\end{equation}
Equation (\ref{shifz}) implies that the combination
on the right-hand side is renor\-malization-group invariant.

\subsection{Classification of models}
\label{sec:class}

The models to be considered  in this paper fall into several distinct categories
characterized by the following features:
\begin{itemize}
\item[(i)]
The geometry of the spatial coordinate $z$: compact versus noncompact (in both
cases the metric is flat for one spatial coordinate);
\item[(ii)] The geometry of the target manifold ${\cal T}$: compact versus
noncompact.
\item[(iii)] 
The metric of ${\cal T}$: flat versus nonflat.
\end{itemize}
Our examples represent almost all combinations of the classes above.
We start with the flat target space. In this case the generic classical Lagrangian
(\ref{sigmaL}) takes the form
\begin{equation}
{\cal L}=\frac{1}{2}\left\{ \partial _\mu\phi^a
\,\partial^\mu\phi^a
+i \bar \psi^a\gamma^\mu \partial_\mu \psi^a  -\frac{\partial
{\cal W}}{\partial \phi^a}
\,\frac{\partial
{\cal W}}{\partial \phi^a} - \frac{\partial^2 {\cal W}}{\partial
\phi^a \partial
\phi^b}\, \bar \psi^a
\psi^b \right\}
\,.
\label{2lag}
\end{equation}
Some of such
models are
known to be exactly integrable.\cite{Tsvelik,Ahn,FI} However,  following
Refs.\,\protect\citebk{1}, \protect\citebk{2}, we
will limit our consideration to the quasiclassical regime assuming that  the
expansion parameter is small. 

\section{Flat target manifold, noncompact space}
\label{box}

In this  section we consider  the \none Landau-Ginzburg models
(\ref{2lag}) with one or two super\-fields with the spatial coordinate $z\in R$.

\subsection{One-superfield models}
\label{sec:onesup}

Our presentation in this section follows Ref.\,\protect\citebk{1}.  Although the
superpotential
${\cal W}(\phi)$ can be arbitrary,  for classification purposes we will discuss
two representative examples: the polynomial (PM) model,
\begin{equation}
{\cal W}_{\rm PM}(\phi ) =\frac{m^2}{4\lambda}\,\phi
-\frac{\lambda}{3}\,\phi^3\,,
\label{PS}
\end{equation}
and the sine-Gordon (SG) model,
\begin{equation}
{\cal W}_{\rm SG}(\phi ) =m v^2\sin\frac{\phi }{v}\;.
\label{SSG}
\end{equation}
The target space is noncompact in the PM case and compact $S^1$  in the SG model.
The classical BPS equation 
\begin{equation}
\frac{ {\rm d} \phi_{0}}{{\rm d} z}= {\cal
W}^\prime(\phi_0)\,,
\end{equation}
has the following solutions:
\begin{eqnarray}
& \phi_{\,0} =\mbox{\large$\frac{m}{2\lambda}$}\,{\rm tanh}\,
\mbox{\large$\frac{m\,z}{2}$}\,,~~\qquad\qquad & ({\rm PM})
\nonumber\\[2mm]
& \phi_{\,0} =v\arcsin[{\rm tanh}(mz)]\,,\qquad &
({\rm SG})
\label{solit}
\end{eqnarray}
interpolating between two neighboring  vacua. 

For infrared regularization the system is placed in a large spatial box, 
i.e., the boundary conditions at
$z=\pm L/2$ are imposed.  The conditions we choose are 
\begin{eqnarray}
&&\left[\partial_z\phi -{\cal W}'(\phi) \right]_{z=\pm L/2}=0\,,\qquad
\left. \psi_1\right|_{z=\pm  L/2}=0\,,
\nonumber\\[0.2cm]
&&\left. \left[\partial_z -{\cal W}''(\phi)\right]
\psi_2\right|_{z=\pm  L/2}=0 \,,
\label{boundary}
\end{eqnarray}
where $\psi_{1,2}$ denote the components of the spinor $\psi_\alpha$.
The first line is nothing but a supergeneralization of the BPS equation, 
$D_1 \Phi(t, z=\pm L/2, \theta)=0$  at the boundary. The second line
is the  consequence of the Dirac equation of motion, if $\psi$ satisfies 
the Dirac equation there are essentially no boundary conditions for 
$\psi_2\,$. Therefore, it is not an independent boundary condition in the
solution of the classical equations of motion. We will use these boundary 
conditions later for the construction of modes in the differential operators 
of the second order.

The above choice is particularly convenient because it is compatible with
the residual supersymmetry in the presence of the BPS soliton.  The boundary
conditions (\ref{boundary}) are consistent with the classical solutions,
both for the spatially constant vacuum configurations  and for the kink. In
particular, the soliton solution $\phi_{\,0}$ of Eq.\,(\ref{solit}) satisfies 
$\partial_z\phi - {\cal W}' =0$ everywhere.  Note that the conditions
(\ref{boundary}) are not periodic.

The next step is to introduce the expansion in modes for deviations from
the soliton solution (\ref{solit}). For the mode expansion we use 
the  second order Hermitean differential operators  $L_2$ and
$\tilde L_2$,
\begin{equation}
L_2 = P^\dagger P\,,\qquad \tilde L_2 = P P^\dagger\,,
\end{equation}
where 
\beq
P= \partial_z -\left.{\cal W}''\right|_{\phi = \phi_0(z)},\qquad
P^\dagger= -\partial_z -\left.{\cal W}''\right|_{\phi = \phi_0(z)}\,.
\eeq
The operator $L_2$ defines the modes of $\chi
\equiv \phi-\phi_0$, and those of the
fermion field $\psi_2$, while $\tilde L_2$ does this job for $\psi_1$. The
boundary conditions for $\psi_{1,2}$ are given in Eq.\,(\ref{boundary}), for
$\phi-\phi_0$ they follow from the expansion  of
the first condition in Eq.\,(\ref{boundary}),
\begin{equation}
\left. \left[\partial_z -{\cal W}''(\phi_0(z))\right]
\chi\right|_{z=\pm  L/2}=0 \,.
\label{chibound}
\end{equation}

It is easy to verify that there is only one zero mode $\chi_0(z)$ for
the operator $L_2$  which has the form,
\begin{equation}
\chi_0\propto \frac{{\rm d}\phi_0}{{\rm d}z}\propto \left.{\cal
W}'\right|_{\phi=\phi_0(z)}\propto
\left\{ \begin{array}{l}
\mbox{\large $\frac{1}{\cosh^2 (mz/2)}
$}
\quad \qquad\
~\mbox{(PM)}\,.
\\[0.4cm]
\mbox{\large $\frac{1}{\cosh\,(mz)}$}
\qquad \qquad
~~~\mbox{(SG)}\,.
\end{array}\right.
\label{chi0}
\end{equation}
This is the zero mode  for the boson field $\chi$ (translational mode)
and for fermion $\psi_2$ (supersymmetric mode). 

The operator $\tilde L_2$ has no zero modes at all. Let us emphasize
that the absence of the zero modes for $\tilde L_2$ is {\em not} because 
the solution of $\tilde L_2\, \tilde\chi =0$ is non-normalizable (we keep 
the size of the box  finite) but
because of the boundary conditions $\tilde \chi (z=\pm L/2)=0$.

The translational and supersymmetric zero modes discussed above imply
that the soliton is described by two collective coordinates: its center 
$z_0$ and a ``fermionic'' center $\eta$,
\begin{equation}
\phi=\phi_0(z-z_0) +{\rm nonzero~modes } \,,\quad \psi_2=\eta\,\chi_0
+{\rm nonzero~modes}\,,
\end{equation}
where $\chi_0$ is the normalized mode given by Eq.\,(\ref{chi0}). The nonzero
modes are those of the operator $L_2$.  As for $\psi_1$ it is given by
the sum over
the nonzero modes of  the operator $\tilde L_2$. 

Substituting the mode expansion
in the supercharges (\ref{supercur}) we arrive at
\begin{equation}
Q_1=2\sqrt{\cal Z}\,\eta +{\rm nonzero~modes } \,,
\qquad Q_2=\sqrt{\cal Z}\,\dot z_0\,\eta +{\rm nonzero~modes } \,.
\label{qq}
\end{equation}
Now we can proceed to the quasiclassical quantization. Projecting  the
canonical equal-time commutation relations for the fields $\phi$ and 
$\psi$ on the zero modes  we get
\begin{equation}
\left[\,p , z_0\right]= -i\,,\qquad \eta^2=\frac 1 2\,,
\end{equation}
where $p={\cal Z}\,\dot z_0$ is the canonical momentum conjugated to
$z_0$. It means that in quantum dynamics of the soliton moduli $z_0$ 
and $\eta$ the operators $p$ and $\eta$ can be realized as
\begin{equation}
p =-i\,\frac{\rm d}{{\rm d} z_0}\,,\qquad \eta=\frac{1}{\sqrt{2}}\,.
\end{equation}
It is clear that we could have chosen $\eta=-\,1/\sqrt{2}$. This 
is the same unobservable ambiguity that was discussed in 
Sec.\,\ref{sec:algeb}, the supercharge $Q_1$ is linear in $\eta.$

Thus, the supercharges depend only on the canonical  momentum $p$,
\begin{equation}
Q_1=\sqrt{2\cal Z}\,,
\qquad Q_2=\frac{p }{\sqrt{2{\cal Z}}}\,.
\label{qonetwo}
\end{equation}
 In the rest frame in which we perform our consideration
$\{Q_1,Q_2\}=0$, and the only value of $p$  consistent with it is $p=0$.
Thus, for the soliton $Q_1=\sqrt{2\cal Z}$, $Q_2=0$ in full agreement
with the general construction discussed in Sec.\,\ref{sec:algeb}.

Note that the representation (\ref{qonetwo}) can be used at nonzero
$p$ as well.  It reproduces the superalgebra (\ref{alg}) in the
nonrelativistic limit, with $p$ having the meaning of the total
spatial momentum $P_1$.

In passing from Eq.\,(\ref{qq}) to (\ref{qonetwo}) we have omitted the
nonzero modes. For each given nonzero eigenvalue there is one bosonic
eigenfunction (in the operator $L_2$), the same eigenfunction in
$\psi_2$ and one eigenfunction in $\psi_1$ (of the operator $\tilde
L_2$).  The quantization of the nonzero modes is quite standard. The
corresponding additional terms in $Q_{1,2}$ can be easily written in
term of the creation and annihilation operators. They describe
excitations of the BPS solitons. These excitations form long
(two-dimensional) multiplets. Both supercharges do not vanish and one
can introduce the fermion number (\ref{Rop}), (\ref{fchar}).

The multiplet shortening guarantees that the equality $M={\cal Z}$ is
not corrected. For the exactly solvable \none
models,\cite{Tsvelik,Ahn,FI} such as that with the superpotential
${\cal W}=mv^2\sin(\phi/v)$, the soliton mass is known exactly.  In
Ref.\,\protect\citebk{1} it was explicitly checked that $M$ is equal
to the matrix element of ${\cal Z}$ (see Eq.\,(\ref{claz}) with the
account for the anomaly (\ref{anoma})) up to two loops.  Moreover, it
was seen that the coupling constant expansion has a finite radius of
convergence (no essential singularity at small coupling).

What lessons can one draw from the considerations of this section?  In
the case of the polynomial model the target space is noncompact, while
the sine-Gordon case can be viewed as a compact target manifold $S^1$.
In these both cases we found one and the same result: short
(one-dimensional) soliton multiplet defying the fermion parity.  It is
clear that this conclusion remains valid for a general choice of the
superpotential ${\cal W}(\phi )$ admitting classical BPS solitons.

We would like to emphasize the following point. Although we started
from a noncompact spatial coordinate technically our analysis was
performed in the finite box (with specific boundary conditions). Thus,
the infrared regularization was guaranteed. However, the theory is not
ultraviolet finite, only super-renormalizable. This circumstance turn
out to be crucial, as we will see in the next section where a finite
model will be considered. We will see there that 
a heavy field which acts as an UV regulator develops a fermion zero mode 
localized on the boundary of the box.

\subsection{Two-superfield model}
\label{sec:2sf}

We  start from the Landau-Ginzburg model with the extended \ntwo
super\-symmetry which is ultraviolet finite theory. Then a soft breaking 
down to \none by a mass term preserves finiteness. Our presentation in this 
section follows Ref.\,\protect\citebk{2}.

The Lagrangian (\ref{2lag}) with
two real superfields $\Phi^a=\{\Phi,\widetilde \Phi\}$
has \ntwo
supersymmetry if the superpotential ${\cal W}(\phi,\widetilde \phi)$ is a
harmonic function,
\begin{equation}
\Delta_\phi {\cal W}\equiv\frac{\partial^2 {\cal W}}{\partial \phi^a \partial
\phi^a}=0\qquad \mbox{for ${\cal N}=2$}
\,.
\end{equation}
 It means, in particular, 
the absence of the anomaly in the central charge -- the
superpotential  is not changed by radiative corrections.
The \ntwo supersymmetry makes the model finite, while in \none it
was superrenormalizable. 
A polynomial example of the harmonic superpotential is 
\begin{equation}
{\cal W}(\phi,\widetilde \phi) = \frac{m^2}{4\lambda} \,\phi -
\frac{\lambda}{3} \,\phi^3 +\lambda\,\phi\,\widetilde\phi^{\,2}\,.
\end{equation}

How can one introduce  breaking  of \ntwo ? To this end,
 consider  a more general case of nonharmonic ${\cal W}\,(\phi_1,\phi_2) $,
\begin{equation}
{\cal W}\,(\phi,\widetilde \phi) = \frac{m^2}{4\lambda} \,\phi
-
\frac{\lambda}{3} \,\phi^3 +\lambda\,\phi\,\widetilde\phi^{\,2}
+\frac{p\,m}{2}\,\widetilde\phi^{\,2} +\frac{q\lambda}{3} 
\,\widetilde\phi^{\,3}\,,
\end{equation}
where
$p$ and $q$ are dimensionless parameters.
For $ p,q\neq 0$, the extended \mbox{\ntwo} supersymmetry
is explicitly broken down
to \none. The parameter $p$ introduces soft breaking of \ntwo
which preserves finiteness of the theory and the absence of the
anomaly.\footnote{~The term $p\,m\,\phi_2^2/2$ leads to a constant in
$\Delta_\phi {\cal W}$ which shifts the superpotential by an unobservable
constant.}
 The
nonvanishing
$q$ breaks the finiteness (the theory stays superrenormalizable,
however) and
introduces the anomaly, $\Delta_\phi {\cal W}=2q\lambda \widetilde\phi$.

The classical solution for the kink is the same as in the one-field  PM
model, see the first line in Eq.\,(\ref{solit}), with  second field
$\widetilde\phi$  vanishing, 
\begin{equation}
\phi_{\rm sol}= \phi_0 (z)=\frac{m}{2\lambda}\tanh \frac{mz}{2}\,,
\qquad
\widetilde\phi_{\rm sol}=0\,.
\label{N2sol}
\end{equation}
It satisfies supersymmetric boundary conditions in the finite box, 
which has the following form in terms of superfields $\Phi^a$:
\begin{equation}
D_1 \Phi^a(t, z=\pm L/2, \theta )=0\,,\qquad a=1,2\,.
\end{equation}
It is a straightforward generalization of the one-field case (\ref{boundary}).

The mode expansion is again based on operators $L_2=P^\dagger P$,
 $\tilde L_2=P P^\dagger$ where the operator 
 \begin{equation}
   P_{ab}=\delta_{ab}\, \partial_z -\partial_a \partial_b{\cal
W}(\phi=\phi_{\rm sol})
\label{pij}
 \end{equation}
now has a matrix form. The matrix is diagonal in our case,
\begin{equation}
 P_{ab}=\left(\begin{array}{cc}
\partial_z -2\lambda\phi_0(z)&0\\[2mm]
0&\partial_z +2\lambda\phi_0(z)+pm
\end{array}\right)\,.
\end{equation}
The zero modes for the fields $\phi$, $\psi$ are the same as in 
Sec.\,\ref{box}.  A new zero mode appears in the field $\widetilde\psi$,
\begin{equation}
\left(\widetilde\psi\right)_{\rm
zero~mode}=\xi\,N \,\frac{\exp\,(-p\, mz)}{\cosh^{2}\,(mz/2)}\,\left(
\begin{array}{l}
0\\[2mm]1
\end{array}\right)\,,
\label{ps2}
\end{equation}
where $\xi$ is the operator coefficient, $N$ is the normalization factor.
At $p=0$ it has the same functional form as the old fermionic
mode in $\psi$. This is not surprising  because  of  \ntwo supersymmetry
at $p=0$.  What is crucial is that the zero mode (\ref{ps2}) is not
lifted  even at nonvanishing $p$. This feature is  due to the Jackiw-Rebbi 
theorem.\cite{JR}
 
 Let us consider first $q=0$ and $|p|<1$; the second condition ensures
 localization of the zero mode (\ref{ps2}).  One boson and two fermion
 zero modes mean that we have two soliton states which are the BPS
 states (within our approximation). They form a reducible multiplet
 which preserves the fermion parity.  The fact that the multiplet is
 not short implies that its BPS nature can be lost. It could be
 demonstrated, for instance, by introducing a nonvanishing $q$.

Indeed, let us show that at $q\neq 0$ the one-loop anomaly makes $Q_2\neq 0$.
The anomalous part in $Q_2$ is
\begin{eqnarray}
Q_2&\!\!\!\!=\!\!\!\!&- \frac{1}{4\pi} \int\! {\rm d}z
\left[\frac{\partial}{\partial\phi_2 }\,
\Delta_\phi {\cal W}\right]
\left(\tilde\psi_2\right)_{\rm zero~mode}\nonumber\\[2mm]
&\!\!\!\!=\!\!\!\!&-\xi\,\frac{q\lambda\,N}{2\pi}\!\! \int\! {\rm d}z
\,\frac{\exp\,(-p\, mz)}{\cosh^{2}\,(mz/2)}=-\xi\,\frac{q\lambda}{\pi
m}\sqrt{\frac{3\,m\, p\,\pi}{2\,(1-4p^2) \tan\, (p\pi)}}
\;,
\end{eqnarray}
where $\xi$ is the second fermion modulus, $\xi^2=1/2$.
Correspondingly, the shift of the soliton mass from ${\cal Z}$ is
\begin{equation}
M-{\cal Z}=Q_2^2=\frac{3\,q^2\lambda^2}{4\pi^2 m}\,\frac{
p\,\pi}{(1-4p^2) \tan \,(p\pi)}\,.
\label{mz}
\end{equation}
Note the  absence of singularity at $p=1/2$.

Moreover, even at $q=0$ when there is no anomaly (and $Q_2$ remains
zero at one loop) we conjecture that a nonvanishing $Q_2$ is generated
by nonperturbative effects. If it is the case, $M-{\cal Z}\propto
\exp(-c/\lambda)$.

Now let us discuss what happens when $|p|\ge 1$.  It is clear that in
this interval the zero mode (\ref{ps2}) delocalizes: depending on the
sign of $p$ it runs to the left or right wall of the box. It becomes
non-normalizable in the limit $L \to \infty$; there is no
normalization problem at finite $L$, however.  If one considers the
entire system which includes the box, the supermultiplet continues to
be reducible even at $|p|\ge 1$, i.e., unprotected against leaving the
BPS bound.

However, physically we would like to limit ourselves to experiments
which are insensitive to the boundaries in the limit of large $L$.
Then we loose one state (associated with the boundaries) as well as
the fermion parity; the multiplet becomes short and BPS saturated at
$L \to \infty$.

In fact, at $|p|\gg 1$, when the mass of the second superfield is
large, this field can be viewed as an ultraviolet regulator for the
one-field model of Sec.\,\ref{sec:onesup}.  The two-field model
demonstrates that the short multiplets appear at a price of running
away from the soliton to the boundaries. The states which run away are
associated with the heavy (regulator) fields.



\section{Circle in the target and coordinate spaces}
\label{sub:circle}

In the model with Lagrangian (\ref{2lag}) with {\bf one} superfield
let us assume that the field $\phi$ lives on the circle $S^1$ of
circumference $2\pi v$. This implies that ${\cal W}'(\phi)$ is
periodic, with the period $2\pi v$. Moreover, we assume that the
spatial coordinate $z$ is also compact and defined on a circle $S^1$
of circumference $L$, i.e. the points $z$ and $z+L$ are identified.

As was shown in Ref.\,\protect\citebk{11}, the BPS saturated solitons are  
possible provided the superpotential ${\cal W}$ is a multivalued function 
such that ${\cal W}'$ is single-valued. Let us take, for instance
\beq
{\cal W}(\phi) = c\phi + w(\phi )\,,\qquad {\cal W}'(\phi)=c +w'(\phi )\,,
\eeq
where $w(\phi )$ is a $2\pi v$ periodic function and $c$ is an
appropriately chosen
numerical coefficient.  
The central charge will be equal to
$2\pi v c$.
As an example one can have in mind $w=mv^2
\sin (\phi/v)$.

The BPS equation
\vspace{-2mm}
\beq
\frac{{\rm d}\phi}{{\rm d}z} = {\cal W}'(\phi )
\eeq

\vspace{-2mm}
\noindent
has an implicit solution
\beq
\int_{\phi(0)}^{\phi(z)} \frac{{\rm d}\phi}{{\cal W}'(\phi )} = z\,. 
\label{aaa}
\eeq
The function ${\cal W}'(\phi )$ must be positive everywhere on the target
space circle. We choose the value of $\phi (0)$ such that ${\cal
W}'(\phi(0) )=
{\rm Max}\{{\cal W}'\}$, it puts the center of the soliton  at
$z=0$.

 The condition of periodicity
\beq
\int_0^{2\pi v} \frac{{\rm d}\phi}{{\cal W}'(\phi )} = L\,,
\label{mmm}
\eeq
fixes the value of $c$,
assuming that Eq.\,(\ref{mmm}) has a solution, which is a generic
situation. 
We  denote the solution $\phi_0 (z)$.

The mode expansion of $\phi - \phi_0$ and $\psi_{1,2}$
is performed in the eigenmodes of differential operators $L_2$ and
$\tilde L_2$,
in the same way it was done
 in the previous section. The only difference is in the
boundary conditions. Now, instead of Eq.\,(\ref{boundary}), we require
periodicity.
 In noncompact space 
the operator $L_2$ had a zero mode while
$\tilde L_2$ had no  zero mode. Now, 
in the compact space,  both have zero modes,
we denote them as $\chi_0$ for $L_2$ and $\tilde\chi_0$ for $\tilde L_2$,
\begin{eqnarray}
\chi_0&\propto &\exp\left\{\int_0^z {\cal W}''(\phi_0 (z)){\rm d}z\right\}
\propto \frac{{\rm d} \phi_0}{{\rm d} z} \propto {\cal W}'(\phi_0 )\,,
\nonumber\\[0.2cm]
\tilde \chi_0&\propto &\exp\left\{-\int_0^z {\cal W}''(\phi_0 (z))
{\rm d}z\right\}
 \propto \frac{1}{ {\cal W}'(\phi_0 )}\,.
\label{zmod}
\end{eqnarray}
Note that while the zero mode  $\chi_0$ (in $\phi$ and $\psi_2$ fields)
is localized
on the kink, the mode $\tilde \chi_0$, i.e. that of $\psi_1$ is localized
off the kink.
The zero mode balance is  the same as for nonzero modes: we have one
bosonic mode and two fermionic. 
Retaining only 
the zero modes we have the following
expansion  for the bosonic and fermionic fields:
\begin{equation}
\phi(z)=\phi_0 (z-z_0)\,,\quad
\psi_1=\xi\,\tilde\chi_0\,,\quad
 \psi_2=\eta\,\chi_0\,,
\end{equation}
where $\xi$ and $\eta$ are the fermion collective coordinates.
This leads to exactly the same supercharges as in Eq.\,(\ref{qq}). 
The difference lies in the quantization relations,
\begin{equation}
\left[\,p, z_0\right]= -i\,,\quad \eta^2=\xi^2=\frac 1 2\,,\quad
\{\eta,\xi\}=0\,.
\end{equation}
Due to $\{\eta,\xi\}=0$ the representation now is two-dimensional. 

In the leading approximation above both soliton states are BPS since
$Q_2=0$.  However, shortly we will show that already at the one-loop
level the supercharge $Q_2$ does not vanish. Thus, the long
(two-dimensional) multiplet is formed. The states are non-BPS, their
mass exceeds the central charge by a two-loop correction.

The easiest way to demonstrate the phenomenon is the explicit
calculation of 
$Q_2$ with account of the anomaly (\ref{anoma}),
\begin{equation}
Q_2=\xi \int {\rm d}z \left[\partial_z \phi_0 -{\cal W}'(\phi_0)-\frac{{\cal
W}'''(\phi_0)}{4\pi}\right] \tilde\chi_0(z)\,,
\end{equation}
where we substituted the classical soliton solution for $\phi$ and 
the zero modes for $\psi$ in the definition (\ref{supercur}). The zero 
mode of $\psi_2$ drops out from $Q_2$ at $p={\cal Z}\dot z_0=0$. 
The term ${\cal W}'''(\phi_0)/4\pi$ is due to the anomaly.  
On the classical solution the first two terms  in the square brackets
cancel each other, only the anomalous term survives. Thus, we see that 
$Q_2\neq 0$,
\begin{equation}
Q_2=-\frac{1}{4\pi}\,\xi\,
\left[\int\frac{{\rm d} \phi}{({\cal W}')^3}\right]^{-1/2}
\int{\rm d} \phi\, \frac{{\cal W}'''}{({\cal W}')^2}\,,
\end{equation}
where we used expression (\ref{zmod}) for the zero mode $\tilde \chi_0$.
It means that the excess of the soliton mass over the central charge is
\begin{equation}
M-{\cal Z}=Q_2^2=\frac{1}{32\pi^2}\,\left[\int\frac{{\rm d} \phi}{({\cal
W}')^3}\right]^{-1}
\left[\int{\rm d} \phi\, \frac{{\cal W}'''}{({\cal W}')^2}\right]^{2}\,.
\end{equation}

Note that 
taking account of the anomaly in the model of Sec.\,\ref{box} 
(in the box) does not lead to
nonvanishing $Q_2$ because of the absence of the zero mode in $\psi_2$. 
Its  effect on $Q_1$,
\begin{equation}
\Delta Q_1=\frac{1}{\sqrt{2}}\int {\rm d}z \left[\frac{{\cal
W}'''(\phi_0)}{4\pi}\right] \chi_0(z)=\frac{1}{\sqrt{{2\cal Z}}}\,
\frac{1}{4\pi}\big[{\cal
W}''(z\to\infty)-{\cal W}''(z\to-\infty)\big]\,,
\end{equation}
amounts to the shift in 
the classical value of ${\cal Z}$ (see Eq.\,(\ref{claz})) 
caused by the anomaly by virtue of the substitution
(\ref{anoma}).

\section{Nonflat  target space: \mathversion{bold} Tr$\,{Q}_{\,1}$ 
as  index of the Dirac operator on the reduced moduli space}
\label{sec:nonflat}
In this section we treat target spaces with nonflat metric.  Our
central point is to show that the index ${\rm Tr} \,Q_1$ introduced
above is, in fact, the index of a Dirac operator defined on the
soliton moduli space.  More exactly, the index defined in
Eq.\,(\ref{Ind}) coincides with the square of the index of a Dirac
operator on the reduced moduli space of solitons (see Sec.\,7.2 for
the definition).  The latter was studied by mathematicians.  Thus, it
is possible to determine in which ${\cal N}=1$ models ${\rm
  Ind}_{\,\cal Z}\,(Q_2/Q_1)=0$, i.e.  the multiplet shortening does
{\em not} take place (in the general situation).  In particular, the
index vanishes provided that the reduced moduli space is not a point,
i.e., its dimension nonvanishing, and is compact.  This happens, for
instance, in the following situation.  In terms of the superpotential
${\cal W}$ the soliton sweeps the interval 
$[{\cal W}(A),\,{\cal W}(B)]$. If there are no other critical points 
in this interval the reduced moduli space is compact.

A representative example of nonflat target space ${\cal T}$ is sphere
$S^{n+1}$ with a superpotential producing only two critical points
coinciding with the poles of the sphere. For instance, ${\cal W}=\cos
\xi $, where $\xi $ is a polar angle, does the job.  The case $n=0$
(the circle $S^1$) was considered in Sec.\,\ref{sec:onesup}; this is
the sine-Gordon model in which we observed short multiplets, i.e.,
${\rm Tr} \,Q_1\neq 0$. However, in the case at hand, when we deal
with a single field, the target space metric is necessarily flat.
Note, that for higher spheres, $n\ge 1$, the metric is necessarily
nonflat.

The first nonflat example is $S^2$.  In this case we deal with two
fermion moduli from the very beginning: hence, $(-1)^F$ is defined and
all representations are even-dimensional. The multiplet shortening
cannot occur. This is obviously true for all odd $n$. The first
example with an odd number of fermion moduli is $S^3$ (i.e. $n=2$),
from which we start.

\subsection{Superpotential on $S^{3}$ target space}
\label{sec:sigma}

In this section we consider solitons in the model with the sphere
$S^3$ as a target space and some specific superpotential. The target
space $S^3$ is symmetric so the theory contains only one running
coupling.  This example is of special interest for us because, as we
will see, it leads to an odd number of fermionic zero modes, similar
to the one-field model of Sec.\,\ref{box}.  We will show, however,
that unlike the one-field model of Sec.\,5.1, in the case of $S^3$
there will be no BPS solitons.

The generic form of the Lagrangian of the sigma model is given 
by Eq.\,(\ref{sigmaL}). The metric in this case is given by 
the following expression for the interval,
\begin{equation}
{\rm d} s_3^2=g_{ab}\,{\rm d}\phi^a {\rm d}\phi^b=\frac{1}{\lambda}\left[{\rm
d}\xi^2 +\sin^2\xi\left({\rm d}\theta^2 +\sin^2\theta {\rm
d}\varphi^2\right)\right]\,,
\end{equation}
where $\lambda$ is the coupling constant and we choose the  angle 
coordinates $\xi$, $\theta$ and $\varphi$ to parameterize $S^3$. 

 The superpotential is 
\begin{equation}
{\cal W}(\phi)=\frac{1}{2}\,M_0\cos \xi\,.
\end{equation}
The superpotential has maximum at $\xi=0$ and minimum at $\xi=\pi$ and no
other critical points.  The critical points are two vacua of the theory. 
Excitations at these vacua form three boson-fermion supermultiplets with 
the mass $\lambda M_0/2$.

In the classical approximation the model has a family of BPS solitons
interpolating between the maximum and the minimum.  The mass of the soliton is
given by the central charge (\ref{cch})
\begin{equation}
M_{\rm sol}={\cal Z}=\Delta {\cal W}=M_0\,.
\end{equation}
Its profile as a function of the coordinate $z$ is
determined by the BPS equations,
\begin{equation}
\frac{{\rm d} \phi^a_{\rm sol}}{{\rm d} z}=g^{ab}\,\partial_b \,{\cal
W}(\phi_{\rm sol})\,.
\end{equation}
 On the soliton trajectories
$\xi$ changes between 0 and $\pi$ at fixed $\theta$ and $\phi$,
\begin{eqnarray}
&&\xi_{\rm sol}(z) =2 \arctan\left(\exp\left[-\frac 1 2 \,\lambda M_0 \,
(z-z_0)\right]\,\right)\,,\nonumber\\[1mm]
&& \theta_{\rm sol}(z)=\theta_0\,,\quad \varphi_{\rm sol} (z)=\varphi_0\,,
\end{eqnarray}
so in
addition to the soliton center $z_0$ there are two  extra moduli,
$\theta_0$ and $\varphi_0$.  We denote the set of all three moduli by
$m^i=\{\theta_0,\varphi_0,z_0\}$ where $i=1,2,3$.

There is a fermionic partner $\eta^{i}$ to each bosonic modulus and the 
 fields  in the soliton sector are represented as 
\begin{eqnarray}
\label{fieldexp}
&&\!\!\!\!\!\phi^a(z, t)=\phi^a_{\rm sol}(z, \, m^i) +\mbox{nonzero modes}\,,
\qquad a=1,2,3\\[1mm]
&&\!\!\!\!\!\psi^a_1(z, t)=\mbox{nonzero
modes}\,,\quad\psi^a_2(z, t)=
 \eta^i \, \frac{\partial \phi^a_{\rm sol} }{\partial m^i}+\mbox{nonzero
modes}\,,\nonumber
\end{eqnarray}
where the time dependence enters through collective coordinates.
Equation (\ref{fieldexp}) implies that we are in the rest frame of the soliton.

Substituting these expressions in the Lagrangian (\ref{sigmaL}) and 
neglecting all nonzero modes we obtain the Lagrangian for dynamics of moduli,
\begin{equation}
L=
\frac{1}{2}\,h_{ij}(m) \left( \dot m^i \,\dot m^j +
i\,\eta^i \,{\cal D}_t\,\eta^j\right)-M_0\,,
\label{lmod}
\end{equation}
where the induced metric $h_{ij}(m)$ refers to the  $S^2\times R $
 geometry of the moduli space,
\begin{equation}
{\rm d} s_m^2= h_{ij}(m) \,{\rm d} m^i \,{\rm d} m^j=
 \frac{4}{\lambda^2\,M_0}\left[{\rm d}\theta_0^2 +\sin^2\!
\theta_0\, {\rm d}\varphi_0^2\right] + M_0\, {\rm d}z_0^2\,.
\label{meth}
\end{equation}
The coordinate on $R$ is $m^3=z_0$  and
$m^{1,2}=\{\theta_0,
\varphi_0\}$ are the angles on $S^2$.
Moreover,  the covariant derivative ${\cal D}_t$ is defined as
\begin{equation}
{\cal D}_t\,\eta^j=\dot \eta^j +\dot m^k \,\tilde\Gamma^j_{kl} \,\eta^l\,.
\end{equation}
This is  in correspondence with the field-theoric definition (\ref{covd}), but 
the   Christoffel symbols $\tilde\Gamma^j_{kl}$ refer, of course, to the moduli 
metric $h_{ij}$. We put the tilde to differentiate from the field theory
ones.

By the same token we get also expressions for the supercharges,
\begin{eqnarray}
&& Q_1=2{\cal Z}\, \eta^1\,,\label{q1}\\[2mm]
&& Q_2= h_{ij}\, \dot m^i \,\eta^j = \dot m_j \,\eta^j\,,\label{q2}
\end{eqnarray}
in terms of  bosonic and fermionic moduli where ${\cal Z}=M_0$.

The next step is to quantize the moduli dynamics.\footnote{~The
  procedure of quantization has a rich literature. In our presentation
  we follow the one given in Witten's lecture.\cite{WL} As far as
  technical details are concerned, we closely follow
  \mbox{Ref.\,\protect\citebkcap{Ali}} where they are thoroughly
  discussed.}  To this end one introduces the canonical momenta
conjugated to the coordinates and imposes commutation relations. The
crucial point is to establish the ordering of noncommuting operators.
The ordering is completely fixed by general covariance in the target
space and supersymmetry. Namely, unlike Hamiltonian, quantization of
the supercharge is uniquely defined.

 For the  bosonic coordinates $m^i$ one has
\begin{equation}
p_i =\frac{\partial L}{\partial \dot m^i}=h_{ij}\,\dot m^j +\frac{i}{2}\,\eta_j\,
\tilde\Gamma^j_{i\,l} \, \eta^l\,,\qquad [p_i\, , m^j]=-i\delta^j_i\,.
\label{comm}
\end{equation}
For the fermion coordinates $\eta^i$,
\begin{equation}
\zeta_i=\frac{\partial L}{\partial \dot \eta^i}=i\,h_{ij}\,\eta^j\,,
\qquad \{\zeta_i\, , \eta^j\}=i\,\delta^j_i\,,
\qquad \{\eta^i\, , \eta^j\}= h^{ij}\,.
\label{antico}
\end{equation}
A subtlety in this case is that the canonical momenta are function of
coordinates and are not independent. The validity of the
anticommutation relations (\ref{antico}) can be verified by
substituting expressions (\ref{fieldexp}) into field-theoretic
commutators. Alternatively, one can check them considering Green
functions in quantum mechanics (see, e.g.,
Ref.\,\protect\citebk{Ali}).

We can realize the algebra of the commutation relations (\ref{comm}), 
(\ref{antico}) in the Hilbert space of two-component spinor wave functions  
$\Psi_a (m)$ with the scalar product 
\begin{equation}
\langle\Phi\, | \,\Psi \rangle =\int\! \Pi {\rm d} m^i
\, \sqrt{h(m)}\; \Phi^\dagger (m) \, \Psi(m) 
\end{equation}
in the following way
\begin{eqnarray}
&& p_i=-i\,\delta_{ab}\, {h}^{-1/4}\,\frac{\partial}{\partial m^i}\,{h}^{1/4}\,,
\qquad (i=1,2,3)\,, \quad (a,b=1,2)\,,\nonumber\\[1mm]
&& \eta^i\equiv\frac{1}{\sqrt{2}}\,\sigma^i\equiv\frac{1}{\sqrt{2}}\,e^i_A
\left(\sigma^A\right)_{ab}\,,~~~
\qquad \qquad (A=1,2,3)\,, 
\label{modsigma}
\end{eqnarray}
where $\sigma^A$ are the Pauli matrices and we introduce frames $e^i_A$,
satisfying the conditions $e^i_A e^i_B \, \delta^{AB}=h^{ij}$. A possible 
choice for $e^i_A$ is
\begin{equation}
e^i_A={\rm diag} \left\{\frac{\lambda\sqrt{{\cal Z}}}{2}
\;,
\frac{\lambda\sqrt{{\cal Z}}}{2\, \sin \theta_0}\;,  
\frac{1}{\sqrt{{\cal Z}}}\right\}\,.
\end{equation}

In fact, what we need is the quantum version of the classical supercharges
presented in Eqs.\,(\ref{q1}),  (\ref{q2}),
\begin{eqnarray}
&& Q_1=2{\cal Z}\, \eta^1\,,\label{qq1}\\[1mm]
&& Q_2= \frac{1}{2}\left(\eta^i \,\pi_i +\pi_i\,\eta^i\right)\,,\label{qq2}
\end{eqnarray}
where the operator of covariant momentum $\pi_i$ (a quantum version of
the velocity operator $\dot m_i=h_{ij}\dot m^j$, see Eq.\,(\ref{comm})) is
defined as 
\begin{equation}
\pi_i=p_i -\frac{i}{4}\,\tilde\Gamma_{j,i\,l} \left[\eta^j,\eta^l\right]\,.
\label{pii}
\end{equation}
In fact, $\pi_i$ reduces to the covariant derivative on the spin manifold,
$\pi_i=-i\nabla_i\,$, the fermion term in Eq.\,(\ref{pii}) represent the spin
connection.  In terms of $\nabla_i$ the supercharge (\ref{qq2}) can be 
rewritten as
\begin{equation}
Q_2= \eta^j \; (-i \nabla_j)\,\equiv \frac{1}{\sqrt{2}}\,\sigma^j\,(-i\nabla_j)\,,
\qquad j=1,2\,.
\label{dirop}
\end{equation}
This is nothing but the Dirac operator $\not \!\! D$ on the manifold.

Let us stress that the dynamics of the moduli $m^3$, $\eta^3$ along
the $R$ direction is factored out, this is just a free motion of the
center of mass (together with its fermionic partner). In particular
$\pi^3\! \equiv p^3$ is conserved and we set it to zero by choosing
the rest frame. It means that the sum in Eqs.\,(\ref{qq2}),
(\ref{dirop}) for $Q_2$ runs only over the $S^2$ coordinates, $i=1,2$.
We will show below that the situation is general: the moduli space
always factorizes as ${\cal R}\otimes {\cal M}$.


The commutators 
\begin{eqnarray}
\left[\pi_i\,, m^j\right]=-i\delta^j_i\,, \qquad
\left[\pi_i\,, \pi_j\right]=-\frac 1 2\,\tilde R_{ij kl}\,\eta^k\,\eta^l\,,
\qquad \left[\pi_i\,, \eta^j\right]=i\,\tilde\Gamma^j_{i\,l} \,\eta^l
\end{eqnarray}
allow one to calculate the commutators of supercharges (\ref{qq1}), (\ref{qq2})
 with the coordinates $m^i$, $\eta^i$,
\begin{eqnarray}
&&\left[Q_1, m^i\right]=0\,,\qquad \left\{Q_1, \eta^1\right\}=2\,,\qquad
\left\{Q_1, \eta^{2,3}\right\}=0\,,\nonumber\\[1mm]
&&\left[Q_2, m^i\right]=-i\,\eta^i\,,\qquad 
\left\{Q_2, \eta^i\right\}=\frac{1}{2}\left(h^{ij}\,\pi_j+ 
\pi_j \,h^{ij}\right)\,.
\end{eqnarray}
These commutators match the classical supersymmetry transformations.

Finally, the algebra of the supercharges $Q_{1,2}$ is
\begin{eqnarray}
&& (Q_1)^2= 2\,{\cal Z}\,,\qquad \{Q_1,Q_2\}=0\label{hamilt1}\,,\\[2mm]
&& (Q_2)^2= H-{\cal Z}=\frac{1}{2}\,h^{-1/4}\,\pi_i\,h^{1/2}\, 
h^{ij}\,\pi_j\,h^{-1/4}
+\frac{1}{8}\,\tilde R\,.
\label{hamilt}
\end{eqnarray}
The expression (\ref{hamilt}) for $(Q_2)^2\!=(1/2)(\sigma^j\;i\nabla_j)^2$ 
is a particular case of the famous Lichnerowicz formula: the
first term can be written as $-h^{ij}\nabla_i\nabla_j/2$, i.e., it
represents the invariant Laplacian in application to spinors, and
$\tilde R$ denotes the scalar curvature for the moduli metric
$h_{ij}$.  In our example it is the curvature of the $S^2$ sphere,
\begin{equation}
\tilde R=\frac{\lambda^2}{2}\,{\cal Z}\,.
\end{equation}

Although our derivation was framed in terms of a concrete metric the
results (\ref{hamilt1}), (\ref{hamilt}) are perfectly general and can
be applied to $\sigma$ model on arbitrary manifold. The geometry of
the moduli space, i.e., $h_{ij}$, depends on both: the field-theoretic
metric $g_{ab}$ and on the form of the superpotential ${\cal W}$.

Let us return to our example. In this case the scalar curvature
$\tilde R$ is clearly a positive constant. This provides a positive
term in the Hamiltonian, the last term in Eq.\,(\ref{hamilt}). The
first term in $H-{\cal Z}$, which coincides (up to fermion terms) with
the invariant Laplacian, is positive definite by itself. Thus, there
can be no zero eigenvalues of $H-{\cal Z}$. In other words, there are
no states which are annihilated by the supercharge $Q_2$. The states
which were BPS saturated at the classical level cease to be BPS at the
quantum level.

It is not difficult to determine a complete spectrum of $H-{\cal Z}$ 
in the $S^2$ case. We limit ourselves to the lowest eigenvalue.
After some simple algebra we get 
\begin{equation}
\frac{4}{\tilde R} \,h^{-1/4}\,\pi_i\,h^{1/2}\, h^{ij}\,\pi _j\,h^{-1/4}
=-\frac{1}{\sin
\theta}\,\frac{\partial}{\partial\theta}\,\sin \theta
\,\frac{\partial}{\partial\theta}+ \frac{1}{\sin^2 \theta}\left(i\,
\frac{\partial}{\partial\varphi} +\frac{\sigma^3}{2} \cos\theta\right)^2 .
\nonumber
\end{equation}
The $\varphi$ dependence is just $\exp(i\,m\,\varphi)$ and for the ground state
$m=0$. The $\theta$ dependence is given by the associated Legendre functions
$P^{\pm1/2}_{1/2}$, so we get the lowest eigenvalue  equal to $3\tilde
R/16$  and doubly degenerate ground state,
\begin{eqnarray}
&&\Psi_{1/2}= \sqrt{\frac{\sin\theta}{\pi}}\left(\begin{array}{cc}1\\ 0
\end{array} \right)\,,\quad
\Psi_{-1/2}=
\sqrt{\frac{\sin\theta}{\pi}}\left(\begin{array}{cc}0\nonumber\\ 1
\end{array} \right)\,,\\[1mm]
&&\left(H-{\cal Z}\right)\Psi_{\pm1/2}= \frac{3}{4}\cdot \frac{\tilde
R}{4}\,\Psi_{\pm1/2}\,.
\end{eqnarray}


The moduli dynamics we have considered is a nice example of the 
${\cal N}\!=\!1/2$ supersymmetric quantum mechanics. The possibility of a
nontrivial ${\cal N}\!=\!1/2$ construction is due to the fact that the
interaction enters through kinetic terms rather than through
potential. The same moduli dynamics can be viewed as a theory based on
the Dirac operator defined on the curved moduli space.  Indeed, the
realization of $ H-{\cal Z}=(Q_2)^2$ by matrix-valued differential
operators in the moduli dynamics allows to interpreted the supercharge
$Q_2$ as the Dirac operator on the moduli manifold.

The Dirac operators on manifolds were extensively studied in the
mathematical literature. In particular, the absence of zero modes of
the Dirac operator on $S^n$ spheres (in our language the absence of
the BPS states) follows from the Lichnerowicz formula (\ref{hamilt})
as was mentioned above. More generally, one can introduce an index of
the Dirac operator $ \not\!\! D$ which counts the difference of left
and right chiral zero modes,
\begin{equation}
{\rm ind}\,(\not \!\! D)={\rm Tr} \left[\,\sigma^3 \exp\left(-\beta\not \!\! D^2
\right)\right]\,,
\label{index}
\end{equation}
where the matrix $\sigma^3$  anticommutes with the Dirac operator $\not \!\!
D$  defined by Eq.\,(\ref{dirop}).  For $S^2$ the Dirac operator has 
no zero modes at all, so the index vanishes. 

The matrix $\sigma^3$ in the definition (\ref{index}) is a realization of
$\gamma^5$ in our $S^2$ case. Moreover,  in the moduli dynamics $\sigma^3$
is a realization of the supercharge $Q_1=\sqrt{2{\cal Z}}\,\sigma^3 \,$, 
therefore the index can be rewritten  as
\begin{equation}
{\rm ind}\,(\not \!\! D)={\rm Tr} \left[\frac{Q_1}{\sqrt{2{\cal
Z}}}\,\exp\left(-\beta\,Q_2^2\right)\right]\,.
\end{equation}

\subsection{ Generic target space}

Let us pass now to the general case: ${\cal T}$ is an arbitrary
Riemann manifold endowed with a metric $g_{ab}(\phi )$ and a
superpotential ${\cal W}(\phi)$. The classical vacua are the critical
points of the superpotential, $\partial_a {\cal W}=0$. Classical BPS
solitons interpolate between vacua $A$ and $B$ and satisfy the first
order differential equations (\ref{AAone}).

We start from  briefly reviewing elements of the Morse theory
(see e.g. Refs.\,\protect\citebk{novem2}, \protect\citebk{novem3}).
For every critical point $A$ the Morse index of this point $\nu(A)$
is defined as the number of the negative eigenvalues in 
the matrix of the second derivatives 
\begin{equation}
H_{ab}(\phi)=\nabla_a \partial_b {\cal W}(\phi)
\end{equation}
at $\phi =A$. At the critical points 
the covariant derivative $\nabla_a$ coincides with the regular $\partial_a$.
For solitons interpolating between two critical points, $\phi =A$ at 
$z\to -\infty$
and $\phi =B$ at $z\to \infty$
one can determine the relative Morse index $\nu_{BA}$,
\begin{equation}
\nu_{BA} = \nu (B) - \nu (A)\,.
\end{equation}
This relative Morse index counts the difference between the numbers 
of the zero modes  of the operators $P$ and $P^\dagger$,
\begin{equation}
\nu_{BA}={\rm ker}\left\{P\right\}- {\rm ker}\left\{P^\dagger\right\}\,,
\end{equation}
where $P$ and $P^\dagger$ are 
\begin{equation}
P_{ab}= g_{ab} D_z - H_{ab}\,,\qquad P^\dagger_{ab}= -g_{ab} D_z - H_{ab}\,.
\end{equation}
Here $D_z$ is defined in Eq.\,(\ref{covd}), and the field $\phi$ is taken to be 
$\phi_{\rm sol}(z)$. 

Note that $\nu_{BA}=0$
in the \ntwo case and its small deformations. Indeed, due to the
harmonicity of ${\cal W}$ in this case, $\Delta_\phi {\cal W}=0$,
which leads to one negative eigenvalue in the matrix of the second
derivatives in each vacua (per pair of fields related by \ntwo).

For the BPS soliton, satisfying Eq.\,(\ref{AAone}), one zero mode
certainly present in $P$ is the translational mode. It corresponds to
the soliton center $z_0$, one of the coordinates in the soliton moduli
space. The same zero mode of $P$ is the fermion zero mode --- the
corresponding modulus $\eta$ is the superpartner of $z_0$.

We will limit ourselves to the case when $ {\rm ker}\,{P^\dagger}=0$.
(Note that even if that is not the case, one can get rid of the zero modes in
$P^\dagger$ by small deformations of the superpotential). Then, the Morse 
index 
\begin{equation}
\nu_{BA}\equiv n+1 \ge 1
\end{equation}
counts  the dimension of the soliton moduli space
${M}^{n+1}$. Thus, we arrive at quantum mechanics of $n+1$ bosonic
and $n+1$ fermionic moduli on ${M}^{n+1}$.

As was mentioned above, one of $n+1$ bosonic moduli is $z_0$, the
coordinate of the soliton center.  This is a cyclic coordinate
conjugated to the generator $P_z$ of the spatial translations, $z_0
\in {\cal R}$.  Note an ambiguity in $z_0$ --- one can add to $z_0$ an
arbitrary function of other moduli. This ambiguity is fixed by the
definition given below, see Eq.\,(\ref{mzero}).  Thus, the moduli space
${M}^{n+1}$ is a direct product
\begin{equation}
{M}^{n+1}={\cal R}\otimes {\cal M}^{n}
\label{RM}
\end{equation}
of ${\cal R}$ and the manifold ${\cal M}^{n}$ with coordinates  $m^1,...,m^{n}$
describing internal degrees of freedom of the soliton.  This manifold ${\cal
M}^{n}$  is what we call the {\em reduced moduli space}. 

It is instructive to elucidate the factorization (\ref{RM}) in more 
detail.\cite{LSV} We must show that the moduli space metric
$h_{ij}$,
\begin{equation}
h_{ij}(m) =\!\int\! {\rm d}z\, g_{ab}(\phi_{\rm sol})\,\,
\frac{\partial \phi_{\rm sol}^a}{\partial m^n}\,\,
\frac{\partial \phi_{\rm sol}^b}{\partial m^k}\,,\qquad i,j=0,1,2,...,n\,,
\label{Aone}
\end{equation}
where $m^0\equiv z_0$, has a block form,
i.e. $h_{0j}=0$ for $j=1,2,...,n$.
Indeed,
\begin{equation}
h_{0j}(m)\, =- \!\int\! {\rm d}z\,\partial_b {\cal W}(\phi_{\rm sol}) \, 
\frac{\partial\phi_{\rm sol}^b}{\partial m^j}
 \, \,
= - \frac{\partial}{\partial m^j}\!\int\!{\rm d}z\left[ \,{\cal
W}(\phi_{\rm sol})-{\cal W}(\phi_{\rm
sol})_{m=m_*}\right]\,,
\label{A2}
\end{equation}
where we use the fact that the soliton solution depends on
the spatial coordinate  only through 
the combination $z-z_0$, to replace
${\partial \phi_{\rm sol}^a}/{\partial m^0}$
by ${\partial \phi_{\rm sol}^a}/{\partial z}$,
which, in turn, can be replaced by 
$\, g^{ac}\,\partial_c \,{\cal W}(\phi_{\rm sol})$ by virtue of 
Eq.\,(\ref{AAone}).
We also regularized 
the integral on the right-hand side
of Eq.\,(\ref{A2})  by subtracting from the integrand
the superpotential at some fixed values of the moduli 
$m = m_*$. 

Considering Eq.\,(\ref{A2}) for $h_{00}$ we get 
\begin{equation}
h_{00}= - \frac{\partial}{\partial m^0}\!\int\!{\rm d}z\left[ \,{\cal
W}(\phi_{\rm sol})-{\cal W}(\phi_{\rm
sol})_{m=m_*}\right]\,.
\end{equation}
Having in mind $h_{00}={\cal Z}$ we define the modulus $m^0$ as  
\begin{equation}
m^0= -\frac{1}{{\cal Z}} \!\int\!{\rm d}z\left[ \,{\cal
W}(\phi_{\rm sol})-{\cal W}(\phi_{\rm
sol})_{m=m_*}\right]\,.
\label{mzero}
\end{equation}
With this definition it is clear that  
\begin{equation}
h_{0j} =\frac{\partial m_0}{\partial m^j}=0\,,\quad (j=1,...,n)\,.
\end{equation}

Thus, the Lagrangian 
describing the moduli dynamics has the form
\begin{equation}
L\left({ M}^{n +1}\,\right)=-{\cal Z}
+\frac{{\cal Z}}{2}\left[(\dot
z_0)^2 +i
\,\eta\,\dot\eta\,
\right] +
L\left({\cal M}^{n }\right)\,,
\label{Athree}
\end{equation}
where $L\left({\cal M}^{n }\right)$
is the Lagrangian of the internal moduli, both bosonic and fermionic,
a sigma-model quantum mechanics on ${\cal M}^{n }$. We see, that the motion of
the center of mass (together with its fermionic partner) is factored out, 
and we only need to consider the dynamics on ${\cal M}^{n}$.

The simplest case $n=0$ was already analyzed in Sec.\,5.1. In this case the
quantum moduli  dynamics is trivial, and the single state BPS multiplet does
exist, ${\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)=1$.  
For $n\ge1$ one must differentiate between even and odd $n$.  

For odd $n$ the total number $n+1$ of the fermion moduli is even.
Under quantization these $n+1$ moduli become $\gamma $ matrices
(multiplied by frames, as in Eq.\,(\ref{modsigma})) satisfying the
Clifford algebra with an even number of generators.  All $\gamma$'s
are multiplied by frames, as in Eq. (93) Taking the product of all
these $\gamma $ matrices we get the matrix $\gamma^{n+2}=\prod
\gamma^i$ which anticommutes with all $\gamma^i$, ($i=1,\ldots,n+1$).
This $\gamma^{n+2}$, an analog of $\gamma^5$ in four dimensions,
represents $(-1)^F$, i.e., all multiplets are long.

Consider now a less trivial case of even $n$ --- only in this category
can one expect to find Tr$\,Q_1\neq 0$.  Quantization of 
$L\left({\cal M}^{n}\right)$ is standard. All operators act in the Hilbert space
of the spinor wave functions $\Psi_\alpha (m)$, where
$\alpha=1,\ldots, 2^{n/2}$. The operators $m^i$ act as multiplication,
while $\dot m_i$ become matrix-differential operators.  The fermion
moduli of the reduced moduli space become $\gamma$ matrices of
dimension $2^{n/2}\times 2^{n/2}$.  The matrix
$\gamma^{n+1}=\prod_{i=1}^{i=n} \gamma^i$ is used to represent the
remaining fermion modulus, a partner of translation.  On the moduli
space ${\cal M}^{n}$ the supercharges (\ref{supercur}) take the form
\begin{equation}
Q_1=\sqrt{2{\cal Z}}\, \gamma^{n+1}\,,\qquad
 Q_2=-\frac{i}{\sqrt{2}}\;\gamma^j\,\nabla_j \,,
\label{qq22}
\end{equation}
where the covariant derivative $\nabla_j$ includes spin connection.
The expression for $Q_2$ is in fact the Dirac operator $i\!\!\not \!
\nabla$ on
${\cal M}^n$. Moreover, the Hamiltonian takes the form,
\begin{equation}
H-{\cal Z}=Q_2^2=\frac{1}{2}\,(i\!\!\not \!\nabla)^2=
-\frac{1}{2}\,\nabla^j\nabla_j+
\frac{1}{8}\,\tilde R\,,
\label{lich}
\end{equation}
where $\tilde R$ is the curvature in the soliton moduli space, and we
again used the Lichnerowicz formula (cf. Eq.\,(\ref{hamilt})).

From Eqs.\,(\ref{qq22}), (\ref{lich}) it is clear that the BPS soliton
states are in correspondence with the zero modes of the Dirac operator
$i\!\!\not \! \nabla$ on ${\cal M}^{n }$.  The index ${\rm
  Ind}_{\,\cal Z}\,(Q_2/Q_1)$ we defined in Eq.\,(\ref{Ind}) becomes
the square of the index of the Dirac operator
\begin{eqnarray}
&&{\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)=\left\{{\rm Ind}\,(i\!\!\not \!\nabla)_{{\cal
M}^{n}}\right\}^2, \nonumber\\[2mm]
&&
{\rm Ind}\,(i\!\!\not \! \nabla)_{{\cal M}^{n}}={\rm Tr}
\left[\,\gamma^{n+1}
\exp\left(\beta\!\!\not \! \nabla^2
\right)\right]_{{\cal M}^{n}}.
\label{IndDir}
\end{eqnarray}

Equation (\ref{lich}) shows that if the curvature $\tilde R$ is
positive everywhere on the soliton moduli space the Dirac operator has
no zero modes, its index vanishes, and so does the index ${\rm
  Ind}_{\,\cal Z}\,(Q_2/Q_1)$. Thus, there are no BPS solitons in this
case. An explicit example is provided by a sigma model on $S^3$.

Moreover, the situation turns out to be general in the hybrid models:
the geometry of the reduced moduli space is similar to spherical, the
index of the Dirac operator vanishes for any compact ${\cal M}^{n}$
with $n\ge 1$.  The proof due to P.~Pushkar' is presented in Appendix.

A comment is in order here concerning the vanishing of the index of
the Dirac operator on even-dimensional reduced moduli spaces
${\cal M}^{2\ell}$ (with integer $\ell$). Naively one might
be tempted to think that since all $\gamma$ matrices are traceless the 
index in Eq.\,(\ref{IndDir})
vanishes automatically, irrespective of the properties
of ${\cal M}^{2\ell}$. It is well known that this naive conclusion is
wrong --- a more careful consideration is necessary.

The moduli dynamics is governed by the Lagrangian
$$
L({\cal M}^{2\ell})=\frac{1}{2}\,h_{ij}(m)\left( \dot{m}^i \dot{m}^j 
+i\eta^i {\cal D}_t \eta^j\right)
$$
where the metric $h_{ij}$ is that for the reduced moduli space given
in Eq.\,(\ref{Aone}) with $i,j = 1,2,...,2\ell$.  As we have already
mentioned, after quantization the wave functions become spinors
$\Psi_{\alpha}(m)$, while $\eta$'s turn out to be $\gamma$ matrices
(times the frames).

The index of the Dirac operator is the regularized trace of
$\gamma^{2\ell+1}$ over the space of wave functions, see
Eq.\,(\ref{IndDir}). Naively, one might think that the space of the
wave functions is a product of the spinor representation of the
$2\ell$-dimensional Clifford algebra and the space ${\cal C}$ of
smooth functions on ${\cal M}^{2\ell}$.  If so, the trace of
$\gamma^{2\ell+1}$ over the space of the wave functions would be
automatically zero.

In fact, the space of wave functions is a product of the two spaces
mentioned above only locally!  The manifold ${\cal M}^{2\ell}$ should
be thought of as covered by open sets, and when we go from one open
set to another, generally speaking we need to rotate the spinor
representation with the ${\rm Spin}(2\ell)$-valued function of $m^i$.
This means that the wave functions $\Psi_{\alpha}(m)$ are sections of
the spinor bundle that is generically nontrivial. The index of the
Dirac operator is one of the characteristics that reflects this
nontriviality. It might be nonzero would ${\cal M}^{2\ell}$ be similar
to $CP_2$ or $K3$.  The central point of the Pushkar' theorem outlined
in Appendix is that the geometry of the soliton reduced moduli space
is similar to spherical and cannot be similar to that of $CP_2$ or
$K3$.

Thus, for $n\ge 1$ the soliton multiplets 
are long and generically non-BPS, $M>{\cal Z}$. If, for 
accidental or other reasons, they are still BPS saturated,  they form a 
reducible representation. For example, in \ntwo models the index  
${\rm Ind}_{\,\cal Z}\,(Q_2/Q_1)$ vanishes while the BPS states do exist. 
From the standpoint of \none
they form a reducible representation for which $(-1)^F$ is well defined.

Our  consideration refers to the case of compact ${\cal M}^{n}$. 
Generally speaking, ${\cal M}^{n}$ may be noncompact.
Noncompact geometry of ${\cal M}^{n}$  may emerge, for example,  
if there is an 
extra critical point $C$ such that
${\cal Z}_{AB}={\cal Z}_{AC}+{\cal Z}_{CB}$.
Physically it means that there is an infinite degeneracy of the quantized 
soliton states. 

\section{Conclusions}
\label{sec:conlusion}

We analyzed a wide class of hybrid models with \none supersymmetry and
central charges in (1+1) dimensions.  For the BPS states only one out
of two supercharges is realized nontrivially which leads to
one-dimensional irreducible representations of the superalgebra. The
non-BPS supermultiplets are two-dimensional.  Our main topic was the
soliton multiplet structure in various models at weak coupling
(quasiclassical approach).

We introduced and thoroughly discussed the index 
Tr$\,Q_1/\sqrt{2{\cal Z}}$ which counts the supershort (single-state) 
supermultiplets.
It was demonstrated that nonvanishing Tr$\,Q_1$ implies the loss of
the fermion parity $(-1)^F$. We showed that Tr$\,Q_1$ is related to
the index of the Dirac operator on the reduced moduli space. The
geometry of this space is similar to spherical. It implies that the
index vanishes except the very special case when the reduced moduli
space is a point.

The vanishing index implies long multiplets which may or may not be
BPS saturated. It is clear that the BPS saturation is not protected
for long multiplets. We demonstrated that indeed quantum corrections
destroy BPS saturation in many cases, by calculating the supercharge
$Q_2$ at one loop. Classically vanishing $Q_2$ becomes nonzero at one
loop.  This leads to $M-{\cal Z}\neq 0$ at two loops.  In special
cases, where $Q_2$ remains zero in perturbation theory, $Q_2 \neq 0$
may be generated nonperturbatively.\cite{Tonnis}

What lessons have we learned from the study of the \none  theories?
The main lesson is that of the fermion quantization in the case when the 
number of fermion zero modes is odd. Let us remind that 
the only consistent approach
to  fermions in field theory is based on Berezin's 
holomorphic quantization which implies the number of the fermion 
degrees of freedom is even. 

How can it be consistent with the fact that the BPS-saturated
irreducible representations of ${\cal N}=1$ centrally extended
superalgebra are one-dimensional?  The resolution is as follows: if
the theory is explicitly regularized both in the ultraviolet and
infrared it becomes a quantum mechanics of a large number of
variables; the number of the fermion variables is necessarily even,
the supermultiplets are reducible and $(-1)^F$ is preserved. Thus, in
fully regularized theory the number of the fermion zero modes is
always even, and so is the number of states in the supermultiplet.
Moreover, the BPS saturation is not protected so we can say that there
are no BPS states in the fully regularized theory.

We gave clear-cut illustrations to this point.  In
Sec.~\ref{sub:circle} the spatial dimension was compactified onto a
circle. The soliton had two zero modes -- one localized on the
soliton, another on the other side of the circle.  Another example is
given by the softly broken \ntwo theory, discussed in
Sec.~\ref{sec:2sf}.  There we could see how the second zero mode
becomes delocalized once the ${\cal N}=2$ breaking parameter becomes
large enough (more exactly the localization of the second zero mode
shifts away from the soliton, to the edge of the box).

In spite of the ``evenness'' of the total number of the fermion modes,
in the limit $L\to\infty$ it may (and actually does) happen that in
the physical subsector of the Hilbert space there remains an odd
number of the fermion moduli.  If some of the modes are localized at
the boundaries of the large box, the corresponding states are
unobservable in any physical local measurement.  What is observable
are the localized states associated with the soliton. If the number of
fermion modes localized on the soliton is odd, we arrive to an
abnormal situation.  An explicit example was given in Sec. 5.  In this
case we get the multiplet shortening, and the BPS saturation is
implemented.  The number of such short multiplets in the physical
subsector is counted by Tr$\,Q_1/\sqrt{2{\cal Z}}$.

Our results 
naturally ``blend in" into a general picture.  The BPS saturation was
studied in detail in the \ntwo ~theories in (1+1) dimensions\,\cite{Ntwo}
and in the \ntwo\!\!, \nfour ~theories in (3+1)-dimensions.  
The assertion that under the full regularization (UV and IR) there are no 
short multiplets is general, it is applicable to higher dimensions and 
higher supersymmetries. Extra states are localized away from the soliton 
center. What is specific for \none in two dimensions, where the number of 
supercharges is minimal, is the odd number of the soliton fermion moduli 
(in higher dimension it is always even). This may lead to the loss of
$(-1)^F$ in the physical sector.

In a broader context, we found another example of a remarkable
phenomenon first discussed by Witten\,\cite{Ed} -- supersymmetry
without the full fermion-boson degeneracy.  If such theories could be
found in four dimensions, this would be ``a dream came true."

Witten's example is in the context of 2+1 supergravity with the conic
geometry.  Out of four supercharges of the model two supercharges
annihilate the BPS solitons. The other two supercharges produce the
fermion zero modes. Without gravity these modes are normalizable which
leads to two-component short supermultiplet of \ntwo. With gravity
switched on the fermion modes become non-normalizable, implying the
single-state supermultiplet.  This means that in the physical sector
of the localized states all supercharges act on the soliton trivially.

In our \none examples of the single-state supermultiplet one of two
supercharges is realized nontrivially, $Q_1=\pm\sqrt{2{\cal Z}}$.  In
terms of modes there is one normalizable fermion mode.  In Witten's
case all fermion zero modes run away to the boundary, while in our
case one mode is localized on the soliton, and the other at the
boundary.  Similar run-away behavior of the modes occurs in the
phenomenon of the fractional charge and other phenomena known in solid
state physics.


\vspace{-3mm}
\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

Our study was stimulated by a comment made by Ken Intriligator who
noted, in
response to Ref.\,\protect\citebk{1}, that there was no doubling of short
multiplets in
\ntwo ~in 1+1. We are grateful to him for this remark. We would like to
thank 
 Paul Fendley, Leonid Glazman, Fred Goldhaber, Anatoly Larkin,
Bob Laughlin, Peter van Nieuwenhuizen, Pyotr Pushkar',  Adam Ritz, Andrei
Smilga,
 Boris Shklovsky, Alexei Tsvelik,  Misha Voloshin, Edward Witten, and Alexander
Zamolodchikov for  valuable discussions.

The work of A.L. was supported in part by RFFI grant 00-02-16530,
Support for Scientific Schools grant 00-15-96-557,  and INTAS grant
99-590, the work of M.S. and A.V. was supported in part by  DOE  grant 
DE-FG02-94ER408.

\vspace*{-3mm}
\section*{Appendix (by P.~Pushkar'): Vanishing of the index of the Dirac 
operator on compact reduced moduli space}
\label{sec:append}
\addcontentsline{toc}{section}{\numberline{}Appendix (by P.~Pushkar'): 
Vanishing of the index of the Dirac operator on compact reduced moduli space}

The proof presented in this Appendix belongs to  P.~Pushkar'.

The index of the Dirac operator on the reduced moduli space ${\cal M}^{n}$ 
is known to be equal to
$\hat{A}$-genus and can be expressed as an
integral of polynomial of Pontriagin classes
along ${\cal M}^{n}$.
In order to show that the index vanishes we will show (in {\em Statement 3})
 that tangent bundle to
the space ${\cal M}^{n}$ of nonparametrized trajectories
is such, that its sum with the trivial bundle is a trivial bundle.
Really, then its Pontriagin classes (of nonzero degree) should vanish.

Here we assume that there are no other critical values between values
of the initial and final critical points of superpotential, thus,
the space ${\cal M}^{n}$ is compact.
We will also assume that dimension of ${\cal M}^{n}$ is nonzero.
Let us take the equi(super)potential surface $L$ defined by ${\cal W}=c$ 
where the constant  $c$ is between the values of ${\cal W}$
at the initial and final points.
The gradient trajectories coming out of the initial point intersect the 
surface $L$
and produce a sphere $S^{\rm initial}$. The antigradient trajectories coming out
of the final point also intersect $L$ and produce another sphere 
$S^{\rm final}$.
The space ${\cal M}^{n}$ is an intersection of these two spheres.
\\[2mm]
{\em Statement 1:} The normal bundles of $S^{\rm initial}$ and 
$S^{\rm final}$  in $L$ are trivial.
\\[2mm]
For instance, to show this for $S^{\rm initial}$
let us move $L$ close to the initial critical
point (it would be a homotopy that should not change the triviality
of the bundle).  In the vicinity of the critical point we can
replace the function by its quadratic approximation -- then the
triviality of the normal bundle becomes obvious.
\\[2mm]
{\em Statement 2:} The normal bundle to ${\cal M}^{n}$ in $S^{\rm initial}$ 
is trivial.
\\[2mm]
Indeed, this bundle is a restriction
(to ${\cal M}^{n}$)  of the normal bundle of $S^{\rm final}$
in $L$, and the latter is trivial due to {\em Statement 1}.
Since the restriction of the trivial bundle is trivial, we
have proved {\em Statement 2}.
\\[2mm]
{\em Statement 3:} The tangent bundle to ${\cal M}^{n}$ plus the trivial bundle
is a trivial bundle.
\\[2mm]
Suppose that ${\cal M}^{n}$ is different from the total sphere 
$S^{\rm initial}$,
then it is a submanifold of the Euclidean space.
The tangent bundle to ${\cal M}^{n}$ plus the normal bundle
(that is trivial due to {\em Statement 2}) gives a restriction
on ${\cal M}^{n}$ of the tangent
bundle to the Euclidean space (that is obviously trivial).
Since the restriction of the trivial bundle is trivial we have proved
{\em Statement 3} for ${\cal M}^{n} \neq S^{\rm initial}$.
If ${\cal M}^{n}$ is a sphere $S^{\rm initial}$, then it could be obviously 
embedded in
the Euclidean space with the trivial normal bundle. This completes the proof
of \mbox{{\em Statement 3}}.

The generation function for Pontriagin classes for a trivial bundle
is a class of degree zero.
The generation function for Pontriagin classes of the sum of
bundles is the product of the generation functions of each bundle.
Thus, the only nonvanishing Pontriagin class on ${\cal M}^{n}$ is of degree
zero,  and the integral of the polynomial of Pontriagin classes
along ${\cal M}^{n}$ will give zero.

\vspace*{-2mm}
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\addcontentsline{toc}{section}{\numberline{}References}

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\end{document}



