ALAG

(Abelian Lagrangian Algebraic Geometry)

by A. L. Gorodentsev and A. N. Tyurin

December 1999

This is a by-product after a series of discussions during our staying at MPI, where the second author has presented his program for linking the Lagrangian and Algebraic geometries via the geometric quantization and the mirror symmetry and the first author tried to explain it in our seminar as far he has understood it himself. In fact, the present paper is just the first chapter of a book that probably will be published in a future. In this chapter we just fix some basic concepts, construct the holomorphic BPU-map, and give the

simplest applications to the Legendrian knots in S^{3}.

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

§ 1 Lagrangian preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 § 2 Abelian preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 § 3 Prequantization equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 § 4 Holomorphic quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 § 5 A ‘toy’ example: the Legendrian knots in S^{3} . . . . . . . . . . . . . . . . 28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Introduction

0.1. Classical mechanics. A natural mathematical framework for the classical Hamiltonian dynamics is presented by the Symplectic geometry (see [A]). A smooth (compact or not) C^{∞}- manifold M is called symplectic, if it is equipped with a symplectic structure, that is, with a closed 2-form ω that induces a skew symmetric isomorphism ω : T M ^{∼}^{}T^{∗}M between the tangent and cotangent bundles (in particular, M should be even dimensional). The symplectic structure provides the commutative algebra of smooth functions C^{∞}(M) with the Lie algebra structure given by the Poisson bracket {f_{1}, f_{2}} =^{def} ω(H_{f_{1}}, H_{f_{2}}) , where H_{f} =^{def} ω^{−1}(df) is the Hamiltonian vector field (or the symplectic gradient) of a smooth function f ∈ C^{∞}(M), that is, the unique vector field H_{f} on M satisfying ω(H_{f}, v) = df(v) for all vector fields v. The Hamiltonian flows defined by Hamiltonian vector fields satisfy the well known Hamilton equations and preserve the symplectic form ω.

Fig. A. ‘Quantum decay’.

0.2. Quantization dreams. Ideally, a quantization is a procedure Q assigning to each symplectic manifold (M, ω) a Hilbert space^{1} Q(M, ω), and to each function f on M a self-adjoint operator Q(M, ω) ^{Q(f)}^{}Q(M, ω) such that Q(f) depends on f linearly, the constant functions are represented by the corresponding scalar homotheties, and also some ‘irreducibility’ and the Dirac commutator condition [Q(f_{1}), Q(f_{2})] = i Q({f_{1}, f_{2}}) hold. Unfortunately (or fortunately, maybe), such a procedure can not exist by Van Hove’s theorem (see [AM]). Mathematical attempts to budge this problem are concentrated in enlarging a classical area and, simultaneously, relaxing some requirements on a correspondence between the quantum and classical objects.

1called the space of wave functions

0.3. The big triple. There are several ways to move amendments to the above quantization rules. The ‘quantum mathematics’ looks today like a radioactive decay via algebraic, analytic, and geometric emanations shown on fig. A. The first two directions are developed especially wide and extensive. Say, algebraic quantization (deformation quantization, ∗-products, e. t. c.) sometimes completely leaves the original setup and forgets the classical phase space (M, ω). In a contrast, the analytical approach rests very close to the origin and produces a lot of special tricks to keep in mind the classical situation.

Fig. B. The Geometrical World.

Geometrically, most of modified quantization rules aim at a choice of a suitable function class that can be quantized in a consistent fashion. This choice is nothing but a choice of some auxiliary structure on the manifold. But although several (rather limited) uniqueness results are known in the framework of such the theories, much more interesting and important phenomenon is that we know also many special situations where the result of a quantization procedure does not depend on the choice of the auxiliary data to some extent. So, there is a heuristic fruitful lurking of a quantization scheme, which is more or less unique and grosso modo obeys the above rules. The main aim of this paper is to develop the Geometrical part of fig. A in order to tie in all three branches together. Such a fusion requires a new power geometry, which could be putted in the kernel of the geometrical picture of the world (see fig. B). We call it ALAG, that means a synthesis of the Algebraic and Lagrangian geometries. Let us describe briefly its basic ingredients and their interaction with different rays on the fig. A.

0.4. Deformation quantization. Let A = C^{∞}(M)[[q]] be an algebra of the formal power series in a variable q over C^{∞}(M). An associative ℂ[[q]]-linear product ∗ on A is called a star product, if ∀ f_{1}, f_{2} ∈ C^{∞}(M)

f_{1} ∗ f_{2} ≡ f_{1} · f_{2} ( mod q)  (0-1)
\frac{1}{q} (f_{1} ∗ f_{2} − f_{2} ∗ f_{1}) ≡ −i \{f_{1}, f_{2}\} ( mod q)  (0-2)

The first condition means that A/qA = C^{∞}(M) and we can write

f_{1} ∗ f_{2} = f_{1} · f_{2} + ∑_{i=1}^{∞} C_{i}(f_{1}, f_{2})q^{i} ,

for some C_{i}(f_{1}, f_{2}) ∈ C^{∞}(M) that are ℂ-linear in f_{j} and satisfy the equation

C_{1}(f_{1}, f_{2}) − C_{1}(f_{2}, f_{1}) = −i{f_{1}, f_{2}} ,  (0-3)

because of the second condition (0-2). The algebra A equipped with a star product, that is, with a collection {C_{i}(f_{1}, f_{2})} satisfying (0-3), is called a deformation quantization of C^{∞}(M).

The concept of deformation quantization has a long story … About twenty years ago Berezin, Bayen, Flato, Frosdal, Lichnerowicz and Sternheimer consider the quantization procedure as a deformation of the usual commutative product of classical observables into a new non commutative associative product ∗_{h} that depend on the Planck constant h and satisfies the requirements listed in n^{∘} 0.2 (see [BFFLS]). Subsequently, the theory flows into the pure algebraic framework combining the Hochschild cohomology theory for the associative algebra deformations and the Chevalley - Eilenberg cohomology theory for the Poisson - Lie algebra of classical observables. The problem of existence and classification of ∗-products on a symplectic manifold was solved by Fedosov and De Wilde - Lecomte (see survey [De]). The same problem for any Poisson manifold was solved by M. Kontsevich who presents an universal formula for the deformation quantization of an arbitrary Poisson structure on ℝ^{n}. His formula expresses a ∗-product as a power series in h in terms of bidifferential operators parametrized by graphs. Actually, it gives a perturbation expansion for a 2-dimensional field theory^{1} on a disc with boundary (see [CF]) and the coefficients in h are given by integrals over suitable configuration spaces.

However, all these results (and their proofs) do not exceed the bounds of the formal algebra, that is, do not yield Hilbert spaces and/or allow to reconstruct either manifolds or auxiliary geometric structures. Only in the Kählerian case there is known Karabegov’s result on a deformation quantization ‘with the separation of variables’ where the algebraic machinery respects a complex structure on a manifold.

1in the topological sigma-model of B type

So, if we do not want to kill the geometry or give rise a ‘non commutative’ one, we have to return to the starting point.

0.5. Geometric Quantization. The original area containing the both classic and quasiclassic systems is the Symplectic geometry. The main heroes here are the Lagrangian submanifolds (see the ‘symplectic creed’ in [We1, p. 5]). Moreover, the symplectic geometry on a given (M, ω) may be treated as a ‘category’ with the Lagrangian submanifolds as the morphisms (see [We1] again) and this viewpoint is very closed to the motivic ideology in the algebraic geometry. On the other hand, Boutet de Monvel and Guillemin has developed a quantization procedure for the contact manifolds. It leads to a remarkable unification of the spectral theory of the Fourier integral operators with the algebraic-geometric theory of the Hilbert polynomials (comp. with n^{∘} 4.4.2 below). So, the advanced ‘symplectic’ developments a posteriori come naturally into some classical algebraic-geometric framework.

0.6. From Symplectic to Algebraic. The straightforward way for sending the Symplectic geometry into the Algebraic one is a Kähler polarization, which equips M with an integrable complex structure I compatible with ω, that is, converts M to a Kähler manifold M_{I} with the Kähler form ω. If we assume, in addition, that the cohomology class [ω] is integer, then there exists a complex line bundle L on M with c_{1}(L) = [ω]. Moreover, this bundle admits a Hermitian metric h and a compatible Hermitian connection a with the curvature 2πi ω. Since the Hodge type of the Kähler form ω is (1,1), we get a natural holomorphic structure on L induced by h and a. So, M_{I} turns to an algebraic variety polarized by L. Now, the Kodaira embedding theorem implies that L^{⊗k} is very ample for k ≫ 0, that is, a natural map into a projective space:

φ_{k} : M_{I} ^{⊂} ^{x↦Ann (x)} ^{}ℙH^{0}(M_{I}, L^{⊗k})^{∗} .  (0-4)

is a holomorphic embedding. So, we are inside the Projective Algebraic Geometry.

On the other side, H^{0}(M_{I}, L^{⊗k}) ⊂ C^{∞}(M, L^{⊗k}) is a subspace in the space of all smooth sections. C^{∞}(M, L^{⊗k}) is normed by the Liouville volume form Ω = ω^{∧n} on M and the Hermitian form ( ∗ , ∗)_{L^{⊗k}} on L^{⊗k} via

⟨ s_{1} , s_{2} ⟩_{k} =^{def} ∫_{M}(s_{1}, s_{2})_{L^{⊗k}} Ω , ∥s∥_{k} =^{def} \sqrt{⟨ s , s ⟩_{k}} ,

where s, s_{1}, s_{2} ∈ C^{∞}(M, L^{⊗k}) . Let us write L^{2}(M, L^{⊗k}) for the completion of C^{∞}(M, L^{⊗k}) w. r. t. this norm. So, we can consider H^{0}(M_{I}, L^{⊗k}) ⊂ L^{2}(M, L^{⊗k}) as the Hilbert subspace.

0.6.1. Quantization of phase space. In the Kähler polarization framework, the Hilbert space of the global holomorphic sections

Q_{k}(M_{I}) =^{def} H^{0}(M_{I}, L^{⊗k}) ⊂ L^{2}(M, L^{⊗k})

is called a wave function space of level k. The orthogonal projector

L^{2}(M, L^{⊗k}) ^{Π_{k}}[figure] H^{0}(M_{I}, L^{⊗k})

is called a Szegö projector. The holomorphic embedding (0-4) composed with the Hermitian isomorphisms H^{0}(M_{I}, L^{⊗k})^{∗} ^{∼}^{}\overline{H^{0}(M_{I}, L^{⊗k})} gives an (antiholomorphic) embedding

M_{I} ^{⊂} ^{\overline{φ}_{k}} ^{}ℙH^{0}(M_{I}, L^{⊗k}) .  (0-5)

0.6.2. Quantization of functions. Each f ∈ C^{∞}(M) gives a Toeplitz operator:

H^{0}(M_{I}, L^{⊗k}) ^{T_{f}^{(k)}}^{}H^{0}(M_{I}, L^{⊗k}) : s ⟼ Π_{k}(f · s) .

So, we get a Berezin - Toeplitz map C^{∞}(M) ^{T_{k}}^{}End(H^{0}(M_{I}, L^{⊗k})) sending f ⟼ T_{f}^{(k)} .

0.6.3. Quasi-classic limit. The Berezin - Toeplitz map is just a linear map from a vector space underlying the (commutative) algebra of functions to the one underlying the (non commutative) matrix algebra, because it preserves neither Lie nor associative algebra structures. However, the whole infinite family {T^{(k)}}_{k≥0} ‘approximates’ the algebra C^{∞}(M) in the following sense. Consider the supremum norm ∥f∥_{∞} on C^{∞}(M) and the corresponding operator norm on End ( H^{0}(M_{I}, L^{⊗k}) ) :

∥T_{f}^{(k)}∥ = sup_{s≠0} \frac{∥T_{f}^{(k)}(s)∥}{∥s∥} .

Then the Bordemann - Meinrenken - Schlichenmaier theorem (see [BMS]) says that the Berezin - Toeplitz maps have nice asymptotics, that is, for every f ∈ C^{∞}(M) there exists a constant K_{f} > 0 such that

∥f∥_{∞} + \frac{K_{f}}{k} ≤ ∥T_{f}^{(k)}∥ ≤ ∥f∥_{∞}

and for every pair f_{1}, f_{2} ∈ C^{∞}(M) we have the following asymptotics as k → ∞

∥ k · i [ T_{f_{1}}^{(k)} , T_{f_{2}}^{(k)} ] − T_{{f_{1},f_{2}}}^{(k)} ∥ ∼ ∥ T_{f_{1}}^{(k)} · T_{f_{2}}^{(k)} − T_{f_{1}·f_{2}}^{(k)} ∥ ∼ O(1/k) .

In other words, although the Berezin - Toeplitz maps are not faithful on any finite level k, they tend to be that as k → ∞. This property is usually referred as a good quasi-classical behavior.

Returning to the deformation quantization, Schlichenmaier has proved that there exists a unique star product {C_{i}(f_{1}, f_{2})} on C^{∞}(M) such that for every pair functions f_{1}, f_{2} and every integer N > 0 there exists a constant K = K(f_{1}, f_{2}, N) satisfying

∥T_{f_{1}}^{(k)} · T_{f_{2}}^{(k)} − ∑_{0≤i≤N} \frac{T_{C_{i}(f_{1},f_{2})}^{(k)}}{k^{i}} ∥ = \frac{K}{k^{N}} .  (0-6)

So, the Berezin - Toeplitz maps provide a unique strict quantization procedure with the good quasi-classical behavior.

0.7. ALAG. The crucial algebraic-geometric fact observed in many examples of a geometric quantization output is the following: although the wave function spaces Q_{k}(M_{I}) do depend a priori on a choice of the complex structure I on M, their projectivizations ℙ ( Q_{k}(M_{I}) ) do not, actually. More precisely, there is a holomorphic vector bundle Q_{k} ^{π_{k}}^{}M over the moduli space M, of polarized complex structures on M, such that its fiber over I ∈ M equals Q_{k}(M_{I}) and there is a natural projective flat connection (see [Hi1]) on this bundle.

This phenomenon has a very simple geometrical reason: certain finite codimensional submanifolds in the space of lagrangian cycles admit some internal^{1} additional structures that make them ‘polarized algebraic varieties’ and put into Algebraic Geometry. These algebraicgeometric structures are universal in the sense that an Algebraic Geometry of any Kähler quantization procedure is naturally induced by them. On the other side, these infinite dimensional algebraic moduli spaces, of Lagrangian cycles with some extra data, can be handled by the standard algebraic-geometric tools. We hope to explain in this paper how it works actually.

1that is, induced only by the symplectic structure on M

0.7.1. The text below is organized as follows. In the first two paragraphs we fix some

(mostly well known) basic concepts from symplectic and differential geometry: moduli of Lagrangian cycles, Darboux - Weinstein uniformization near a smooth cycle, interaction between symplectic and almost Hermitean structures, Abelian connections and characters of π_{1}, Jacobians e. t. c. The third paragraph introduces the special submanifolds and extra structures coming from the moduli space of Lagrangian cycles: Bohr - Sommerfeld cycles, Planckian cycles, Berry bundles, half densities e. t. c. We construct here the holomorphic structures on the spaces of half weighted Planckian cycles P^{hw} and half weighted Bohr - Sommerfeld cycles of fixed volume B_{t}^{hw}. In § 4 we give a detailed explanation of what is the Kähler quantization procedure and construct the holomorphic BPU-maps Bhw_{k,t} [figure] ℙ ( H^{0}(L^{⊗k})^{∗} ). This is the central point of the present paper. The last § 5 illustrates our construction in the simplest possible example, where the initial symplectic system (M, ω) is the Rimannian sphere S^{2} ≃ ℂℙ_{1} equipped with the standard Kähler structure and the Lagrangian cycles are just the immersed circles. The half weighted Planckian cycles space P^{hw} turns here to the space of parametrized knots in S^{3}, which is fibered over S^{2} by Hopf. So, although this case is ‘almost trivial’ with the ALAG-viewpoint, it gives the reasonable geometric envelop for the whole of the Legendrian knots theory, which is so popular now, because of plays especially important role in many CFT-questions.

0.8. Acknowledgments. We would like to express our gratitude to the Max-Planck-Institute für Mathematik in Bonn for the hospitality and very nice atmosphere for a good work.

§ 1. Lagrangian preliminaries.

1.1. Symplectic manifold. Let us fix the notation (M, ω) for a smooth symplectic 2ndimensional manifold M with the symplectic structure presented by a smooth closed 2-form ω giving a skew isomorphism

ω : T M ^{∼}^{}T M^{∗}  (1-1)

between tangent and cotangent bundles on M. Such a manifold M may be considered as a phase space for a classical mechanical system like in [A].

1.2. Lagrangian cycles. Let us fix a smooth compact oriented n-dimensional C^{∞}-manifold S and also some homotopy class Map_{la}(S, M), of smooth maps S ^{φ}^{}M such that φ^{∗}(ω) = 0. Such the maps are called Lagrangian. They form an infinite dimensional space with a natural action of the identity component of the diffeomorphism group Diff_{0}(S) via h : φ ↦ φ ∘ h^{−1}, where h ∈ Diff_{0}(S), φ ∈ Map_{la}(S, M). The orbit space 𝔏 =^{def} Map_{la}(S, M)/Diff_{0}(S) is called a space of Lagrangian cycles (of a given topological type).

A Lagrangian cycle S ^{f}^{}M is called smooth (resp. smoothly immersed), if φ is a smooth embedding (resp. a smooth immersion). In the first case, f(S) ⊂ M is called a smooth Lagrangian submanifold of M and the condition φ^{∗}ω = 0 means that ω restricts to zero on T S. But in the both cases, the symplectic structure isomorphism (1-1) identifies the normal bundle ( N_{M/S}) = φ^{∗}(T M)/T S with the cotangent bundle T^{∗}S ≃ φ^{∗}(T^{∗}M)/Ann (T S). To simplify the notations, we will often write S for φ(S) as well as soon the mistakes will be impossible (and for embedded or immersed cycles only).

1.2.1. Methodological remark. We will not investigate any details connected with the topology of 𝔏, its singularities, stability conditions, and so on. We use the quotient notation Map_{la}(S, M)/Diff_{0}(S) rather as a very convenient tool to activate the geometrical intuition. All our ‘global’ constructions always can be formulated in terms of some Diff_{0}(S) - invariant objects on Map_{la}(S, M). So, we treat 𝔏 rather in a framework of the invariant theory and, after S. Donaldson, interpret special structures on the space of Lagrangian cycles in terms of moment maps and symplectic and/or Kähler reduction.

On the other hand, each smooth Lagrangian cycle gives a smooth point on 𝔏 together with a standard open neighborhood that can be modelled by the following very tame example.

1.3. Regular local example: the cotangent bundle. Let M be the total space of the cotangent bundle T^{∗}S ^{p} [figure] S on a smooth manifold S. Then there is a tautological 1-form η on M. It is called an action form and sends a tangent vector w ∈ T_{(ξ,x)}T^{∗}S, where x ∈ S, ξ ∈ T_{x}^{∗}S, to

η(w) =^{def} p^{∗}ξ(w) .  (1-2)

The differential of the action form ω =^{def} dη gives a canonical symplectic structure on M. If we take some local coordinates (q_{1}, q_{2}, …, q_{n}) on a small ball U ⊂ S and coordinate T^{∗}S|_{U} by ( p_{1}, p_{2}, …, p_{n} , q_{1}, q_{2}, …, q_{n}), where p_{i} are linear coordinates along ∂/∂q_{i} in the fibers of the cotangent bundle, then

η(p, q) = ∑_{j=1}^{n} p_{j}dq_{j} ,  ω(p, q) = ∑_{j=1}^{n} dp_{j} ∧ dq_{j} .  (1-3)

So, such coordinates are the symplectic ones. They are called canonical. One checks immediately that any smooth q-coordinates change on S induces a symplectic (p, q)-coordinates change on M.

Clearly, all fibres of the canonical projection M [figure] S and also the zero section S ^{⊂} ^{s_{0}}^{}M are Lagrangian submanifolds. It is not difficult to describe all Lagrangian sections S ^{⊂} ^{s}^{}M. Namely, any section is presented by a differential form α_{s} = s^{∗}(η), such that

s(x) = (x, α_{s}(x)) ∈ M = T^{∗}S .

1.3.1. PROPOSITION. A section S ^{⊂} ^{s}^{}T^{∗}S is Lagrangian iff the corresponding form α_{s} ∈ C^{∞}(X, T^{∗}S) is closed.

Proof. X ^{s}^{}M is Lagrangian iff 0 = s^{∗}(ω) = s^{∗}(dη) = ds^{∗}(η) = dα_{s}. □

In fact, the zero section S ^{⊂} ^{s_{0}}^{}M gives the universal local model for a smooth Lagrangian cycle because of the famous Darboux - Weinstein theorem discussed in the next section.

1.4. Local structure of 𝔏 near a smooth cycle. Let S ^{φ}^{}M be a smooth Lagrangian immersion. The tangent space T_{φ}Map(S, M) to the space of all smooth maps at a given map S ^{φ}^{}M coincides with the space C^{∞}(S, φ^{∗}T M) of smooth sections of the pulled back tangent bundle on M: a small deformation of φ given by a local vector field v near φ(S) is the shift by its integral flow. Such a shift preserves the condition φ^{∗}(ω) = 0 iff φ^{∗}(L_{v}ω) = 0 and is tangent to Diff_{0}(S)-orbit of φ iff v|_{φ(S)} is tangent to φ(S). So,

T_{φ}(Map(S, M)/Diff_{0}(S)) = { v ∈ C^{∞} ( S, N_{M/S}) | φ^{∗}(Lie_{v}ω) = 0 } ,

where v is considered as a class of local vector fields near φ(S) modulo such the fields tangent to φ(S) (this interpretation for the sections of the normal bundle is correct, because any tangent field on φ(S) can be extended onto a small tube around φ(S)).

1.4.1. PROPOSITION. If S ^{φ}^{}M is a smooth Lagrangian immersion, then T_{φ}𝔏 is naturally identified with the space Z_{dr}^{1} of all closed 1-forms on S. Proof. Sending a normal vector field v to the 1-form ξ = φ^{∗}(i_{v}ω) ∈ C^{∞}(S, T^{∗}S), we identify the normal bundle N_{M/S} with the cotangent bundle T^{∗}φ(S) (like in n^{∘} 1.2). Under this identification, the tangent fields v ∈ T_{φ}𝔏 go to the closed 1-forms, because of

dξ = dφ^{∗}(i_{v}ω) = φ^{∗}(di_{v}ω) = φ^{∗}(L_{v}ω + i_{v}dω) = φ^{∗}(L_{v}ω) = 0 .
□

1.4.2. The Darboux - Weinstein theorem (see [GS, Ch.4, § 1], [We3, § 6], and [We4,

§ 4]) implies that for any smoothly immersed Lagrangian cycle S ^{⊂} ^{φ}^{}M there exist a tube neighborhood U of the zero section in the cotangent bundle T^{∗}S and a smooth immersion U ^{⊂} φ̂^{}M such that φ̂^{∗}ω = dη is the canonical symplectic form on T^{∗}S and φ is the restriction of φ̂ on the zero section. Moreover, If φ is the smooth embedding, then we can take U = T^{∗}S and φ̂ being a symplectomorphism between the whole of T^{∗}S and some smooth open tubular neighborhood of f(S) in M.

Further, the immersion T^{∗}S ⊃ U φ̂^{}M induces (by sending a closed 1-form to its graph) a smooth map

T_{S}𝔏 ⊃ 𝔘 ^{ε}^{}𝔏 ,  (1-4)

where 𝔘 ⊂ Z_{dr}^{1}(S) consist of closed 1-forms ξ whose graphs lay inside U. It follows from the Weinstein’s results (see [We3, § 6], [We4, § 4] already cited) that the image of (1-4) contains an open neighborhood of S in 𝔏 considered in suitable topology. So, we can release any small enough deformation of a smoothly immersed Lagrangian cycle as a graph of a closed 1-form laying in a small neighborhood of zero in Z_{dr}^{1}(S). We will refer the map (1-4) as the exponential map.

1.4.3. Isodrastic deformations. Let b_{1} = dim H^{1}(S, ℝ). The subspaces of exact 1-forms

B_{dr}^{1}(S) ⊂ Z_{dr}^{1}(S) = T_{S}𝔏

form a plane distribution of codimension b_{1} in the tangent bundle to the space of Lagrangian cycles. The exponential map (1-4) shows that (locally, nearby a smooth cycle) this distribution is integrable to a foliation whose leafs have codimension b_{1}: if we identify a small neighborhood of a given cycle with a small neighborhood of zero in Z_{dr}^{1}, then small deformations tangent to Z_{dr}^{1} will be just the shifts by exact forms. Experts call such the deformations isodrastic (exhausting linguistic comments may be found in [We2]). Clearly, the graphs of two closed 1-forms may be obtained from each other by an isodrastic deformation iff these forms are cohomologous.

1.5. Global geometric model: almost Kähler manifolds. Main efforts of this work will be directed towards a studying of an interaction between the previous local cotangential model for 𝔏 and the following main geometric source of the symplectic structures.

1.5.1. Almost Hermitian manifolds. A smooth real 2n-dimensional manifold M is called almost Hermitian, if its tangent bundle has a structure of complex Hermitian bundle of rank n. This means that T M admits a linear operator T M ^{I}^{}T M, a Rimannian metric g : T M ^{∼}^{}T M^{∗}, and a non degenerate skew symmetric form ω : T M ^{∼}^{}T M^{∗} such that I^{2} = −Id, ω(v, Iw) = g(v, w), ω(v, Iw) = −ω(Iv, w), and g(v, Iw) = −g(Iv, w). Actually, the triple (I, g, ω) is recovered from any two of its elements.

Certainly, any holomorphic complex manifold is almost Hermitian as well. On the other side, one can develop a kind of “pseudo holomorphic” geometry on any almost Hermitian manifold. Since the complexified tangent bundle ℂ ⊗ T M = T_{+}M ⊕ T_{−}M splits onto direct sum ±i-eigensubspaces for I, we can define a pseudo holomorphic function M ^{f}^{}ℂ as a one whose differential df annihilates T_{−}M. Similarly, we can consider pseudo holomorphic submanifolds, pseudo holomorphic complex vector bundles e. t. c.

1.5.2. PROPOSITION. Let V ⊂ M be a pseudo holomorphic (possible, singular) subvariety in an almost complex manifold M (that is, V is presented locally as a zero set of some system of pseudo holomorphic functions) and T C_{x}V be the tangent cone to V at an arbitrary point x∈V. Then ω has non degenerate restriction onto T C_{x}V, that is, for any non zero v ∈T C_{x}V there is u∈T C_{x}V such that ω(v, u) ≠ 0. Proof. Since V is pseudo holomorphic, the complex structure operator I preserves T C_{x}V and, hence, acts on T C_{x}V bijectively, because of I^{−1} = −I. So, since the symmetric form g is positive, for any non zero v ∈ T C_{x}V we have ω(v, Iv) = g(v, v) > 0 and can take u = Iv ∈ T C_{x}V. □

1.5.3. DEFINITION. An almost Hermitian manifold M is called almost Kähler, if its skew symmetric form ω is closed, that is, defines the symplectic structure on M.

In some sense, the Lagrangian geometry of almost Kähler manifolds is perpendicular to their pseudo holomorphic geometry.

1.5.4. COROLLARY. A non zero section of a pseudo holomorphic line bundle can not vanish along a Lagrangian cycle. Proof. The zero set of a pseudo holomorphic section is a pseudo holomorphic (2 n − 2)- dimensional variety in M. If it contains an n-dimensional Lagrangian subvariety, then the restriction of ω should be degenerated along it. This contradicts with n^{∘} 1.5.2. □

1.5.5. Tame example: tangent bundles. Let X be an oriented smooth n-dimensional manifold. Then there is a canonical complex structure on its tangent bundle T X. Namely, the tangent space T T X splits canonically as T X ⊕ T X, where the first summand is the ‘vertical’ kernel of the differential of the canonical projection T X [figure] X and the second one is the ‘horizontal’ image of the differential of the zero section X ^{⊂} ^{}T X. In terms of this splitting, the multiplication by i ∈ ℂ at a point (v, q) ∈ T X, where v ∈ T_{x}X, q ∈ X, takes a tangent vector w = (w_{1}, w_{2}) ∈ T_{(}v, q)T X to (−w_{2}, w_{1}). This complex structure is, clearly, integrable: each local coordinate system (y_{1}, y_{2}, …, y_{n}) on X near q ∈X produces the holomorphic local coordinates (z_{1}, z_{2}, …, z_{n}) on T X, where z_{ν} =^{def} x_{ν} + i y_{ν} and x_{ν} is linear coordinate along ∂/∂y_{ν} in the fiber T_{q}X. One can check immediately that smooth coordinate changes on X produce the holomorphic coordinate changes on T X and also that the transition functions for the tangent bundle produce the holomorphic gluing rules for the complex charts just described. In terms of these holomorphic coordinates, the multiplication by i ∈ ℂ on T_{(}v, q)T X sends

( \frac{∂}{∂x}, \frac{∂}{∂y} ) ⟼ ( \frac{∂}{∂y} , −\frac{∂}{∂x} )

On the other side, the cotangent bundle T^{∗}X has the canonical symplectic structure ω = dη whose symplectic coordinates are of the form (dy, ∂/∂y) (see n^{∘} 1.3). So, if there is an isomorphism of bundles T X ^{β}^{}T^{∗}X, then we get immediately an integrable weak Kähler structure on the both T X, T^{∗}X, that is, a triple (I, g, ω) which satisfy all conditions from n^{∘} 1.5.1 except for positivity of the symmetric form g(u_{1}, u_{2}) = ω(u_{1}, Iu_{2}).

For example, if X is equipped with a Rimannian metric T X ^{G}^{}T^{∗}X, then its cotangent bundle M = T^{∗}X admits an integrable almost complex structure

I : (ξ, v) ⟼ (Gv, −G^{−1}ξ)

compatible with the canonical symplectic structure ω = dη from (1-3). In this case, the symmetric form g(u_{1}, u_{2}) = ω(u_{1}, Iu_{2}) is positive as well and we get an integrable almost Kähler structure on T X.

§ 2. Abelian preliminaries.

2.1. Complex line bundles. In this paragraph we would like to fix some basic notations and concepts concerning complex line bundles, which have an Abelian structure group: U(1) or its complexified version GL_{1}(ℂ) = ℂ^{∗}. Topologically, a complex line bundle L [figure] X on a smooth manifold X is defined up to an isomorphisms by the cohomology class of the transition functions between local trivializations of L. These transition functions form the C̆ech 1-cocycle with coefficients in the sheaf of smooth ℂ^{∗}-valued functions C^{∞}(X, ℂ^{∗}). Its cohomology class is called a first Chern class (see, say [GH]):

c_{1}(L) ∈ H^{1}(X, C^{∞}(X, ℂ^{∗})) =^{δ} H^{2}(X, ℤ) ,

where the isomorphism δ is induced by the exponential exact triple of sheaves

0 ^{}ℤ ^{}C^{∞}(X, ℂ) ^{f↦e^{2πif}}^{}C^{∞}(X, ℂ^{∗}) ^{}0

because C^{∞}(X, ℂ) is flabby.

2.2. Hermitian line bundles. Each complex Hermitian line bundle L [figure] X is associated via tautological representation of U(1) with a principal U(1)-bundle P ^{p}[figure] X, that is, L = P ×_{U(1)} ℂ. So, P has inverse transition functions and is naturally identified with the bundle of unit circles in L^{∗} (in particular, P and L are uniquely recovered from each other).

We identify the Lie algebra 𝔰𝔲(1) with ℝ by sending t ⟼ 2πi t. So, the exponential map looks like

ℝ ^{t↦e^{2πit}} [figure] U(1)

and the tautological representation of 𝔰𝔲(1) sends t ∈ ℝ to the skew Hermitian operator z ⟼ 2πi tz. In particular, the ℝ-valued differential forms may be considered as 𝔲(1)-valued via this representation. We denote by ν the canonical invariant 1-form on U(1) such that

∫_{U(1)} ν = 1 and denote by u the dual invariant vector field satisfying ν(u) = 1. Corresponding

invariant vertical 1-form and vector field on P will be denoted by ν and u as well.

2.2.1. A Hermitian connection a on a principal U(1)-bundle P ^{p} [figure] X is a differential 1-form α = α_{a} on P such that

Lie_{u}α = 0 and α(u) = 1 .

The first condition means that α is U(1)-invariant; the second says that T P ^{α} [figure] T U(1) gives an U(1)-equivariant splitting of the following canonical exact triple of vector bundles on P:

0 ^{}T U(1) [figure: .; .; .; .; .; .; .; .; α] T P ^{dp}^{}p^{∗}T X ^{}0 .

The kernel hyperplanes of α in T P are called horizontal. They are identified with the fibers of p^{∗}T X by dp. So, the space of all Hermitian connections A_{h}(L) is an affine space over C^{∞}(X, T^{∗}X), because the difference between two splittings is a linear map p^{∗}T X ^{}T U(1), that is, a pull back of a 1-form on X. The connection lifts any vector field v on X to a unique U(1)-invariant horizontal vector field ṽ on P such that dp(ṽ) = v. Namely, if w is any U(1)-invariant vector field on X projected to v by dp, then ṽ = w − α(w)u.

2.2.2. Covariant derivative. In terms of the associated Hermitian line bundle L, the connection a is presented by a covariant derivative ∇ = ∇_{a}, which is a differential operator

C^{∞}(X, L) ^{∇_{a}}^{}C^{∞}(X, T^{∗}X ⊗ L)

sending a local unitary basic section σ over a small open disk U ⊂ X to ∇σ = 2πi ξ_{σ} ⊗ σ, where ξ_{σ} = σ∗_{P}α is a pull back of the connection form along a local section U ^{⊂} ^{σ_{P}}^{}P induced^{1} by σ. Then any other local section fσ, where f ∈ C^{∞}(U, ℂ), is differentiated as

∇(fσ) = (df + 2πi fξ_{σ}) ⊗ σ .  (2-1)

and a local unitary base change σ ⟼ σ′ = hσ with h ∈ C^{∞}(U, U(1)) takes ξ_{s} to

ξ_{σ′} = ξ_{σ} − \frac{i}{2π} \frac{dh}{h} = ξ_{σ} − \frac{i}{2π} d log h .  (2-2)

In particular, the local covariantly constant sections fσ are given by the differential equation

d log f = −2πi ξ_{σ}  (2-3)

Of course, it is locally solvable iff dξ_{σ} = 0.

2.2.3. Curvature. It follows from (2-2) that the differentials dξ_{σ} do not depend on a local unitary base choice, that is, there is a global closed 2-form ω_{a} ∈ C^{∞}(X, Λ^{2}T^{∗}X) such that ω_{a}|_{U} = dξ_{σ} for any unitary section σ over any small open ball U. The base change formula (2-2) shows also that the cohomology class [ω_{a}] ∈ H^{2}(X, ℝ) is integer and coincides with the first Chern class:

1actually, this section is defined only up a U(1)-action but it does not affect on α

[ω_{a}] = c_{1}(L) ∈ H^{2}(X, ℤ) ⊂ H^{2}(X, ℝ) .

A 𝔰𝔲(1)-valued form F_{a} =^{def} 2πi ω_{a} is called a curvature form of a. But we will work mostly with the original real valued 2-form ω_{a} and refer it as the real 2-form associated with the curvature of a. In terms of the principal bundle P ^{p} [figure] X, the pull back p^{∗}ω_{a} = dα_{a} sends a pair of U(1)-invariant vector fields v_{1}, v_{2} on P to

dα(v_{1}, v_{2}) = (Lie_{v_{1}}α − d ( α(v_{1}) ))(v_{2}) = Lie_{v_{1}}α(v_{2}) = −α([v_{1}, v_{2}]) .

In terms of the covariant derivative, the second order differentiation

C^{∞}(X, L) ^{∇_{a}^{2}}^{}C^{∞}(X, Λ^{2}T^{∗}X ⊗ L) ,

induced by ∇_{a} via Leibnitz rule is nothing but a C^{∞}(X)-linear operator multiplying the sections by 2πi ω_{a} = F_{a}, because of ∇^{2}σ = 2πi dξ_{σ} ⊗ σ.

Let us remark finally that the curvature presents a moment map for the natural action of the gauge group C^{∞}(X, U(1)) on the space of Hermitian connections. Indeed, a gauge transformation X ^{⊂} ^{h}^{}U(1) takes α to α + p^{∗}f^{∗}ν and preserves the curvature, because the basic unitary invariant vertical form ν on P is closed.

2.2.4. Flat Hermitian connections. A Hermitian connection is called flat if its curvature is the zero form. In this case, L is topologically trivial, that is has a global never vanishing section X ^{⊂} ^{σ}^{}L. Moreover, the equation (2-3) becomes solvable over any simply connected open set U and its solutions are given by the formula

f = c e^{−2πig} ,  (2-4)

where c ∈ ℂ is an arbitrary constant and g is a smooth function on U (defined up to an additive constant) such that dg = ξ_{σ}. So, L admits a covariant trivialization over any simply connected open set. Such the trivialization by some f from (2-4) is unitary iff |c| = 1. So, the unitary covariant trivializations form a principal U(1)-homogeneous space and these spaces over different U’s are glued together precisely into the principal bundle P associated with L.

Let us take the universal covering X̃ ^{π}[figure] X and fix some covariant trivialization for the pulled back bundle π^{∗}L [figure] X̃ by some global covariantly constant unitary section X̃ ^{⊂} σ̃^{}π^{∗}L. Identifying C^{∞}(X̃, ℂ) with C^{∞}(X̃, π^{∗}L) by taking f ↦ fσ̃, we get a representation of π_{1}(X) in the space of the constant unitary global sections of π^{∗}L. The equivalence class of this representation does not depend on the choice of σ̃ and is uniquely defined by its character π_{1}(X) ^{χ}^{}U(1). In terms of L, this character gives the holonomy of the connection a, that is, χ(γ) equals a fiber shift under the covariant parallel transport along γ. The cohomology class of χ in H^{1}(X, U(1)) is preserved by the gauge transformations and goes to 0 = c_{1}(L) in the long exact cohomology sequence

0 ^{}H^{1}(X, ℤ) ^{}H^{1}(X, ℝ) ^{}H^{1}(X, U(1)) ^{}H^{2}(X, ℤ) ^{}⋯

associated with the exact triple of the constant sheaves

0 ^{}ℤ ^{}ℝ ^{t↦e^{2πit}} ^{}U(1) ^{}0 .

So, the cohomology classes of characters χ, that is the gauge classes of the flat Hermitian connections on L, are naturally parametrized by the real torus

J_{S} =^{def} H^{1}(S, ℝ)/H^{1}(S, ℤ)

called a Jacobian of X.

2.3. Supercycles. A Lagrangian cycle S ^{}M equipped with a flat U(1)-connection a on the trivial complex line bundle C = S × ℂ is called a supercycle or a brane supported by S. We consider the sections of C as functions on S. There is a distinguished supercycle (S, a_{0}) whose connection form α_{0} = ν coincides with the unitary vertical form ν on P and the covariant differentiation ∇_{0} = d coincides with the external differential. So, we can identify the affine space of flat connections A_{fh}(C) with the underlying vector space Z_{dr}^{1}(S) by sending α_{a} and ∇_{a} to α_{a} − α_{0} and ∇_{a} − ∇_{0}. We define a space of Lagrangian supercycles as an orbit space

𝔏^{s} = Map_{la}(S, M) × A_{fh}(C)/Diff_{0}(S) ,  (2-5)

where A_{fh}(C) is the space of flat Hermitian connections and Diff_{0}(S) acts diagonally. It is fibered over 𝔏 via forgetful map

𝔏^{s} ^{(S,a)↦S}[figure] 𝔏 .

Of course, this map is nothing but the tangent bundle 𝔏^{s} ≃ T 𝔏. So, the supercycle space 𝔏^{s} has the natural integrable complex structure of a tangent bundle (see n^{∘} 1.5.5). If we identify a small neighborhood of S ∈ 𝔏 with the small tube around the zero section S ^{⊂} ^{}T^{∗}S and represent Lagrangian cycles nearby S by the closed 1-forms (see n^{∘} 1.4.2), then the complex coordinate system on 𝔏^{s} near (S, α) consists of ξ_{1} + i ξ_{2} that runs trough Z_{dr}^{1}(S, ℂ) and corresponds to a supercycle supported by the graph of ξ_{1} and equipped with the flat connection α + ξ_{2}, where ξ_{1}, ξ_{2} are real closed 1-forms on S.

2.3.1. Isodrastic deformations of supercycles. The tangent space T_{(S,a)} = Z_{dr}^{1}(S, ℂ)

contains a complex subspace B_{dr}^{1}(S, ℂ), of exact forms. We refer this distribution of complex planes as a complex isodrastic distribution. The same arguments as in n^{∘} 1.4.3 show that it is locally integrable near each smooth supercycle to some complex foliation.

2.4. Jacobian bundles. Let C = S × ℂ be the trivial complex line bundle, as before. A semidirect product G =^{def} Map(S, U(1)) ⋊ Diff_{0}(S) , where Diff_{0}(S) acts on Map(S, U(1)) as hγh^{−1} =^{def} γ ∘ h^{−1}, is the group of diffeomorphisms L ^{∼}^{}L sending fibers to fibers by linear unitary transformations. It acts naturally on Map_{la}(S, M) × A_{fh}(C) and the orbit space

J =^{def} Map_{la}(S, M) × A_{fh}(C)/G

is called a Jacobian bundle over 𝔏. Its fiber over a given S consists of the gauge classes of flat U(1)-connections on C (see n^{∘} 2.2.4) and is naturally identified with the Jacobian

J_{S} = H^{1}(S, ℝ)/H^{1}(S, ℤ) .

If we consider ℂ^{∗}-connections on C instead of the Hermitian ones and factorize them by ℂ^{∗}-gauge group C^{∞}(S, ℂ), then we get a complex Jacobian bundle

J^{ℂ} [figure] 𝔏

whose fiber over a Lagrangian cycle S is a complex torus

J_{S}^{ℂ} =^{def} H^{1}(S, ℂ)/H^{1}(S, ℤ) .

This torus parametrizes the ℂ^{∗}-valuated characters π_{1}(S) χ̃^{}ℂ^{∗}. It is called a complex Jacobian of S. There is a natural surjection J_{S}^{ℂ} ^{r} [figure] J_{S} sending

χ̃ ⟼ χ̃/|χ̃| .

2.4.1. Remark. Since T_{S}𝔏 = Z_{dr}^{1}(S) and T J_{S} = H^{1}(S, ℝ), there is a canonical linear map from the base tangent space to the fiber tangent space of the real Jacobian bundle. This map sends a closed form to its cohomology class. The existence of such the map will play maybe hidden but important role in almost all our constructions.

On the other side, the Jacobian bundle may be equipped, certainly, with the Gauss - Manin connection, which comes from a local uniform base choice in the lattice H^{1}(S, ℤ) via the exponential map. In terms of this connection, say the complex isodrastic deformations may be described as follows. Each infinitesimal deformation ξ = (ξ_{1} + i ξ_{2}) ∈ T_{(S,a)}𝔏^{s} of a supercycle (S, a) defines a flat ℂ^{∗}-connection α + ξ_{1} + i ξ_{2} whose ℂ^{∗}-gauge class gives a point χ̃_{ξ} ∈ J_{S}^{ℂ}. A family of supercycles is complex isodrastic, if the real projection of this character r(χ̃_{ξ}) = χ̃_{ξ}/|χ̃_{ξ}| ∈ J_{S} is a covariantly constant section of the real Jacobian bundle w. r. t. the Gauss - Manin connection. As soon as we will not study the global topology of 𝔏 in this paper, we will not need such type constructions and stop here their further development.

§ 3. Prequantization equipment.

3.1. Prequantization data. Suppose that the cohomology class of the symplectic form ω on M is integer [ω] ∈ H^{2}(M, ℤ) ⊂ H^{1}(M, ℝ) , that is, [ω] = c_{1}(L) presents the first Chern class of some complex line bundle L [figure] M. We fix a Hermitian structure on L and equip it with a Hermitian connection a with the curvature F_{a} = 2πi ω; so, the real 2-form associated with this curvature (in accordance with our agreements in n^{∘} 2.2.3) coincides with the symplectic structure: ω_{a} = ω. Let P_{k} ^{p_{k}} [figure] M be the principal U(1)-bundle associated with L^{⊗k} via tautological representation. Then for all k, m ∈ ℕ we have the commutative diagram

[figure: P_{k}; μ_{m,k}; P_{mk}; M; p_{mk}; p_{k}]  (3-1)

where P_{k} ^{μ_{m,k}}[figure] P_{mk} is an unramifided covering of degree m that takes z ↦ z^{m} in each fiber. We denote by α_{k} the (real valued) connection form induced by a on P_{k}. The curvature of this connection equals

F_{k} = 2πik ω , k ∈ ℕ ,  (3-2)

that is, has associated real 2-form ω_{k} = kω. We refer the quadruple (M, ω, P_{k}, α_{k}) as a level k prequantization data on the phase space M. The corresponding complex Hermitian line bundle L^{⊗k} is called a level k prequantization line bundle on (M, ω).

3.2. Bohr - Sommerfeld cycles. Let S ^{φ}^{}M be a Lagrangian cycle. It follows from (3-2) that φ^{∗}L^{⊗k} is the trivial line bundle with a flat connection on S, that is a supercycle (see n^{∘} 2.3). Taking its gauge for each Lagrangian cycle S ∈ 𝔏, we get a section of the Jacobian bundle 𝔏^{⊂} ^{j_{k}}^{}J (see n^{∘} 2.4). The zero set of this section is denoted by B_{k} and is called a space of level k Bohr - Sommerfeld cycles (or BS_{k}-cycles for shortness).

In other words, a Lagrangian cycle S ^{φ}^{}M is BS_{k}-cycle iff the connection φ^{∗}α_{k} on the pull back φ^{∗}L^{⊗k} has the trivial holonomy, that is, admits a covariant trivialization φ^{∗}L^{⊗k} ≃ S × ℂ. Since the covariant trivialization of φ^{∗}L^{⊗k} trivializes all the tensor powers φ^{∗}L^{⊗mk} as well, we have the natural inclusions

B_{k} ^{⊂} ^{β_{k,m}} ^{}B_{mk} ∀ k, m ∈ ℤ  (3-3)

3.3. Planckian cycles and Berry bundle. The connection 1-form α_{k} presents a contact structure^{1} on P_{k} with the volume element

α_{k} ∧ (dα_{k})^{∧n} = k^{n} α_{k} ∧ (p^{∗}ω)^{∧n}  (3-4)

Let Map_{le}(S, P_{k}) be the space of all smooth maps S φ̃^{}P_{k} such that φ̃^{∗}(α_{k}) = 0 (this means that φ(S) ⊂ P is the Legendrian submanifold w. r. t. the contact structure α_{k}). The orbit space

P_{k} =^{def} Map_{le}(S, P_{k})/Diff_{0}(S)

is called a space of level k Planckian cycles. We write φ = p_{k} ∘ φ̃ : S ^{}M for the composition of a Legendrian map S φ̃^{}P_{k} with the projection P_{k} ^{p_{k}} [figure] M. This composition has to be a BS_{k}-cycle, because of φ^{∗}ω = dφ̃^{∗}α_{k}/k = 0. So, S φ̃^{}P_{k} is just a covariant constant section of φ^{∗}P_{k}, or equivalently, a covariant constant unitary section of φ^{∗}L^{⊗k} over S. Since such the sections form a principal homogeneous U(1)-space (see n^{∘} 2.2.4), we get the principal U(1)-bundle

P_{k} ^{π_{k}} [figure] B_{k}  (3-5)

called the principal Berry bundle of level k. The Hermitian line bundle associated with P_{k} via tautological representation is called the Berry line bundle on B_{k} and is denoted by L_{k}. The diagrams (3-1) and (3-3) show that we have the non ramified degree m covering, which takes each covariant section to its m-th power:

P_{k} μ̃_{m,k}[figure] β_{m,k}^{∗}P_{mk} ∀ k, m ∈ ℕ .  (3-6)

In other words, β_{k,m}^{∗} L_{mk} = L_{k}^{⊗m}.

3.3.1. PROPOSITION. A tangent space Tφ̃P_{k} at a smooth Planckian cycle S ^{⊂} φ̃^{}P_{k} is naturally identified with C^{∞}(S). Under this identification, the differential dμ̃_{k,m} acts on C^{∞}(S) as the scalar multiplication by m.

1see [A, App. 4] about the contact structures

Proof. The same arguments as in n^{∘} 1.4 show that the tangent space

Tφ̃(Map_{le}(S, P_{k})/Diff_{0}(S)) = { v ∈ C^{∞} ( S, N_{P_{k}/S} ) | φ̃^{∗}(Lie_{v}α_{k}) = 0 }

that is, a tangent vector to P_{k} at φ̃ is a class of vector fields (modulo the fields tangent to S̃) defined near S̃ on P_{k} such that Lie_{v}α_{k} = 0. Since α_{k} annihilates the fields tangent to S̃, the contraction with α_{k} gives a well defined map Tφ̃P_{k} ^{}C^{∞}(S) taking

v ⟼ f_{v} =^{def} φ̃^{∗}α_{k}(v) ∈ C^{∞}(S) .

Moreover, the horizontal component of v, that is dp(v − α_{k}(v) · u) ∈ T_{pφ}𝔏 ≃ Z_{dr}^{1}(S) (where u is the basic unitary invariant vector field on P_{k}) coincides with df_{v}/k, since for any local vector field w tangent to S̃ on P_{k} we have:

ω( dp(v − α_{k}(v) · u) , dp(w) ) = p^{∗}ω(v, w) = \frac{1}{k} dα_{k}(v, w) =

= \frac{1}{k} [Lie_{v}α_{k} + d(α_{k}(v))] (w) = \frac{1}{k} df_{v} (w)

Conversely, any f ∈ C^{∞}(S) produces (after an extension on a small neighborhood of S̃) a local vector field v_{f} on P_{k} such that f_{v_{f}} = α_{k}(v_{f}) = f: the vertical component of v_{f} equals fu and the horizontal component of v_{f} is the horizontal lifting of ω^{−1}(df̂)/k from M on P_{k}, where f̂ is an extension of f to a small neighborhood of φ(S) in M. □

3.3.2. Local structure of P near a smooth cycle. Let a smooth Planckian cycle S ^{⊂} φ̃ P_{k}

be given by a covariant constant section S ^{⊂} ^{σ}^{}φ^{∗}P_{k} over a smoothly immersed Bohr - Sommerfeld cycle S ^{φ}^{}M. Then the previous proof identifies the differential

Tφ̂P_{k} ^{dπ|φ̂}[figure] T_{φ}B_{k} ,

of the Berry bundle P_{k} ^{π}[figure] B_{k}, with the divided by k external differentiation of functions on S

C^{∞}(S, ℝ) ^{\frac{1}{k} d}[figure] B_{dr}^{1}(S) .

In particular, the space B_{k} ⊂ 𝔏 is isodrastic, T_{S}B_{k} ≃ B_{dr}^{1}(S), and under this identification, the differential dβ_{k,m} acts on B_{dr}^{1}(S) as the scalar multiplication by m.

Further, by the Darboux - Weinstein theorem (see n^{∘} 1.4.2), φ is continued to a symplectic immersion U ^{⊂} φ̂^{}M, where U is some neighborhood of the zero section S ^{⊂} ^{}T^{∗}S. Let us denote by S_{f}^{(k)} ⊂ T^{∗}S the graph of a closed 1-form df/k ∈ B_{dr}^{1}(S). A map

df ^{}φ̂( S_{f}^{(k)} )

coordinates some neighborhood of φ in B_{k} by a small ball around the zero in B_{dr}^{1}(S) (see n^{∘} 1.4.2). Now we can lift this coordinatization on a small ball around the zero in C^{∞}(S, ℝ) = Tφ̃P_{k} and get the coordinates for a neighborhood of φ̃ in P_{k}. Namely, let Pk^{U} = φ̂^{∗}P_{k} be the pull back of P_{k} on U along the Darboux - Weinstein immersion U ^{⊂} φ̂^{}M and let U ^{⊂} σ̂^{}Pk^{U} be the covariant constant section whose restriction on S ⊂ U is the initial covariant trivialization S ^{⊂} ^{σ}^{}φ^{∗}P_{k}. We send each f ∈ C^{∞}(S, ℝ) such that S_{f}^{(k)} ⊂ U to BS_{k}-cycle S_{f}^{(k)} over which Pk^{U} is trivialized by the section

σ_{f} =^{def} e^{if} σ̂|_{S{(k) \atop f}} .  (3-7)

So, we have got the real smooth functions on S as the local coordinates near φ̃ in P_{k}. We call them DW-coordinates.

3.3.3. Complex DW-coordinates on T P_{k}. Recall, that the tangent bundle T P_{k} admits the canonical integrable complex structure (see n^{∘} 1.5.5): for any smooth Planckian cycle S ^{⊂} φ̃^{}P_{k} and any tangent vector g ∈ C^{∞}(S, ℝ) = Tφ̃P_{k} the multiplication by i on

T_{(φ̃,g)} ( T P_{k}) = T P_{k} ⊕ T P_{k} = C^{∞}(S, ℝ) ⊕ C^{∞}(S, ℝ)

sends (ψ_{1}, ψ_{2}) ⟼ (ψ_{2}, −ψ_{1}), where the first component runs through the ‘vertical’ linear fibers of the tangent bundle T P_{k} [figure] P_{k} and the second one runs through the ‘horizontal’ summand T P_{k} included into T T P_{k} by the differential of the zero section P_{k} ^{⊂} ^{}T P_{k}. In other words, complex coordinates on T P_{k} are the complex valued smooth functions ψ = ψ_{1} + iψ_{2} ∈ C^{∞}(S, ℂ) whose real part ψ_{1} is interpreted as linear coordinate in the fiber of the tangent bundle and imaginary part ψ_{2} is interpreted as DW-coordinate on P_{k}. We will refer these coordinates as the complex DW-coordinates on T P_{k}.

3.4. Half weighted Planckian cycles. Our next task is to combine the natural complex structure on T P and the natural symplectic structure on T^{∗}P in order to get an integrable Kähler structure on a suitable vector bundle over P.

3.4.1. Measures. Let 𝔐 = 𝔐(S) be an equivalence class of measures on the source manifold S induced by its manifold structure (see [M, § 2-6]). It contains a subclass 𝔐_{0} ⊂ 𝔐 of smooth measures whose pairwise Radon - Nikodym derivatives are the smooth functions on S. Let us recall that the equivalence λ ∼ μ of two measures λ, μ on S means that λ(B) = 0 ⇔ μ(B) = 0 for any Borel subset B ⊂ S, or by the Radon - Nikodym theorem, that there are two real integrable positive functions called the Radon - Nikodym derivatives:

ϱ_{μλ} = \frac{dμ}{dλ} ∈ L^{1}(S, λ) and ϱ_{λμ} = \frac{dλ}{dμ} ∈ L^{1}(S, μ)

such that ∫_{B} dμ = ∫_{B} ϱ_{μλ} dλ and ∫_{B} dλ = ∫_{B} ϱ_{λμ} dμ for any Borel subset B ⊂ S. Of course,

ϱ_{μλ}ϱ_{λζ} = ϱ_{μζ} and ϱ_{μλ}ϱ_{λμ} = 1 almost everywhere w. r. t. any measure from 𝔐. One writes usually λ = ϱ_{λμ}μ for two equivalent measures.

3.4.2. Half weighings. By the definition, half weight ϑ = f\sqrt{μ} on S is an equivalence class of pairs (f, μ), where μ ∈ 𝔐, f ∈ L_{ℝ}^{2}(S, μ), and the equivalence (f_{1}, μ_{1}) ∼ (f_{2}, μ_{2}) means that f_{1}\sqrt{ϱ_{μ_{1}λ}} = f_{2}\sqrt{ϱ_{μ_{2}λ}} almost everywhere w. r. t. any λ ∈ 𝔐. A half weight f\sqrt{μ} is called smooth, if f ∈ C^{∞}(S) and μ ∈ 𝔐_{0}. The straightforward check^{1} shows that the half weights on S form the real closed normed vector space ℌ(S) with the norm induced by a scalar product

( f_{1} \sqrt{μ_{1}} , f_{2} \sqrt{μ_{2}} ) =^{def} ∫_{S} f_{1}f_{2} \sqrt{ϱ_{μ_{1}λ}ϱ_{μ_{2}λ}} dλ  (3-8)

1see, for example, [M, § 2-6] or [Go, sec 4.1], where our half weights are referred as σ-functions

and the linear combinations defined as t_{1}·f_{1} \sqrt{μ_{1}}+t_{2}·f_{2} \sqrt{μ_{2}} =^{def} (t_{1}f_{1} \sqrt{ϱ_{μ_{1}λ}}+t_{2}f_{2} \sqrt{ϱ_{μ_{2}λ}}) \sqrt{λ}, where λ in the both formulas is an arbitrary measure from 𝔐. The smooth half weights form a dense subspace ℌ^{∘} (S) ⊂ ℌ(S).

There is a natural action of Diff_{0}(S) on ℌ(S) via (f, μ) ^{h}^{}( f ∘ h^{−1} , ( h^{−1} )^{∗} μ ) and ℌ^{∘}, clearly, goes to itself under this action. The orbit spaces w. r. t. the corresponding diagonal actions:

𝔏^{hw} =^{def} Map_{la}(S, M) × ℌ(S)/Diff_{0}(S) P_{k}^{hw} =^{def} Map_{le}(S, P_{k}) × ℌ(S)/Diff_{0}(S) .

are called spaces of half weighted Lagrangian and Planckian cycles (of level k) respectively. If we replace ℌ by ℌ^{∘} in these definitions, then we get the dense subspaces 𝔏^{∘} _{k}^{hw} ⊂ 𝔏_{k}^{hw} and P^{∘} _{k}^{hw} ⊂ P_{k}^{hw} called the spaces of smoothly half weighted cycles. The forgetful maps 𝔏^{hw} ^{F} [figure] 𝔏

and P_{k}^{hw} F̃ [figure] P_{k} (and their smooth versions) provide these spaces with the vector bundle structures. These bundles may be arranged in the following commutative diagram:

[figure: P_{k}^{hw}; B_{k}^{hw}; ⊂; 𝔏^{hw}]

[figure: P; F̃=π^{∗}F |_{B{hw \atop k}}; π; B_{k}; F |_{B{hw \atop k}}; ⊂; 𝔏 .; F]

(3-9)

3.5. Kähler structure on P^{∘} _{k}^{hw}. For any fixed half weight f \sqrt{μ} the tangent space TS̃P_{k} = C^{∞}(S) is included into T_{f\sqrt{μ}}ℌ ≃ ℌ by sending

f_{1} ⟼ ff_{1} \sqrt{μ} .  (3-10)

All together these maps produce an inclusion T P_{k} ^{⊂} ^{}P_{k}^{hw} as a dense set. Restricting ourselves by the smooth half weights, we get the identification T P_{k} ^{∼}^{}P^{∘} _{k}^{hw}, which allows to push down onto P^{∘} _{k}^{hw} the canonical integrable complex structure on T P_{k} described in n^{∘} 3.3.3. Namely, consider a smooth Planckian cycle S ^{⊂} φ̃^{}P laying over a BS_{k}-cycle S ^{⊂} ^{φ}^{}M (like in n^{∘} 3.3.2) and half weighted by smooth ϑ = f\sqrt{μ}. Then we have the splitting:

T_{(φ̃,ϑ)} P^{∘} _{k}^{hw} ≃ T_{ϑ} ℌ^{∘} (S) ⊕ Tφ̃ P^{∘} _{k}^{hw} ≃ C^{∞}(S) ⊕ C^{∞}(S)  (3-11)

and can coordinate a small neighborhood of (φ̃, ϑ) in P^{∘} _{k}^{hw} by smooth complex valued functions ψ = ψ_{1}+iψ_{2} ∈ C^{∞}(S, ℂ) as follows. The imaginary part ψ_{2} produces the Planckian cycle with the DW-coordinate ψ_{2} as it was explained in n^{∘} 3.3.2: to produce it we should continue the section S ^{⊂} ^{σ}^{}φ^{∗}P k, which defines the Planckian cycle ψ̃, to the covariant constant section U ^{⊂} σ̂^{}Pk^{U}, where U ⊂ T^{∗}S is the Darboux - Weinstein neighborhood of the zero section immersed in M via U ^{⊂} φ̂^{}M and Pk^{U} = φ̂^{∗}P_{k}; then the Planckian cycle being constructed is the graph S_{ψ_{2}}^{(k)} ⊂ U, of closed 1-form dψ_{2}/k, and Pk^{U} is trivialized over this graph by the section

σ_{ψ_{2}} =^{def} e^{iψ_{2}} σ̂|_{S_{ψ_{2}}^{(k)}} .

The real part ψ_{1} gives the half weight on S_{ψ_{2}}^{(k)} as follows. Let S_{ψ_{2}}^{(k)} ^{pr} [figure] S be the restriction of the canonical projection T^{∗}S [figure] S. Then

pr^{∗}ϑ = (pr ∘ f) · \sqrt{pr^{∗}μ}

is a well defined half weight on S_{ψ_{2}}^{(k)}. So, in accordance with (3-10) we have to equip S_{ψ_{2}}^{(k)} with the half weight ψ_{1}pr^{∗}ϑ. We refer just described complex coordinates ψ_{1} + iψ_{2} ∈ C^{∞}(S, ℂ) on P^{∘} _{k}^{hw} as the the complex DW-coordinates. In fact, they correspond to some Kähler structure on P^{∘} _{k}^{hw}. To define it, let us describe now two its remaining ingredients.

3.5.1. The symplectic and Riemannian structures on P_{k}^{hw}. The product of two half

weights ϑ_{1} = f_{1}\sqrt{μ_{1}} and ϑ_{2} = f_{2}\sqrt{μ_{2}} is nothing but a measure multiplied by f_{1}f_{2}, that is

ϑ_{1}ϑ_{2} =^{def} f_{1}f_{2}\sqrt{ϱ_{μ_{1}}ϱ_{μ_{2}λ}} λ ,

which does not depend on the choice of λ ∈ 𝔐 (see [M, § 2-6]). So, we have a well defined integration w. r. t. the product of two half weights

∫_{S} f dϑ_{1}ϑ_{2} =^{def} ∫_{S} ff_{1}f_{2}\sqrt{ϱ_{μ_{1}λ}ϱ_{μ_{2}λ}} dλ  (3-12)

for any f ∈ C^{∞}(S) and any half weights ϑ_{1} = f_{1}\sqrt{μ_{1}}, ϑ_{2} = f_{2}\sqrt{μ_{2}}. In particular, each half weight ϑ produces a linear form

κ(ϑ) : f ^{}∫_{S} f dϑ^{2}  (3-13)

on the tangent space TS̃P_{k} ≃ C^{∞}(S).

In other words, ℌ(S) double covers T ∗ ; S̃P_{k}, because κ is quadratic in ϑ. The differential of this quadratic covering sends ϑ_{1} ∈ ℌ(S) to the linear form

dκ_{ϑ}(ϑ_{1}) : f ^{}2 ∫_{S} f dϑϑ_{1} .

on C^{∞}(S) = TS̃P_{k}. So, dκ identifies T^{∗}P_{k} with P_{k}^{hw}. Now, consider an action form H =^{def} κ^{∗}η, where η is the canonical action 1-form on the cotangent bundle T^{∗}P_{k} (see n^{∘} 1.3). In terms of the splitting (3-11), H sends v = (ϑ_{1}, f_{1}) ∈ T_{(S,ϑ_{0})}P^{hw} ≃ ℌ(S) ⊕ C^{∞}(S) to

H(v) = 2 ∫_{S} f_{1} dϑ_{0}ϑ_{1} ,  (3-14)

and its differential Ω = dH acts on T_{(S,ϑ_{0})}P^{hw} by the rule

Ω( ( ϑ_{1}, f_{1} ) , ( ϑ_{2}, f_{2} ) ) = ∫_{S} f_{2} dϑ_{0}ϑ_{1} − ∫_{S} f_{1} dϑ_{0}ϑ_{2} ,  (3-15)

which is a straightforward infinite dimensional generalization^{1} of (1-3) with the summation replaced by the integration over S. In terms of the complex DW-coordinates from n^{∘} 3.5, the real tangent vector, along (ψ_{1}, 0), and an imaginary tangent vector (0, ψ_{2}) are paired by Ω as

Ω( ( ψ_{1}, 0 ) , ( 0, ψ_{2} ) ) = ∫_{S} ψ_{1}ψ_{2} dϑ_{0}^{2}

1We would like to remark here that this infinite dimensional symplectic form was commonly used for S = S^{1} in the context of the Korteweg - de Vries equation (see [DKN] and also our ‘toy example’ in § 5).

Such the symplectic form has appeared also in the Hamiltonian formulation of the motion of vortex patches for planar incompressible fluids.

that coincide with L^{2}-product of ψ_{1} and ψ_{2} w. r. t. the measure ϑ_{0}^{2}. So, Ω is compatible with the complex structure on P^{∘} _{k}^{hw}, that is, produces the Kähler triple whose Riemannian metric G acts on T_{(S,ϑ_{0})}P^{hw} = C^{∞}(S) ⊕ ℌ(S) by the rule

G( (ϑ_{1}, f_{1}) , (ϑ_{2}, f_{2}) ) = ∫_{S} f_{2} dϑ_{1}ϑ_{0} + ∫_{S} f_{1} dϑ_{2}ϑ_{0} ,  (3-16)

So, we have proved the following:

3.5.2. THEOREM. The space P_{k}^{hw}, of half weighted Planckian cycles of level k, has the natural structure of an infinite dimensional Kähler variety; the symplectic form and the Riemannian metric induced by this structure are given by the formulas (3-15) and (3-16) respectively; the complex coordinates near a given S̃ ∈ P_{k}^{hw} are the smooth complex valued functions on S. □

3.6. Half weighted Bohr - Sommerfeld cycles of fixed volume. The natural U(1)-action on P_{k}, which changes a covariant trivialization of L over a given S ⊂ M, has the standard lifting on T P_{k} and on P_{k}^{hw} as well. This action preserves the Kähler structure: z ∈ U(1) ⊂ ℂ just multiplies by z the complex DW-coordinates. In particular, the both forms (3-15), (3-16) are U(1)-invariant (in fact, it is clear without the coordinates, because our identification T_{S}P_{k} ≃ C^{∞}(S) did not depend on the choice of a trivialization of P over S, see n^{∘} 3.3.1).

Since the volume function P_{k}^{hw} ^{vol}^{}ℝ, which sends (S̃, ϑ) to ∫_{S} ϑ^{2}, is U(1)-invariant, we

can separately factorize by U(1)-action each level hypersurface P_{k,t}^{hw} that consists of weighted Planckian cycles whose underlying BS_{k}-cycle has the fixed volume t. The corresponding orbit space B_{k,t}^{hw} =^{def} P_{k,t}^{hw}/U(1) is called a space of half weighted BS_{k}-cycles of volume t.

3.6.1. THEOREM. The space B_{k,t}^{hw}, of half weighted BS_{k}-cycles of volume t, also has a natural structure of an infinite dimensional Kähler variety. This structure is induced by the Kählerian structure on P_{k}^{hw}; in particular, its symplectic form and the Riemannian metric remain to be given by the formulas (3-15) and (3-16) respectively. Proof. In terms of the decomposition (3-11), the tangent space T_{(S̃,ϑ_{0})}P_{k,t}^{hw} consists of v =

(f, ϑ) such that ∫_{S} dϑϑ_{0} = 0, because in the DW-coordinates an infinitesimal volume in-

crement along v is measured by this integral. Since the U(1)-factorization takes (f, ϑ) to (df, ϑ) ∈ T_{(S,ϑ_{0})}B_{k,t}^{hw}, the both forms (3-15), (3-16) remain to be well defined, because the

conditions ∫_{S} dϑ_{1}ϑ_{0} = ∫_{S} dϑ_{2}ϑ_{0} = 0 annihilate an arbitrariness in lifting df to f. □

3.6.2. Remark towards the symplectic reduction. Of course, the above construction is

nothing but taking the symplectic quotient P_{k}^{hw}//U(1). Indeed, in our commutative case, the U(1)-invariant function vol gives a moment map for the U(1)-action on P_{k}^{hw}; factorizing its level hypersurfaces, we just follow the general machinery of producing the symplectic quotient from the moment map (see [At] and [AB2] for a detailed explanations of what’s it in general setup).

3.7. Prequantization equipment of B_{k,t}^{hw}. The Kählerian manifold B_{k,t}^{hw} is quipped with his own prequantization data. Namely, there is the principal U(1)-bundle

P_{k,t}^{hw} ^{π_{t}^{hw}} [figure] B_{k,t}^{hw} ,  (3-17)

obtained by the restriction of P_{k}^{hw} = F̃^{∗}P [figure] B_{k}^{hw} (upper left horizontal row on the diagram

(3-9)) on B_{k,t}^{hw} and equipped the U(1)-connection 𝔞 =^{def} H|_{P_{k,t}^{hw}}, which is the action form (3-14)

restricted on the Planckian cycles of volume t. By the construction, its curvature F_{𝔞} = 2πi Ω, that is ω_{𝔞} = Ω as required in n^{∘} 3.1. Moreover, this prequantization data has the canonical Kähler polarization described in n^{∘} 3.6.1. So, it is ready for applying the Kählerian quantization procedure described in the next paragraph.

3.7.1. Remark on weighting rules. The prequantization data (B_{k,t}^{hw}, Ω, P_{k,t}^{hw}, 𝔞) can be used

for producing a prequantization equipment for the (non weighed) Berry bundle P_{k,t} ^{π_{t}} [figure] B_{k,t}. Although there is no canonical U(1)-connection here, one can fix some one by pulling back the ‘universal’ connection 𝔞 from P_{k,t}^{hw} along a suitable section P_{k} ^{⊂} ^{}P_{k}^{hw}. A choice of such a section is called a Planckian half weighting rule. A Planckian half weighting may be lifted in turn from a Bohr - Sommerfeld one, which is just a section

B_{k} ^{⊂} ^{}B_{k}^{hw} ,  (3-18)

For example, if M is an almost Kählerian manifold, then the Riemannian metric g on M has non degenerate restriction on each Lagrangian cycle S and gives there a canonical measure μ_{L,g}. Taking S̃ with the half weight class ϑ_{L,g} = (1, μ_{L,g}), we get the natural half weighting rule called the Kähler half weighting.

§ 4. Holomorphic quantization.

4.1. Kähler polarization. To quantize a prequantization data (M, ω, L^{⊗k}, α_{k}) (see n^{∘} 3.1) we have to polarize it (see [T2]). By the definition, a complex (or Kähler) polarization is a choice of a complex structure I on M such that M_{I} turns to a Kähler manifold with the Kähler form ω. Since the curvature of the Hermitian connection α_{k} has the Hodge type (1, 1) w. r. t. I (see [GH, ch. 3]), the line bundle L turns automatically to a holomorphic line bundle on M_{I}.

4.1.1. Wave function spaces. As soon a complex polarization is fixed, a complex quantization procedure provides the collection of wave function spaces:

Q_{k}(M_{I}) = H^{0}(M_{I}, L^{⊗k}) ,  (4-1)

where H^{0}(M_{I}, L^{⊗k}) is the space of I-holomorphic sections of the line bundle L^{⊗k}. The integer index k is called a level of the quantization procedure (see the survey [K]). We will consider in this paper only the case of compact M where the wave function spaces (4-1) are finite dimensional. But we would like to outline here that all our constructions remains correct even in a non compact case, where the spaces of wave functions became infinite dimensional, for example, when M = T^{∗}X is the cotangent bundle of a configurations space X.

4.1.2. Hardy space and Szegö projector. Consider the initial principal U(1)-bundle P = P_{1} on M_{I} as the boundary for the bundle of unit disks inside L_{−1} = L^{∗}, the Hermitian dual to the prequantization line bundle L:

P = ∂D , where D ⊂ L^{∗} [figure] M_{I} .  (4-2)

The total space D is a strictly pseudoconvex domain and its complex structure depends strongly on a choice of a complex structure I on M, that is, on a choice of a complex polarization. Let L^{2}(P) be the L^{2}-completion of the space C^{∞}(P, ℂ), of smooth ℂ-valued functions on P w. r. t. the Hermitian scalar product defined by the contact structure volume form (3-4):

(f, g) =^{def} ∫_{P} f\overline{g} α_{k} ∧ (dα_{k})^{∧n} .  (4-3)

A closed subspace H ⊂ L^{2}(P) that consists of functions admitting a holomorphic continuation onto D is called a Hardy space on P. The orthogonal projector

Π: L^{2}(P) [figure] H  (4-4)

is called the Szegö projector.

4.1.3. U(1) weight decomposition of the Hardy space. The action of U(1) on the prin-

cipal bundle P commutes with the Szegö projector and produces the weight decomposition H = ⨁_{k∈ℤ} H_{k}, where k-th isotype component H_{k} consists of fiberwise homogeneous polynomials of degree k on L^{∗}, that is, can be naturally identified with H^{0}(M, L^{⊗k}). So, the weight decomposition of H w. r. t. U(1)-action looks like

H = ⨁_{k} H^{0}(L^{⊗k}) .  (4-5)

Under this identification, the Hermitian structure on H coming from L^{2}(P) goes to the Hermitian structure on H^{0}(M, L^{⊗k}) induced by the Hermitian scalar product on L^{⊗k} and the integration through M:

(f, g)_{k} =^{def} ∫_{M}(f, g)_{L^{⊗k}} ω^{∧n} ,  (4-6)

because of the volume form ω^{∧n} on M is induced by the volume form α_{k} ∧(dα_{k})^{∧n} = α_{k} ∧ω^{∧n} on P. We write Π_{k} for the orthogonal Szegö projectors L^{2}(P) ^{Π_{k}}[figure] H^{0}(L^{⊗k}).

4.1.4. Toeplitz operators. Each function f on M produces a series of Toeplitz operators H^{0}(L^{⊗k}) ^{Q_{k}(f)}^{}H^{0}(L^{⊗k}) as follows. To calculate Q_{k}(f)ψ for ψ ∈ H^{0}(L^{⊗k}), we consider ψ as an element of H ⊂ L^{2}(P), then multiply it by f, and then apply the Szegö orthogonal projector onto H^{0}(L^{⊗k}). Briefly,

Q_{k}(f)(ψ) = Π_{k}(fψ) ∈ H^{0}(L^{⊗k}).  (4-7)

All Toeplitz operators on H form a Poisson algebra, which is a quantum deformation of the Poisson algebra of smooth functions on the original symplectic manifold M in the sense of n^{∘} 0.4.

4.1.5. Projective embedding of M_{I}. The holomorphic line bundle L^{⊗k} defines a rational (that is, meromorphic) map M_{I} ^{}ℙ ( H^{0}(L^{⊗k})^{∗} ) by sending x ∈ M_{I} to the hyperplane

Ann (x) = { ψ ∈ H^{0}(L^{⊗k}) | ψ(x) = 0 } .

More precisely, there is an exact sequence of sheaves on M

0 ^{}I_{x} ⊗ L^{⊗k}  ^{}L^{⊗k} ^{res}^{}O_{M}|_{x}  ^{}0

where res is the restriction epimorphism and I_{x} is the ideal sheaf of x. Its long exact cohomology sequence

⋯ ^{}H^{0}(I_{x} ⊗ L^{⊗k}) ^{}H^{0}(L^{⊗k}) ^{}ℂ ^{}H^{1}(I_{x} ⊗ L^{⊗k}) ^{}⋯

shows that H^{0}(I_{x} ⊗ L^{⊗k}) is a hyperplane if H^{1}(I_{x} ⊗ L^{⊗k}) = 0. Since H^{1}(I_{x} ⊗ L^{⊗k}) = 0 for k ≫ 0 by Serre’s classical A & B Theorems, the map

φ_{k}: M_{I} ^{x↦Ann (x)} ^{}ℙ ( H^{0}(L^{⊗k})^{∗} )  (4-8)

is an embedding for k ≫ 0. This standard construction reduces the study of M_{I} to the projective algebraic geometry.

4.1.6. The space of states. The Hermitian structure (4-6) gives an antiholomorphic isomorphism

H^{0}(L^{⊗k}) ^{∼}^{}\overline{H^{0}(L^{⊗k})}^{∗} .  (4-9)

Applying its projectivization after the holomorphic embedding (4-8), we get the complex conjugate (antiholomorphic) embedding

\overline{φ}_{k}: M_{I} ^{⊂} ^{}ℙH^{0}(L^{⊗k})  (4-10)

In the complex quantization context, vectors in H^{0}(L^{⊗k}) are called states and vectors \overline{φ}(x), where x ∈ M, are called coherent states.

4.1.7. Important warning. Although the principal prequantization bundle P_{k} ^{p_{k}} [figure] M of level k is nothing but the pull back of the degree k Hopf circle bundle over the projective space ℙH^{0}(M_{I}, L^{⊗k}) along (4-10), the curvature 1-form α_{k} on P_{k} does not coincide, in general, with the pull back of the Fubini - Study invariant 1-form on the Hopf bundle.

4.2. Borthwick - Paul - Uribe map. Now, let S ^{⊂} φ̃^{}P_{k} be a Planckian cycle, that is a BS_{k}-cycle S ^{⊂} ^{φ}^{}M together with a fixed covariant constant unitary section S ^{⊂} ^{σ}^{}φ^{∗}L^{⊗k}. For any holomorphic section s ∈ H^{0}(M_{I}, L^{⊗k}) consider a smooth complex valued function ψS̃ ∈ C^{∞}(S, ℂ) such that φ^{∗}s = ψS̃ · σ in C^{∞}(S, φ^{∗}L^{⊗k}). Note that the linear map

H^{0}(M, L^{⊗k})^{⊂} ^{s↦ψS̃} ^{}C^{∞}(S, ℂ)  (4-11)

is injective, because of n^{∘} 1.5.4, and holomorphic w. r. t. the standard complex structure on C^{∞}(S, ℂ), which is nothing but the fiber of the holomorphic tangent bundle T_{(φ̃,ϑ)} P^{∘} _{k}^{hw}, where P^{∘} _{k}^{hw} is considered with the intrinsic holomorphic structure defined in n^{∘} 3.5. So, we have an inclusion of coherent sheaves on P^{∘} _{k}^{hw}:

0 ^{}H_{k} ⊗ O ∘_{P} ; {hw \atop k}  ^{}T ∘_{P} ; {hw \atop k} ,  (4-12)

where the first is the sheaf of holomorphic sections of the trivial vector bundle with the fiber H_{k} = H^{0}(M_{I}, L^{⊗k}) and the second is the sheaf of holomorphic vector fields on P^{∘} _{k}^{hw}.

4.2.1. THEOREM. A map P^{∘} ^{hw} ^{\overline{𝔪}_{k}} ^{}H^{0}(M, L^{⊗k})^{∗} that sends a half weighted Planckian

cycle (φ̃, ϑ) to ℂ-linear form taking s ∈ H^{0}(M, L^{⊗k}) to ∫_{S} ψS̃ dϑ^{2} = ∫_{S} ψS̃ f^{2} dμ ∈ ℂ is

antiholomorphic with the surjective differential.

Proof. In terms of the complex DW-coordinates on P^{∘} _{k}^{hw} near (φ̃, ϑ), a linear functional

\overline{𝔪}_{k}(ψ_{1} + iψ_{2}) on H^{0}(M, L^{⊗k}) takes s to ∫_{S} ψS̃ψ_{1}e^{−iψ_{2}} dϑ^{2}, because P_{k} is trivialized by e^{iψ_{2}} σ̂

over S_{ψ_{2}}^{(k)} (see n^{∘} 3.5) and s is holomorphic, that is, α_{k}(s) = 0. So, the differential d\overline{𝔪}_{k} sends ψ_{1} + iψ_{2} to the functional

s ⟼ ∫_{S} ψS̃(ψ_{1} − iψ_{2}) dϑ^{2} ,  (4-13)

which ℂ-antilinearly depends on ψ_{1} +iψ_{2} as required. Moreover, the right hand site is nothing but the standard Hermitean pairing on the Hilbert space L^{2}(S, ℂ) w. r. t. the measure ϑ^{2}, that is, any ℂ-linear form on the finite dimensional subspace H^{0}(M, L^{⊗k}) ⊂ L^{2}(S, ℂ) has the form (4-13) for suitable ψ_{1} + iψ_{2}. So, d\overline{𝔪}_{k} is surjective at every point. □

4.2.2. Holomorphic BPU_{k}-map. The map \overline{𝔪}_{k} from the theorem n^{∘} 4.2.1 is called the antiholomorphic BPU_{k}-map. Its composition with the Hermitian structure isomorphism (4-9), that is, the holomorphic map P^{∘} _{k}^{hw} ^{𝔪_{k}}^{}H^{0}(M, L^{⊗k}), is called a holomorphic BPU_{k}- map. It sends a half weighted Planckian cycle (φ̃, ϑ) to the holomorphic section 𝔪_{k}(φ̃, ϑ) ∈ H^{0}(M, L^{⊗k}) such that

∫_{M} ( s , 𝔪_{k}(φ̃, ϑ) )_{L^{⊗k}} dϱ = ∫_{S} ψS̃ dϑ^{2}

for any s ∈ H^{0}(M, L^{⊗k}). It follows from (4-13) that the differential T_{(φ̃,ϑ)} ^{d𝔪_{k}}[figure] H^{0}(M, L^{⊗k}) sends ψ ∈ C^{∞}(S, ℂ) to the section s_{ψ} ∈ H^{0}(M, L^{⊗k}) such that

∫_{M} ( s, s_{ψ})_{L^{⊗k}} dϱ = ∫_{S} ψS̃\overline{ψ} dϑ^{2} .

So, the Hermitean dual map H_{k} ^{d𝔪_{k}^{∗}}[figure] C^{∞}(S, ℂ) (w. r. t. the standard Hermitean structures on the both spaces) is nothing but the inclusion (4-11).

4.2.3. Symplectic reduction of the BPU_{k}-map w. r. t. U(1)-action. Recall (see [At])

that the projectivization ℙV = V/ℂ^{∗}, of a Hermitean vector space V, may be treated as symplectic reduction ℙV = V//U(1) w. r. t. the moment map v ↦ ∥v∥. So, the symplec-

tic quotient H_{k}//U(1) w. r. t. the moment map s ↦ ∫_{M}(s, s)_{L^{⊗k}}dϱ is the space of states

ℙH_{k}. Recall also that in n^{∘} 3.6 we have considered the natural U(1)-action on P^{∘} _{k}^{hw} with

the moment map (φ̃, ϑ) ↦ ∫_{S} dϑ^{2}. The corresponding symplectic quotient P^{∘} _{k}^{hw}//U(1) is the

space of half weighted BS_{k}-cycles of fixed volume B^{∘} _{k,t}^{hw}. Now, the holomorphic BPU_{k}-map

P^{∘} _{k}^{hw} ^{𝔪_{k}}^{}H^{0}(M, L^{⊗k}) is U(1)-equivariant, because its differential is. Hence, it is factorizible to the holomorphic map

𝔪_{k,t} : B^{∘} _{k,t}^{hw} ^{}ℙH_{k}

with the surjective differential. This map assigns holomorphically a state to each half weighted BS_{k}-cycle of a fixed volume t. We call it BPU_{k,t}-map. By the construction, this map is nothing but the projective map induced by the holomorphic Bery line bundle L_{k,t} associated with the principal Bery bundle P^{∘} _{k,t}^{hw} ^{π_{t}} [figure] B^{∘} _{k,t}^{hw}, that is, L_{k,t} = 𝔪_{k,t}^{∗} ( O_{ℙH_{k}}(1) ).

4.3. Action of symplectomorphisms of M. The intrinsic infinite dimensional prequantization data (B_{k,t}^{hw}, Ω, P_{k,t}^{hw}, 𝔞) described in n^{∘} 3.7 is invariant with respect to the natural action of the identity component Sym_{0} in the symplectomorphism group Sym = Sym(M, ω). Recall that two half weighted BS_{k}-cycles (S_{1}, ϑ_{1}), (S_{2}, ϑ_{2}) ∈ B^{∘} _{k,t}^{hw} are called isotopic if there is a one

parameter family of symplectomorphisms g_{t} ∈ Sym such that g_{0} = Id_{M} and g_{1}((S_{1}, ϑ_{1})) = (S_{2}, ϑ_{2}). The BPU_{k}-map is, clearly, invariant under the symplectomorphisms, that is, produces the family of invariants for the isotopy classes. We would like to make here some remarks towards more strong factorization that is an infinite dimensional analog of the symplectic reduction w. r. t. to the Sym_{0}-action. Since the BPU_{k}-map, which is quite similar to what we have in mind, produces a finite dimensional factor, it is reasonable to wait that

dim ( B^{∘} _{k}^{hw}//Sym_{0} ) < ∞.

The Lie algebra 𝔰𝔶𝔪, of the symplectomorphism group, is the algebra of Hamiltonian vector fields on M. Its Lie coalgebra may be treated as

𝔰𝔶𝔪^{∗} = C_{2n−1}(M)/dC_{2n}(M) ,

where C_{j}(M) is the space of degree 2n − j currents with a compact support (it is dual of the space of differential forms of degree j on M). The moment map B^{∘} _{k,t}^{hw} ^{μ}^{}𝔰𝔶𝔪^{∗} is defined by sending a half weighted BS_{k}-cycle (S, ϑ) to the functional μ_{(S,ϑ)} that takes v ∈𝔰𝔶𝔪 to

⟨ μ_{(S,ϑ)} , v ⟩ =^{def} ∫_{S} ξ_{v} dϑ^{2} ,

where ξ_{v} = ωξ is a 1-form corresponding to the vector field v on M. Clearly, μ is equivariant w. r. t. the coadjoint action of Sym on 𝔰𝔶𝔪^{∗} and {μ_{(S,ϑ)}(v), μ_{(S,ϑ)}(w)} = μ_{(S,ϑ)}([v, w]) for any pair v, w ∈ 𝔰𝔶𝔪. So, we have:

4.3.1. PROPOSITION. The orbits of Sym_{0} in B^{∘} _{k,t}^{hw} are the isotopy components of B^{∘} _{k,t}^{hw} and any such a component is diffeomorphic to the homogeneous space

ℙ ( H^{0}(L^{k}) ) × (Sym_{0}/Stab_{(S,ϑ)}) ,

where (S, ϑ) is any point of the component. □

In other words, there exists a smooth bijective map from S × ℙ ( H^{0}(L^{k}) ) to the coadjoint orbit of μ_{(S,ϑ)}. Is it a diffeomorphism? A quite fruitful application of this construction will be considered in § 5 where we will deal with the loop space LS^{2} and the space of Legendrian knots in S^{3}.

4.4. Analytical viewpoint on the BPU_{k}-map. Originally, the BPU_{k}-map was discovered in a pure analytical framework, see [BPU]. To allow the reader to compare our geometrical approach with the original one, we trace out very briefly some of the analytical key points in this section, although we will not use it in what follows.

4.4.1. Analytical definition of the BPU_{k}-map The analytical background of the BPU_{k}-

map construction is based on the remarkable possibility to continue the Szegö projector L^{2}(P, ℂ) ^{Π_{k}}[figure] H^{0}(M, L^{⊗k}) to some space of distributions D(H) ⊃ L^{2}(P, ℂ), which includes the delta functions δ_{p} of all points p∈P. This continuation uses the Toeplitz structure concept developed in [BG] and sketched in n^{∘} 4.4.2 below.

Applying the continued Szegö projector to δ_{p} with p∈S̃ ⊂ P, we get a family of holomorphic sections {Π_{k} ( δ_{p})}_{p∈P_{k}} ⊂ H^{0}(M_{I}, L^{⊗k}) and the BPU_{k}-map P^{∘} _{k}^{hw} ^{𝔪_{k}}^{}H^{0}(M_{I}, L^{⊗k}) just

takes their overage along S, that is, sends each (S̃, ϑ) ∈P^{∘} _{k}^{hw} to 𝔪_{k}(S, ϑ) = ∫_{S} Π_{k}(δ_{p})dϑ.

Now, if we write K_{k}(p_{1}, p_{2}) for the Schwartz kernel of the continued Π_{k}, then we have K_{k}(p_{1}, p_{2}) = Π_{k} ( δ_{p_{2}} ) (p_{1}), where s(p_{k}(p)) =^{def} ⟨ s , Π_{k}(δ_{p}) ⟩ for s ∈ H^{0}(M_{I}, L^{⊗k}). Taking here s = Π_{k}(δ_{p}), we get K_{k}(p, p) = ∥Π_{k} ( δ_{p}) ∥^{2}, which is a function of p∈P_{k} with the L^{2}-norm

∥K_{k}(p, p)∥^{2} = ∫_{P_{k}} K_{k}(p, p) α_{k} ∧ (dα_{k})^{∧n} = rk H^{0}(M_{I}, L^{⊗k}) ∼ ( \frac{k}{2π} )^{n}

for k large enough by the Riemann - Roch theorem.

4.4.2. Generalized Toeplitz operators. Let us cut the zero section out the cotangent bundle T^{∗}P_{k} and consider a section P_{k} ^{⊂}^{α_{k}}^{}T^{∗}P_{k} \{0} given by the curvature form. An open real positive cone over its image

Σ =^{def} { t α_{k}|_{p} ⊂ T^{∗}P_{k} \ {0} | p∈P_{k} , t∈ℝ , t > 0 }  (4-14)

is a symplectic submanifold w. r. t. the standard symplectic structure on the cotangent bundle. The pair (Π, Σ), where Π is the Szegö projector (4-4), is called a Toeplitz structure of Boutet de Monvel and Guillemin (see [BG]). For any pseudodifferetial operator A of order n on P_{k} an operator T_{A} = Π ∘ A ∘ Π : L^{2}(P) ^{}H is called a generalized Toeplitz operator. A symbol of the generalized Toeplitz operator T_{A} is the principal symbol p(A) restricted on the symplectic submanifold (4-14) : σ(T_{A}) =^{def} p(A)|_{Σ} . It is well defined, although A is not uniquely recovered from p(A). Moreover, the symbols of the generalized Toeplitz operators obey the standard rules for pseudodifferetial operator symbols:

σ ( T_{A_{1}}T_{A_{2}} ) = σ ( T_{A_{1}} ) σ ( T_{A_{2}} )
σ ( [T_{A_{1}}, T_{A_{2}}] ) = i {σ ( T_{A_{1}} ) , σ ( T_{A_{2}} )} ,

where the last Poisson bracket is with respect to the restriction of the standard symplectic form to Σ. For example, a usual (non generalized) zero order Toeplitz operator T_{f} coming from the multiplication by f ∈ L^{2}(P_{k}) has the symbol σ(T_{f}) = p_{Σ}^{∗}(f), where p_{Σ} is the composition of the projections Σ [figure] P_{k} ^{p_{k}} [figure] M .

4.4.3. Extended Szegö projector. Of course, the conormal bundle of a smooth Planckian cycle N^{∗}S̃ = Ann (T S̃) ⊂ T^{∗}P_{k} is Lagrangian w. r. t. the standard symplectic structure on T^{∗}P_{k}. Hence, it defines a series of Hermitian distributions spaces I^{m}(P_{k}, N^{∗}S) ⊂ D(P_{k}) of half integer order m ∈ \frac{1}{2}ℤ (see [BG]) and L^{2}(P_{k}) ⊂ I^{m}(P_{k}, N^{∗}S), certainly. Each Hermitean distribution also has a symbol, which is a symplectic spinor on N^{∗}S. Using composition theorems from [BG] for Hermitian distributions, the Szegö projector (4-4) can be continued to the map

I^{m}(P_{k}, N^{∗}S̃) ^{Π}^{}D(H)

as follows. The Schwartz kernel of Π lays in I^{\frac{1}{2}} (P_{k} × P_{k}, Σ_{2}), where

Σ_{2} = { (p_{1}, tα(p_{1})) ; (p_{2}, −tα(p_{2})) } ⊂ T^{∗}P_{k} × P_{k} .

is an isotropic submanifold, which is composable with N^{∗}S (see [BG, Th. 9.4]), that is, one can apply the Fourier integral operator associated with Σ_{2} to the Lagrangian distributions on N^{∗}S. This gives the map I^{m}(P_{k}, N^{∗}S̃)^{⊂} ^{}D(H) whose image J^{m− \frac{n}{2}} (P_{k}, S) ⊂ D(H) is called the space of Legendrian distributions of order m − \frac{1}{4} dim M. Finally, J^{\frac{1}{2}} (P_{k}, S) may be identified with the space of half weights: each element of J^{\frac{1}{2}} (P_{k}, S) produces a half weight on S associated with its symbol, which is a symplectic spinor on the Legendrian submanifold S̃ ⊂ P_{k} (see [BG] and [BPU]), and wise versa, a Hermitean distribution is recovered from a half weight ϑ on S as

s(ϑ) = T_{Lie_{u_{k}}}^{m− \frac{1}{2}} Π(δ_{S}) ,

where δ_{(S,ϑ)} is the delta function defined by a pair (S, ϑ) and T_{Lie_{u_{k}}} is the generalized first order Toeplitz operator induced by the Lie differentiation along the standard U(1)-invariant vertical unitary vector field u_{k} on P_{k} (its symbol is the real function t, the coordinate along the rulers on the conic submanifold (4-14)).

§ 5. A ‘toy’ example: the Legendrian knots in S^{3}.

5.1. Standard prequantization data on Riemannian sphere. We are going to illustrate the constructions from the last two paragraphs in the simplest non trivial case, where (M, ω) is the Riemannian sphere S^{2} = ℂℙ_{1} = ℙ(ℂ^{2}) equipped with the standard Kähler structure on ℂℙ_{1} and the standard very ample line bundle of homogeneous degree 1 forms L = O(1) as the prequantization line bundle. So, the principal prequantization bundle P ^{p} [figure] ℂℙ_{1}, that is the bundle of unit circles in the tautological line bundle L^{∗} = O(−1), is nothing but the Hopf bundle S^{3} ^{p} [figure] ℂℙ_{1} whose total space is the unit sphere S^{3} ⊂ ℂ^{2} w. r. t. the standard Hermitean structure: its fiber over ℓ ⊂ ℂ^{2} is the unit circle ℓ ∩ S^{3}. The homotopy class of this bundle gives the standard negative generator {−1} ∈ π_{3}(S^{2}) ≃ ℤ and the other negative elements of this group are exhausted by our level k principal prequantization bundles P_{k}, which are the twisted Hopf bundles S^{3} ^{p_{k}} [figure] S^{2}, of the unit circles in O(−k). The standard 𝔲(1)-valued connection form on P = P_{1} is usually written as of the U(2)-invariant 1-form

2πi α = ( ∂ − \overline{∂} ) log ϱ ,

restricted on S^{3} ⊂ ℂ^{2}, where ℂ^{2} ^{ϱ}^{}ℝ is the Hermitean length ϱ(v) = ∥v∥ = \sqrt{(v, v)}. The differential

dα = \frac{i}{2π} ∂\overline{∂} log ϱ = p^{∗}ω

is the pull back of the standard Kähler form on ℂℙ_{1} (see, for example, [GH, ch. 0 , § 7]). The cohomology class [ω] coincides with c_{1}(L) and gives the standard ‘positive’ generator for the group H^{2}(S^{2}, ℤ) ≃ ℤ.

5.2. Lagrangian cycles, or unicoursal curves on S^{2}. We take the simplest source manifold for the Lagrangian cycles, namely, the circle S = S^{1}. Since any smooth map S^{1} ^{φ}^{}S^{2} is Lagrangian just because of the dimension reasons, the space of smooth Lagrangian immersions Map_{la}(S^{1}, S^{2}) turns to the space of loops LS^{2} and the space of Lagrangian cycles 𝔏 = Map_{la}(S^{1}, S^{2})/Diff_{0}(S^{1}) is just the space of (unparameterized) unicoursal curves φ(S^{1}) ⊂ S^{2}. Further, the tangent space to the loop space is the space of smooth sections of the pulled back tangent bundle: T_{φ} ( LS^{2} ) = C^{∞}(S^{1}, φ^{∗}T S^{2}) . This 2-bundle is equipped with the non degenerate skew form φ^{∗}ω, which gives the natural symplectic structure

Ω_{LS^{2}}(σ_{1}, σ_{2}) = ∫_{S^{1}} φ^{∗}ω^{−1}(σ_{1}) ∧ φ^{∗}ω^{−1}(σ_{2})  (5-1)

on T ( LS^{2} ). It is, clearly, Diff_{0}(S^{1})-invariant and gives the symplectic structure on 𝔏. It is closed and weakly non degenerated but a priori does not define the isomorphism between the tangent and cotangent bundles, in particular, only the supersmooth functions (see [Br, def. 3.1.5]) on 𝔏 will define the Hamilton vector fields.

5.2.1. Darboux - Weinstein coordinates. Following n^{∘} 1.4, we identify the normal bundle to S^{1} along φ with the cotangent bundle on S^{1}, which is trivial:

N_{φ}S^{1} = φ^{∗}T S^{2}/T S^{1} ≃ φ^{∗}T^{∗}S^{2}/Ann (T S^{1}) = T^{∗}S^{1} = S^{1} × ℝ .  (5-2)

Let us fix this trivialization by taking the rotation invariant 1-form ν_{S^{1}} with ∫_{S^{1}} ν_{S^{1}} = 1 as

the base of T^{∗}S^{1}. Note that this form produces the smooth measure μ_{0}, whose class will be used below for our half weighings.

The Darboux - Weinstein construction (see n^{∘} 1.4.2) continues any smooth immersion S^{1} ^{}S^{2} to an symplectic immersion U φ̂^{}S^{2}, where U ⊂ T^{∗}S^{1} is some neighborhood around the zero section considered with the canonical symplectic structure φ̂^{∗}ω = dη on the cotangent bundle T^{∗}S^{1}. Thus, the graphs of closed 1-forms laying in U form the standard chart near φ on 𝔏. A local coordinatization of 𝔏 by such the charts from Z_{dr}^{1}(S^{1}) is called the DW-coordinatization.

5.2.2. Remark on extra symmetries. The case we are deal with now has some special extra symmetries that are lacking in the general setup. They are:

• the triviality of the cotangent bundle (5-2) ;

• the natural inclusion U(1) ⊂ Diff_{0}(S^{1}), which allows to consider the source manifold S^{1} as the principal U(1)-space;

• the natural coincidence between U(1) and the Jacobian J(S^{1}) = H^{1}(S^{1}, ℝ)/H^{1}(S^{1}, ℝ).

We will outline in special remarks the additional effects added in the general picture by these coincidences.

5.3. Planckian cycles, or Legendrian knots in S^{3}. For each k the pull back φ^{∗}L^{⊗k} along any loop S^{1} ^{φ}^{}S^{2} is the trivial complex line bundle on S^{1} with the flat U(1)- connection φ^{∗}α_{k}. Its gauge class is described by a character π_{1}(S^{1}) = ℤ ^{χ_{α_{k}}}^{}U(1) such that χ_{α_{k}}(1) ∈ U(1) is the fiber twist in φ^{∗}L^{⊗k} under the one round parallel transport along S^{1} (in particular, χ_{α_{k}} = k·χ_{α_{1}}). The map j_{k} : LS^{2} ^{⊂} ^{φ↦k·χ_{α}(1)} ^{}J gives a section of the Jacobian bundle^{1} (see n^{∘} 2.4). The zero set of this section is the codimension 1 subspace B_{k} ⊂ 𝔏 of level k Bohr - Sommerfeld cycles. Clearly, ∀ k, m ∈ ℤ we have the natural inclusions (3-3) : B_{k} ^{⊂} ^{β_{k,m}} ^{}B_{mk}.

A Planckian cycle is a class of Legendrian maps S^{1} φ̃^{}S^{3} w. r. t. the contact structure α_{k} on S^{3} ≃ P_{k} modulo Diff_{0}(S^{1}). Usually, a smooth map of this kind is called a Legendrian knot in S^{3}. In our terms, such a knot is nothing but a covariant constant global section S^{1} ^{⊂} ^{σ}^{}φ^{∗}P_{k} over a BS_{k}-cycle S^{1} ^{φ}^{}S^{2}, that is, a covariant constant global unitary section of φ^{∗}L^{⊗k} over S^{1}. So, Legendrian knots have got the levels and the knots of level k form a principal U(1)-bundle P_{k} ^{π_{k}} [figure] B_{k} . called the Berry bundle. The corresponding Hermitian line bundles L_{k} over B_{k} satisfy the conditions β_{k,m}^{∗} L_{mk} = L_{k}^{⊗m}.

1in our case J is the principal U(1)-bundle over LS^{2}, because of the third coincidence in n^{∘} 5.2.2 :

H^{1}(S^{1}, ℝ)/H^{1}(S^{1}, ℤ) = U(1)

The tangent space Tφ̃P_{k} at a smooth Legendrian knot S^{1} ^{⊂} φ̃^{}P_{k} ≃ S^{3} of level k is naturally identified with C^{∞}(S^{1}). Extra symmetries listed in n^{∘} 5.2.2 make this to be almost trivial. Indeed, the tangent space Tφ̃P_{k} is the space of smooth sections of the normal bundle

Nφ̃ = φ̃^{∗}T^{∗}P_{k}/T S^{1} = T^{∗}S^{1} ⊕ T^{∗}U(1) = S^{1} × ℝ × ℝ ,

where the right equality comes from (5-2) and the left one is induced by the connection α_{k} (in general case it holds only locally). Any f ∈ C^{∞}(S^{1}) gives the section τ_{f} ∈ C^{∞}(S^{1}, Nφ̃) as follows. The vertical component of τ_{f} equals kf · ν_{U(1)}, where ν_{U(1)} is the basic 1-form u_{U(1)}

on U(1) with ∫_{U(1)} ν_{U(1)} = 1. The horizontal component of τ_{f} is k df ∈ C^{∞}(T^{∗}S^{1}) (and equals

the angle derivative of kf multiplied by the standard 1-form ν_{S^{1}} on S^{1} fixed in (5-2)).

5.4. Half weights. Let μ_{0} be the rotation invariant measure on S^{1} induced by ν_{S^{1}}. Then a smooth half weight ϑ = f\sqrt{μ} on S^{1} is an equivalence class of pairs (f, μ), where f ∈ C^{∞}(S^{1}), μ is a measure obtained from μ_{0} by multiplication by a smooth function, and the equivalence (f_{1}, μ_{1}) ∼ (f_{2}, μ_{2}) means that

f_{1}\sqrt{\frac{dμ_{1}}{dλ}} = f_{2}\sqrt{\frac{dμ_{2}}{dλ}}

almost everywhere w. r. t. any measure λ equivalent to μ_{0}. The space of smooth half weights is denoted by ℌ^{∘}=ℌ^{∘} (S^{1}) and the orbit spaces

𝔏^{∘}^{hw} =^{def} Map_{la}(S^{1}, S^{2})× ℌ^{∘} (S^{1})/Diff_{0}(S^{1})  P^{∘} _{k}^{hw} =^{def} Map_{le}(S^{1}, S^{3})× ℌ^{∘} (S^{1})/Diff_{0}(S^{1}) .

(w. r. t. the natural diagonal actions) are called spaces of half weighted unicoursal curves and Legendrian knots respectively.

5.4.1. Complex structure. For any fixed half weight f \sqrt{μ} the tangent space Tφ̃P_{k} =

C^{∞}(S^{1}) is identified with T_{f\sqrt{μ}} ℌ^{∘} ≃ ℌ^{∘} by sending a function f_{1} to the half weight ff_{1} \sqrt{μ}.

This gives the canonical integrable complex structure on P^{∘} _{k}^{hw}. Namely, consider a smooth knot S^{1} ^{⊂} φ̃^{}S^{3} laying over a BS_{k}-cycle S^{1} ^{⊂} ^{φ}^{}S^{3} and half weighted by ϑ = f\sqrt{μ}. Then we have the splitting (3-11) :

T_{(φ̃,ϑ)} P^{∘} _{k}^{hw} ≃ T_{ϑ} ℌ^{∘} (S^{1}) ⊕ Tφ̃ P^{∘} _{k}^{hw} ≃ C^{∞}(S^{1}) ⊕ C^{∞}(S^{1}) .

Now DW-construction identifies a small neighborhood of (φ̃, ϑ) in P^{∘} _{k}^{hw} with a ball around the zero in the space of smooth complex valued functions ψ = ψ_{1} + iψ_{2} ∈ C^{∞}(S^{1}, ℂ). Indeed, the imaginary part ψ_{2} produces the Legendrian knot, which is the section (dψ_{2}, e^{iπψ_{2}}) of the bundle

φ̂^{∗}P_{k}|_{U} ⊂ T^{∗}S^{1} × U(1)

The real part equips this knot with a half weight ψ_{1}ϑ = ψ_{1}f\sqrt{μ}.

5.4.2. Kählerian structure on the space of half weighted Legendrian knots is given by the skew symmetric form whose action on T_{(φ̃,ϑ_{0})} P^{∘} _{k}^{hw} is given by

Ω( ( ϑ_{1}, f_{1} ) , ( ϑ_{2}, f_{2} ) ) = ∫_{S^{1}} f_{2} dϑ_{0}ϑ_{1} − ∫_{S^{1}} f_{1} dϑ_{0}ϑ_{2} .

This form is the differential of the action 1-form H ( (ϑ_{1}, f_{1}) ) = 2 ∫_{S^{1}} f_{1} dϑ_{0}ϑ_{1}. The symmetric form corresponding to the above complex and symplectic structures acts on T_{(S^{1},ϑ_{0})}P^{hw} = C^{∞}(S) ⊕ ℌ(S) as

G( (ϑ_{1}, f_{1}) , (ϑ_{2}, f_{2}) ) = ∫_{S^{1}} f_{2} dϑ_{1}ϑ_{0} + ∫_{S^{1}} f_{1} dϑ_{2}ϑ_{0} .

5.4.3. Half weighted Bohr - Sommerfeld unicoursal curves of fixed length are obtained

by the symplectic reduction of the half weighted Legendrian knots w. r. t. the natural U(1)- action on the above data. The moment map for this action is given by the length functional

(φ̃, ϑ) ⟼ ∫_{S^{1}} dϑ^{2}

Its t-th level hypersurface factorized by U(1) is denoted by B^{∘} _{k,t}^{hw}. The previous complex

structure and the forms Ω and G remains to be well defined on T B^{∘} _{k}^{hw} whose fiber over a

given (φ̃, ϑ_{0}) consists of all pairs (ϑ, df) such that f ∈ C^{∞}(S^{1}) and ∫_{S^{1}} dϑϑ_{0} = 0.

To compare this Kählerian structure with the classical results, let us note that the fiber of the weight forgetful projection

B^{∘} _{k,t}^{hw} [figure] B_{k,t}  (5-3)

is naturally identified with Diff_{0}(S^{1})/U(1), because each half weighting ϑ = f\sqrt{μ} produces a well defined up to rotations parametrization of the curve by the arc length w. r. t. the measure ϑ^{2} = f^{2}μ. The complex structure on Diff_{0}(S^{1})/U(1) is well known (see [K], for example). More generally, let F_{2m} ⊂ C^{∞}(S^{1}) consist of functions having precisely 2m zeros (this space is a vector bundle over the 2m-th symmetric power of S^{1}). These F_{2n} give the orbits for the natural action of Diff (S^{1}) on C^{∞}(S^{1}) (see for example [K]). So, the space of, say, positively half weighted unicoursal curves is nothing but LS^{2}/U(1) = F_{0}; in particular, it forms precisely one Diff_{0}(S^{1})-orbit.

On the other hand, there is an almost complex structure on B_{k,t} defined as follows. Identifying S^{1} with U(1) like in n^{∘} 5.2.2, we get the complex structure on the loop space LS^{3}, just because the normal bundle of any loop S^{1} φ̃^{}S^{3} splits as Nφ̃ = T^{∗}S^{1} ⊕ T^{∗}U(1) and Tφ̃LS^{2} coincides with the space of smooth sections C^{∞}(S^{1}, Nφ̃). Restricting this complex structure on the space of Legendrian knots P_{k} ⊂ LS^{3} we get there an almost complex structure, which is invariant w. r. t. the diagonal action ofU(1) on Nφ̃ = T^{∗}S^{1} ⊕ T^{∗}U(1). Its symplectic reduction produces the required complex structure on B_{k,t} = P_{k,t}/U(1). Now, the point is that the forgetful map (5-3) is pseudo holomorphic (and has pseudo holomorphic fibers) w. r. t. the complex structures just described.

5.5. BPU_{k}-map. Let V_{k} = H^{0}(ℂℙ_{1}, L^{⊗k}) be the space of homogeneous binary forms of degree k. It is equipped with the standard Hermitean structure (actually, if we fix some basis { 0 , 1 , ∞ } on ℂℙ_{1}, then V_{k} turns to the space of usual polynomials of degree k in one complex variable). Then, each level k Legendrian knot S^{1} ^{⊂} φ̃^{}S^{3} = P_{k} defines the inclusion V_{k} ^{⊂} ^{}C^{∞}(S^{1}, ℂ) by sending a holomorphic section s ∈ H^{0}(ℂℙ_{1}, L^{⊗k}) to the function ψφ̃ such that φ^{∗}s = ψφ̃ · σ, where S^{1} ^{⊂} ^{σ}^{}φ^{∗}L^{⊗k} is the trivialization defined by the Legendrian knot. The map P^{∘} ^{hw} ^{\overline{𝔪}_{k}} ^{}V_{k}^{∗} sending a half weighted Legendrian knot (φ̃, ϑ) to the form \overline{𝔪}_{k}(φ̃, ϑ)that takes s ∈ H^{0}(M, L^{⊗k}) to

s ⟼ ∫_{S} ψφ̃ dϑ^{2} = ∫_{S} ψφ̃ f^{2} dμ

is antiholomorphic and has the surjective differential. Its composition with the Hermitian dualization V_{k}^{∗} ^{∼}^{}\overline{V}_{k} gives the holomorphic map P^{∘} _{k}^{hw} ^{𝔪_{k}}^{}V_{k} called the BPU_{k}-map. It sends a half weighted Legendrian knot (φ̃, ϑ) to 𝔪_{k}(φ̃, ϑ) ∈ V_{k} defined by the identity

⟨ s , 𝔪_{k}(φ̃, ϑ) ⟩_{V_{k}} = ∫_{S} ψφ̃ dϑ^{2}

which has to hold for any s ∈ V_{k}. Of course, this map is invariant w. r. t. the natural action of the symplectomorphism group Sym_{0} = Sym_{0}(S^{2}) and is equivariant w. r. t. the standard U(1)- action on P^{∘} _{k}^{hw} and V_{k}. The symplectic reduction of the BPU_{k}-map w. r. t. the U(1)-action gives the holomorphic map

𝔪_{k,t} : B^{∘} _{k,t}^{hw} ^{}ℙ(V_{k}) ,

which is Sym_{0}-invariant as well and assigns holomorphically an algebraic cycle of degree k on ℂℙ_{1} to each isotopy class of the half weighted Bohr - Sommerfeld unicoursal curves of fixed volume. So, these isotopy classes, that is, the orbits of Sym_{0} in B^{∘} _{k,t}^{hw}, are diffeomorphic to homogeneous spaces

ℙ ( V_{k}) × (Sym_{0}/Stab_{(φ,ϑ)}) ,

where (φ, ϑ) is any representative for a component. Let us recall, finally, that each isotopy component of the Legendrian knots space produces a series of well known polynomials: Alexander, V. Jones, Kaufmann, HOMFLY, e. t. c. The BPU_{k}-map holomorphically takes each Legendrian knot to a polynomial from V_{k}. How it interacts with the polynomials just mentioned?

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