Abstract
The group acts naturally on by simultaneous right and left multiplication. We study the Kähler metrics invariant under this action using a global Kähler potential. The volume growth and various curvature quantities are then explicitly computable. Examples include metrics of positive, negative and zero Ricci curvature, and the 1lump metric of the model on a sphere.
We then look at the holomorphic quantization of these metrics, where some physically satisfactory results on the dimension of the Hilbert space can be obtained. These give rise to an interesting geometrical conjecture, regarding the dimension of this space for general Stein manifolds in the semiclassical limit.
Part I
1 Introduction
Among the geometrical procedures for quantization, holomorphic quantization is a particularly simple and natural one, and can be used whenever the classical system “lives” on a complex Kähler manifold. In this paper the complex manifold under study will be , and we will consider the Kähler metrics on this manifold which are invariant under a natural action of the group , namely the action defined by simultaneous right and left multiplication of the matrix in by the matrices in .
In the first part of the paper a purely classical study of these Kähler metrics is carried out. We find that each of these metrics has a global invariant Kähler potential, which is essentialy unique, and is in fact a function of only one real variable. We then use this potential to compute explicitly several properties of the Kähler manifold. These include the scalar curvature, a potential for the Ricci form, the volume and volume growth, the geodesic distance from the submanifold , and a simple criterion for completeness. Choosing particular functions as Kähler potentials we get metrics with positivedefinite, negativedefinite and zero Ricci tensor; the Ricciflat one being just the usual Stenzel metric on .
A significant application of the above results, which was in fact the original motivation for this paper, is a closer study of the metric on the moduli space of one lump on a sphere. These lumps are a particular kind of soliton that appear in sigma models, and have been widely studied [2, 13]. In particular, the special case of one lump on a sphere has been studied by Speight in [10, 11], where the author also examines general invariant Kähler metrics on and finds some of the results mentioned above. The approach there however is rather different, since it is based on the choice of a particular frame for , instead of using the perhaps more natural Kähler potentials.
The second part of the paper examines some aspects of holomorphic quantization on the manifold with the Kähler metrics described above. We basically look at two things: the nature and dimension of the quantum Hilbert space, and the quantum operators corresponding to the classical symmetries of the metric.
Regarding the latter point, we start by finding the moment map of the action. This map encodes the classical symmetries of the system and, through the usual prescriptions of geometric quantization, subsequently enables us to give an explicit formula for the operators corresponding to these symmetries. Regarding the first point, i.e. the dimension of the Hilbert space, the story is a bit more involved, and we will now spend a few lines describing the motivation and the results.
If you apply holomorphic quantization to a compact Kähler 2manifold, it is a consequence of the HirzebruchRiemannRoch formula that the dimension of the Hilbert space is finite and grows asymptotically as when , where is the volume of the manifold. This result is also physically interesting, since it agrees with some predictions of semiclassical statistical mechanics. Trying to see what happens on the noncompact with our invariant metrics, we were thus led to compute the dimension of the Hilbert space. The results obtained can be summarized as follows.
The Hilbert space in our setting is essentially the space of squareintegrable holomorphic functions on , where squareintegrable means with respect to some metricdependent measure on . Furthermore all these holomorphic functions can be seen as restrictions of holomorphic functions on . Defining the subspace of the holomorphic functions which are restrictions of polynomials in , we then find that as whenever both members are finite. The exact dimension of , which we also compute, depends on the particular invariant metric one puts on ; its asymptotic behaviour however does not. This leads us to conjecture that, as in the compact Kähler case, also for general Stein manifolds (i.e. complex submanifolds of ) this asymptotic behaviour of is “universal” – see the discussion of section 8.
2 The invariant Kähler metrics
We start by considering the action of the group on the complex manifold defined by
(1) 
This is clearly a smooth action which acts on through biholomorphisms. A detailed study of and its orbits is done in Appendix A. For example one finds there that all the orbits except one have real dimension , the exceptional one being , which has dimension . For the purposes of this section, however, it is enough to quote the following result :
Proposition 2.1.
Any smooth invariant function can be written as a composition , where is defined by , and is a smooth even function.
We are now interested in studying Kähler metrics and forms over . To begin with, the wellknown diffeomorphism implies that the de Rham cohomology of and are the same. In particular every closed form on is exact. On the other hand, regarding as the set of complex matrices, we have that is the hypersurface given as the zero set of the polynomial . Since the derivative of this polynomial is injective on the zero set, is a complex submanifold of . It then follows from standard results in complex analysis of several variables (see th. and of [6]) that is a Stein manifold with Dolbeault groups (except for ).
From all this we get the following lemma:
Lemma 2.2.
Any closed (1,1)form on can be written , where is a smooth function on . If is real, then can also be chosen real.
Proof.
This is just like the usual proof of the local lemma. As argued above, the closedness of implies its exactness, hence for some . Since (because is a (1,1)form) and , we have that and for some smooth functions on . Defining we thus get . If is real, then is a real potential for . ∎
Having done this preparatory work, we now head on to the main result of this section.
Proposition 2.3.
Suppose is a closed invariant form. Then one can always write , where and are as in proposition and is smooth. The function is unique up to a constant. Furthermore, the hermitian metric on associated with is positivedefinite iff on and on .
Proof.
By the previous lemma for some . Now, for any , the invariance of and the holomorphy of imply that
Hence by averaging over if necessary (recall that is compact), one may assume that the potential is invariant. The first part of the result then follows from proposition .
To establish the second part, recall that the associated hermitian metric is defined by
(2) 
where is the complex structure on . Since both and are invariant (the last one because is holomorphic), we conclude that also is invariant. Now consider the complex submanifold consisting of the diagonal matrices in . It follows from lemma A.1 of Appendix A that intersects every orbit of . Hence, by the invariance, is positivedefinite on iff it is positivedefinite at every point of . To obtain the condition for positiveness over we now use a direct computation.
Take the neighbourhood and the complex chart of defined by
(3) 
Note that and that is a chart of adapted to . Defining we have that and
A direct calculation using the chain rule now shows that, on a point diag, we have
(4) 
and hence
(5) 
Thus at points of such that (i.e. ), we have and the positivedefiniteness of is equivalent to . On the other hand, since and the chart are defined over all of , continuity implies that at a point of with (i.e. ) we must have
where it was used that
Thus at this point the positivedefiniteness of is equivalent to . This establishes the last part of the proposition.
To end the proof we finally note that formula (5) implies the uniqueness of , and hence the uniqueness of up to a constant. ∎
Roughly speaking, this proposition guarantees the existence of invariant potentials for invariant Kähler forms. A particular feature of these potentials, which will be crucial for the explicit calculations later on, is that they are entirely determined by their values on the diagonal matrices, since every orbit of the action contains one of these. Having this in mind, we now end this section by presenting a technical lemma which will prove useful on several occasions.
Lemma 2.4.
Suppose is a smooth invariant function on , and consider the submanifold of diagonal matrices in . If is a smooth function on such that , then on the submanifold .
Proof.
The hypothesis is that . Writing as and using the expression for the laplacian in polar coordinates, we have on
But the invariance implies that only depends on ; since the same is assumed for , we get
Now, invariance also implies that , thus
∎
3 Curvature and completeness
Throughout this section will be the Kähler form of a invariant Kähler metric on . Thus according to proposition we can write
(6) 
where is smooth and satisfies all the conditions of proposition .
The first task now is to calculate the Ricci form associated to this Kähler metric. More precisely, we will obtain a potential for expressed in terms of the function .
Proposition 3.1.
The Ricci form of the metric with Kähler form is given by
Proof.
The invariance of the metric implies the invariance of the Ricci form . Thus, by proposition , has a global invariant potential . Now consider the chart for defined in the proof of the same proposition. According to a standard result, if in this chart
then the Ricci form is given by
In particular, over the complex submanifold of diagonal matrices we have
But (5) gives us over , and so we compute that
Since this function only depends on , by lemma we get that
Finally the invariance of guarantees that this expression is valid all over . Thus we conclude that has the stated form. ∎
The next step is the computation of the scalar curvature. Note that the invariance of the metric implies the invariance of this function.
Proposition 3.2.
The scalar curvature of the Riemannian metric associated with the Kähler form is
Proof.
Let us call . The same calculations that led to formula (4) now give , so that
(7) 
Writing and
(8) 
The invariance of then shows that this formula is valid all over . ∎
In the last part of this section we will make contact with a paper by Patrizio and Wong [9]: this will give us almost for free some results about the completeness of the invariant metric associated to .
To make contact one just needs to note that the linear transformation on defined by the matrix
takes the standard hyperquadric to , and the norm function on to the function on . Therefore all the results in [9] valid for can be restated here for . In particular we have that

The function is plurisubharmonic exhaustion on , and solves the homogeneous MongeAmpère equation on ([9], th. 1.2).

Suppose is a strictly plurisubharmonic function on . Then with respect to the metric defined by , the distance in between the level sets and is ([9], th. 3.3)
(9) Furthermore, the distanceminimizing geodesics between these level sets are the integral curves of the vector field , where is the gradient vector field of (one can check directly that on ).
As a consistency check, we remark that the strict plurisubharmonicity of together with proposition garantees that on ; thus the integral formula for the distance is well defined. It is now more or less straightforward to prove the following proposition.
Proposition 3.3.
The metric on with Kähler form is complete if and only if
Proof.
By HopfRinow, the metric is complete iff the closed bounded sets of are compact. So suppose that and that is a closed and bounded subset of . Then for big enough we have
But is just the usual norm on restricted to , thus is also closed and bounded in , and so is compact.
Conversely, if , then itself is a closed bounded set which is not compact, and thus the metric is incomplete. ∎
4 Volume and integration
The purpose of this section is to study the integrals over of invariant functions, where is as in (6). More precisely, we want to prove the following result.
Proposition 4.1.
Let be a smooth invariant function on , which by proposition can be written , and let be the open submanifold . Then we have that
(10) 
Notice that is the volume form of the metric on associated with , so with the particular choice we get the volume of . Remark also that with or , where is the scalar curvature given by proposition , the integral on the righthand side is trivially computable. Thus taking into account the restrictions on imposed by propositions and , one gets the following corollary.
Corollary 4.2.
For the Kähler metric on associated with , the volume of and the integral of the scalar curvature over are, respectively,
In particular has finite volume iff is bounded.
We now embark on the proof of proposition . To start with, it will be convenient to restate here some results used in [10, 11] to study the lump metric.
Consider the Pauli matrices
so that is a basis for the Lie algebra . Associated to each is a leftinvariant form on , and is a global trivialization of the cotangent bundle of . Then according to [10, 11] and the references therein we have that :

There is a diffeomorphism defined by

The usual action of on is taken by to the action on given by
(11) where is the usual double covering; explicitly has components .

Regarding the and the as forms defined over , the action acts on these forms by .

The Euler angles define an oriented chart of with dense domain such that, on this domain,
(12)
The plan now is to use the diffeomorphism to compute the integrals on , instead of . Since the trivialize the cotangent bundle of , the pullback by of the volume form on can be written
for some nonvanishing function on . Moreover, must be invariant under , because the volume form on is invariant under . But notice now that, under ,
because . For the same reason, also is invariant, and hence is invariant too. This fact together with the invariance of implies the invariance of the function . From the formula (11) for the action it is then clear that only depends on .
The computation of the function is now straightforward. First we have
On the other hand, using the chart (3) and (4), at the point of we also have
Finally a tedious calculation that we will not reproduce shows that
and so we get
Having calculated the volume form on , the rest of the proof of proposition goes on smoothly.
Call as usual and . A quick calculation shows that , and so we have an explicit relation . From this relation it is clear that , where is the open ball, centered at the origin of , with radius such that . Hence, for any invariant function on we have
Using the value of and the relation , a change of variables in the last integral shows that it coincides with
The first integral can be computed using (12). Namely we have
Putting these two results together we finally obtain the formula stated in proposition .
5 Examples
5.1 The onelump metric
The socalled moduli space of degree lumps on a sphere, which we will call , is just the group of rational maps . Identifying , this group is the same as the group of projective transformations
In the physics literature, is the space of minimal energy static solutions of the sigmamodel defined on the Lorentzian spacetime with as target space. The kinetic energy functional of this sigmamodel induces a certain Riemannian metric on , which is also very natural geometrically. It can be defined in the following way.
Let be a one parameter family of projective transformations, i.e. a curve on , and call its tangent vector at . For each , is a curve in , and we call its tangent vector at . Then the Riemannian metric on is defined by
(13) 
where is the FubiniStudy metric on and is the associated volume form. In informal terms, one may say that the squaredlength of an infinitesimal curve in is just the average over of the squaredlengths of the infinitesimal curves in ; thus the measure of “displacement” in is how much the image points of are moved. Using the fact that transformations in are isometries of , it is not difficult to check that right and left multiplication in by elements of are in fact isometries of .
Now consider the usual chart of the projective space , and let be any complex chart of defined on a neighbourhood of the point . In these charts we have
where the last two equalities are standard properties of the FubiniStudy metric. Calling the local potential of the FubiniStudy metric we get