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\begin{document}
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\begin{titlepage}
\noindent



\vskip 3.0cm

\begin{center}

%{\Large\bf Composite Systems with Superconformal  }\\
{\Large\bf Many--Body Superconformal Systems  }\\
\bigskip

%{\Large\bf Symmetry from Hamiltonian Reductions }\\
{\Large\bf from Hamiltonian Reductions }\\
\bigskip

\vskip 1cm

%{\Large
{\large Stefano Bellucci${}^{a,}$}\footnote{bellucci@lnf.infn.it},
{\large Anton Galajinsky${}^{a,b,}$}\footnote{galajin@mph.phtd.tpu.edu.ru}
and {\large Sergey Krivonos${}^{c,}$}\footnote{krivonos@thsun1.jinr.ru}


\vskip 1.0cm

{\it ${}^a$INFN--Laboratori Nazionali di Frascati, C.P. 13,
00044 Frascati, Italy}\\

\vskip 0.2cm
{\it ${}^b$Laboratory of Mathematical Physics, Tomsk Polytechnic University, \\
634050 Tomsk, Lenin Ave. 30, Russian Federation}

\vskip 0.2cm

{\it ${}^c$ Bogoliubov Laboratory of Theoretical Physics, JINR, \\
141980 Dubna, Moscow Region, Russian Federation}

\end{center}

\vskip 1cm

\begin{abstract}
We propose a new variant of the Hamiltonian reduction for constructing  $n$--particle (super)conformal
theories in one dimension starting from a direct sum of $(n-1)n/2+1$ (super)conformal
mechanics.  As an application we explicitly build
$N=4$ superconformal Calogero--type models both in superfields and in components.
\end{abstract}

\vspace{0.5cm}

PACS: 04.60.Ds; 11.30.Pb\\
Keywords: $N=4$ superconformal many--body systems, Hamiltonian reduction

\end{titlepage}
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\vspace{0.5cm}

\noindent
{\bf 1. Introduction}\\

Systems describing pairwise interactions on $n$ particles in one
dimension are important for several reasons. At the classical
level they provide interesting examples of integrable models (see
Ref. \cite{per} for a review) and exhibit intriguing relations with
semisimple Lie algebras \cite{per} and Hamiltonian reductions of
$2d$ Yang--Mills theory \cite{gor}. At the quantum level they turn out
to be completely solvable (see e.g. Refs. \cite{op2},\cite{vas} and references
therein).

Apart from the mathematical concern in the chain of
one--dimensional many--body systems, some of them admit
interesting physical applications. It has been known for a long
time that the $n$--particle Calogero model \cite{cal} characterized by the
potential $\sum_{i<j}\frac{g}{{(x_i-x_j)}^2}$, with $g$ being a
dimensionless coupling constant, exhibits conformal invariance
\cite{reg}. Exploiting the latter, Gibbons and Townsend argued
recently \cite{gib} that an $N=4$ superconformal extension of the
Calogero model may be relevant for a microscopic description of
the extremal Reissner--Nordstr\"om black hole, at least near the
horizon. Unfortunately, beyond the two--particle case \cite{gal}
no such model has been constructed so far. Generally, one works
within the framework of supersymmetric (quantum) mechanics, where
an attempt to incorporate superconformal generators leads one to
contradictory equations \cite{wyl}. Surprisingly enough, in $d=2$
the situation turns out to be simpler \cite{ghosh} and an $N=4$
superconformal Calogero model can be constructed. However, a naive
dimensional reduction to one dimension breaks $N=4$ down to $N=2$.
It is worth mentioning also the approach of Ref.~\cite{dprt}, where an
$N=4$ supersymmetric multidimensional quantum mechanics involving
an arbitrary potential has been constructed. However, no specific
choice for the potential that would reproduce the Calogero model
has been presented.

In the present letter we suggest a simple method for constructing
variant $N=4$ superconformal Calogero--type models
out of a composite system containing one free particle (describing the center
of mass) and $\frac{n(n-1)}{2}$ copies of the $N=4$ superconformal
mechanics~\cite{ikl,ikL}. The approach appeals to a
specific Hamiltonian reduction
where one imposes constrains on an original (simple)
model in order to generate a nontrivial potential for a resulting
reduced system. As will be discussed in detail below, the main
point is not to destroy the standard structure of a kinetic term
when implementing the reduction.

In the next section we illustrate the method and show how the
reduction works for the purely bosonic Calogero model. In Sect. 3
we turn to treat the supersymmetric case and reproduce the $N=2$
superconformal extension constructed earlier by Freedman and Mende
in \cite{FM}. Sect. 4 is split into three subsections related
to three different $N=4$ off--shell superconformal multiplets known.
In particular, we construct an $N=4$ superconformal
extension for a complexification of the Calogero model, build a
many--body generalization of the $N=4$ superconformal mechanics of
Ref. \cite{ikL} and discuss problems related to an $N=4$
superconformal extension of the real Calogero model. Throughout
the paper, we choose to work in superfields.




\vspace{0.5cm}

\noindent
{\bf 2. The Calogero model from multidimensional conformal mechanics}

\vspace{0.5cm}
Before treating in detail the entire superconformal case, it is worth illustrating how
the reduction works for bosonic systems. Our goal in this section will be to get the Calogero model
with the action functional
\be\label{Cal1}
S=\int dt \left(\sum_{i=1}^n \frac{{\dot q}_i^2}{2}
-\sum_{i<j}\frac{g}{{(q_i-q_j)}^2}\right),
\ee
starting from a multidimensional conformal mechanics.

The idea which underlies our construction is to start from a model which is the sum of $n(n-1)/2$
conformal mechanics \cite{aff}
\be\label{cm0}
S_0=\int dt  \sum_{i<j}\left( \frac{1}{2} {\dot x_{ij}}{\dot x_{ij}} -\frac{\hat g}{x_{ij}^2} \right)\;, \quad x_{ij}=-x_{ji}
 \;,\; i,j =1,\ldots,n \;.
\ee
The latter action involves the same number of interaction terms as that in \p{Cal1} but, for the time being, all of them
are independent. In order to bring these terms to the form familiar form the Calogero model, let us impose the
constraints\footnote{Despite of the fact that $n$ variables $q_i$
explicitly appear in the constraints, only $(n-1)$ variables $x_{ij}$ in \p{cm1} prove to be linearly
independent.}
\be\label{cm1}
x_{ij}=\frac{1}{\sqrt{n}}\left( q_i-q_j \right) \; ,
\ee
which reduce the number of independent degrees of freedom from $n(n-1)/2$ down to $(n-1)$.  After substituting \p{cm1} into the
action \p{cm0} we get the proper interaction terms, provided the identification
\be
g=n{\hat g} \;.
\ee
However, the kinetic term is non-standard
\be\label{cm2}
S_{kin}=\int dt  \sum_{i<j} \frac{1}{2n} (\dot q_{i} - \dot q_{j} )^2 \;.
\ee
In order to bring this to a diagonal form, let us add to the  action \p{cm0} an extra term
which describes a free motion of the center of mass
\be\label{cm3}
S_1= S_0 +\int dt \frac{{\dot x_0}^2}{2} \;,
\ee
where
\be\label{cm1a}
 x_0=\frac{1}{\sqrt n} \sum_i q_i \;.
\ee
With such an amendment the action \p{cm3} coincides with that of the Calogero model.
Thus, we derived the $n$-particle Calogero model starting from $n(n-1)/2+1$ copies of the conformal
mechanics (a free particle and $n(n-1)/2$ copies with the same coupling constant).

Being a composite system, the action \p{cm3} exhibits
invariance which is the direct product of $n(n-1)/2+1$ $SO(1,2)$ groups.
The constraints \p{cm1},\p{cm1a} which specify the reduction
hold invariant with respect to the "diagonal" $SO(1,2)$ transformations realized as follows
\be\label{cm4}
\delta t = a+ bt+ct^2 \equiv f(t)\;,\quad \delta q_i =\frac{1}{2} \dot f  q_i \;.
\ee
This residual symmetry is just the conformal symmetry of the Calogero model.

Worth mentioning also is that the reduction scheme outlined above can be put in rigorous
Hamiltonian terms. For example, for the three--particle case it suffices to impose only one  constraint
$x_{12}+x_{23}+x_{31}=0$. This can be added to the original action \p{cm3} with a Lagrange multiplier $\pi$.
Applying the Dirac procedure, one finds a constraint system which, along with the former one, involves also
$p_{12}+p_{23}+p_{31}=0$, $p_\pi=0$ and $\pi+\sum_{i<j} \frac{2g}{3{(x_{ij})}^3}=0$. Here $p_\pi$ and $p_{ij}$
stand for momenta canonically conjugate to the variables $\pi$ and $x_{ij}$, respectively. The entire constraint system
proves to be second class. After introducing the Dirac bracket the variables $(\pi,p_\pi)$ and, say $(x_{13},p_{13})$, can
be eliminated from the consideration, brackets for the remaining variables having a non--canonical form. A passage to
the new variables $q_i$, $x_0=\frac{1}{\sqrt{3}}(q_1+q_2+q_3)$, $x_{ij}=\frac{1}{\sqrt{3}}(q_i-q_j)$  (and the same for the momenta)
proves to diagonalize the bracket and leads one to the three--particle Calogero model.

Obviously, the construction is general enough and can be applied to any $n$--particle system with a potential of the form
$\sum_{i<j} V(q_i-q_j)$. In order to get the action for such a system,
one should start with $n(n-1)/2$ copies of one-particle models involving the
potential $V(x)$ and a free particle (the center of mass). After implementing the reduction \p{cm1}, \p{cm1a}
one generates the $n$-particle system with the potential $\sum_{i<j} V(q_i-q_j)$.
For example, the $n$--particle Calogero model with the harmonic oscillator potential
\be\label{harmcal}
S=\int dt \left(\sum_{i=1}^n \frac{{\dot q}_i^2}{2} -\sum_{i<j} \o^2 {(q_i-q_j)}^2
-\sum_{i<j}\frac{g}{{(q_i-q_j)}^2}\right),
\ee
can be obtained from a composite system which contains a free particle and $\frac{n(n-1)}{2}$ copies of the
conformal mechanics
modified by the inclusion of the harmonic oscillator potential
\be
S=\int dt \left(\frac{{{\dot x}_0}^2}{2} +  \sum_{i=1}^{n(n-1)/2}( \frac{{\dot x_i}^2}{2}  -\frac{g/n}{x_i^2} -n\o^2 x_i^2) \right).
\ee
One should stress that all symmetries of the one-particle system which are realized {\it homogeneously} and {\it linearly} on the
coordinate $x(t)$, will hold in the resulting $n$-particle system because they preserve the reduction constraints.


The fact that one can treat the $n$--particle Calogero model as a specific Hamiltonian reduction of a simpler
composite system is not just a matter of aesthetics. It is well known that the conformal mechanics
can be endowed with $N=2$~\cite{ap,FR} and
$N=4$ supersymmetries~\cite{FR,ikl}.
The trick described above suggests a quite new and intriguing possibility to
construct a superconformal generalization of the $n$--particle Calogero model. It suffices to
start with a composite system involving a superconformal extension of a free particle and $\frac{n(n-1)}{2}$
copies of the superconformal mechanics~\cite{ikl} and apply an appropriate Hamiltonian reduction.
In the next section we treat in detail the $N=2$ case. $N=4$ models are discussed in Sect. 4.

\vspace{0.5cm}

\noindent
{\bf 3. The $N=2$ superconformal  Calogero model}

\vspace{0.5cm}

We next wonder how to incorporate $N=2$ supersymmetry in the reduction scheme outlined above.
As we already mentioned, in order to preserve symmetries characterizing one--particle
constituents of a composite system, one should apply the reduction
to coordinates which transform homogeneously and linearly. For the supersymmetric case this means
that one is forced to work with superfields because in many cases after elimination of
auxiliary fields supersymmetry transformations become nonlinear.
Thus, the starting point is the $N=2$ superconformal mechanics \cite{ap,ikl}, whose superfield action
reads
\be\label{n2scm}
S_0=\frac{1}{2}\int dt d\theta d\bar\theta \left( D X \Db X -2 g \log | X | \right) \;.
\ee
Here $X(t,\theta,\tb)$ is a general bosonic $N=2$ real superfield and the covariant derivatives are defined as
\be
D=\frac{\partial}{\partial \theta}+i\tb \frac{\partial}{\partial t}\;,\;
\Db=\frac{\partial}{\partial \tb}+i\theta \frac{\partial}{\partial t}\;,\;
\left\{ D,\Db\right\}=2i\frac{\partial}{\partial t} \;.
\ee
The action \p{n2scm} is invariant with respect to the following transformations
\be\label{n2tr}
\delta t=E-\frac{1}{2}\tb\Db E-\frac{1}{2}\theta D E\;,\;\delta\theta=-\frac{i}{2}\Db E\;,\;
\delta\tb = -\frac{i}{2} D E\;,\;\delta X= \frac{1}{2}{\dot E}X\;,
\ee
where the superfunction $E(t,\theta,\tb )$ collects all the infinitesimal parameters of the $N=2,d=1$
superconformal group
\be
E(t,\theta,\bar\theta ) = f(t)-2i\left(\varepsilon+\beta t\right) \tb -2i\left(\bar\varepsilon + \bar\beta t\right)\theta
+\theta\tb h \;.
\ee
Here $f(t)$ is the same as in \p{cm4}, $h$ is a $U(1)$ rotation parameter, and
$\varepsilon$ and $\beta$ correspond to Poincar\'e and conformal supersymmetries, respectively.

In order to construct an $N=2$ superconformal $n$-particle Calogero model, let us consider a composite
system which involves an $N=2$ superconformal extension of a free particle action
and $n(n-1)/2$ copies of the $N=2$ superconformal mechanics
\be\label{n2cal1}
S=\frac{1}{2}\int dt d\theta d\bar\theta\left[ D X_0\Db X_0 +\sum_{i<j} \left( D X_{ij} \Db X_{ij} -2 g \log |X_{ij}| \right)\right] \;,
\ee
where
\be
X_{ij}=-X_{ji} \;,\quad  i,j =1,2,\ldots , n.
\ee
Much alike the bosonic case, we impose the reduction constraints which express $n(n-1)/2+1$ original
superfields $X_0,X_{ij}$ in terms of $n$ independent new ones $V_i$
\be\label{n2hamred}
X_0=\frac{1}{\sqrt{n}}\sum_i V_i \;\quad X_{ij}=\frac{1}{\sqrt{n}} \left( V_i-V_j \right) \;.
\ee
Upon substitution of \p{n2hamred} into \p{n2cal1} one gets the action of an $N=2$ superconformal $n$-particle
Calogero model
\be\label{n2cal2}
S=\frac{1}{2}\int dt d\theta d\bar\theta \left( \sum_{i=1}^n D V_{i} \Db V_{i} -2 g \sum_{i<j} \log |V_i-V_j| \right) .
\ee
Eliminating $n$ auxiliary fields $F_i=\left[\Db,D\right]V_i|_{\theta=0}$ with the use of their equations of motion
one can rewrite the action functional \p{n2cal2} in terms of the physical degrees of freedom
%$\left\{ q_i, \psi_i,\bar\psi_i\right\}$
\be\label{n2cal3}
S=\frac{1}{2}\int dt \left[ \sum_{i=1}^n ({\dot{q_{i}}}^2 -i\bar\psi_i{\dot\psi_i} +
 i\dot{\bar\psi_i}\psi_i) - \sum_{i<j} \frac{g^2-g(\psi_i-\psi_j)(\bar\psi_i-\bar\psi_j)}{(q_i-q_j)^2} \right] ,
\ee
where
\be
q_i={V_i}|_{\theta=0}\;,\quad \psi_i=iD {V_i}|_{\theta=0} \; \quad {\bar\psi}_i=i\Db {V_i}|_{\theta=0}.
\ee
Being reformulated in Hamiltonian terms, the latter theory proves to coincide with that obtained by
Freedman and Mende in the framework of many--body supersymmetric quantum mechanics \cite{FM}.

Two comments are in order. Firstly, since each of the superfields $X_0,X_{ij}$ transform linearly and homogeneously
with respect to the proper $SU(1,1|1)$ transformations, the constraints \p{n2hamred}
hold invariant under the residual $SU(1,1|1)$ group acting in a uniform way on all the superfields
\be\label{n2tr1}
\delta V_i =\frac{1}{2}{\dot E} V_i \;.
\ee
Thus, the actions \p{n2cal2}, \p{n2cal3} possess $N=2$ superconformal symmetry by construction. Secondly,
one can employ the supersymmetric reduction, in order to build supersymmetric many--body systems with a potential of
a more general form $\sum_{i<j} \Phi(V_i-V_j)$. It suffices to start with a proper number of one-particle
systems and perform the Hamiltonian reduction \p{n2hamred} in that framework. All symmetries which
are realized linearly and homogeneously will survive the reduction.

\vspace{0.5cm}

\noindent
{\bf 4. Calogero models with $N=4$ superconformal symmetry}

\vspace{0.5cm}
The situation becomes more complicated when one tries to extend the analysis, in order to incorporate
$N=4$ superconformal symmetry. First of all, let us remind that the $N=4$ superconformal group in $d=1$ is the exceptional
supergroup $D(2,1|\alpha)$ \cite{FrSorba}. Only for specific values of the parameter, namely $\alpha=-1,0$ with the notation
from Ref. \cite{bils}, this algebra
is isomorphic to $su(1,1|2)\times su(2)$. Generally, only the $su(1,1|2)$ factor is taken into account,
while the $su(2)$ subalgebra is kept explicitly broken. Secondly, three different
$N=4$ off-shell supermultiplets are known, which can be used for constructing an $N=4$ superconformal Calogero model.
The first multiplet contains one real scalar, four real fermions and three real auxiliary fields \cite{ikl}.
We shall call this the {\it 4a} multiplet. The second multiplet involves
one complex scalar, four real fermions and one complex auxiliary scalar \cite{ap,FR}. We will call this the {\it 4b} multiplet.
Finally, it was recognized recently that
a multiplet with three real scalars, four real fermions and one real auxiliary field \cite{IS,BP,MSS}
 underlies
a new version \cite{ikL} of $N=4$ superconformal mechanics (for the case of a vanishing coupling constant, see also Ref.~\cite{bgik}).
 We will call the latter multiplet the $4c$ multiplet.
Thus, taking different multiplets one may expect to be able to construct at least three different versions of the
$N=4$ superconformal Calogero model. Finally, it is important to stress that in all the cases
basic $N=4$ superfields are subject to constraints. Therefore, one should check that the Hamiltonian reduction is
compatible with such conditions. Keeping all this in mind, we proceed to construct
$N=4$ superconformal extensions of the $n$-particle Calogero model.

\vspace{0.5cm}

\noindent
{\bf 4.1 The {\it 4b} multiplet}

\vspace{0.5cm}

We start with the {\it 4b} multiplet, this being the simplest multiplet to handle.
The superfield action of the $N=4$ superconformal mechanics for this multiplet reads \cite{ap,FR,ikl}
\be\label{4baction}
S=\frac{1}{2} \left( \int dt d^4\theta Y{\bar Y} -g\int dt d^2\theta \log |Y| -g \int dt d^2 \tb \log |{\bar Y}|\right) \;,
\ee
where the $N=4$ superfields $Y,{\bar Y}$ are constrained to obey the chirality conditions
\be
\Db^a Y= D_a {\bar Y}=0\;.
\ee
The covariant derivatives entering the equations above read
\be
D_a=\frac{\partial}{\partial \theta^a}+i\tb_a\frac{\partial}{\partial t}\;,\;
\Db^a=\frac{\partial}{\partial \tb_a}-i\theta^a \frac{\partial}{\partial t}\;,\quad a=1,2 \;.
\ee
The action \p{4baction} is invariant with respect to the $SU(1,1|2)$ transformations
\be\label{4btr}
\delta t=E-\frac{1}{2}\theta^2 D_a E+\frac{1}{2}\tb_a\Db^a E\;,\;
\delta \theta^a=\frac{i}{2}\Db^a E\;,\; \delta\tb_a=-\frac{i}{2}D_a E\;,\quad \delta Y =\dot{E_L}(t_L,\theta)Y\;,
\ee
where we made use of the notation $t_L=t+i\theta \bar\theta$, $t_R=t-i\theta \bar\theta$.
The superfunction $E$ collects all the parameters of the $SU(1,1|2)$ group
\bea
&& E=f-2i\left( \varepsilon(t)\tb -\theta\bar\varepsilon (t)\right) +\frac{1}{2}\left( \theta \tau^k \tb\right) b^k -2
 \left( \dot\varepsilon \tb +\theta\dot{\bar\varepsilon}\right)\theta\tb+\frac{1}{2}\left(\theta\tb\right)^2 \ddot{f} \;,
 \nonumber \\
&& f=a+bt+ct^2\;, \; \varepsilon^a (t) =\varepsilon^a+\beta^a t\;,
\eea
while the chiral superfunction $E_L$ is defined as follows
\be
E_L=\frac{1}{2}f(t_L)+2i\theta^a{\bar\varepsilon}_a(t_L)\;.
\ee
Owing to the homogeneous character of the transformation law \p{4btr} for $Y$ and ${\bar Y}$, one can apply the Hamiltonian reduction.
As before, the starting point is the sum of a kinetic term for the superfield $Y_0$ $(g=0)$ and $n(n-1)/2$ extra terms
of the form \p{4baction} with $Y_{ij}=-Y_{ji}$.
The reduction constraints read
\be\label{n4hamred}
Y_0=\frac{1}{\sqrt{n}}\sum_i V_i \;\quad Y_{ij}=\frac{1}{\sqrt{n}} \left( V_i-V_j \right) \;,
\ee
where  $V_i$ denote $n$ new independent superfields which obey
the same chirality conditions \be \Db^a V_i= D_a {\bar V_i}=0\;.
\ee The resulting action functional has the form \be\label{4bCal}
S_{4b}=\frac{1}{2} \left( \int dt d^4\theta  \sum_i V_i{\bar V_i}
-g\int dt d^2\theta \sum_{i < j}\log |V_i-V_j| -
g \int dt d^2 \tb \sum_{ i < j} \log | {\bar V_i} -{\bar V_j}| \right) \;.
\ee
Notice that by construction the action holds invariant with respect to the residual "diagonal" $SU(1,1|2)$ transformations,
which are realized in the same way as in Eq. \p{4btr} above, for all the superfields $V_i,{\bar V_i}$.
In order to clarify the status of this model, we eliminate auxiliary fields and go over to the
physical components
%$\left\{ q_i,{\bar q_i}, \chi_{ia},\bar\chi_i{}^a\right\}$
\be
q_i={V_i}|_{\theta=0}\;,\quad \bar{q_i} ={\bar V_i}|_{\theta=0}\;,\quad \chi_{ia}=-i D_a {V_i}|_{\theta=0} \;,
 \quad {\bar\chi}_i^a=i\Db^a {\bar V_i}|_{\theta=0},
\ee
the component action having the form
\bea\label{4bCalcomp}
 S_{4b}=&&\frac{1}{2}\int dt\left[ \sum_i \left( {\dot q_i}{\dot {\bar q_i}} +\frac{1}{2}{\dot\chi_i}{\bar\chi_i}\right)-
 \sum_{i<j}\left(  \frac{g^2}{(q_i-q_j)(\bar q_i -\bar q_j)} -\frac{g (\chi_i-\chi_j)(\chi_i-\chi_j)}{4(q_i-q_j)^2} -\right.\right. \nonumber \\
 && \left. \left. \frac{g(\bar\chi_i-\bar\chi_j)(\bar\chi_i-\bar\chi_j)}{4(\bar q_i -\bar q_j)^2}
 \right) \right] \;.
\eea
Thus, the model \p{4bCalcomp} describes an $N=4$ superconformal extension of a complexification of the
$n$--particle Calogero model.

\vspace{0.5cm}

\noindent
{\bf 4.2 The {\it 4c} multiplet}

\vspace{0.5cm}

This model is based on a recently proposed version \cite{ikL} of
an $N=4$ superconformal mechanics. In the bosonic sector this
system contains a dilaton field together with two variables
parameterizing the two-sphere $S^2\sim SU(2)/U(1)$. It was
suggested in \cite{ikL} that this variant of superconformal
mechanics may be a disguised form of a charged superparticle
moving in $AdS_2\times S^2$ background. Inspired by the
application found for the $n$--particle Calogero model in the
context of black hole physics \cite{gib}, we extend the system to
the $n$-particle case.

The starting point is the superfield action functional
\be\label{4cactionA}
S=-\int dt d^2\theta\left[ \frac{1}{4}\left(
v^2+4\rho\bar\rho\right)^{\frac{1-2\alpha}{2\alpha}}\left( D v \Db
v + D\rho \Db{\bar \rho}\right) +
 g \log \left| \frac{v+\sqrt{v^2+4\rho\bar\rho}}{2}\right| \right]
 \;,
\ee which depends on the parameter $\alpha$ characterizing the
supergroup $D(2,1|\alpha)$ and holds invariant with respect to the
latter.

In the last line $v$ is a general bosonic $N=2$ superfield, while the
$N=2$ superfields $\rho$ and ${\bar \rho}$ obey the chirality
conditions \be D \bar\rho=\Db \rho =0. \ee
Altogether  $v$, $\rho$ and ${\bar \rho}$ form
an irreducible $N=4$ supermultiplet \cite{IS,BP,MSS}. The
fact that the superfields $v,\rho,\bar\rho$ transform
homogeneously under the $D(2,1|\alpha)$ supergroup allows one to immediately
apply the reduction. Omitting the details we expose
the resulting action functional
\bea\label{4cCalg}
S&=&-\int dt
d^2\theta\left[ \frac{1}{4}\left( Y_0^2+4 X_0\bar
X_0\right)^{\frac{1-2\alpha}{2\alpha}}\left( D Y_0 \Db Y_0 + D X_0
\Db{\bar X_0}\right) +
     \right.  \\
&& \left. \frac{1}{4}\sum_{i<j} \left( Y_{ij}^2+4 X_{ij}\bar X_{ij}\right)^{\frac{1-2\alpha}{2\alpha}}
\left( D Y_{ij} \Db Y_{ij} + D X_{ij} \Db{\bar X_{ij}}\right)+
 g \log \left| \frac{Y_{ij}+\sqrt{Y_{ij}^2+4X_{ij}\bar X_{ij}}}{2}\right| \right] \;, \nonumber
\eea where \be Y_0=\frac{1}{\sqrt{n}}\sum_{i=1}^n
v_i\;,\;X_0=\frac{1}{\sqrt{n}}\sum_{i=1}^n \rho_i\;,\;
Y_{ij}=\frac{1}{\sqrt{n}} \left(
v_i-v_j\right)\;,\;X_{ij}=\frac{1}{\sqrt{n}}\left(
\rho_i-\rho_j\right)\;, \ee
and the superfields
$\bar\rho_i(\rho_i)$ are chiral (anti-chiral).

As formulated in Eq. \p{4cCalg} above, the action functional exists
for a generic value of the parameter $\alpha$ {}\footnote{A singular
value of the parameter $\alpha=0$ is equivalent to $\alpha=-1$,
see e.g. Ref. \cite{ikL}.}. However, only for $\alpha=1/2$ the kinetic
term has the standard flat form (for other values of $\alpha$
there appears a nontrivial metric \cite{ikL}). Thus, in what
follows we concentrate on this particular case, the action
functional being drastically simplified
\bea\label{4caction}
S=&& -\int dtd^2\theta\left[\frac{1}{4} \sum_i \left( D v_i \Db v_i + D \rho_i \Db{\bar \rho_i}\right) + \right.
\nonumber \\
&& g\sum_{i<j} \left. \log \left|
\frac{v_i-v_j+\sqrt{(v_i-v_j)^2+4 (\rho_i-\rho_j)(\bar \rho_i
-\bar\rho_j)}}{2}\right| \right] \;. \eea Notice that for a
vanishing $\rho_i$, the action \p{4caction} reduces to that of the $N=2$
Calogero model \p{n2cal2}, while for $v_i=0$ the interaction term
disappears and one ends up with a collection of $n$ free
particles. For completeness of the presentation we write down
the  component action (with all auxiliary fields eliminated by means of their equations of motion)
\bea
S& =& \int dt \sum_i\left( \frac{1}{2} {\dot v_i}^2 +2 {\dot \rho_i}{\dot{\bar\rho_i}}\right) -
 \sum_{i<j} \left(    \frac{8g^2}{v_{ij}^2+4\rho_{ij}{\bar\rho_{ij}}}+
 \frac{8ig \left( \rho_{ij} \dot{\bar\rho_{ij}}-\dot\rho_{ij}\bar\rho_{ij} \right) }
 { \sqrt{v_{ij}^2+4\rho_{ij}{\bar\rho_{ij}}} \left( v_{ij}+\sqrt{ v_{ij}^2+4\rho_{ij}{\bar\rho_{ij}}}\right)} \right.+ \nonumber \\
&& \left. \frac{4g}{(v_{ij}^2+4\rho_{ij}{\bar\rho_{ij}})^{3/2}}\left( v_{ij}\xi_{ij}{\bar\xi_{ij}} -v_{ij}\psi_{ij}{\bar\psi_{ij}}+
   2\rho_{ij}\xi_{ij}{\bar\psi_{ij}}+2{\bar\rho_{ij}\psi_{ij}{\bar\xi_{ij}}} \right) \right)\;,
 \eea
where
\be
\rho_{ij} = \rho_i-\rho_j\;, \quad v_{ij}=v_i-v_j \;,\quad \psi_{ij} =\psi_i-\psi_j\;,\quad \xi_{ij}=\xi_i-\xi_j
\ee
and the physical components $\left\{ v_i,\rho_i,\psi_i,\xi_i \right\}$ are defined as follows
\be
v_\i=v_i|_{\theta=0}\; ,\quad \rho_i=\rho_i|_{\theta=0}\;,\quad \xi_i=Dv_i|_{\theta=0}\;, \quad \psi_i=D\rho_i|_{\theta=0}\;.
\ee
Thus, the model constructed in this subsection can be viewed as an $N=4$
superconformal Calogero model in three dimensions.


\vspace{0.5cm}

\noindent {\bf 4.3 Troubles with the {\it 4a} multiplet}

\vspace{0.5cm}

Being the most interesting multiplet in the context of the conjecture of
Ref. \cite{gib}, this case turns out to be the most difficult to
handle. In the framework of the Hamiltonian formalism, the problem
has been revealed already in Ref. \cite{wyl} (see also the related
work \cite{gal}). In order to clarify how the problem looks
in our setting, let us recall that the action of the $N=4$
superconformal mechanics for the case at hand reads~\cite{ikl, ikp}
\be\label{4a}
S=\frac{1}{16}\int dt d^4 \theta \left( e^u
-\frac{1}{8} \theta^a_\alpha \theta_{\beta a} m^{\alpha\beta} u
\right).
\ee
Here $m^{\alpha\beta}$ is a constant $SU(2)$ vector,
which plays the role of a coupling constant, whereas the $N=4$ superfield $u$
is constrained to obey the equation
\be\label{2acon}
D_{a(\alpha}D^\alpha_{b)}u=0 \;.
\ee
Because the superfield $u$
transforms homogeneously with respect to the $N=4$ superconformal
group $su(1,1|2)$, one still can apply the reduction procedure. As
an outcome of this one finds the action functional
\be\label{4aCal}
S=\frac{1}{16}\int dt d^4 \theta \left[ e^{Y_0}
+\sum_{i<j}\left( e^{Y_{ij}} -\frac{1}{8} \theta^a_\alpha
\theta_{\beta a} m^{\alpha\beta} Y_{ij}\right) \right],
\ee
 where, as before,
 \be
 Y_0=\frac{1}{\sqrt{n}}\sum_{i=1}^n v_i\;,\;
Y_{ij}=\frac{1}{\sqrt{n}} \left( v_i-v_j\right),
\ee
and all the superfields $v_i$ satisfy the constraints \p{2acon}. A simple
inspection then shows that the kinetic term of the component action
includes a non-trivial coupling to a specific  metric and
thus the model fails to reproduce the Calogero model in the
bosonic limit. A way out could be to perform the reduction not in
terms of the superfield $u$, but rather $e^{\frac{1}{2}u}$. In
other words, one can pass to the new variable
$w=e^{\frac{1}{2}u}$ in the action \p{4a} and then perform the
same reduction as before for the newly introduced superfields.
Unfortunately, this step proves to be incompatible with the
constraints \p{2acon} which are of the second order in the covariant
derivatives. Thus, for the {\it 4a} multiplet the reduction method
fails to produce an $N=4$ superconformal extension of a real
Calogero model.


\vspace{0.5cm}

\noindent {\bf 5. Concluding remarks }

\vspace{0.5cm}

To summarize, in this paper we implemented a new variant of Hamiltonian reduction
for constructing one--dimensional many--body theories with extended superconformal symmetry starting
from a direct sum of one--particle systems. It should be stressed that all symmetries which were
realized homogeneously and linearly for the one--particle models hold in the resulting
many--body theories. The point crucial for the whole scheme was to preserve the standard structure of a
kinetic term when putting an original composite system on a constraint surface. This was
achieved  by adding to the action a free particle describing the center of mass motion.

As was mentioned above, the range of possible applications of the specific reduction
exploited in this work is actually larger than just the (super) conformal systems.
Whenever the reduction preserves the standard structure of a kinetic term, a many--body
system with the general potential $\sum_{i<j}V(x_i-x_j)$ can be constructed.
Using this variant of the Hamiltonian reduction  we built two different versions of an $N=4$ superconformal
extension of the $n$--particle Calogero model. Both extensions contain only bilinear combinations
of fermions which enter the Hamiltonian . Moreover, these variants can be viewed as the two-, and
three-dimensional extensions of the Calogero model.
The additional bosonic degrees of freedom describe angular variables and their appearance seems to be
unavoidable.
As a byproduct, we also constructed a new $n$-particle system which is invariant with
respect to the $D(2,1|\alpha)$ group. It seems interesting to understand whether this system is
integrable.

Another interesting question is whether the reduction proposed in this work allows one to generate
solutions for a reduced $n$--particle model proceeding from those characterizing an initial
simple composite system. We hope also that this kind of the Hamiltonian reduction will be useful for two-dimensional
field theories, perhaps with some modifications.

\vspace{0.5cm}
\noindent{\bf Acknowledgements}\\

We thank M.A. Olshanetsky  for clarifying  discussions.
This work was partially supported by INTAS grant No 00--254 (S.B., A.G. and S.K.),
NATO Collaborative Linkage grant PST.CLG. 979389 (S.B. and A.G.), RFBR grant No 03--02--16193,
grant E02-2.0-7 from the Ministry of Education of Russian Federation (A.G.) and grants DFG No.436
RUS 113/669, RFBR-DFG 02-02-04002 (S.K.).


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