Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M 2d by removing a skeleton M 0 of lower dimension such that what remains is diffeomorphic to R 2d (cell decomposition) which we identify with Cd and embed in CPd . A local Poisson structure and Berezin-type quantization are induced from CPd . Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.


Introduction
Berezin quantization [1] is a method of defining a star product on the symbol of operators acting on a Hilbert space (with a reproducing kernel) on a Kähler manifold under certain conditions such that the star product satisfies the correspondence principle.The literature on subsequent work after [1] on Berezin quantization is vast.We mention that in [6] the conditions have been relaxed considerably.Another direction this field has expanded is Berezin-Toeplitz quantization, see for instance [2], [9].Some quantum systems donot come from quantizing classical systems (which are expected to have a symplectic structure) but there is a semi-classical limit of the quantum system.For instance there is a semi-classical limit of the quantum system of spin, see for instance, [12] (S → ∞ in Radcliffe's notation).We wish to include systems which donot have symplectic structure (or group action) and study if they are a semi-classical limit of some quantum system as goes to zero.This is the motivation of considering manifolds which have no symplectic or Poisson structure.We induce local Poisson structure on the manifold by embedding parts of it (i.e.removing sets of measure zero) into CP n or C n (depending on whether we expect a finite dimensional or an infinite dimensional Hilbert space) and induce the Berezin quantization from one of these two spaces.
The other motivation of the work is that sometimes the Hilbert space of the problem turns out to be different from what the actual manifold of parameter space should prescribe.The Hilbert space could be just obtained from geometric quantization of C n , or CP n , whereas the parameter space is not C n or CP n .Roughly speaking in these two cases (namely C n or CP n ), the Hilbert space consists of polynomials.For some situations this could be a at least a good approximation, for example the Quantum Hall Effect (where polynomials suffice for lowest Landau levels [17]).The global holonomy needs to be calculated, which we explain in the example of the torus.
In this article we show that a Berezin quantization can be achieved on a compact even dimensional manifold M 2d by removing a skeleton of lower dimension such that what remains is diffeomorphic to R 2d which we identify with C d and embed in CP d .We get an induced Berezin quantization from CP d .In other words, we obtain a Hilbert space with a reproducing kernel and a star product on the symbol of bounded linear operators on the Hilbert space which satisfy the correspondence principle.The Berezin quantization depends on the diffeomorphism of M \ M 0 to R 2d but if we choose a different diffeomorphism of M \ M 0 to R 2d then we obtain a quantization with another reproducing kernel with star product on symbols which satisfy the correpondence principle.These two quantizations need not be equivalent in the sense that there maynot be a natural map between the Hilbert spaces which preserve the reproducing kernel.
The set of meausre zero which we remove is the lower dimensional skeleton in cell decomposition so that what remains is a top dimensional cell which we identify with U 0 ⊂ CP n , one of the homogeneous charts.We pull back the polynomials on U 0 to X for the quantization.However we have to keep track of the cell decomposition because of global holonomy.The loops may pass through the sets of measure zero in M 0 which we have removed.But this can be handled if we remember the lower dimensional skeleton we had removed.Thus, even though we remove a set of measure zero, it plays an important role in detemining the global holonomy.We illustrate with the torus.
In this context we recall that in [4] we had considered totally real submanifolds of CP n and defined pull back operators and their CP n -symbols and showed that they satisfied the correspondence principle.
In this article we also exhibit Berezin-Toeplitz quantization on a compact complex manifold.
This work is part of Kohinoor Ghosh's thesis [8].
It has many interesting applications in harmonic analysis and non-commutative geometry.This is work in progress.
Role of CP n or C n can taken by other appropriate manifolds too.

Review of Berezin quantization on CP n
This section is a review based on ideas from [1].In Berezin [1], the quantization on CP n is achieved thinking of it as a homogeneous space.In this section we give an explicit path to the quantization using a local description.
Let Φ F S be a local Kähler potential for the Fubini-Study Kähler form Ω F S on CP n .Let us recall how this looks in local coordinates.
Let Φ F S (µ, μ) = ln 1 + n i=1 |µ| 2 be the Kähler potential and the Kähler metric G is given by g F S ij = ∂ 2 ΦF S ∂µi∂ μj .The Fubini-Study form is given by Ω F S = n i,j=1 Ω F S ij dµ i ∧dμ j , where the Kähler metric G and the Kähler form Ω F S are related by Ω F S (X, Y ) = G(IX, Y ).
The coefficients of the inverse matrix Ω ij F S appears in the definition of the Poisson bracket of two functions t and s: Let H ⊗m be the m-th tensor product of the hyperplane bundle H on CP n .Then recall that mΩ F S is its curvature form and mΦ F S is a local Kähler potential.Let Γ hol be holomorphic sections on it.Let {ψ i } N i=1 be an orthonormal basis for it.On U 0 the sections of H ⊗m are functions since the bundles are trivial when restricted to U 0 .
Proof.By linearity, it is enough to check this for Ψ = Ψ I0 a basis element.
Proposition 2.2.Resolution of identity property: In particular, Let Â be a bounded linear operator acting on H.Then, as in [1], one can define a symbol of the operator as One can show that one can recover the operator from the symbol by the formula [1]: Let Â1 , Â2 be two such operators and let Â1 • Â2 be their composition.Then the symbol of Â1 • Â2 will be given by the star product defined as in [1]: where recall 1 c(m) = U0 e −mΦF S (ν,ν) dV (ν).This is the symbol of Â1 • Â2 .

It is easy to show ([8])
Proposition 2.4.We have φ F S is non-positive on S and has a zero and a nondegenerate critical point ( as a function of ν) at ν = µ.

Berezin-type quantization on compact even dimensional manifolds
Let M 2d be an even dimensional compact smooth manifold.We do not consider any symplectic structure or Poisson structure or group action on it.To obtain a Berezin-type quantization on it, first we embed the manifold (after perhaps removing a subset of measure zero) in CP d and then induce a local Poisson structure on the embedded submanifold and induce the Berezin quantization from CP d .The Hilbert space of quantization is expected to be of finite dimension (since M is compact) and for that we choose CP d and not C d .
Let M 2d be a compact topological manifold.Then by [5], there exists a skeleton M 0 of dimension at most 2d − 1 such that X = M \ M 0 is homeomorphic to R 2d .We assume M 2d is equipped with a differentiable structure such that M \ M 0 is diffeomorphic to R 2d with standard smooth structure.
Then symbols and star product can be defined for Â via Â and correspondence principle follows.Now we elaborate this.
The symbol of Â is defined to be Ã(p, q) ≡ A(z, w) where z = τ (p), w = τ (q).Suppose we have two operators Â1 and Â2 .Then Ã1 * Ã2 is defined on In general the algebra of operators will depend on the diffeomorphism.
Then we can see that the star product satisfy the correspondence principle.The proof is exactly same as the previous section with n = d.
Proof.Set n = d in the previous section.The proof follows essentially from Lemma (2.1) in [1] as elaborated in [8].
3.1.Equivalence of two Berezin quantizations: On a smooth (complex) manifold M 2d \M 0 , let there be a local Poisson structure and a Berezin-type quantization defined as above induced from CP d .there are two diffeomorphisms (biholomorphisms if M \ M 0 is complex) which induce two such quantizations.Then there are two Hilbert spaces with reproducing kernels and star products on symbols of bounded linear operators which satisfy the correspondence principle.Suppose there exists a smooth (or biholomorphic) bijective map ψ from M \ M 0 to M \ M 0 which preserve the local Poisson structures.If ψ induces an isomorphism (i.e. a bijective linear map that preserves innerproduct) between the two Hilbert spaces such that the reproducing kernel maps to the corresponding reproducing kernel then we shall say the two Berezin quantizations are equivalent.

Our method of quantization for the torus
Let L be a line bundle on CP 1 .CP 1 is homeomorphic to the sphere of radius 1 and let N, S be the north and the south poles and E the equatorial circle.Let U N = S 2 \ N and U S = S 2 \ S be two charts on the sphere such that U N is homeomorphic to the equatorial plane using the stereographic projection from N and U S being the same using stereographic projection from S. The transition function t N S of the line bundle L when restricted to the equator, winds the equatorial circle E to r times U (1) ≡ S 1 , r ∈ Z.This winding number characterises smooth line bundles on the sphere.For the transition function of H, the hyperplane line bundle, let the winding number be r 0 .Then for L = H ⊗m the winding number is q = r 0 m.(As an aside, the set of holomorphic sections of H ⊗m are in one to one correspondence with polynomials of degree ≤ m in one complex variable-for more details, see [11], p 500).
Let iθ 1 be the imaginary valued connection 1-form for H (curvature proportional to the Fubini-Study form ω F S ).Let iθ = miθ 1 the connection 1-form on H ⊗m .Let m = 2s be an even integer.Let ψ be a section of H ⊗m on sphere which satisfies (d + miθ 1 )ψ = 0. On integration on any closed loop C 1 on the sphere, ψ = exp(−im C1 θ 1 )ψ 0 where ψ 0 = ψ(t 0 ).The phase factor is called holonomy and is well defined along this path because the curvature of the line bundle ω F S = dθ 1 belongs to the integral cohomolgy H 2 (S 2 , Z), in [18], p 158.
Let U u be the upper hemisphere of the sphere with boundary E. U u is the interior of U u which is diffeomorphic to a disc.As before let ψ be a section of H ⊗m and E be parametrized by t such that E 1 and E 2 are parametrised by 0 e. E 2 with the reverse direction.
We note that exp , where E 1 2 is half way of E 2 and E 2 2 is the other half of E 2 .Let A and B be the two representatives of the homology of the torus, M 0 = A∪B.
Let us identify points on E with A ∪ B such that one-fourths of the equator E 1  1 is identified with A and the other half E 2  1 is identified with B (using a quotient map).Similarly, E 2 = E 1 2 + E 2 2 , such that E 1 2 and E 2 2 are identifed with −A, −B.This is in keeping with the cell decomposition of the torus.E 1 and E 2 are identified with loops A + B and −A − B.
Take a loop C on the torus such that be a closed loop of the torus, as before, parametrised by 0 ≤ t ≤ 1 and ψ be a section of H ⊗m , m = 2s, an even integer.Then ψ(1) = exp(−i k1A θ − i k2B θ)ψ 0 where ψ 0 = ψ(0).Then the phase factor due to holonomy is exp θ) which is well defined (because we can translate the question to that on the sphere).
If p ∈ X = T 2 \ (A ∪ B), β is a loop on the torus T 2 which is contained entirely in T 2 \ (A ∪ B) one can easily show there is well defined global holonomy, after the identification of X with U u .If β is a loop starting and ending at p ∈ T 2 that intersects A or B, by our identification of A and B with E 1 1 and E 2 1 , the holonomy on β is also well defined (as we can translate the question to that on the sphere).

Toeplitz quantization on compact complex manifolds
Let M be a compact complex manifold of dimension d.Let M 0 be a set of measure zero such that z 2 , ..., z d ]} be one of the homogeneous charts of CP d .Thus we have an embedding, ǫ, which maps M \ M 0 onto U 0 ⊂ CP d .Note that CP d is endowed with Fubini-Study metric.
Recall that the volume element on CP d restricted to U 0 is given by dV CP d = dV (µ) = |dµ∧dμ| (1+|µ| 2 ) d+1 .Let H be the hyperplane line bundle on CP d and H ⊗m be the m-th tensor power of H and H m be the Hilbert space of square integrable holomorphic sections of H ⊗m restricted on U 0 .Let H m X denotes ǫ * (H m ).M \ M 0 has an induced volume form as follows.Let Σ = ǫ(M \ M 0 ) and h(ζ)dS(ζ) = dV Σ (ǫ(ζ)), where h > 0 is a smooth function.Note that all pullback sections in H m X are square integrable w.r.t. the measure hdS on X = M \ M 0 .Let f, g be a smooth function on CP d restricted to U 0 and let f , g be the smooth functions on M \M 0 , which are pulled back by ǫ, i.e., for µ ∈ M \M 0 , f (µ) .= f (ǫ(µ)), similarly g(µ) . = g(ǫ(µ)).We claim f is a unique function on that its complement is of measure zero.Since f 1 − f 2 is smooth, it extends to all of CP d and is identically 0.
Recall for CP d (restricted to U 0 ), m-th level Toeplitz operator of f , denoted by T m f , defined on H m , defined as T m f (s) = Π m (f s), where Π m is the projection map from square integrable smooth sections onto H m and s ∈ H m .Let s = ǫ * s.One can show that given s, s is unique.This is because if s = ǫ * s 1 = ǫ * s 2 .Then s 1 − s 2 = 0 on Σ and thus on CP d .But s 1 , s 2 are global holomorphic sections of H m and can be extended to all of CP d .Thus s 1 − s 2 ≡ 0.
Thus we have where the first norm is in X = M \ M 0 and second norm is in CP d .For a functions f ∈ C ∞ (X), we define a set of operators for M \ M 0 , defined on Since f s = ǫ * (f s) for a unique f s ∈ H m , we have that T m f (s) is well defined.We know from Toeplitz quantization of CP d (see [2]), that, Proof.To prove the first equality, Rest follows from the previous two propositions.This ends the proof.
If we have two biholomorphisms ǫ 1 and ǫ 2 from M \ M 0 to U 0 ⊂ CP d we have an equivalent Toeplitz quantization because we can define an equivalence of the Hilbert spaces and the Poisson bracket is also preserved.[8].

A conjecture
Let M be a compact integral Kähler manifold with Kähler form ω. Let L be a geometric quantum bundle whose curvature is proportional to ω.Let us consider L ⊗m whose curvature is proportional to mω.
Theorem Then motivated by the theorem above we have the following conjecture.