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as is well known , three - dimensional pure einstein gravity is a topological theory and has no propagating degrees of freedom .
the solutions are locally flat apart from conical singularities at the location of matter sources in no cosmological constant case  @xcite
. moreover , cosmological gravity and supergravity are solvable in many cases  @xcite .
three - dimensional supergravity has been investigated in various situations intensively .
as particular examples , the non - linear sigma models coupled to @xmath2-extended supergravity were studied  @xcite .
the geometry of the target manifolds parametrized by scalar fields were classified completely : the target space is riemanian , khler , and quaternionic for @xmath3 and @xmath4 respectively .
for @xmath5 it generally decomposes into two separate quaternionic spaces , associated with inequivalent supermultiplets . for @xmath6
there is a symmetric space for any given number of supermultiplets . for @xmath7 and @xmath8
there are only theories based on a single supermultiplet associated with coset spaces with the exceptional isometry groups @xmath9 and @xmath10 respectively .    moreover , the three - dimensional maximal ( @xmath11 gauged supergravity were constructed  @xcite . the duality relation between the gauge fields and the scalar field plays an important role in the derivation of the ungauged @xmath12 supergravity : in order to expose its global @xmath13 isometry , all vector fields obtained by dimensional reduction of the @xmath14 supergravity in an eight - torus must be dualized into the scalar fields .
moreover , in order for the supergravity to gauge , the chern - simons term ( rather than the yang - mills term ) must be required in the lagrangian .
recently the supersymmetric vortex solutions in @xmath0-gauged @xmath15 supergravity whose scalar sector is an arbitrary khler manifold with @xmath0 isometry were constructed  @xcite .
it has been found that the einstein equations and the matter field equations of the model can be recast into a set of self - duality equations for a specific eighth - order choice of the higgs potential which reduces to the sixth - order potential of the flat space model when the newton gravitational coupling constant is set to zero .
the abelian chern - simons higgs model in three - dimensional minkowski space - times and its vortex solutions were studied by  @xcite .
it was shown that the model with a specific sixth - order higgs potential admits topologically stable vortex solutions and non - topological soliton solutions with nonzero flux and charge , which satisfy the bogomolnyi - type or the first order self - duality equations  @xcite . in the bogomolnyi limit
, there is a ( first - order ) phase transition point between the symmetric ( non - topological solutions ) and the asymmetric phase ( topological vortex solutions ) .    in this paper , we construct the non - topological solitons , which do not preserve the supersymmetry , in three - dimensional @xmath0-gauged @xmath1 supergravity and find the region where the solutions can exist for various value of a cosmological constant .
a further motivation for studying the non - topological solitons is that this model provided us with the simplest model of relativistic stars as self - gravitating systems .
rotating boson stars with large self - coupling constant in three dimensions were studied in  @xcite , for example .
the study of the non - topological soliton will lead to a new aspect of the self - gravitating systems .
the organization of this paper is as follows ; in the next section we introduce the model and the basic ingredients . for the construction of the @xmath0-gauged @xmath16 , @xmath1 supergarvity lagrangian
, we refer the reader to the existing literature  @xcite . in section  [ sec:3 ] , we restrict ourselves to the simple case of the khler manifold and a circularly symmetric ansatz for the metric which approaches asymptotically the btz configuration  @xcite , which is asymptotically anti - de sitter space ( rather than asymptotically flat ) and does not preserve the supersymmetry in general . in this setting
we obtain numerically the non - topological solitons with zero - winding number case for various values of a cosmological constant . in the section  [ sec:4 ]
, we show the numerical results and discuss the relation among the physical parameters for various values of a cosmological constant .
in this section we consider ungauged @xmath1 three - dimensional supergarvity . assuming a @xmath0 isometry of the khler potential
, we apply the noether procedure to obtain the @xmath0-gauged @xmath1 supergarvity  @xcite .
the field content of the ungauged theory is the following .    *
the @xmath1 supergravity multiplet @xmath17 contains a graviton  @xmath18  and two gravitini which are assembled into one complex spinor @xmath19 . *
the @xmath1 scalar multiplet @xmath20 contains the fermions @xmath21 and scalars @xmath22 with @xmath23 , respectively in the complex notation .
the matter sector is obtained by @xmath24 copies of the @xmath1 scalar multiplet .
the scalar fields define a khler manifold of real dimension @xmath25 , characterized by its khler potential @xmath26 .
it is assumed that the ungauged @xmath1 supergarvity is invariant under the global @xmath0 isometry . as in the maximally gauged supergravity theories  @xcite ,
the chern - simons term is required , in order to gauge the @xmath0 khler isometry .
note that this term is topological and hence does not introduce new propagating degrees of freedom in the gauged theory .
moreover , the @xmath27-dependent terms , due to the chern - simons terms , give rise to extra terms in supersymmetry transformations . in order to compensate these terms ,
the extra yukawa - type bilinear fermionic terms and a scalar potential must be added to the lagrangian
.    then the bosonic parts of the @xmath0-gauged @xmath16 , @xmath1 supergarvity lagrangian is given by : @xmath28 where @xmath29 is a covariant derivative and @xmath30 denotes the khler metric @xmath31 .
@xmath32 is an abelian field strength and @xmath27 is a gauge coupling constant .
@xmath33 is a real scalar potential , which will be mentioned in the next section .
we restrict ourselves to the simple case of the lagrangian  ( [ lagrangian ] ) considered in the preceding section .
we adopt a single complex scalar field case , that is @xmath34 .
moreover we assume that the @xmath35 is a function of @xmath36 only and the khler potential is @xmath37 in order to obtain the canonical kinetic term for the scalar field .
we thus find that the @xmath38 is unity .
then , the eighth - order scalar potential is reduced to : @xmath39,\ ] ] where @xmath40 and @xmath41 are arbitrary real constant parameters .
we assume the three - dimensional metric for a circularly symmetric spacetimes as @xmath42 where @xmath43 , @xmath44 , and @xmath45 are functions of the radial coordinate @xmath46 only .    for the single complex scalar field , we also assume the following dependence on the coordinates : @xmath47 where @xmath48 is a function of the radial coordinate @xmath46 only and @xmath49 is a constant .
we impose the boundary condition that @xmath50 approaches zero when the radial coordinate @xmath46 is taken to infinity , in order to obtain the non - topological soliton .    for the vector field @xmath51 , we choose the gauge , in order to require finite energy solution for the scalar field at radial infinity , in which @xmath52 where @xmath53 and @xmath54 are functions of the radial coordinate @xmath46 only .    varying the lagrangian with respect to the gauge field , the scalar field and the metric yields equations of motion .
the equation of motion for the chern - simons term induces the first order equations @xmath55 and two of these equations take the form @xmath56 and the third one ( @xmath57 ) is automatically satisfied .
the scalar field equation is reduced to the form : @xmath58    the einstein equation is also reduced to : @xmath59,\\   \frac{\delta}{r}\delta'&=&-2\kappa^2\left   [ \frac{e^{2\delta}}{\delta}g^2r^2   ( w+\omega p)^2+\delta(r')^2\right],\\   \frac{\delta}{2r^2}(r^3e^{\delta}\omega')'&= &   2\kappa^2g^2r^2pe^{\delta}\frac{w+\omega p}{r}.\end{aligned}\ ] ]    we set the metric which approaches asymptotically the btz solution when the scalar field falls into a vacuum of the negative value at radial infinity . in this setting , one can find that the cosmological constant @xmath60 is obtained to be @xmath61 and as analyzed in  @xcite , the physical parameters ( the btz mass , the angular momentum and the magnetic flux ) is the following : @xmath62 where @xmath63 is a sufficiently large value for radial coordinate @xmath46 .
these equations unfortunately can not be solved analytically .
then we will find the numerical solution . in the next section ,
we discuss the relation among the physical parameters for various values of a cosmological constant .
in this paper , we have numerically constructed the non - topological solitions in three - dimensional @xmath0-gauged @xmath64@xmath65 supergravity .
the non - topological soliton does not preserve the supersymmetry , in general , because the metric is described by the btz solution at radial infinity .
we have found that the region of the solutions can exist for the non - topological solitons in fig .  1 , shown with the btz mass @xmath66 , the angular momentum @xmath67 and the magnetic flux @xmath68 for various values of the cosmological constant @xmath60 . in fig .  2 , we find two branches in a sequence of the solutions which are obtained numerically by the different conditions ( values of the scalar field and the vector field at the origin ) .
one branch is the solutions from the origin to the maximal value of the @xmath66 , the other is from the point @xmath69 to the maximal value . as studied in  @xcite , one can also compare the results with black hole physics . in the case of the black hole
, it is known that a black hole with a larger angular momentum than the critical one has a naked singularity and can not be the usual black hole .
moreover there is the relation @xmath70 which is satisfied by the extreme black hole .
if the non - topological soliton becomes a black hole , the soliton with @xmath71 can become the usual black hole .
the non - topological soliton with @xmath72 can not be the usual black hole and the soliton with a large mass can certainly become an extreme black hole .
thus , the region of the solutions for the non - topological solitons is plausible as self - gravitating systems because all of the maximal values for the soliton s mass exist in the area @xmath71 . in fig  3 ,
if we fit the plotted line with a solid line , one can find that the @xmath67 is proportional to the square of the @xmath68 .
this is the same results as in the minkowski space - time case , as known in  @xcite .
therefore , the relation between the angular momentum and the magnetic flux is not dependent of the cosmological constant .
there is a phase transition point between the symmetric and the asymmetric phase in the abelian chern - simons higgs model in minkowski space - times . on the other hands , in the @xmath0-gauged @xmath64@xmath65 supergravity
, differed from the usual phase transition , there is not a non - topological soliton in the transition point because the metric approaches asymptotically anti de - sitter space - time
. it may be crucial in cosmological situation though this model has still three - dimensions .    in the present paper ,
we have treated the soliton with zero - winding number case , i.e. @xmath73 only .
we will further investigate the binding energy , the stability of the soliton .
the authors would like to thank n.  kan for useful advice .
m.  baados , c.  teitelboim and j.  zanelli , phys .  rev
.  lett .  * 69 * , 1849 ( 1992 ) ; m.  baados , m.  henneaux , c.  teitelboim and j.  zanelli , phys .  rev .  *
d48 * , 1506 ( 1993 ) . for a review of btz black holes , s.  carlip , class .
* 12 * , 2853 ( 1995 ) gr - qc/9506079 .
we construct numerical solutions for non - topological solitons in three - dimensional @xmath0-gauged @xmath1 supergravity . 
 we find the region of the solutions showing with the btz mass , the angular momentum and the magnetic flux and discuss the relation among the physical parameters for various values of a cosmological constant .