aa r X i v : . [ m a t h . S G ] D ec GEOMETRIC QUANTIZATION OF DIRAC MANIFOLDS
Yuji HIROTA
Abstract
We define prequantization for Dirac manifolds to generalize known procedures for Poisson and(pre) symplectic manifolds by using characteristic distributions obtained from 2-cocycles associatedto Dirac structures. Given a Dirac manifold ( M , D ), we construct a Poisson structure on the space ofadmissible functions on ( M , D ) and a representation of the Poisson algebra to establish the prequan-tization condition of ( M , D ) in terms of a Lie algebroid cohomology. Additional to this, we introducea polarization for a Dirac manifold M and discuss procedures for quantization in two cases where M is compact and where M is not compact. Mathematics Subject Classification(2000).
Keywords.
Dirac structures, Lie algebroids, prequantization, geometric quantization.
In classical mechanics, a state of a system is described by a pair of position and momentum, andthe time evolution of the system is controlled by Hamilton’s equation. On the other hand, in quantummechanics, a quantum state is given as a point in a complex Hilbert space, and the time evolution ofquantum system is described by Shr¨odinger’s equation. In addition, physical quantities such as position,momentum and energy are given by self-adjoint operators on the Hilbert space describing the quantumsystem, and the time evolution of the physical quantity is defined by Heisenberg’s equation. For instance,we consider the motion of N -particles in R and let ( q j , p j ) ( j = , ,
3) be the coordinates of position andmomentum variables. Then, the space C ∞ ( R N ) of smooth functions on the phase space R N ≃ R N × R N is Poisson algebra by { F , G } : = X j (cid:16) ∂ G ∂ p j ∂ F ∂ q j − ∂ F ∂ p j ∂ G ∂ q j (cid:17) . This leads to the following relations: { q j , q k } = { p j , p k } = , { q j , p k } = δ j , k ( ∀ j , k = , , · · · , N ) . Defining self-adjoint operators ˆ q j and ˆ p j in a Hilbert space L ( R N ) as ˆ q j : = q j · , ˆ p j : = − √− ~ ∂∂ q j for positions q j and momenta p j , one can get the following relations, called the canonical commutationrelations: [ ˆ q j , ˆ q k ] = [ ˆ p j , ˆ p k ] = , [ ˆ q j , ˆ p k ] = i ~ δ j , k ( ∀ j , k = , , · · · , N ) , where [ · , · ] means a commutator [ A , B ] : = AB − BA for operator algebras A , B . In other words, one canobtain quantum objects from classical objects by choosing a Hilbert space proper for corresponding tothe classical theory and by constructing self-adjoint operators on it.Such a procedure to determine a quantum theory which corresponds to a given classical theory iscalled a quantization. Mathematically, a quantization is a procedure to construct a representation of a1oisson algebra on the space of functions on a proper Hilbert space for a given manifold. Quantizationis a meaningful subject for the study of mathematics and physics, and is an extremely interesting one atwhich mathematics and physics intersect as well. It is known that there are several kinds of quantizations,such as canonical quantization, Feynman’s path integral quantization, geometric quantization, Moyalquantization, Weyl-Wigner quantization and so on. Among those, geometric quantization consists of twoprocedures: prequantization and polarization. Prequantization assigns to a given symplectic manifold S a Hermitian line bundle L → S with a connection whose curvature 2-form is the symplectic structure.Then, a Poisson subalgebra of C ∞ ( S ) acts faithfully on the space Γ ∞ ( S , L ) of smooth sections of L . Onthe other hand, a polarization is the procedure which reduces Γ ∞ ( S , L ) to a subspace A ⊂ Γ ∞ ( S , L )appropriate for physics so that a subalgebra of C ∞ ( S ) may still act on A .The study of geometric quantization in symplectic geometry goes back to the theory by B. Kostantand J. Souriau (see [17, 21]). Later, a target for quantization was extended to a presymplectic manifold,and its quantization was studied by many researchers in [12, 13, 26]. After that, J. Huebschmann ex-tended the target from a (pre)symplectic manifold to a Poisson manifold and investigated algebraicallyits quantization in [15]. Besides, I. Vaisman studied the quantization of a Poisson manifold in terms ofHermitian line bundles in [27], and D. Chinea, J. Marrero and M. de Leon did it in terms of S -bundlesin [6]. In the case where the target is a twisted Poisson manifold, its geometric quantization is studiedby F. Petalidou in [19]. Lastly, in [32], A. Weinstein and M. Zambon studied a prequantization of Diracmanifolds which is a generalization of (pre) symplectic and Poisson manifolds in terms of Dirac-Jacobistructures appeared in [29, 16].Dirac manifolds were introduced by T. Courant for the purpose of unifying approaches to the geom-etry of Hamiltonian vector fields and their Poisson algebras, which are thought of generalizations of bothpresymplectic manifolds and Poisson manifolds (see [7]). The purpose of this paper is to unify knownprequantization approach for Poisson and (pre) symplectic manifolds by introducing a prequantizationprocedure for Dirac manifolds. Using characteristic distributions from a 2-cocycle associated to ( M , D ),we define a Poisson bracket for admissible functions on ( M , D ) and give a representation of the Poissonalgebra in terms of a connection theory of Lie algebroids to describe a condition for prequantization of( M , D ). Our approach to prequantization for Dirac manifolds is di ff erent from the one discussed in [32].The paper is organized as follows: In Section 2, we review the fundamentals of Dirac manifolds. InSection 3, after reviewing the Lie algebroid cohomology and the connection theory of Lie algebroids,we introduce the first Dirac-Chern class of line bundles over Dirac manifolds and show that it does notdepend on a choice of connections. In Section 4, we introduce a prequantization for Dirac manifolds. Wedefine a Poisson structure on the space of admissible functions associated to a singular distribution for agiven Dirac manifold ( M , D ) and construct a map from the Poisson algebra to the space of sections of acomplex line bundle over ( M , D ). We provide the necessary and su ffi cient condition for the map to be arepresentation of the Poisson algebra. Lastly, we formulate the condition for prequantization of ( M , D )to be realized in terms of Lie algebroid cohomology for ( M , D ). In Section 5, we introduce polarizationsfor Dirac manifolds and develop the quantization process of them, basing on the discussion in Section 4.Throughout the paper, every smooth manifold is assumed to be paracompact, and all maps are as-sumed to be smooth. We denote by Γ ∞ ( M , E ) the space of smooth sections of a smooth vector bundle E → M . Especially, if E = T M , we often write X ( M ) for Γ ∞ ( M , T M ). We use Ω k ( M ) and X k ( M ) for Γ ∞ ( M , ∧ k T ∗ M ) and Γ ∞ ( M , ∧ k T M ), respectively. 2
Dirac manifolds
Let M be a finite dimensional smooth manifold. We define symmetric and skew-symmetric opera-tions on the vector bundle T M : = T M ⊕ T ∗ M over M as h X ⊕ ξ, Y ⊕ η i + : = (cid:8) ξ ( Y ) + η ( X ) (cid:9) ∈ C ∞ ( M )and ~ X ⊕ ξ, Y ⊕ η (cid:127) : = [ X , Y ] ⊕ ( L X η − i Y d ξ ) ∈ Γ ∞ ( M , T M )for all X ⊕ ξ, Y ⊕ η ∈ Γ ∞ ( M , T M ). Here L X η stands for the Lie derivative of η by X and i Y d ξ forthe contraction of d ξ with Y . A subbundle D ⊂ T M is called a Dirac structure on M if the followingconditions are satisfied:(D1) h· , ·i + | D ≡ D has rank equal to dim M ;(D3) ~ Γ ∞ ( M , D ) , Γ ∞ ( M , D ) (cid:127) ⊂ Γ ∞ ( M , D ).A smooth manifold M together with Dirac structure D ⊂ T M is called a Dirac manifold, denoted by( M , D ). In addition to the natural pairing h· , ·i + , one defines a skew-symmetric pairing h· , ·i − as h X ⊕ ξ, Y ⊕ η i − : = { ξ ( Y ) − η ( X ) } ∈ C ∞ ( M ) . The following formula can be shown by direct calculation and Condition (D3).
Lemma 2.1
It holds that ( L X ξ )( X ) + ( L X ξ )( X ) + ( L X ξ )( X ) = for any X ⊕ ξ , X ⊕ ξ , X ⊕ ξ ∈ Γ ∞ ( M , D ) . Given a Dirac structure D ⊂ T M , there are natural projections( ρ T M ) p : = pr | D p : D p → T p M and ( ρ T ∗ M ) p : = pr | D p : D p → T ∗ p M for each point p ∈ M . The singular distribution M ∋ p ( ρ T M ) p ( D p ) ⊂ T p M is called the characteristicdistribution. It is known that the characteristic distribution is integrable in the sense of [10] and [24].The corresponding singular foliation is called the characteristic foliation. For a discussion of singulardistributions and the integrability, we refer to [22, 23] and [24]. It is easy to check thatker ρ T M = D ∩ T ∗ M and ker ρ T ∗ M = D ∩ T M , (2.2)where D ∩ T ∗ M : = D ∩ ( { } ⊕ T ∗ M ) , D ∩ T M : = D ∩ ( T M ⊕ { } ). Here, we remark that, for each p ∈ M , D p ∩ T p M (resp. D p ∩ T ∗ p M ) are thought of as a subspace of either T p M ⊕ T ∗ p M or T p M (resp. T ∗ p M ).The following proposition is easily checked. 3 roposition 2.2 ([ ]) Given a Dirac manifold ( M , D ) , one has the characteristic equations ρ T M ( D ) = ( D ∩ T ∗ M ) ◦ and ρ T ∗ M ( D ) = ( D ∩ T M ) ◦ , where the symbol ◦ stands for the annihilator. For each p ∈ M , we define a bilinear map Ω p on the subspace ( ρ T M ) p ( D p ) ⊂ T p M as Ω p ( X p , Y p ) : = ξ p ( Y p ) ( ∀ Y p ∈ ( ρ T M ) p ( D p )) , (2.3)where ξ p is an element in T ∗ p M such that X p ⊕ ξ p ∈ D p . It is shown that Ω is well-defined, and is apresymplectic form on ρ T M ( D ) by Lemma 2.1 and Proposition 2.2 (see [7]). The symbol Ω ♭ denotesthe bundle map induced from Ω . That is, Ω ♭ is the map Ω ♭ : ρ T M ( D ) → ρ T M ( D ) ∗ which assigns Ω ♭ ( X ) = Ω ( X , · ) to X ∈ ρ T M ( D ). One easily finds that ker Ω ♭ = D ∩ T M .In the same way, one also obtains a skew-symmetric tensor fields Π : ρ T ∗ M ( D ) × ρ T ∗ M ( D ) → C ∞ ( M )by Π p ( ξ p , η p ) : = ξ p ( Y p ) ( p ∈ M ) , (2.4)where Y p is a vector in T p M such that Y p ⊕ η p ∈ D p . The form Π defines a map, denoted by Π ♯ , from thesubspace ρ T ∗ M ( D ) = ( D ∩ T M ) ◦ to ( ρ T ∗ M ( D )) ∗ by( ρ T ∗ M ) p ( D p ) ∋ η p (cid:8) ξ p ξ p (cid:0) Π ♯ p ( η p ) (cid:1) : = ξ p ( Y p ) (cid:9) ∈ ( ρ T ∗ M ( D )) ∗ for each p ∈ M . Letting X ⊕ ξ, Y ⊕ η be smooth sections of D , we have X = Π ♯ ( ξ ) , Y = Π ♯ ( η ) and ~ Π ♯ ( ξ ) ⊕ ξ, Π ♯ ( η ) ⊕ η (cid:127) = [ Π ♯ ( ξ ) , Π ♯ ( η )] ⊕ { ξ, η } ∈ Γ ∞ ( M , D ) , where { ξ, η } : = L Π ♯ ( ξ ) η − i Π ♯ ( η ) d ξ . This implies that Π ♯ ( { ξ, η } ) = [ Π ♯ ( ξ ) , Π ♯ ( η )] ( ∀ X ⊕ ξ, Y ⊕ η ∈ Γ ∞ ( M , D ) ) . Example 2.1
Suppose that M be a (pre)symplectic manifold with a (pre)symplectic form ω . The 2-form ω induces the bundle map ω ♭ : X ( M ) −→ Ω ( M ) , X i X ω. One can obtain the subbundle graph ( ω ♭ ) in T M as graph ( ω ♭ ) p : = { X p ⊕ i X p ω p ∈ T p M ⊕ T ∗ p M | X p ∈ T p M } ( p ∈ M ) and can verify that graph( ω ♭ ) satisfies the three conditions (D1) – (D3) in the above. Therefore, ( M , graph ( ω ♭ )) is a Dirac manifold. Similarly, any symplectic manifold M defines a Dirac structure on M. Example 2.2
Similarly to Example 2.1, any Poisson manifold ( P , π ) defines a Dirac structure. Indeed,the 2-vector field π induces the bundle map π ♯ : Ω ( P ) −→ X ( P ) , α β π ( β, α ) } . and the subbundle graph ( π ♯ ) given by graph ( π ♯ ) p : = { π ♯ ( ξ p ) ⊕ ξ p ∈ T p P ⊕ T ∗ p P | ξ p ∈ T ∗ p P } ( p ∈ P ) . It can be easily verified that ( P , graph ( π ♯ )) is a Dirac manifold. xample 2.3 We let F ⊂ T M be a regular distribution and assume that F is involutive. Then, the vectorbundle F ⊕ F ◦ → M is a Dirac structure on M, where F ◦ denotes the annihilator of F in T ∗ M. Example 2.4
Let Q be a submanifold of a Dirac manifold ( M , D ) . If either D q ∩ ( T q Q ⊕ T ∗ q M ) orD q ∩ ( T q Q ) ◦ has constant dimension at each point q ∈ Q, Q has a Dirac structure D Q defined by ( D Q ) q = D q ∩ ( T q Q ⊕ T ∗ q M ) D q ∩ ( T q Q ) ◦ . Example 2.5
Let η be a 1-form on M. A subbundle ( D η ) p : = { X p ⊕ i X p ( d η ) p ∈ T p M ⊕ T ∗ p M | X p ∈ T p M } ( p ∈ M ) satisfies the three conditions (D1) – (D3) . Therefore, ( M , D η ) is a Dirac manifold. Example 2.6
We define a 2-form on T ∗ M × R as the pullback of the canonical symplectic form ω onT ∗ M by the projection pr on the first factor. Then, a subbundle ( D ω ) ( z , t ) = n (cid:16) X z , f ( z , t ) ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:17) ⊕ pr ∗ (i X z ω ) (cid:12)(cid:12)(cid:12) (cid:16) X z , f ( z , t ) ddt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t (cid:17) ∈ T ( z , t ) ( T ∗ M × R ) o (( z , t ) ∈ T ∗ M × R ) turns out to be a Dirac structure over T ∗ M × R by noting that the vector field X on T ∗ M is pr -related to (cid:0) X , f ddt (cid:1) on T ∗ M × R . Example 2.7
Let ( M , ω, η ) be an almost cosymplectic manifold. That is, M is a k + -dimensionalmanifold equipped with a 2-form ω and a 1-form η such that η ∧ ω k is a volume form on M. If ω isclosed, then a subbundle ( D ω,η ) p = { X p ⊕ i X p ( ω + d η ) | X p ∈ T p M } ( p ∈ M ) is a Dirac structure over M. Example 2.8
A Jacobi manifold ( M , π, E ) is a smooth manifold equipped with a bivector field π anda vector field E on M which satisfy [ π, π ] SN = E ∧ π and [ E , π ] SN = , where [ · , · ] SN denotes theSchouten-Nijenhuis bracket. Let us consider a subbundle L of ( T M × R ) ⊕ ( T ∗ M × R ) over M byL p = (cid:8) ( π ♯ α p + f ( p ) E p , − α p ( E p )) ⊕ ( α p , f ( p )) | ( α p , f ( p )) ∈ T ∗ p M × R (cid:9) ( p ∈ M ) . The subbundle L gives rise to a Dirac structure ˜ L ⊂ T ( M × R ) over M × R by ˜ L ( p , t ) = n (cid:0) π ♯ α p + f ( p ) E p , − α p ( E p ) ∂ t (cid:1) ⊕ e t (cid:0) α p , f ( p )( dt ) t (cid:1) (cid:12)(cid:12)(cid:12) ( α p , f ( p ) ∂ t ) ∈ T ∗ p M × T t R o (( p , t ) ∈ M × R ) , where ∂ t = ∂∂ t (cid:12)(cid:12)(cid:12) t (see Section 5 in [16]). Example 2.9
Let ( M , π, E ) be a Jacobi manifold of dimension n and z a point in M where E z , .Suppose that u , · · · , u n − are functions on a neighborhood U z such that ( du ) p , · · · , ( du n − ) p are linearlyindependent at each p ∈ U z and du i ∈ E ◦ ( i = , · · · , n − . A subbundleD π, E = span (cid:8) ( π ♯ ( du ) + u E ) ⊕ du , · · · , ( π ♯ ( du n − ) + u n − E ) ⊕ du n − , E ⊕ (cid:9) of T U z is a Dirac structure over U z (see the subsection 4.2 in [7]). .2 Admissible functions A smooth function f on a Dirac manifold ( M , D ) is said to be admissible if there exists a vector field X f ∈ X ( M ) such that X f ⊕ d f is a smooth section of D (see [7]). We note that the vector field X f isnot uniquely determined. As easily checked by the case of a Dirac manifold induced by a presymplecticstructure (see Example 2.1), X f is uniquely defined up to elements of ker Ω . Example 2.10
Consider the presymplectic structure ω = dx ∧ dx + dx ∧ dx on R and a functionf ( x , x , x , x ) = x + k ( x + x ) ( k ∈ R ) . Then, vector fields written in the formX = k ∂∂ x + ϕ ( x ) ∂∂ x + ϕ ( x ) ∂∂ x − (2 x + ϕ ( x )) ∂∂ x (cid:0) ϕ , ϕ ∈ C ∞ ( R ) (cid:1) , turn out to satisfy that ω ♭ ( X ) = d f . Therefore, f is an admissible function on ( R , graph ( ω ♭ )) . Given a Dirac manifold ( M , D ), we denote the space of admissible functions on ( M , D ) by C ∞ adm ( M , D ).For any admissible functions f , g ∈ C ∞ adm ( M , D ), one defines their bracket { f , g } ′ as { f , g } ′ : = X g f . (2.5)It can be shown that the bracket (2.5) is both well-defined and skew-symmetric in the same way as thecase of Ω . If f , g are admissible, there exist vector fields X f and X g on M such that ( X f , d f ) , ( X g , dg ) ∈ Γ ∞ ( M , D ). Then, the simple computation yields ~ X g ⊕ dg , X f ⊕ d f (cid:127) = ( − [ X f , X g ]) ⊕ d { f , g } ′ ∈ Γ ∞ ( M , D ) . This implies that the bracket { f , g } ′ , also, is admissible and satisfies the equation X { f , g } ′ + [ X f , X g ] = . (2.6)The next proposition can be shown by using (2.6) (see [7]). Proposition 2.3 ( C ∞ adm ( M , D ) , {· , ·} ′ ) forms a Poisson algebra. To carry out the procedure of prequantization for Dirac manifolds, the notion of cohomology forDirac manifold is needed. Before proceeding the discussion, let us recall the definition of Lie algebroidand its cohomology.
Definition 3.1
A Lie algebroid over M is a smooth vector bundle A → M with a bundle map ♯ : A → T M, called the anchor map, and a Lie bracket [ · , · ] on the space Γ ∞ ( M , A ) of smooth sections of A suchthat [ α, f β ] = (cid:0) ( ♯α ) f (cid:1) β + f [ α, β ] (3.1) for any f ∈ C ∞ ( M ) and α, β ∈ Γ ∞ ( M , A ) . A simple example is a tangent bundle
T M over a smooth manifold M : the anchor map ♯ is the identitymap, and the bracket [ · , · ] is the usual Lie bracket of vector fields. This is called the tangent algebroid of M . As is well-known, Poisson manifolds define the structure of Lie algebroid on their cotangent bundles.6 xample 3.1 (Cotangent algebroids) If ( P , π ) is a Poisson manifold, then a cotangent bundle T ∗ P is aLie algebroid : the anchor map is the map π ♯ induced from ̟ , π ♯ : T ∗ P −→ T P , α (cid:8) β
7→ h β, ̟ ♯ ( α ) i = π ( β, α ) (cid:9) and the Lie bracket is given by { α, β } : = L π ♯ ( α ) β − L π ♯ ( β ) α + d (cid:0) π ( α, β ) (cid:1) = L π ♯ ( α ) β − i π ♯ ( β ) d α as in the part immediately before the subsection 2.2. The Lie algebroid ( T ∗ P → P , {· , ·} , ̟ ♯ ) is called acotangent algebroid. For other examples and the fundamental properties of Lie algebroids, see [5] and [10].Let ( A → M , [ · , · ] , ♯ ) and ( A → M , [ · , · ] , ♯ ) be Lie algebroids. A Lie algebroid morphismfrom A to A is a vector bundle morphism Φ : A → A with a base map ϕ : M → M which satisfies ♯ (cid:0) Φ ( α ) (cid:1) = ϕ ∗ (cid:0) ♯ ( α ) (cid:1) , (cid:0) ∀ α ∈ Γ ∞ ( M , A ) (cid:1) , and, for any smooth sections α, β ∈ Γ ∞ ( M , A ) written in the forms Φ ◦ α = X i ξ i ( α ′ i ◦ ϕ ) , Φ ◦ β = X j η j ( β ′ j ◦ ϕ ) , where ξ i , η j ∈ C ∞ ( M ) and α ′ i , β ′ j ∈ Γ ∞ ( M , A ), Φ ◦ [ α, β ] = X i , j ξ i η j ([ α ′ i , β ′ j ] ◦ Φ ) + X j (cid:0) L ♯ ( α ) η j (cid:1) ( β ′ j ◦ Φ ) − X i (cid:0) L ♯ ( β ) ξ i (cid:1) ( α ′ i ◦ Φ ) (3.2)For further discussion of Lie algebroid morphisms, we refer to [10] and [18].Concepts in Lie algebroid theory often appear as generalizations of standard notions in Poissongeometry and di ff erential geometry. The following theorem is an analogue of the splitting theorem by A.Weinstein which states that any Poisson manifold is locally a direct product of symplectic manifold withanother Poisson manifold (see [30]). The splitting theorem for Lie algebroids appears in [9, 11, 31]. Werefer to [10] for the proof of this theorem. Theorem 3.2 (The splitting theorem [9, 11, 31])
Let ( A → M , ♯, [ · , · ]) be a Lie algebroid. For eachpoint m ∈ M, there exist a local coordinate chart with coordinates ( x , · · · , x r , y · · · , y s ) centered at m,where r = rank ♯ p and r + s = dim M, and a basis of local sections { α , · · · , α r , β , · · · , β s } over an openneighborhood of m such that [ α j , α k ] = , [ α j , β k ] = , [ β j , β k ] = X ℓ f ℓ jk ( y ) β ℓ ,♯α j = ∂∂ x j , dx j ( ♯β k ) = L ∂∂ xj ♯β k = for all possible indices j , k , ℓ . Here, f ℓ jk ( y ) are smooth functions depending only on the variables y = ( y , · · · , y s ) . A -connections given bellow generalizes the usual one of connections on vector bun-dles (see [8]). Definition 3.3
Let ( A → M , ♯, [ · , · ]) be a Lie algebroid over M and E a vector bundle over M. An R -bilinear map ∇ A : Γ ∞ ( M , A ) × Γ ∞ ( M , E ) −→ Γ ∞ ( M , E ) , ( α, s ) A α sis called an A-connection if it satisfies (1) ∇ Af α s = f ∇ A α s ;(2) ∇ A α ( f s ) = f ∇ A α s + (cid:0) ( ♯α ) f (cid:1) sfor any f ∈ C ∞ ( M ) , α ∈ Γ ∞ ( M , A ) and s ∈ Γ ∞ ( M , E ) . The notion of ordinary connection is the case where A is the tangent algebroid T M . We denote by ∇ an ordinary connection, that is, ∇ : X ( M ) × Γ ∞ ( M , E ) −→ Γ ∞ ( M , E ) , ( X , s ) X s . When E → M is a complex vector bundle, an A -connection on E is defined as an A -connection which is C -linear on Γ ∞ ( M , E ).Similarly to the case of usual connection theory on vector bundles, one can define the curvature ofan A -connection. The curvature R A ∇ of an A -connection ∇ A is the map R A ∇ : Γ ∞ ( M , A ) × Γ ∞ ( M , A ) → End R ( Γ ∞ ( M , E ))given by the usual formula R A ∇ ( α, β ) = ∇ A α ◦ ∇ A β − ∇ A β ◦ ∇ A α − ∇ A [ α, β ] for any α, β ∈ Γ ∞ ( M , A ).For each k ∈ N ∪{ } , consider the exterior bundle ∧ k A ∗ over M . A smooth section of ∧ k A ∗ is called an A -di ff erential ℓ -form. One defines a multilinear map, called a A -exterior derivative, d A : Γ ∞ ( M , ∧ ℓ A ∗ ) → Γ ∞ ( M , ∧ ℓ + A ∗ ) as( d A θ ) ( α , · · · , α ℓ + ) = ℓ + X j = ( − j + ♯α j (cid:0) θ ( α , · · · , b α j , · · · , α ℓ + ) (cid:1) + X j < k ( − j + k θ (cid:0) [ α j , α k ] , α · · · , b α j , · · · , b α k , · · · , α ℓ + (cid:1) for any α , · · · , α k + ∈ Γ ∞ ( M , A ).The following proposition can be verified by a direct computation. Proposition 3.4
The di ff erential operator d A has the following properties: (1) d A ◦ d A = For any A-di ff erential k-form θ and A-di ff erential ℓ -form ϑ ,d A ( θ ∧ ϑ ) = ( d A θ ) ∧ ϑ + ( − k θ ∧ ( d A ϑ ) . Γ ∞ ( M , ∧ • A ∗ ) , d A ) forms a chain complex. The cohomology of( Γ ∞ ( M , ∧ • A ∗ ) , d A ) is called the Lie algebroid cohomology, or A -cohomology (see [5]). By the definition,the k -th cohomology group, denoted by H k L ( M , A ), is given by H k L ( M , A ) = ker { d A : Γ ∞ ( M , ∧ k A ∗ ) −→ Γ ∞ ( M , ∧ k + A ∗ ) } im { d A : Γ ∞ ( M , ∧ k − A ∗ ) −→ Γ ∞ ( M , ∧ k A ∗ ) } . We denote by [ α ] the cohomology class of α ∈ ker { d A : Γ ∞ ( M , ∧ k A ∗ ) −→ Γ ∞ ( M , ∧ k + A ∗ ) } . A Diracstructure D over M can be regarded as a Lie algebroid D → M with the bracket ~ · , · (cid:127) and the anchor map ♯ = ρ T M = pr | D . The cohomology of ( M , D ) is defined as the Lie algebroid cohomology H • L ( M , D ) ofthe Lie algebroid ( D → M , ρ T M , ~ · , · (cid:127) ).Let φ ⊕ Q be any D -di ff erential ℓ -form of ( M , D ), where φ ∈ Ω ℓ ( M ) and Q ∈ X ℓ ( M ). Then, theexterior derivative d D ( φ ⊕ Q ) for φ ⊕ Q is given by( d D ( φ ⊕ Q )) (cid:0) X ⊕ ξ , · · · , X ℓ + ⊕ ξ ℓ + (cid:1) = ℓ + X j = ( − j + X j (cid:16) φ (cid:0) X , · · · , b X j , · · · , X ℓ + (cid:1) + (cid:0) ξ ∧ · · · ∧ b ξ j ∧ · · · ξ ℓ + (cid:1) ( Q ) (cid:17) + X j < k ( − j + k φ (cid:0) [ X j , X k ] , X · · · , b X j , · · · , b X k , · · · , X ℓ + (cid:1) + X j < k ( − j + k (cid:16)(cid:0) L X j ξ k − i X k d ξ j (cid:1) ∧ ξ ∧ · · · ∧ b ξ j ∧ · · · ∧ b ξ k ∧ · · · ∧ ξ ℓ + (cid:17) ( Q ) , where X ⊕ ξ , · · · , X ℓ + ⊕ ξ ℓ + are smooth sections of D . Especially, if f is a smooth function on M , d D f is calculated to be ( d D f )( X ⊕ ξ ) = X f . (3.3)As noted in the subsection 2.1, one has a skew-symmetric form Π ♯ : ρ T ∗ M ( D ) → ( ρ T ∗ M ( D )) ∗ by Π ♯ ( ξ j ) = X j for X j ⊕ ξ j ∈ Γ ∞ ( M , D ) ( ∀ j = , · · · , ℓ + L X j ξ k − i X k d ξ j = { ξ j , ξ k } , we get ( d D ( φ ⊕ Q )) (cid:0) X ⊕ ξ , · · · , X ℓ + ⊕ ξ ℓ + (cid:1) = ( d φ ) ( X , · · · , X ℓ + ) + ( ∂ Q ) ( ξ , · · · , ξ ℓ + ) , where ∂ : X • ( M ) → X • + ( M ) denotes the contravariant exterior derivative (see I. Vaisman [28]):( ∂ Q ) ( α , · · · , α ℓ + ) = ℓ + X j = ( − j + Π ♯ ( α j ) (cid:16) Q ( α , · · · , b α j , · · · , α ℓ + ) (cid:17) + X j < k ( − j + k Q (cid:0) { α j , α k } , α · · · , b α j , · · · , b α k , · · · , α ℓ + (cid:1) , for any α , · · · , α ℓ + ∈ Ω ( M ). As a result, we get the following lemma: Lemma 3.5
Let ( M , D ) be a Dirac manifold. The D-exterior derivative d D : Γ ∞ ( M , ∧ • D ∗ ) → Γ ∞ ( M , ∧ • + D ∗ ) has the decomposition of exterior di ff erentials d and ∂ :d D ( φ ⊕ Q ) = d φ + ∂ Q . ρ T M : D → T M has the natural extension to a map ∧ ρ T M : Γ ∞ ( M , ∧ D ) → X ( M )by ( α ∧ α ) ( ∧ ρ T M ( ϑ )) : = ρ ∗ T M ( α ) ∧ ρ ∗ T M ( α ) ( ϑ )for any α , α ∈ Ω ( M ) and ϑ ∈ Γ ∞ ( M , ∧ D ). The dual map ( ∧ ρ T M ) ∗ : Ω ( M ) → Γ ∞ ( M , ∧ D ∗ ) of ∧ ρ T M is explicitly given by (cid:0) ( ∧ ρ T M ) ∗ σ (cid:1) ( ψ , ψ ) = σ (cid:0) ∧ ρ T M ( ψ ∧ ψ ) (cid:1) = σ (cid:0) ρ T M ( ψ ) , ρ T M ( ψ ) (cid:1) for any pair of sections ψ , ψ ∈ Γ ∞ ( M , D ) and any σ ∈ Ω ( M ). Since the de Rham cohomology group H • dR ( M ) of M is isomorphic to the ˇCech cohomology group H • ( M , R ), one gets a homomorphism( ∧ ρ T M ) ∗ : H ( M , R ) −→ H ( M , D ) , [ σ ] [( ∧ ρ T M ) ∗ σ ] . (3.4)from the second cohomology group H ( M , R ) of M to the Lie algebroid cohomology group H ( M , D ) of( M , D ). It is easy to check that the composition map of ( ∧ ρ T M ) ∗ and the exterior derivative d : Ω ( M ) → Ω ( M ) commutes with the map d D ◦ ρ T M ∗ : Ω ( M ) → Γ ∞ ( M , ∧ D ∗ ). Proposition 3.6 ( ∧ ρ T M ) ∗ ◦ d = d D ◦ ρ T M ∗ . Let ( A → M , ♯, ~ · , · (cid:127) ) be a Lie algebroid and Φ : M ′ → M a smooth map. Assume that thedi ff erential d Φ of Φ is transversal to the anchor map ♯ : A → T M in the sense thatim ♯ Φ ( x ) + im ( d Φ ) x = T Φ ( x ) M , ( ∀ x ∈ M ′ ) . (3.5)Here, im ♯ Φ ( x ) stands for the image of ♯ Φ ( x ) . This assumption leads us to the following condition:im ( id x × ♯ Φ ( x ) ) + T ( x . Φ ( x )) (cid:0) graph( Φ ) (cid:1) = T x M ′ ⊕ T Φ ( x ) M , ( ∀ x ∈ M ′ ) , (3.6)where id x means the identity map on T x M ′ . The condition ensures that the preimage( id × ♯ ) − T (cid:0) graph( Φ ) (cid:1) = a x ∈ M ′ n ( V ; α ) (cid:12)(cid:12)(cid:12) V ∈ T x M ′ , α ∈ A Φ ( x ) , ( d Φ ) x ( V ) = ♯ Φ ( x ) α ( Φ ( x )) o (3.7)is a smooth subbundle of ( T M ′ × A ) | graph( Φ ) over graph( Φ ) ≈ M ′ . The vector bundle (3.7) is a Liealgebroid whose anchor map is the natural projection pr ( V ; α ) : = V and whose Lie bracket is given by h(cid:16) V ; X j f j ⊗ α j (cid:17) , (cid:16) V ′ ; X k g k ⊗ β k (cid:17)i = (cid:16) [ V , V ′ ] ; X j , k f j g k ~ α j , β k (cid:127) + X k L V g k ⊗ β k − X j L V ′ f j ⊗ α j (cid:17) for any section written in the form (cid:16) V ; X j f j ⊗ α j (cid:17) (cid:0) V ∈ T x M ′ , f j ∈ C ∞ ( M ′ ) , α j ∈ A Φ ( x ) , ( d Φ ) x ( V ) = f j ( x ) ♯ Φ ( x ) α j ( Φ ( x )) (cid:1) . Φ ! A (see [14]). We remarkthat f ! A has rank rank( Φ ! A ) = rank A − dim M + dim M ′ .Let Φ : M ′ → ( M , D ) be a smooth map from a smooth manifold M ′ to a Dirac manifold ( M , D ) whichsatisfies the condition (3.5). Given a D -di ff erential ℓ -form ϑ , we define a Φ ! D -di ff erential ℓ ( ℓ > Φ ∗ ϑ , called the pull-back of ϑ , as( Φ ∗ ϑ ) x (cid:0) (cid:0) V ; ( d Φ ) x ( V ) , ξ (cid:1) , · · · , (cid:0) V ℓ ; ( d Φ ) x ( V ℓ ) , ξ ℓ (cid:1) (cid:1) : = ϑ Φ ( x ) (cid:0) (cid:0) ( d Φ ) x ( V ) , ξ (cid:1) , · · · , (cid:0) ( d Φ ) x ( V ℓ ) , ξ ℓ (cid:1) (cid:1) for any V j ∈ T x M ′ and ξ j ∈ T ∗ Φ ( x ) M ( j = , , · · · , ℓ ). If ℓ =
0, the pull-back of 0-form f ∈ C ∞ ( M ) isdefined as Φ ∗ f : = f ◦ Φ . By using Lemma 3.5, it can be easily verified that Φ ∗ and d D commute witheach other, that is, Φ ∗ ◦ d D = d D ◦ Φ ∗ . Applying Theorem 3.2 to a Dirac structure D → M , we find that, for each p ∈ M , there exist localcoordinates ( x , · · · , x r , y · · · , y s ) centered at p ( r = rank ( ρ T M ) p and r + s = dim M ) and a basis of localsections (cid:18) ∂∂ x ⊕ λ (cid:19) , · · · , (cid:18) ∂∂ x r ⊕ λ r (cid:19) , ( Y ⊕ µ ) , · · · , ( Y s ⊕ µ s ) (3.8)over an open neighborhood W of p which satisfy Y k = s X j = h jk ( y ) ∂∂ y j , L ∂∂ xj λ k = i ∂∂ xk d λ j , L ∂∂ xj µ k = i Y k d λ j . for all possible indices j , k , where we note that det (cid:0) h jk ( y ) (cid:1) ≤ j , k ≤ s =
0. Let us consider the pull-back ofthe Lie algebroid D → M along the projection pr p : M × R → M : D : = pr p ! D = a ( p , t ) ∈ M × R n ( X p , f ( p , t ) ( ∂/∂ t ) t ; X p ⊕ ξ p ) (cid:12)(cid:12)(cid:12) X p ⊕ ξ p ∈ D p , f ∈ C ∞ ( M × R ) o . Noting that rank( D ) = rank D − dim M + dim ( M × R ) = dim ( M × R ), D has the local basis of thesmooth sections on W × R (cid:18) ∂∂ x j , ; ∂∂ x j ⊕ λ j (cid:19) , ( Y k , ; Y k ⊕ µ k ) , (cid:18) , ∂∂ t ; ⊕ (cid:19) (1 ≤ j ≤ r , ≤ k ≤ s )induced by (3.8). We denote their dual basis by γ , · · · , γ r , δ , · · · , δ s , dt ∈ Γ ∞ ( W , D ∗ ) , that is, they are the local smooth sections of D ∗ such that γ j (cid:18) ∂∂ x k , ; ∂∂ x k ⊕ λ k (cid:19) = δ j ( Y k , ; Y k ⊕ µ k ) = j = k )0 ( j , k ) , dt (cid:18) , ∂∂ t ; ⊕ (cid:19) = γ j ( Y k , ; Y k ⊕ µ k ) = δ j (cid:18) ∂∂ x k , ; ∂∂ x k ⊕ λ k (cid:19) = , t ( Y k , ; Y k ⊕ µ k ) = dt (cid:18) ∂∂ x k , ; ∂∂ x k ⊕ λ k (cid:19) = j , k . A smooth section α = ( X , f ( p , t ) ∂/∂ t ; X ⊕ ξ ) ∈ Γ ∞ ( W × R , D ) is written in the form X j , k (cid:18) u j ∂∂ x j + v k Y k (cid:19) , f ( p , t ) ∂∂ t ; X j , k (cid:18) u j ∂∂ x j + v k Y k (cid:19) ⊕ s X k (cid:0) u k λ k + v k µ k (cid:1) . Then, by a simple computation, one finds that u j = γ j ( α ) , v k = δ k ( α ) , f ( p , t ) = dt ( α ) (1 ≤ j ≤ r , ≤ k ≤ s ) . Accordingly, from (3.3), the D -exterior derivative of a smooth function F on M × R is represented as d D F = r X j ∂ F ∂ x j γ j + s X k ( Y k F ) δ k + ∂ F ∂ t dt . Proposition 3.7
Let ( M , D ) be a Dirac manifold and D → M the pull-back of Lie algebroid D alongthe natural projection pr p : M × R → M. Then, pr p induces the isomorphism pr ∗ p : H • L ( M , D ) (cid:27) −→ H • L ( M × R , D ) . The inverse is the homomorphism ι ∗ : H • L ( M × R , D ) → H • L ( M , D ) induced from the inclusion map ι : M → M × R , p ( p , .Proof. Since pr p ◦ ι = id , it holds that ι ∗ ◦ pr ∗ p = id . Therefore, it is su ffi cient to show pr ∗ p ◦ ι ∗ = id on H • L ( M × R , D ) for the proof. For simplicity, we may assume that M is an Euclidean space R dim M and W is a star-shaped open set with respect to the origin ∈ R dim M . We remark that any D -di ff erential ℓ -form ω can be written in the form ω = X I , I ′ f I , I ′ ( p , t ) γ I ∧ δ I ′ + X J , J ′ g J , J ′ ( p , t ) dt ∧ γ J ∧ δ J ′ , where I , I ′ and J , J ′ run over all sequences with 1 ≤ i < i < · · · < i c ≤ r , ≤ i ′ < i ′ < · · · < i ′ c ′ ≤ s ( c + c ′ = ℓ ) and 1 ≤ j < j < · · · < j d ≤ r , ≤ j ′ < j ′ < · · · < j ′ d ′ ≤ s ( d + d ′ = ℓ − ℓ . we define an operator S ℓ from Γ ∞ ( W × R , ∧ ℓ D ∗ ) to Γ ∞ ( W × R , ∧ ℓ − D ∗ ) as S ℓ ( ω ) : = X J , J ′ Z t g J , J ′ ( p , t ) dt ! γ J ∧ δ J ′ . Then, by Proposition 3.4, we have d D ( S ℓ ( ω )) = X J , J ′ X j ∂∂ x j (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) γ j + X k Y k (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) δ k + ∂∂ t (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) dt γ J ∧ δ J ′ − X J , J ′ (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) ( d D γ J ) ∧ δ J ′ + X J , J ′ (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) γ J ∧ ( d D δ J ′ ) .
12n the other hand, the D -exterior derivative of ω is calculated to be d D ( ω ) = X I , I ′ X j ∂ f I , I ′ ∂ x j ( p , t ) γ j + X k ( Y k f I , I ′ ) δ k + ∂ f I , I ′ ∂ t ( p , t ) dt γ I ∧ δ I ′ − X I , I ′ f I , I ′ ( p , t ) ( d D γ I ) ∧ δ I ′ + X I , I ′ f I , I ′ ( p , t ) γ I ∧ ( d D δ I ′ ) + X J , J ′ X j ∂ g J , J ′ ∂ x j ( p , t ) γ j + X k ( Y k g J , J ′ ) δ k + ∂ g J , J ′ ∂ t ( p , t ) dt dt ∧ γ J ∧ δ J ′ + X J , J ′ g J , J ′ ( p , t ) dt ∧ ( d D γ J ) ∧ δ J ′ − X J , J ′ f J , J ′ ( p , t ) dt ∧ γ J ∧ ( d D δ J ′ ) . Therefore, S ℓ ( d D ( ω )) = X I , I ′ (cid:18)Z t ∂ f I , I ′ ∂ t ( p , t ) dt (cid:19) γ I ∧ δ I ′ − X J , J ′ X j (cid:18)Z t ∂ g J , J ′ ∂ x j ( p , t ) dt (cid:19) γ j ∧ γ J ∧ δ J ′ − X J , J ′ X k (cid:18)Z t ( Y k g J , J ′ ) dt (cid:19) δ k ∧ γ J ∧ δ J ′ + X J , J ′ (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) ( d D γ J ) ∧ δ J ′ − X J , J ′ (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) γ J ∧ ( d D δ J ′ ) . As a result, we have that d D ( S ℓ ( ω )) + S ℓ ( d D ( ω )) = X I , I ′ (cid:18)Z t ∂ f I , I ′ ∂ t ( p , t ) dt (cid:19) γ I ∧ δ I ′ + ∂∂ t (cid:18)Z t g J , J ′ ( p , t ) dt (cid:19) dt ∧ γ J ∧ δ J ′ = X I , I ′ f I , I ′ ( p , t ) γ I ∧ δ I ′ − X I , I ′ f I , I ′ ( p , γ I ∧ δ I ′ + X J , J ′ g J , J ′ ( p , t ) dt ∧ γ J ∧ δ J ′ = ω − X I , I ′ f I , I ′ ( p , γ I ∧ δ I ′ . (3.9)Here, we recall again that the pull-backs pr ∗ p : Γ ∞ ( M , ∧ ℓ D ∗ ) → Γ ∞ ( M × R , ∧ ℓ D ∗ ) and ι ∗ : Γ ∞ ( M × R , ∧ ℓ D ∗ ) → Γ ∞ ( M , ∧ ℓ D ∗ ) are given by(pr p ∗ ϑ ) (cid:18) X , f ∂∂ t : X ⊕ ξ (cid:19) , · · · , (cid:18) X ℓ , f ℓ ∂∂ t : X ℓ ⊕ ξ ℓ (cid:19)! : = ϑ (cid:16) X ⊕ ξ , · · · , X ℓ ⊕ ξ ℓ (cid:17) , pr p ∗ f = f ◦ pr p ( f ∈ C ∞ ( M ))and ( ι ∗ ω ) ( X ⊕ ξ , · · · , X ℓ ⊕ ξ ℓ ) : = ω (cid:16) ( X , : X ⊕ ξ ) , · · · , ( X ℓ , : X ℓ ⊕ ξ ℓ ) (cid:17) ,ι ∗ F = F ◦ ι ( F ∈ C ∞ ( M × R )) 13espectively. By a simple computation we get ω − (pr ∗ p ◦ ι ∗ ) ω = ω − X I , I ′ f I , I ′ ( p , γ I ∧ δ I ′ . (3.10)From (3.9) and (3.10) it follows that d D ◦ S ℓ + S ℓ ◦ d D = id − (pr ∗ p ◦ ι ∗ ) . Since d D ◦ ( ι ◦ pr p ) ∗ = ( ι ◦ pr p ) ∗ ◦ d D , it turns out that ω = d D (cid:0) ω − (pr ∗ p ◦ ι ∗ ) ω (cid:1) for any D -di ff erential ℓ -form ω such that d D ω =
0. That is, [ ω ] = H ℓ L ( M × R , D ). This completes the proof. (cid:3) We let q L : L → M be a complex line bundle over a Dirac manifold ( M , D ) and { ( U j , ε j ) } j be a familyof pairs which gives local trivializations of L . That is, { U j } j is an open covering of M and ε j are nowherevanishing smooth sections on U j such that the map U j × C −→ q L − ( U j ) , ( x , z ) z ε j ( x )for each j is a di ff eomorphism.A D -connection ∇ D : Γ ∞ ( M , D ) × Γ ∞ ( M , L ) → Γ ∞ ( M , L ) , is also considered as a map from Γ ∞ ( M , L ) to Γ ∞ ( M , D ∗ ) ⊗ C ∞ ( M ) Γ ∞ ( M , L ) by Γ ∞ ( M , L ) ∋ s ψ
7→ ∇ D ψ s } ∈ Hom C ∞ ( M ) (cid:0) Γ ∞ ( M , D ) , Γ ∞ ( M , L ) (cid:1) . On each U j , ∇ D ε j is written as ∇ D ε j = π √− σ j ⊗ ε j , by using a smooth section σ j ∈ Γ ∞ ( U j , D ∗ ). Since the transition function g jk on U j ∩ U k ( , ∅ ) is given by g jk ( x ) : = ε k ( x ) /ε j ( x ) ( x ∈ U j ∩ U k ), we have ε k ( x ) = g jk ( x ) ε j ( x ). It follows from a simple computation,that ∇ D ε k = ( 2 π √− g jk σ j + d D g jk ) ⊗ ε j . (3.11)On the other hand, ∇ D ε k = π √− g jk σ k ⊗ ε j . (3.12)It immediately follows from (3.11) and (3.12) that σ j − σ k = √− π d D g jk g jk . (3.13)As a result, one gets a D -di ff erential 2-form τ defined on the whole of M by τ = d D σ j = d D σ k ( U j ∩ U k , ∅ ). It is easy to verify that τ satisfies (cid:0) R D ∇ ( ψ , ψ ) (cid:1) ( ε j ) = π √− τ ( ψ , ψ ) ε j ( ∀ ψ , ψ ∈ Γ ∞ ( M , D ))for each j . That is, τ is the curvature 2-section of ∇ D (see Remark 3.1 below). Obviously, τ defines asecond D -cohomology class [ τ ] ∈ H ( M , D ). 14 roposition 3.8 The cohomology class [ τ ] determined by the curvature 2-section τ does not depend onthe choice of the D-connection ∇ D .Proof. Let ∇ ′ be another D -connection on L → M whose curvature is R ′ and σ ′ j the corresponding localsections in Γ ∞ ( U j , D ∗ ). Denoting by τ ′ the curvature 2-section corresponding to R ′ , we have τ ′ − τ = d D σ ′ j − d D σ j = d D ( σ ′ j − σ j ) (3.14)on each U j . We define a C -linear map b ∇ as b ∇ : Γ ∞ ( M , L ) −→ Γ ∞ ( M , D ∗ ) ⊗ C ∞ ( M ) Γ ∞ ( M , L ) , s ( ∇ ′ − ∇ D ) s . On U j , the following holds b ∇ ψ ε j = ( ∇ ′ ψ − ∇ D ψ ) ε j = π √− σ ′ j ( ψ ) ε j − π √− σ j ( ψ ) ε j = π √− σ ′ j − σ j )( ψ ) ε j for any ψ in Γ ∞ ( U j , D ). Putting b σ j = σ ′ j − σ j for each j , we find that, by (3 . b σ j − b σ k = ( σ ′ j − σ ′ k ) − ( σ j − σ k ) = √− π d D g jk g jk − √− π d D g jk g jk = U j ∩ U k , ∅ . Accordingly, there exists a D -di ff erential 1-form b σ over the whole of M given by b σ = b σ j = b σ k on U j ∩ U k ( , ∅ ). Therefore, it follows from (3.14) that τ ′ − τ = d D b σ j = d D b σ. This shows that [ τ ] = [ τ ′ ] in H ( M , D ). (cid:3) Definition 3.9
Let L → M be a complex line bundle over a Dirac manifold ( M , D ) and ∇ D any D-connection on L. The cohomology class [ τ ] ∈ H ( M , D ) by the D-di ff erential 2-form τ which correspondsto the curvature of ∇ D is called the first Dirac-Chern class of L → M. We denote the first Dirac-Chernclass of L by c D ( L ) . We assume that the line bundle L → M has a Hermitian metric h . A D -connection ∇ D is called aHermitian D -connection with respect to h if ρ T M ( ψ ) (cid:0) h ( s , s ) (cid:1) = h ( ∇ D ψ s , s ) + h ( s , ∇ D ψ s )for any smooth section s , s of L and any smooth section ψ of D . The following proposition can beshown in a way similar to the case of the ordinary connections on Hermitian line bundles (see [17]). Proposition 3.10
The curvature 2-section τ of ∇ D is a real D-di ff erential 2-form. Remark 3.1
Let A be any Lie algebroid over M. In general, an A-connection ∇ A on a vector bundle π : E → M is considered as an R -linear map Γ ∞ ( M , E ) to Γ ∞ ( M , A ∗ ) ⊗ C ∞ ( M ) Γ ∞ ( M , E ) satisfying thecondition (2) in Definition 3.3. Let { V λ } λ be an open covering which gives local trivializations of E ands , · · · , s r ( r = rank E ) be smooth sections such that s ( p ) , · · · , s r ( p ) is a basis for the fiber π − ( p ) forevery p ∈ V λ . One can verify that there exists a matrix θ = ( θ jk ) of local sections of A ∗ over V λ such that ∇ A s k = X j θ jk ⊗ s j ( θ jk ∈ Γ ∞ ( V λ , A ∗ ) ) . he matrix θ is called a connection 1-section (see [11]). In the same manner as the ordinary connectiontheory, the curvature R A ∇ of ∇ A is written as (cid:0) R A ∇ ( α , α ) (cid:1) ( s k ) = X j κ jk ( α , α ) s j ( ∀ α , α ∈ Γ ∞ ( V λ , A )) on each V λ , where κ jk ∈ Γ ∞ ( V λ , ∧ A ∗ ) . The matrix κ = ( κ jk ) is called the curvature 2-section of ∇ A ( see [ ]) . Ω -compatible Poisson structures Let ( M , D ) be a Dirac manifold. As mentioned in Section 2, ( M , D ) has the presymplectic structure Ω ♭ : ρ T M ( D ) → ρ T M ( D ) ∗ by (2.3). We define a singular distribution V as V : = ker Ω ♭ = D ∩ T M . Here, we remark again that D p ∩ T p M is thought of as a subspace of either T p M ⊕ T ∗ p M or T p M at each p ∈ M . We consider a subset H of D whose fibers H p are subspaces of D p satisfying V p ⊕ H p = D p ( ∀ p ∈ M ) , (4.1)and fix it. We denote a singular distribution ρ T M ( H ) ⊂ T M by H T M . Note that dim ( H T M ) p = rank ( Ω ♭ ) p for any p ∈ M . Since ker Ω ♭ ∩ H T M = { } , it turns out that H T M is isomorphic to b H T M : = im Ω ♭ | H TM = im Ω ♭ by the restriction map Ω ♭ | H TM : H T M (cid:27) −→ b H T M . We denote its inverse map ( Ω ♭ | H TM ) − : b H T M → H
T M by Θ ♯ . Then, it can be easily verified that Θ ♯ ◦ Ω ♭ | H TM = id H TM and Ω ♭ | H TM ◦ Θ ♯ = id b H TM . (4.2)As mentioned in Section 2, there exists a bundle map Π ♯ p defined as V p ◦ ∋ η p (cid:8) ξ p ξ p (cid:0) Π ♯ p ( η p ) (cid:1) : = ξ p ( Y p ) (cid:9) ∈ T p M / ( D p ∩ T p M ) . We here remark that V p ◦ is the annihilator of V p in T ∗ p M . From the definition of Ω and Proposition 2.2,the image im Ω ♭ of Ω ♭ turns out to beim Ω ♭ p = ρ T ∗ M ( D p ) = ( D p ∩ T p M ) ◦ = V p ◦ . So, we have b H T M = V ◦ , and find that Θ ♯ = Π ♯ | b H TM . By Proposition 2.2, any admissible function f ∈ C ∞ adm ( M , D ) satisfies d f ∈ b H T M . This allows us to define a vector field H f : = Θ ♯ ( d f ) ∈ H T M . f is admissible, there exists a vector field X f such that X f ⊕ d f ∈ Γ ∞ ( M , D ). It is easy to see that(( H f ) p − ( X f ) p ) ⊕ ∈ V p ⊂ D p at each m ∈ M . It follows from this that( H f ) p ⊕ ( d f ) p = (cid:0) ( H f ) p − ( X f ) p (cid:1) ⊕ + ( X f ) p ⊕ ( d f ) p ∈ D p . So, it turns out that H f ⊕ d f ∈ Γ ∞ ( M , D ). For any f , g ∈ C ∞ adm ( M , D ), let us define their bracket { f , g } as { f , g } : = H g f . (4.3)It is easily verified that, for any f , g ∈ C ∞ adm ( M , D ), { f , g } = Ω ( H f , H g ) . Since H f ⊕ d f , H g ⊕ dg ∈ Γ ∞ ( M , D ), we have that[ H g , H f ] ⊕ d { f , g } = ~ H g ⊕ dg , H f ⊕ d f (cid:127) ∈ Γ ∞ ( M , D ) . So, d { f , g } , also, is the admissible function. This implies that one can define the operator {· , ·} : C ∞ adm ( M , D ) × C ∞ adm ( M , D ) −→ C ∞ adm ( M , D ) , as (4.3), which is both bilinear and skew-symmetric. Furthermore, it turns out that this bracket {· , ·} coincides with {· , ·} ′ defined by (2.5) since ( H g − X g ) f = H g − X g ) ⊕ ∈ V . Consequently, weobtain the following proposition: Proposition 4.1 ( C ∞ adm ( M , D ) , {· , ·} ) forms a Poisson algebra, which does not depend on the choice of H . Moreover, it holds that [ H f , H g ] + H { f , g } = (4.4) for any f , g ∈ C ∞ adm ( M , D ) . Following [26], we say that the bracket {· , ·} by (4.3) is an Ω -compatible Poisson structure. Example 4.1
Let us consider a Dirac manifold ( R , graph ( ω ♭ )) by the presymplectic form in Example2.10. The presymplectic form Ω is entirely ω and written in the matrix form Ω ♭ = ω ♭ = − −
11 0 0 00 0 0 01 0 0 0 . From this, one can write the Dirac structure as graph ( ω ♯ ) = span ( ∂∂ x ⊕ ( dx + dx ) , ∂∂ x ⊕ ( − dx ) , ∂∂ x ⊕ , ∂∂ x ⊕ ( − dx ) ) . The subspace V = graph ( ω ♭ ) ∩ T R is given by V = span ( ∂∂ x − ∂∂ x , ∂∂ x ) . hen, we can take a subspace H as H = span ( ∂∂ x ⊕ ( dx + dx ) , ∂∂ x ⊕ ( − dx ) ) . One easily checks that graph ( ω ♭ p ) = V p ⊕ H p . The vector field H f for the admissible function f ( x ) = x + k ( x + x ) ( k ∈ R ) is given by H f = k ∂∂ x − x ∂∂ x . Example 4.2
Let ( M , F ⊕ F ◦ ) be a Dirac manifold obtained from an involutive distribution F ⊂ T M ofconstant rank (see Example 2.3). Since any vector field X ∈ F is embedded in F ⊕ F ◦ with X ֒ → ( X , ) ,one finds that V = F ⊕ { } (cid:27) F. It follows from this that H = { } ⊕ F ◦ (cid:27) F ◦ . If f ∈ C ∞ adm ( M , F ⊕ F ◦ ) ,the vector field H f for f is given by H f = . Example 4.3
Consider the Dirac manifold ( R , graph ( π ♯ )) induced by a Poisson bivector π = G ( x ) ∂/∂ x ∧ ∂/∂ x , where G ( x ) = G ( x , x ) is a smooth function on R . We remark that the Dirac structure graph ( π ♯ ) is written in the form graph ( π ♯ ) = span ( G ( x ) ∂∂ x ⊕ dx , − G ( x ) ∂∂ x ⊕ dx ) and any smooth function on ( R , graph ( π ♯ )) is admissible. The distribution V is given by V = graph ( π ♯ ) ∩ T R = { } and consequently, H is H = graph ( π ♯ ) . For a smooth function h, the vector field H h is repre-sented as H h = G ( x ) ∂ h ∂ x ∂∂ x − ∂ h ∂ x ∂∂ x ! . Example 4.4
Let D η be a Dirac structure obtained by a contact manifold ( R n + , η = dz − P ni = y i dx i ) (see Example 2.5). It is easily checked that V is the subbundle generated by the Reeb vector field ∂∂ z .Accordingly, H can be taken as H = n X i = a i (cid:16) ∂∂ x i ⊕ dy i (cid:17) + n X j = b j (cid:16) ∂∂ y j ⊕ ( − dx j ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) a i , b j ∈ C ∞ ( R n + ) ( ∀ i , j = , · · · , n ) . Then, the vector field H f for f ∈ C ∞ ( R n + ) is represented asH f = n X i = ∂ f ∂ y i ∂∂ x i − ∂ f ∂ x i ∂∂ y i ! . Example 4.5
Consider a Dirac structure graph ( ω ) ⊂ T M over M = R induced from a presymplecticform ω = F dx ∧ dx + G dx ∧ dx + H dx ∧ dx , where F , G and H are smooth functions on M suchthat ∂ F ∂ x ( p ) + ∂ G ∂ x ( p ) + ∂ H ∂ x ( p ) = ∀ p ∈ M ) . That is, graph ( ω ) = span n ∂∂ x ⊕ ( F dx − H dx ) , ∂∂ x ⊕ ( G dx − F dx ) , ∂∂ x ⊕ ( H dx − G dx ) o . t each point p = ( x , x , x ) where F ( p ) , , G ( p ) , , H ( p ) , , the distribution V is given by V = span { G ∂∂ x + H ∂∂ x + F ∂∂ x } . Accordingly, we can take H as the subspace generated by (cid:16) F ∂∂ x − H ∂∂ x (cid:17) ⊕ ( GHF ( Fdx − Hdx ) + F + H F ( Gdx − Fdx ) ) and (cid:16) G ∂∂ x − F ∂∂ x (cid:17) ⊕ ( − F + G F ( Fdx − Hdx ) − GHF ( Gdx − Fdx ) ) . Then, the inverse map Θ ♯ of Ω ♭ restricted to H T M is represented in the matrix form Θ ♯ = F ( F + G + H ) − GH F + G − ( F + H ) GH ! . with respect to the basis { F dx − H dx , G dx − F dx } ⊂ b H T M and n F ∂∂ x − H ∂∂ x , G ∂∂ x − F ∂∂ x o ⊂H T M . If f is an admissible function, f shall satisfy the condition that G ∂ f ∂ x + H ∂ f ∂ x + F ∂ f ∂ x = . Then,the vector field H f for f is given byH f = − F ( F + G + H ) ( ( F + G ) ∂ f ∂ x + GH ∂ f ∂ x ) (cid:16) F ∂∂ x − H ∂∂ x (cid:17) − F ( F + G + H ) ( GH ∂ f ∂ x + ( F + H ) ∂ f ∂ x ) (cid:16) G ∂∂ x − F ∂∂ x (cid:17) . On the other hand, at each point p = ( x , x , x ) where F ( p ) , , G ( p ) , , H ( p ) = , V is spanned byG ∂∂ x + F ∂∂ x . Define H as H = span ( ∂∂ x ⊕ ( G dx − F dx ) , (cid:16) G ∂∂ x − F ∂∂ x (cid:17) ⊕ {− ( F + G ) dx } ) , and we find that Θ ♯ is given by Θ ♯ ( dx ) = − F + G (cid:16) G ∂∂ x − F ∂∂ x (cid:17) , Θ ♯ ( G dx − F dx ) = ∂∂ x . Therefore, the vector field H f for a function f with G ∂ f ∂ x + F ∂ f ∂ x = is represented asH f = − F ∂ f ∂ x ∂∂ x − F + G ∂ f ∂ x (cid:16) G ∂∂ x − F ∂∂ x (cid:17) Lastly, at each point p = ( x , x , x ) where F ( p ) , , G ( p ) = , H ( p ) = , V = span n ∂∂ x o . Define H as H = span ( ∂∂ x ⊕ F dx , ∂∂ x ⊕ ( − F dx ) ) , and we find that Θ ♯ ( dx ) = − F ∂∂ x , Θ ♯ ( dx ) = F ∂∂ x . If f is a function which satisfies ∂ f ∂ x = , the vector field H f is given byH f = F ∂ f ∂ x ∂∂ x − ∂ f ∂ x ∂∂ x ! . .2 Quantizable Dirac manifolds We let ( M , D ) be a Dirac manifold and fix a singular distribution H ⊂ D for V = ker Ω . Supposethat there exists a line bundle L q L → M over ( M , D ) with a D -connection ∇ D whose curvature is R D ∇ . For H , we define a map ˆ : C ∞ adm ( M , D ) → End C ( Γ ∞ ( M , L )) from the Poisson algebra ( C ∞ adm ( M , D ) , {· , ·} ) toLie algebra (End C ( Γ ∞ ( M , L )) , [ · , · ]) asˆ f s : = −∇ DH f ⊕ d f s − π √− f s ( ∀ s ∈ Γ ∞ ( M , L )) (4.5)for each f ∈ C ∞ adm ( M , D ). Proposition 4.2
The map ˆ : C ∞ adm ( M , D ) → End C ( Γ ∞ ( M , L )) is a representation of C ∞ adm ( M , D ) on Γ ∞ ( M , L ) , that is, it holds that [ { f , g } = [ ˆ f , ˆ g ] (4.6) for all f , g ∈ C ∞ adm ( M , D ) if and only ifR D ∇ (cid:0) H f ⊕ d f , H g ⊕ dg (cid:1) = π √− Λ ( H f ⊕ d f , H g ⊕ dg ) , (4.7) where Λ is the skew-symmetric pairing Λ ( · , · ) : = h· , ·i − in Section 2.Proof . Using Proposition 4.1, we have that[ ˆ f , ˆ g ] s = ˆ f (ˆ gs ) − ˆ g ( ˆ f s ) = ˆ f ( −∇ DH g ⊕ dg s − π √− gs ) − ˆ g ( −∇ DH f ⊕ d f s − π √− f s ) = ∇ DH f ⊕ d f ◦ ∇ DH g ⊕ dg s − ∇ DH g ⊕ dg ◦ ∇ DH f ⊕ d f s − π √− n g ∇ DH f ⊕ d f s − f ∇ DH g ⊕ dg s + ∇ DH g ⊕ dg ( f s ) − ∇ DH f ⊕ d f ( gs ) o = R D ∇ (cid:0) H f ⊕ d f , H g ⊕ dg (cid:1) s − ∇ DH { f , g } ⊕ d { f , g } s − π √− { f , g } s = [ { f , g } s + R D ∇ (cid:0) H f ⊕ d f , H g ⊕ dg (cid:1) s − π √− { f , g } s for any admissible function f , g on ( M , D ) and any smooth section s of L → M . The bracket { f , g } iscalculated to be { f , g } = (cid:0) { f , g } − { g , f } (cid:1) = (cid:0) d f ( H g ) − dg ( H f ) (cid:1) = h H f ⊕ d f H g ⊕ dg i − . From this, we immediately get (4.7) as the necessary and su ffi cient condition for the map ˆ to preservetheir brackets. (cid:3) Definition 4.3
A Dirac manifold ( M , D ) is said to be prequantizable if there exists a Hermitian linebundle ( L , h ) over M with a Hermitian D-connection ∇ D in the sense thatH f (cid:0) h ( s , s ) (cid:1) = h (cid:0) ∇ DH f ⊕ d f s , s (cid:1) + h (cid:0) s , ∇ DH f ⊕ d f s (cid:1) , (4.8) which satisfies the condition (4.7) . The line bundle is called the prequantum bundle. Λ again. We find that Λ : Γ ∞ ( M , D ) × Γ ∞ ( M , D ) → C ∞ ( M ) is closed with regard to the di ff erential operator d D . Indeed, by Lemma 2.1 and the Cartanformula, d D Λ is calculated to be( d D Λ ) (cid:0) X ⊕ ξ, Y ⊕ η, Z ⊕ ζ (cid:1) = X (cid:0) η ( Z ) − ζ ( Y ) (cid:1) − Y (cid:0) ξ ( Z ) − ζ ( X ) (cid:1) + Z (cid:0) ξ ( Z ) − η ( X ) (cid:1) − Λ (cid:0) ~ X ⊕ ξ, Y ⊕ η (cid:127) , Z ⊕ ζ (cid:1) − Λ (cid:0) ~ Y ⊕ η, Z ⊕ ζ (cid:127) , X ⊕ ξ (cid:1) − Λ (cid:0) ~ Z ⊕ ζ, X ⊕ ξ (cid:127) , Y ⊕ η (cid:1) = X (cid:0) η ( Z ) − ζ ( Y ) (cid:1) − Y (cid:0) ξ ( Z ) − ζ ( X ) (cid:1) + Z (cid:0) ξ ( Z ) − η ( X ) (cid:1) − ( L X η )( Z ) − ( L Y ζ )( X ) − ( L Z ξ )( Y ) + ( d ξ )( Y , Z ) + ( d η )( Z , X ) + ( d ζ )( X , Y ) + ξ ([ Y , Z ]) + η ([ Z , X ]) + ζ ([ X , Y ]) = X ⊕ ξ, Y ⊕ η and Z ⊕ ζ of D . Accordingly, the D -di ff erential 2-form Λ defines the secondcohomology class [ Λ ] in the Lie algebroid cohomology. Additional to this, we have that Λ ( X ⊕ ξ, Y ⊕ η ) = (cid:8) ξ ( Y ) − η ( X ) (cid:9) = (cid:8) Ω ( X , Y ) − Ω ( Y , X ) (cid:9) = Ω ( X , Y ) = (( ∧ ρ T M ) ∗ Ω ) (cid:0) X ⊕ ξ, Y ⊕ η (cid:1) . That is, it holds that
Λ = ( ∧ ρ T M ) ∗ Ω . (4.9) Theorem 4.4
A Dirac manifold ( M , D ) is prequantizable if and only if the D-cohomology class [ Λ ] of Λ lies in the image ι ∗ ( H ( M ; Z )):[ Λ ] ∈ ( ∧ ρ T M ) ∗ (cid:16) ι ∗ ( H ( M , Z )) (cid:17) ⊂ H ( M , D ) , (4.10) where ι ∗ is the map from H • ( M , Z ) to H • ( M , R ) induced from the inclusion ι : Z ֒ → R .Proof . We assume that [ Λ ] ∈ ( ∧ ρ T M ) ∗ (cid:16) ι ∗ ( H ( M , Z )) (cid:17) . (4.11)Let { W j } j be a contractible open covering of M and β ∈ Ω ( M ) a closed 2-form on M . By Poincar´e’slemma, there exist 1-forms α j ∈ Ω ( W j ) such that β | W j = d α j on each W j . We here remark that W j ∩ W k is also contractible whenever W j and W k are so. Accordingly,by using Poincar´e’s lemma again, one can write α j − α k = dw jk for some function w jk ∈ C ∞ ( W j ∩ W k ) on W j ∩ W k , ∅ . As a result, we obtain D -di ff erential 1-forms σ j ∈ Γ ∞ ( W j , D ∗ ) which satisfy Λ | W j = d D σ j on each W j and find that the functions { w jk } j , k satisfy σ j − σ k = d D w jk . (4.12)21n W j ∩ W k , ∅ by using Proposition 3.6.We remark that f jk ℓ : = w jk + w k ℓ − w j ℓ are constant functions which take the values in Z (see [17])and define functions c jk ∈ C ∞ ( W j ∩ W k ) as c jk : = exp( − π √− w jk ). By (4.12) , we have σ j − σ k = √− π d D c jk c jk . In addition, we can verify that the functions { c jk } j , k satisfy the cocycle condition: c jk c k ℓ = exp ( − π √− w jk + w k ℓ )) = exp ( − π √− f jk ℓ ) exp ( − π √− w j ℓ ) = c j ℓ on W j ∩ W k ∩ W ℓ ( , ∅ ). Consequently, one can obtain a line bundle L → M whose transition functionsare { c jk } j , k and on which { σ j } j determine a connection ∇ D with curvature Λ .We define a Hermitian metric h on L as h p ( s , s ) : = z z , for any section s ( p ) = ( p , z ) , s ( p ) = ( p , z ) ∈ W j × C on each open set W j of the trivialization, and fix H for the singular distribution V = ker Ω ♭ . Then, ∇ D turns out to be a Hermitian connection in the senseof (4.8). Indeed, letting s , s be smooth sections locally written in the form s ( p ) = g ( p ) ε j ( p ) , s ( p ) = g ( p ) ε j ( p ) ( g ( p ) , g ( p ) ∈ C ), where ε is the nowhere vanishing section, and f ∈ C ∞ adm ( M , D ), we have (cid:0) H f (cid:0) h ( s , s ) (cid:1)(cid:1) ( p ) = (cid:0) H f (cid:0) h ( g ε j , g ε j ) (cid:1)(cid:1) ( p ) = (cid:0) H f ( g g ) (cid:1) ( p ) = ( H f g )( p ) g ( p ) + g ( p )( H f g )( p ) . Recall again that H f is the vector field which is uniquely determined for f with respect to H .On the other hand, h (cid:0) ∇ DH f ⊕ d f s , s (cid:1) ( p ) + h (cid:0) s , ∇ DH f ⊕ d f s (cid:1) ( p ) = h (cid:16) ( H f g ) ε j + π √− g σ j ( H f ) ε j , g ε j (cid:17) ( p ) + h (cid:16) g ε j , ( H f g ) ε j + π √− g σ j ( H f ) ε j (cid:17) ( p ) = ( H f g ) p g ( p ) − π √− g ( p ) σ j ( H f ⊕ d f ) g ( p ) + g ( p )( H f g ) p + π √− g ( p ) g ( p ) σ j ( H f ⊕ d f ) . From the assumption, each connection 1-section σ j is real. Accordingly, h (cid:0) ∇ DH f ⊕ d f s , s (cid:1) ( p ) + h (cid:0) s , ∇ DH f ⊕ d f s (cid:1) ( p ) = ( H f g ) p g ( p ) + g ( p )( H f g ) p . Therefore, we have that H f (cid:0) h ( s , s ) (cid:1) = h (cid:0) ∇ DH f ⊕ d f s , s (cid:1) + h (cid:0) s , ∇ DH f ⊕ d f s (cid:1) . This results in that ( M , D ) is prequantizable.Conversely, we suppose that ( M , D ) is prequantizable, that is, there is the prequantization bundle( L , ∇ D ) over M . Note that the D -di ff erential 2-form which corresponds to R D ∇ is Λ : R D ∇ = π √− Λ . It is well-known that the isomorphism classes of Hermitian line bundles over M are classified by thesecond cohomology classes through the map which assigns to the isomorphism class of a line bundle22 → M the first Chern class c ( K ) ∈ H ( M , Z ) of K (see [17] and [33]). According to this, one obtainsan ordinary Hermitian connection ∇ on L whose curvature is R . The curvature form F ∇ correspondingto R satisfies c ( L ) = [ F ∇ ] ∈ H ( M , Z ) . The map R : Γ ∞ ( M , D ) × Γ ∞ ( M , D ) → End C ( Γ ∞ ( M , L )) defined as R : = R ◦ ( ρ T M × ρ T M ) = ( ∧ ρ T M ) ∗ R is the curvature of a D -connection ∇ : = ∇ ◦ ( ρ T M × id ) on L . Then, the D -di ff erential 2-form τ corresponding to R is represented as τ = ( ∧ ρ T M ) ∗ F ∇ by using F ∇ . Using Proposition 3.8, we findthat [ Λ ] = c D ( L ) = [ τ ] = [( ∧ ρ T M ) ∗ F ∇ ] ∈ ( ∧ ρ T M ) ∗ (cid:16) ι ∗ ( H ( M , Z )) (cid:17) . This completes the proof of Theorem 4.4. (cid:3)
Remark 4.1
We would like to remark that the formula (4.5) and Proposition 4.2 are quite di ff erent fromLemma 6.1 in [32], though they might look similar to it. The prequantization formula discussed in [32]acts on local sections s of a line bundle L satisfying the condition ∇ DV ⊕ s = ∀ V ⊕ ∈ V ) , that is, it isa representation of C ∞ adm ( M , D ) on the space { s | s is a local section of L such that ∇ DV ⊕ s = ∀ V ⊕ ∈ V ) } . On the other hand, (4.5) acts on global sections of L which do not necessarily require the condition andis a representation of C ∞ adm ( M , D ) on Γ ∞ ( M , L ) . The quantization procedure in (4.5) generally depends on the choice of H . In other words, thereare as many quantization procedures of ( M , D ) as there are the choice of H . Suppose that we take H ′ di ff erent from H . Let ˆ f ′ be the prequantization procedure for f ∈ C ∞ adm ( M , D ) with respect to H ′ . Then,we have that ( ˆ f − ˆ f ′ ) s : = ∇ D ( H ′ f − H f ) ⊕ s ( s ∈ Γ ∞ ( M , L )) . From this formula, it turns out that ˆ f ′ coincides with ˆ f on the space { s ∈ Γ ∞ ( M , L ) | ∇ DV ⊕ s = ∀ V ⊕ ∈ V ) } . If V = ker Ω ♭ ⊂ T M is a subbundle, then it is integrable and its leaf space M red can be endowed with aPoisson structure Π red (Corollary 2.6.3 in [7]). Denote the natural projection from M to M red by q V anddefine a subbundle D red ⊂ T M as( D red ) p = (cid:8) ( dq V ) p ( X ) ⊕ ξ (cid:12)(cid:12)(cid:12) X ∈ T p M , ξ ∈ T ∗ [ p ] M red , X ⊕ ( dq V ) ∗ [ p ] ξ ∈ D p (cid:9) , where [ p ] : = q V ( p ) ∈ M red . It can be shown that D red is a Dirac structure over M red and coincides witha Dirac structure graph ( Π red ) induced from Π red . As we shall see in Example 4.8, a Poisson manifoldwhose second cohomology class [ π ] is integral has the prequantization bundle. In a word, we can obtainthe following: Proposition 4.5
Let ( M , D ) be a Dirac manifold. If V is a subbundle of T M, there is a prequantizationprocedure on the leaf space M red , which does not depend on the choice of H .
23e end the subsection with some examples.
Example 4.6 (Symplectic manifolds) We let ( M , ω ) be a symplectic manifold and consider the Diracstructure graph ( ω ♭ ) ⊂ T M ⊕ T ∗ M induced from the symplectic form ω (see also Example 2.1). It isverified that the skew-symmetric 2-cocycle Ω by (2.3) is entirely ω . Therefore, it follows from the non-degeneracy of ω that V p = ker Ω ♭ p = ker ω p = { } ( ∀ p ∈ M ) . Accordingly, we can take graph ( ω ♭ ) as a subbundle H satisfying (4.1). Then, for any smooth function fon M, there exists a unique vector field H f such that d f = ω ♭ ( H f ) . Therefore, the Ω -compatible Poissonstructure coincides with the natural Poisson structure induced from ω . In this case, the skew-symmetricpairing Λ is written as Λ (cid:0) X ⊕ ω ♭ ( X ) , Y ⊕ ω ♭ ( Y ) (cid:1) = ω ( X , Y ) and the integrability condition (4.10) is given by [ ω ] ∈ ι ∗ ( H ( M , Z )) . Example 4.7 (Presymplectic manifolds) As discussed in Example 2.1, given a presymplectic manifold ( M , ω ) , one obtains a Dirac manifold ( M , graph ( ω ♭ )) . Similarly to Example 4.6, Ω is entirely ω . Thesingular distribution V = ker ω ♭ is given by V = { X ∈ T M | ω ♭ ( X ) = } ⊂ graph ( ω ♭ ) . We let H ⊂ graph ( ω ♭ ) be a distribution which satisfies (4.1) and fix it. If f is a function such that there exists a vectorfield X which satisfies i X ω = d f , one can get a unique vector field H f which belongs to H and a Poissonalgebra ( C ∞ adm ( M , graph ( ω ♭ )) , {· , ·} ) . According to Theorem 4.4, ( M , graph ( ω ♭ )) is prequantizable as aDirac manifold if and only if the 2-form ω satisfies [ ω ] ∈ ι ∗ ( H ( M , Z )) . Example 4.8 (Poisson manifolds) Let us consider the case of a Poisson manifold ( P , π ) . As seen inExample 2.2, ( P , π ) defines a Dirac manifold ( P , graph ( π ♯ )) . One easily finds that V is given by V = graph ( π ♯ ) ∩ T P = { } . This permits us to take a subbundle graph ( π ♯ ) = { π ♯ ( α ) ⊕ α | α ∈ T ∗ P } as H .Obviously, every smooth function is admissible function. The skew-symmetric pairing Λ is written as Λ (cid:0) π ♯ ( α ) ⊕ α, π ♯ ( β ) ⊕ β (cid:1) = π ( α, β ) . Then, the integrability condition (4.10) indicates that [ π ] ∈ H ( P , π ) is an integral cohomology class,where H • LP ( P , π ) denotes the Lichnerowicz-Poisson cohomology (see [10, 28]). Recall that [ π ] is said tobe integral if there exists a closed 2-form β such that [ β ] ∈ ι ∗ ( H ( P , Z )) which satisfies [ ∧ π ♯ ( β )] = [ π ] .Here, ∧ π ♯ denotes a map from Ω ( P ) to X ( P ) by (cid:0) ( ∧ π ♯ ) φ (cid:1) ( α , α ) : = φ ( π ♯ ( α ) , π ♯ ( α )) ( α , α ∈ Ω ( P )) Consequently, ( P , π ) is prequantizable as a Poisson manifold if and only if ( P , graph ( π ♯ )) is prequantiz-able as a Dirac manifold. Example 4.9 (Contact manifolds) Let M be a contact manifold with a contact 1-form η . As seen inExample 2.5, a subbundle D η = graph ( d η ♭ ) defines a Dirac structure over M. In this case, the integra-bility condition (4.10) is represented as [ d η ] ∈ ι ∗ ( H ( M , Z )) . Especially, a standard contact manifold ( R n + , dz − P ni = y i dx i ) turns out to be prequantizable as a Dirac manifold by d η = P i dx i ∧ dy i . Example 4.10
Let us consider a Dirac manifold ( T ∗ M × R , D ω ) (see Example 2.6). For any section (cid:0) X , f ddt (cid:1) ⊕ pr ∗ (i X ω ) , (cid:0) Y , g ddt (cid:1) ⊕ pr ∗ (i Y ω ) of D ω , Λ is written as Λ (cid:16)(cid:0) X , f ddt (cid:1) ⊕ pr ∗ (i X ω ) , (cid:0) Y , g ddt (cid:1) ⊕ pr ∗ (i Y ω ) (cid:17) = ω ( X , Y ) . ote again that ω is the canonical symplectic form on T ∗ M. Hence, the integrability condition (4.10)is equivalent to [ ω ] ∈ ι ∗ ( H ( T ∗ M , Z )) . From this, it follows that the Dirac manifold ( T ∗ M × R , D ω ) isprequantizable. Example 4.11 (Almost cosymplectic manifolds) Let ( M , D ω,η ) be a Dirac manifold over an almostcosymplectic manifold ( M , ω, η ) with d ω = discussed in Example 2.7. A 2-form Ω associated to D ω,η is ω + d η . Hence, the singular distribution V is given by V = { X ∈ T M | i X ( ω + d η ) = } . Take a singulardistribution H so that V and H satisfy (4.1). Then, similarly to the case of presymplectic manifolds,one can obtain a unique vector field belonging to H and a Poisson algebra on the space of smoothfunctions f such that i X ( ω + d η ) = d f for some vector field X on M. The Dirac manifold ( M , D ω,η ) isprequantizable if and only if ω and η satisfy [ ω + d η ] ∈ ι ∗ ( H ( M , Z )) . α -density bundles Let V be an n -dimensional vector space over C and α a positive number. A function κ : V ×· · ·× V → C on n -copies of V is called an α -density over V if it satisfies κ ( Av , · · · , Av n ) = | det A | α κ ( v , · · · , v n ) ( v , · · · , v n ∈ V )for any invertible linear transformation A : V → V . Denoting the set of all α -densities over V by H ( α ) ( V ),we can check easily that H ( α ) ( V ) is a vector space over C . Since A ∈ GL( V ) acts transitively on bases in V , an α -density is determined by its value on a single basis. For an alternating covariant n -tensor ω , themap | ω | ( α ) : V × · · · × V → C defined as | ω | ( α ) ( v , · · · , v n ) : = | ω ( v , · · · , v n ) | α ( v , · · · , v n ∈ V )is an α -density over V . If ω is nonzero, H ( α ) ( V ) is a 1-dimensional vector space spanned by | ω | ( α ) . So,any element κ ∈ H α ( V ) is represented as κ = c | ω | ( α ) for some c ∈ C .Let M be a smooth manifold and α a positive number. The vector bundleH α : = a m ∈ M H ( α ) ( T p M )over M is called the α -density bundle of M . Especially, H / is called the half-density bundle. Let( U λ ; ( x λ , · · · , x n λ )) be local coordinate chart on M and ω λ = dx λ ∧ · · · ∧ dx n λ . Then, a local trivializationon U λ is defined to be the map Φ λ : π − ( U λ ) −→ U λ × C , Φ λ (cid:16) z | ( ω λ ) p | (cid:17) = ( p , z ) . Letting ( U µ ; ( x µ , · · · , x n µ )) be another chart with U λ ∩ U µ , ∅ and ω µ = dx µ ∧ · · · ∧ dx n µ , we have that Φ λ ◦ Φ − µ ( p , z ) = Φ λ (cid:16) z | ( ω µ ) p | (cid:17) = Φ λ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det ∂ x k µ ∂ x j λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α | ( ω λ ) p | ! = p , z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det ∂ x k µ ∂ x j λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ! . α is a complex line bundle whose transition functions are the square roots of the absolute valuesof the determinants of the matrices ∂ x k µ ∂ x j λ ( p ) ≤ j , k ≤ n ( n = dim M )by the coordinate transformations x j λ = x j λ ( x µ , · · · , x n µ ) ( j = , · · · , n ). A section of H α is called an α -density over M . When α = /
2, a section of H / is called the half-density. As in the linear case, any α -density κ over U can be written in the form κ = f | ω | ( α ) for some complex-valued function f . It iseasily verified that H α ⊗ H β (cid:27) H α + β . Accordingly, for any half densities κ , κ over M , we get a 1-density κ ⊗ κ .Suppose that ( U , φ ) is a local coordinate chart on M and κ is a half density over M such that thesupport supp κ of κ is contained in U . The integral of κ over M is defined as Z p κ : = Z φ − ( U ) ( φ − ) ∗ κ. We here remark that the right-hand side is represented as Z φ − ( U ) ( φ − ) ∗ κ = Z φ − ( U ) f | dx ∧ · · · ∧ dx n | (1 / . If κ is any density over M , the integral of κ over M is defined as Z p κ : = X j Z p ̺ j κ, where { ̺ j } j means a partition of unity subordinate to smooth atlas of M . The pairings h· , ·i ± and the bracket ~ · , · (cid:127) on Γ ∞ ( M , T M ) are naturally extended to operations on thespace Γ ∞ ( M , T M ⊗ C ) by (cid:10) ( X ⊕ ξ ) + √− X ′ ⊕ ξ ′ ) , ( Y ⊕ η ) + √− Y ′ ⊕ η ′ ) (cid:11) ± = n ξ ( Y ) ± η ( X ) − ξ ′ ( Y ′ ) ∓ η ′ ( X ′ ) + √− (cid:0) ξ ( Y ′ ) ± η ( X ′ ) + ξ ′ ( Y ) ± η ′ ( X ) (cid:1) o and ~ ( X ⊕ ξ ) + √− X ′ ⊕ ξ ′ ) , ( Y ⊕ η ) + √− Y ′ ⊕ η ′ ) (cid:127) = (cid:16) [ X , Y ] − [ X ′ , Y ′ ] + √− (cid:0) [ X ′ , Y ] + [ X , Y ′ ] (cid:1)(cid:17) ⊕ (cid:16) L X η − i Y d ξ − L X ′ η ′ + i Y ′ d ξ ′ + √− (cid:0) L X η ′ − i Y d ξ ′ + L X ′ η − i Y ′ d ξ (cid:1) (cid:17) , respectively. If a complex subbundle D ⊂ T M ⊗ C whose sections are closed under the extended bracket ~ · , · (cid:127) is maximally isotropic with respect to the extended symmetric pairing h· , ·i + , D is called a complexDirac structure. Let ( M , D ) be a Dirac manifold and Ω the 2-cocycle associated to it. Then, the com-plexification D C : = D ⊗ R C of D turns out to be the complex Dirac structure. We introduce the notion ofpolarization for Dirac manifold as follows: 26 efinition 5.1 A complex subbundle
P ⊂ D C is called a (complex) polarization of ( M , D ) if it satisfiesthe following conditions: (P1) V p ⊗ R C ⊂ P p ( ∀ p ∈ M );(P2) Λ ( ˜ X ⊕ ˜ ξ, ˜ Y ⊕ ˜ η ) = (cid:0) ∀ ˜ X ⊕ ˜ ξ, ˜ Y ⊕ ˜ η ∈ Γ ∞ ( M , P ) (cid:1) ;(P3) ~ Γ ∞ ( M , P ) , Γ ∞ ( M , P ) (cid:127) ⊂ Γ ∞ ( M , P ) . The condition (P1) can be written in the explicit form Ω ( X , Y ) − Ω ( X ′ , Y ′ ) + √− (cid:0) Ω ( X ′ , Y ) + Ω ( X , Y ′ ) (cid:1) = , where ˜ X ⊕ ˜ ξ = ( X + √− X ′ ) ⊕ ( Ω ♭ ( X ) + √− Ω ♭ ( X ′ )) and ˜ Y ⊕ ˜ η = ( Y + √− Y ′ ) ⊕ ( Ω ♭ ( Y ) + √− Ω ♭ ( Y ′ )). Example 5.1
Let us consider a Dirac manifold ( R n , graph ( ω ♭ )) induced by the standard symplecticform ω = P j dq j ∧ d p j . Then, a complex subbundle P by P = span (cid:26) ∂∂ q j ⊕ d p j (cid:12)(cid:12)(cid:12) j = , · · · , n (cid:27) defines a polarization of ( R n , graph ( ω ♭ )) . On the other hand, P = span (cid:26) ∂∂ p j ⊕ dq j (cid:12)(cid:12)(cid:12) j = , · · · , n (cid:27) , also, is a polarization of ( R n , graph ( ω ♭ )) . Example 5.2
Consider a Poisson structure π = x ∂∂ x ∧ ∂∂ x + x ∂∂ x ∧ ∂∂ x + x ∂∂ x ∧ ∂∂ x on R . A complexsubbundle P of graph ( π ♯ ) C by P = span (cid:26) (cid:26)(cid:16) x ∂∂ x − x ∂∂ x (cid:17) + √− (cid:16) x ∂∂ x − x ∂∂ x (cid:17)(cid:27) ⊕ ( dx + √− dx ) (cid:27) . turns out to be a polarization of ( R , graph ( π ♯ )) by a simple computation. Example 5.3
Let ( M , ω, J ) be a K ¨ ahler manifold, where J denotes an almost complex structure on M.Define a complex subbundle P of graph ( ω ♭ ) C as P = { ( X + √− JX ) ⊕ (i X ω + √− JX ω ) | X ∈ X ( M ) } . By a simple computation, we can check that the conditions (P1) and (P2) in Definition 5.1 are satisfied.From the fact that J is integrable, if follows that (P3) holds. Therefore, the subbundle P is a polarizationof ( M , graph ( ω ♭ )) . Example 5.4
Let us consider a Dirac manifold ( R , D η ) from a contact 1-form η = dx − x dx . Asdiscussed in Example 4.4, V = ker Ω ♭ is given by V = span { ∂∂ x } . A complex subbundle P of D η C by P = span (cid:26) (cid:16) ∂∂ x − ∂∂ x (cid:17) ⊕ ( dx + dx ) , ∂∂ x ⊕ (cid:27) satisfies the conditions (P1) – (P3) in Definition 5.1. Hence, P is a polarization of ( R , D η ) .
27s discussed in the previous section, for a singular distribution D ∩ T M ⊂ D , we choose a subset H ⊂ D which satisfies (4.1) and fix it. Suppose that ( M , D ) is prequantizable and endowed with apolarization P . Let L → M be its prequantum bundle with the Hermitian metric h on L and ∇ D aHermitian D -connection with respect to h . We remark that ∇ D : Γ ∞ ( M , D ) → End C ( Γ ∞ ( M , L )) has anatural extension to a map ∇ D : Γ ∞ ( M , D C ) → End C ( Γ ∞ ( M , L )). Using the extension ∇ D , we define amap δ : Γ ∞ ( M , D C ) × Γ ∞ ( M , L ⊗ H / ) → Γ ∞ ( M , L ⊗ H / )as δ ψ ( s ⊗ κ ) : = ( ∇ D ψ s ) ⊗ κ + s ⊗ ( L ρ TM ( ψ ) κ ) , for any ψ ∈ Γ ∞ ( M , D C ) , s ⊗ κ ∈ Γ ∞ ( M , L ⊗ H / ). It is easily verified that δ is a D -connection on L ⊗ H / .Then, the representation (4.5) of C ∞ adm ( M , D ) can be extended to a mapˆ : C ∞ adm ( M , D ) → End C ( Γ ∞ ( M , L ⊗ H / ))by setting ˆ f ( s ⊗ κ ) = − δ H f ⊕ d f ( s ⊗ κ ) − π √− f ( s ⊗ κ ) . The ˆ f is also represented as ˆ f ( s ⊗ κ ) = ( ˆ f s ) ⊗ κ − s ⊗ ( L H f κ ) . (5.1)Since ( M , D ) is prequantizable, we can check that [ { f , g } ( s ⊗ κ ) = [ ˆ f , ˆ g ] ( s ⊗ κ )for all f , g ∈ C ∞ adm ( M , D ) in the same manner as the proof for Proposition 4.2. Lemma 5.2
It holds that δ ψ (cid:0) ˆ f ( s ⊗ κ ) (cid:1) = ˆ f (cid:0) δ ψ ( s ⊗ κ ) (cid:1) − δ ~ ψ, H f ⊕ d f (cid:127) ( s ⊗ κ ) for any ψ ∈ Γ ∞ ( M , D C ) , s ⊗ κ ∈ Γ ∞ ( M , L ⊗ H / ) and f ∈ C ∞ adm ( M , D ) .Proof. For any smooth section ψ = ( X , ξ ) of D C , s ⊗ κ of L ⊗ H / and any admissible function f , wehave that δ ψ ◦ ˆ f ( s ⊗ κ ) = − δ ψ (cid:16) δ H f ⊕ d f ( s ⊗ κ ) + π √− f ( s ⊗ κ ) (cid:17) = − δ ψ (cid:16) ( ∇ DH f ⊕ d f s ) ⊗ κ + s ⊗ L H f κ (cid:17) − π √− δ ψ (cid:0) f ( s ⊗ κ ) (cid:1) = − ( ∇ D ψ ◦ ∇ DH f ⊕ d f s ) ⊗ κ − ( ∇ DH f ⊕ d f s ) ⊗ L X κ − ( ∇ D ψ s ) ⊗ L H f κ − s ⊗ ( L X ◦ L H f κ ) − π √− X f ) s ⊗ κ − π √− f (( ∇ D ψ s ) ⊗ κ ) − π √− f ( s ⊗ L X κ ) . On the other hand,ˆ f ◦ δ ψ ( s ⊗ κ ) = ˆ f (cid:16) ( ∇ D ψ s ) ⊗ κ + s ⊗ L X κ (cid:17) = ˆ f (cid:0) ( ∇ D ψ s ) ⊗ κ (cid:1) + ˆ f ( s ⊗ L X κ ) = − δ H f ⊕ d f (cid:0) ( ∇ D ψ s ) ⊗ κ (cid:1) − π √− f (cid:0) ( ∇ D ψ s ) ⊗ κ (cid:1) − δ H f ⊕ d f (cid:0) s ⊗ L X κ (cid:1) − π √− f ( s ⊗ L X κ ) = − ( ∇ DH f ⊕ d f ◦ ∇ D ψ s ) ⊗ κ − ( ∇ D ψ s ) ⊗ L H f κ − π √− f (cid:0) ( ∇ D ψ s ) ⊗ κ (cid:1) ( ∇ DH f ⊕ d f s ) ⊗ L X κ − s ⊗ ( L H f ◦ L X κ ) − π √− f ( s ⊗ L X κ )It follows from h X ⊕ ξ, H f ⊕ d f i + = Λ (cid:0) X ⊕ ξ, H f ⊕ d f (cid:1) = (cid:0) ξ ( H f ) − d f ( X ) ) = − X f . By using (4.4) and (4.7) these equations yield( δ ψ ◦ ˆ f − ˆ f ◦ δ ψ ) ( s ⊗ κ ) = − (cid:8) ( ∇ D ψ ◦ ∇ DH f ⊕ d f − ∇ D ( H f ⊕ d f ) ◦ ∇ D ψ ) s (cid:9) ⊗ κ − s ⊗ ( L X ◦ L H f − L H f ◦ L X ) κ − π √− X f ) s ⊗ κ = − n(cid:16) R D ∇ (cid:0) ψ, H f ⊕ d f (cid:1) + ∇ D ~ ψ, H f ⊕ d f (cid:127) (cid:17) s o ⊗ κ − s ⊗ L [ X , H f ] κ − π √− X f ) s ⊗ κ = − π √− n(cid:16) Λ (cid:0) X ⊕ ξ, H f ⊕ d f (cid:1) + ( X f ) (cid:17) s o ⊗ κ − (cid:0) ∇ D ~ ψ, H f ⊕ d f (cid:127) s (cid:1) ⊗ κ − s ⊗ L [ X , H f ] κ = − δ ~ ψ, H f ⊕ d f (cid:127) ( s ⊗ κ )This completes the proof. (cid:3) For a polarization P , we define the subalgebra S ( P ) of ( C ∞ adm ( M , D ) , {· , ·} ) as S ( P ) : = (cid:8) f ∈ C ∞ adm ( M , D ) (cid:12)(cid:12)(cid:12) ∀ ψ ∈ Γ ∞ ( M , P ) : ~ H f ⊕ d f , ψ (cid:127) ∈ Γ ∞ ( M , P ) (cid:9) . On the other hand, we define a subset H of Γ ∞ ( M , L ⊗ H / ) as H : = { s ⊗ κ ∈ Γ ∞ ( M , L ⊗ H / ) | ∀ ψ ∈ Γ ∞ ( M , P ) : δ ψ ( s ⊗ κ ) = } and assume that H , { } . Obviously, H is a linear space over C , and moreover it follows from Lemma5.2 that ˆ f ( H ) ⊂ H for every f ∈ S ( P ). This leads us to the following result: Theorem 5.3 If H , { } , a map ˆ : S ( P ) → End C ( H ) byf (cid:8) s ⊗ κ
7→ − δ H f ⊕ d f ( s ⊗ κ ) − π √− f ( s ⊗ κ ) (cid:9) . is a representation of S ( P ) on H . We proceed with the discussion in the following two cases.
Suppose that M is compact. The linear space H has the inner product h· , ·i defined as h s ⊗ κ , s ⊗ κ i : = Z M h ( s , s ) κ κ for every s ⊗ κ , s ⊗ κ ∈ H . By taking the completion of H , one obtains a Hilbert space H . Theoperator √− f for f ∈ S ( P ) turns out to be self-adjoint with respect to h· , ·i . Indeed, we have that h ˆ f ( s ⊗ κ ) , s ⊗ κ i + h s ⊗ κ , ˆ f ( s ⊗ κ ) i h ˆ f ( s ) ⊗ κ − s ⊗ L H f κ , s ⊗ κ i + h s ⊗ κ , ˆ f ( s ) ⊗ κ − s ⊗ L H f κ i = h ˆ f ( s ) ⊗ κ , s ⊗ κ i − h s ⊗ L H f κ , s ⊗ κ i + h s ⊗ κ , ˆ f ( s ) ⊗ κ i − h s ⊗ κ , s ⊗ L H f κ i = Z M n h ( ˆ f ( s ) , s ) + h ( s , ˆ f ( s )) o κ κ − Z M h ( s , s ) ( L H f κ ) κ − Z M h ( s , s ) κ ( L H f κ ) = − Z M H f (cid:0) h ( s , s ) (cid:1) κ κ − Z M h ( s , s ) ( L H f κ ) κ − Z M h ( s , s ) κ ( L H f κ ) = − Z M L H f (cid:16) h ( s , s ) κ κ (cid:17) for any s ⊗ κ , s ⊗ κ ∈ H . In the last equality, the symbol L H f ( · ) denotes a Lie derivative of a half-density. We refer to [34] for the tensor analysis of Lie derivatives. According to [25], it holds that Z M L V κ = V and every density κ on M . From this it follows that h ˆ f ( s ⊗ κ ) , s ⊗ κ i + h s ⊗ κ , ˆ f ( s ⊗ κ ) i = . This directly implies that √− f is a self-adjoint operator. Then, the condition (4.6) holds up to theconstant factor √− Suppose that M is not compact. We let Q be the subbundle of D such that Q C = P ∩ P (5.2)and assume that a singular distribution M ∋ p ( ρ T M ) p ( Q p ) ⊂ T p M defines a regular foliation F whose leaf space N = M / F is a Hausdor ff manifold. We denote by q F the natural projection from M to N . For any f ∈ S ( P ) and X ⊕ ξ ∈ Q , the vector field ~ H f ⊕ d f , X ⊕ ξ (cid:127) is tangent to each q F -fiber: ~ H f ⊕ d f , X ⊕ ξ (cid:127) = [ H f , X ] ⊕ L H f ξ ∈ Γ ∞ ( M , Q ). Accordingly, it holds that [ H f , X ] ∈ ρ T M ( Q ) for anyvector field X ∈ ρ T M ( Q ). This means that H f is a lift of a smooth vector field on N . Let H / N be a halfdensity bundle over N and κ N a half density. Then, from (5.1) it holds thatˆ f ( s ⊗ ( q F ) ∗ κ N ) = ( ˆ f s ) ⊗ ( q F ) ∗ κ N − s ⊗ ( L H f ( q F ) ∗ κ N ) = ( ˆ f s ) ⊗ ( q F ) ∗ κ N − s ⊗ ( q F ) ∗ ( L q F ∗ ( H f ) κ N )for any s ⊗ ( q F ) ∗ κ N ∈ Γ ∞ ( M , L ⊗ ( q F ) ∗ H / N ). This enables us to consider the half densities of M whichare transversal to F . Since L X (( q F ) ∗ κ N ) = ( q F ) ∗ ( L π ∗ X κ N ) = X , ξ ) ∈ Q , we have that L X (cid:16)(cid:0) ( q F ) ∗ κ N (cid:1) ⊗ (cid:0) ( q F ) ∗ κ N (cid:1)(cid:17) = ( q F ) ∗ ( L q F ∗ X κ N ) ⊗ ( q F ) ∗ κ N + ( q F ) ∗ κ N ⊗ ( q F ) ∗ ( L q F ∗ X κ N ) = q F ) ∗ κ N , ( q F ) ∗ κ N ∈ ( q F ) ∗ H / N . In other words, the tensor field (( q F ) ∗ κ N ) ⊗ (( q F ) ∗ κ N ) on M is invariant under the flow of X ∈ ρ T M ( Q ). Therefore, there exists a 1-density ν N of N onto which(( q F ) ∗ κ N ) ⊗ (( q F ) ∗ κ N ) projects. As a result, we let H be the linear subspace of H consisting of theelements in Γ ∞ ( M , L ⊗ ( q F ) ∗ H / N ) which have compact support in N , and assume that H , { } . Wedefine the inner product h· , ·i on H as h s ⊗ ( q F ) ∗ κ N , s ⊗ ( q F ) ∗ κ N i : = Z N h ( s , s ) ν N s ⊗ ( q F ) ∗ κ N , s ⊗ ( q F ) ∗ κ N ∈ H . Replacing H with H , we obtain a Hilbert space from H and find that √− f for f ∈ S ( P ) is a self-adjoint operator on End C ( H ) in a way similar to the compactcase. Acknowledgments.
The author would like to express his deepest gratitude to Emeritus Professor Toshi-aki Kori of Waseda University for useful discussion and various support. He also would appreciateProfessor Alan Weinstein and the referees very much for helpful comments, pointing out typos andsuggesting many improvements to the first version of the manuscript. Lastly, he wishes to thank KeioUniversity and Waseda University for the hospitality while part of the work was being done.
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Yuji Hirota