Thus, to each function - Real Academia de Ciencias Exactas, Físicas

RAC
Rev. R. Acad. Cien. Serie A. Mat.
VOL . 95 (1), 2001, pp. 65–83
Geometrı́a y Topologı́a / Geometry and Topology
On the geometric prequantization of brackets
M. de León, J. C. Marrero and E. Padrón
Abstract. In this paper we consider a general setting for geometric prequantization of a manifold
endowed with a non-necessarily Jacobi bracket. The existence of a generalized foliation permits to define
a notion of prequantization bundle. A second approach is given assuming the existence of a Lie algebroid
on the manifold. Both approaches are related, and the results for Poisson and Jacobi manifolds are
recovered.
Precuantización geométrica de corchetes
Resumen. En este artı́culo se considera un marco general para la precuantización geométrica de una
variedad provista de un corchete que no es necesariamente de Jacobi. La existencia de una foliación
generalizada permite definir una noción de fibrado de precuantización. Se estudia una aproximación
alternativa suponiendo la existencia de un algebroide de Lie sobre la variedad. Se relacionan ambos
enfoques y se recuperan los resultados conocidos para variedades de Poisson y Jacobi.
1. Introduction
Since the seminal results by Kostant and Souriau [14, 26] a lot of work has been done in order to develop a
geometric theory of quantization. The inspiration behind these ideas was to develop a method to quantize a
classical system and to obtain the quantum system reproducing the Dirac scheme for canonical quantization.
The procedure starts with a phase space which in the most favourable case is a symplectic manifold
. Then, we associate to
a Hilbert space, which at the first step is the space of sections
of a complex line bundle
over . Thus, to each function
(an observable) we attach
an operator
,
, where
is the Hamiltonian vector field
defined by , and
is a covariant derivative on .
is said to be a prequantization bundle of
if
, that is, the commutator of the operators corresponds to the Poisson bracket of
the observables. This condition can be translated as the existence of a covariant derivative such that its
curvature is . The condition is just fullfilled for integral symplectic manifolds.
In constrained Hamiltonian systems and other physical instances, there appear more general phase
spaces, endowed with a non-symplectic Poisson bracket, or even, a Jacobi bracket. An approach to this
problem is the use of symplectic and contact groupoids, and there is an extensive list of results due mainly
to Karasev, Maslov, Weinstein, Dazord, Hector and others (see [4], [5], [6], [11] and [31]; see also [29] and
the references therein).
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Presentado por Pedro Luis Garcı́a Pérez
Recibido: 3 de Noviembre 1999. Aceptado: 8 de Marzo 2000.
Palabras clave / Keywords: Jacobi manifolds, Poisson manifolds, Lie algebroids, H-Chevalley-Eilenberg cohomology,
Lichnerowicz-Jacobi cohomology, Lichnerowicz-Poisson cohomology, foliated cohomology, foliated covariant derivatives, geometric
prequantization.
Mathematics Subject Classifications: 53D50, 53D17, 17B66
c 2001 Real Academia de Ciencias, España.
@
65
M. de León, J. C. Marrero and E. Padrón
On the other hand, in [10] Huebschmann extended the Kostant-Souriau geometric quantization procedure of symplectic manifolds to Poisson algebras and, particularly, to the Lie algebra of functions of a
Poisson manifold. In [28] (see also [29]), Vaisman obtains essentially the same quantization of a Poisson manifold
straightforwardly, without resorting to any special algebraic machinery. He introduced
contravariant instead of covariant derivatives on , in order to take account that the Poisson bracket is
given by a contravariant 2-vector. This permits to obtain similar results for Poisson manifolds. Here, the
traditional de Rham cohomology has to be substitutted by the so-called Lichnerowicz-Poisson cohomology
(LP-cohomology). The LP-cohomology has a clear lecture in the Chevalley-Eilenberg cohomology of the
Lie algebra of functions on
through the representation
.
The above results were recently extended for a Jacobi manifold
(see [19]). In this case one has
to consider the Lie algebroid associated to
and the so-called Lichnerowicz-Jacobi cohomology (LJcohomology) of . The Lie algebroid (respectively, the LJ-cohomology) has been introduced in [12]
(respectively, in [18, 19]). The LJ-cohomology is the cohomology of a subcomplex of the H-ChevalleyEilenberg complex. The cohomology of this last complex is just the cohomology of the Lie algebra of
functions on relative to the representation defined by the Hamiltonian vector fields (for a detailed study of
the cohomologies of another subcomplexes of the H-Chevalley-Eilenberg complex, we refer to [15, 16, 17]).
The purpose of the present paper is to consider a more general setting which extends in some sense
the precedent ones. We consider a manifold
endowed with a skew-symmetric bracket
of functions
which satisfies the Jacobi identity, a rule that assigns a vector field
to each function such
that
, and, in addition, we assume that the generalized distribution defined by the
vector fields
is in fact a generalized foliation. Note that
does not satisfy necessarily any local
property, so that it is not in principle a Jacobi bracket.
Even in this general setting it is still possible to define the corresponding H-Chevalley-Eilenberg cohomology, and the de Rham and the -foliated cohomologies are related with it. By introducing a suitable
definition of -foliated derivatives, we give a first definition of prequantization bundle, and obtain a characterization of a quantizable manifold
(Theorem 3.9) in the setting of foliated forms.
If the existence of a Lie algebroid
is assumed, then we can define a new cohomology by
using the representation of
on
and, under certain conditions, this cohomology is related
with the precedent ones in a natural way. A second definition of prequantization bundle is given in this
context making use of a convenient notion of
-derivative, and the corresponding characterization of
quantizable manifold is obtained (Theorem 4.8). It should be remarked that this second notion of prequantization bundle is more general that the precedent one. Finally, we discuss the cases of Poisson and Jacobi
manifolds recovering the results previously obtained in [28] and [19], respectively.
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All the manifolds considered in this paper are assumed to be connected.
A Jacobi structure on an -dimensional manifold
is a pair
vector field on
satisfying the following properties:
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where
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Here
denotes the Schouten-Nijenhuis bracket ([3, 29]). The manifold
endowed with a Jacobi
structure is called a Jacobi manifold. A bracket of functions (the Jacobi bracket) is defined by
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is skew-symmetric, satisfies the Jacobi identity and
Thus, the space
of
real-valued functions on
endowed with the Jacobi bracket is
a local Lie algebra in the sense of Kirillov (see [13]). Conversely, a structure of local Lie algebra on
66
On the geometric prequantization of brackets
V9 Z; W&
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9 Z;
\Yj
defines a Jacobi structure on
(see [8, 13]). If the vector field identically vanishes then
is a derivation in each argument and, therefore,
defines a Poisson bracket on
and
is
a Poisson manifold. Jacobi and Poisson manifolds were introduced by Lichnerowicz ([20, 21]; see also [3],
[22] and [29]).
Examples of Poisson structures are symplectic and Lie-Poisson structures (see [20] and [30]).
Other interesting examples of Jacobi manifolds, which are not Poisson manifolds, are contact manifolds
and locally conformal symplectic manifolds which we will describe below.
A contact manifold is a pair
, where
is a
-dimensional manifold and is a -form
on
such that
at every point (see, for example, [1], [2], [21] and [22]). If
is the isomorphism of
-modules of the space of vector fields
on onto the space
of -forms
defined by
, then the vector field
is called the Reeb
vector field. A contact manifold
is a Jacobi manifold. In fact, the vector field is the Reeb vector
is defined by
field and the -vector on
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An almost symplectic manifold is a pair
, where
is an even dimensional manifold and is a
non-degenerate 2-form on . An almost symplectic manifold is said to be locally conformal symplectic
(l.c.s.) if for each point
there is an open neighborhood such that
, for some
function
(see, for example, [8] and [27]). So,
is a symplectic manifold. An almost
symplectic manifold
is l.c.s. if and only if there exists a closed 1-form such that
The
1-form is called the Lee 1-form of . It is obvious that the l.c.s. manifolds with Lee 1-form identically
zero are just the symplectic manifolds.
In a similar way that for contact manifolds, we define a -vector and a vector field on
which are
given by
for all
, where
is the isomorphism of
by
. Then
is a Jacobi manifold.
The contact and l.c.s. manifolds are called the transitive Jacobi manifolds (see [7]).
Now, let
be a Jacobi manifold. Define a homomorphism of
by
for
.
This homomorphism can be extended to a homomorphism, which we also denote by
on
onto the space of -vectors
by putting:
of -forms
-modules defined
-modules
(4)
, from the space
(5)
for
and
We also denote by
the corresponding vector bundle morphism.
If
the vector field
is a
real-valued function on a Jacobi manifold
defined by
(6)
is called the Hamiltonian vector field associated with . It should be noticed that the Hamiltonian vector
field associated with the constant function is just . A direct computation proves that (see [21] and [24])
(7)
Now, for every
we consider the subspace
fields evaluated at the point . In other words,
of
generated by all the Hamiltonian vector
. Since is involutive, one easily
67
M. de León, J. C. Marrero and E. Padrón
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follows that defines a generalized foliation, which is called the characteristic foliation in [7]. Moreover,
the Jacobi structure of
induces a Jacobi structure on each leaf. In fact, if
is the leaf over a point of
and
then
is a contact manifold with the induced Jacobi structure. If
,
is a
l.c.s. manifold (for a more detailed study of the characteristic foliation of a Jacobi manifold we refer to [7]
and [8]). If
is a Poisson manifold then the characteristic foliation of
is just the canonical symplectic
foliation of
(see [20] and [30]).
Next, we will recall the definition of the Lie algebroid associated to a Jacobi manifold (see [12]).
Let
be a differentiable manifold. A Lie algebroid structure on a differentiable vector bundle
is a pair that consists of a Lie algebra structure
on the space
of the global cross
sections of
and a vector bundle morphism
( the anchor map) such that:
is a Lie algebra homomorphism.
1. The induced map
2. For all
and for all
one has
(8)
A triple
is called a Lie algebroid over
(see [23], [25] and [29]).
Now, let
be a Jacobi manifold. In [12], the authors obtain a Lie algebroid structure on the
jet bundle
as follows. It is well-known that if
is the cotangent bundle of , the space
can be identified with the product manifold
in such a sense that the space
of the global cross sections of the vector bundle
can be identified with
Now, we consider on
the bracket
given by (see [12])
(9)
where
is the prolongation mapping defined by
(10)
We have (see [12])
Theorem 1 Let
be a Jacobi manifold and
the bracket on
(9). Then, the triple
is a Lie algebroid over
where
is the vector bundle morphism
defined by
(11)
for
mapping
Moreover, if we consider on
the Jacobi bracket then the prolongation
(12)
is a Lie algebra homomorphism.
Remark 1 If
vector bundle
then the canonical projection
in such a sense that
defines a vector subbundle of the
for
where
(respectively, ) is the fibre of
(respectively,
) over . Note
that can be identified with the cotangent bundle
and that, under this identification, the restriction
of the anchor map
to is just the vector bundle morphism
68
On the geometric prequantization of brackets
3. H-Chevalley-Eilenberg cohomology, foliated covariant derivatives and prequantization
In this section we will assume that
is a differentiable manifold which satisfies the following conditions:
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which is
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and
,A : % V W&
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(13)
is called the Hamiltonian vector field associated with .
If
is a point of
, we will denote by
the subspace of
A vector field on
is said to be tangent to
fields tangent to is denoted by
if
defined by
for all
 SD! P Ÿ r ¢ Ÿ The space of the vector
3. The involutive generalized distribution
is completely integrable. Thus,
teristic foliation.
P
defines a generalized foliation on
which is called the charac-
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3.1.
H-Chevalley-Eilenberg cohomology
We consider the cohomology of the Lie algebra
relative to the representation defined
by the Hamiltonian vector fields, that is, to the representation given by
This cohomology is denoted by
and it is called the H-Chevalley-Eilenberg cohomology associated to
(see [16, 17, 18, 19] for the case of a Jacobi manifold). In fact, if
is the real vector
space of the -multilinear skew-symmetric mappings
then
Im
where
is the linear differential operator defined by
(14)
69
M. de León, J. C. Marrero and E. Padrón
for
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and
Let
be the identity map. We will denote by
Yí
0
to the -coboundary
From (14), it follows that
for
(15)
Using (13) and (14), we obtain the following relation between the de Rham cohomology and the HChevalley-Eilenberg cohomology.
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Theorem 2 Let
for
be the homomorphism of
and
Then
. Thus, if
corresponding homomorphism in cohomology
-modules given by
(16)
µRE induces a homomorphism of complexes
is the de Rham cohomology of
we have the
Next, we will show the relation between the -foliated cohomology of
and the H-Chevalley Eilenberg cohomology.
(for the definition of the -foliated cohomolFirst, we will introduce the -foliated cohomology of
ogy associated to a regular foliation on a differentiable manifold, we refer to [9]).
be the space of the -foliated -forms on . An element of
is a mapping
Let
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
is a point of
2. If
where
the restriction of
‡
to the leaf
of
over
is a -form on
are
-differentiable local vector fields defined on an open subset
of
and
are tangent to in then the function
is
-differentiable,
is given by
for
We can consider the linear differential operator
given by
(17)
for
and
70
and
It is clear that
This fact allows us to introduce the differential complex
The cohomology of this complex is denoted by
and it is called the -foliated cohomology of
Now, using (13), (14) and (17), we prove the following result which relates the cohomologies
On the geometric prequantization of brackets
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Theorem 3 Let
be the homomorphism of
-modules defined by
(18)
for
and
Then
induces a homomorphism of complexes
and we have the corresponding homomorphism in cohomology
Moreover, the following diagram is commutative
where
and the cohomology
is the canonical homomorphism between the de Rham cohomology of
Let
be a complex line bundle over . Denote by
the space of the cross sections of
and by
the fibre over
.
If
is the leaf of over
then it is clear that the projection
defines a complex line bundle over . If is a point of , we will denote by Lin
the
space of the -linear maps of
onto
.
Definition 1 A
-foliated covariant derivative
on
is a map
Lin
which satisfies the following conditions:
1. If
then
Lin
Lin
and the map
is a covariant derivative on
.
2. If is an open subset of ,
is a
-differentiable local section of
and
is a
-differentiable local vector field defined in which is tangent to , then the map
given by
for
is a
-differentiable local section of
Let be a Hermitian metric on
said to be Hermitian if
for
A
.
-foliated covariant derivative
,
and
It is clear that a (Hermitian) covariant derivative on
covariant derivative.
on
induces a (Hermitian)
is
-foliated
71
M. de León, J. C. Marrero and E. Padrón
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i
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P
b
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P
Ü ° P ib
G  J 2 ÚD!ª ' P

P
Definition 2 Let
be a complex line bundle over
on
The curvature of is the mapping
If
is a point of
and
a
-foliated covariant derivative
given by
(19)
, we have that
where
is the curvature of the covariant derivative
on
Thus, there exists a globally defined complex -foliated -form
such that
.
(20)
Since the -foliated -form
completely determines to the map
, we will say also that
curvature of
Proceeding as in the case of a usual covariant derivative (see, for instance, [14]), we prove
Theorem 4 Let
derivative on
Then:
1. The complex
be a complex line bundle over
with curvature
and that
-foliated -form
2. The cohomology class
is the
Suppose that is a -foliated covariant
is the complex -foliated cohomology of
defines a cohomology class in
does not depend of the
3. If is a Hermitian metric on
then
is purely imaginary.
and
-foliated covariant derivative.
is a Hermitian
-foliated covariant derivative
ib 2 "!Õ 2 <!( 89 A:D;%z*ê > ê:­ ê: > ê A: W&
=ê [%$Ðd W
# & ,CD!T ê #w
,%?'*) + #h.¹0$243 #Q
' P
2 “D!Õ2 D! b b
Av'&Q
RE P  ° P hDY í ! Ýh° Þ Ã 0 ~
2  <!Û' '
2
Ô
D
Ô
!
P
3
RE P ™ 0$2  ,% Yí b
3.3.
Prequantization
over
we will denote by End
For a complex line bundle
endomorphisms of
Then, we introduce the following definition.
Definition 3 A complex line bundle
over
the space of the -linear
is said to be a prequantization bundle if
(21)
with
defined by
(22)
where is a -foliated covariant derivative on
if there exists a prequantization bundle
Let
The manifold
is said to be quantizable
over
be the -coboundary in the H-Chevalley-Eilenberg complex given by
be the homomorphism defined by (18).
From (13), (19), (20), (21), (22) and Definition 1, we deduce
and let
Lemma 1 The manifold
is quantizable if and only if there exist a complex line bundle
over
and a -foliated covariant derivative on
such that the curvature
purely imaginary and
Next, we will obtain another characterization. For this purpose, we recall the following result.
72
of
is
On the geometric prequantization of brackets
) ( '
2 < !¯3 ' R 2 µD! R
0
 % 0$2  (
0
2 Ú"!÷ *( 
2
¯
"
!
'
R
 % 0$3 2  (
°
°
ãÝ
ÕRE  ~
cD! ÝhÞ Ã ~
0  v
‡
P
RE  ‰% Yí . ã Ý RE P ™‡,
F% ã Ý © . RE P i‡–
W
Šb
Theorem 5 [14] (i) If
is a complex line bundle over ,
and is a Hermitian covariant derivative then the curvature
and
is a Hermitian metric on
of is purely imaginary
is an integral closed -form.
(ii) If is an integral closed -form then there exist a complex line bundle
Hermitian metric on
and a Hermitian covariant derivative with curvature
over , a
such that
.
Now, if
is the homomorphism given by (16) and
is the H-ChevalleyEilenberg cohomology operator (see (14)) then, using Lemma 1 and Theorems 3, 4 and 5, we prove
Theorem 6 The manifold
is quantizable if and only if there exist an integral closed -form
and a -foliated -form such that
on
4. H-Chevalley-Eilenberg cohomology, Lie algebroids and prequantization
2 R I R < !Õ ¬­*“R D! ¢ 2 R º7%R D!(b R º Q/ W&
hD! UR ¬Œ> E
 R Ÿ ,% P Ÿ ¬
R Ÿ % 2R … €   b
ºDAwv ™  V%y aBb
2 R *“R D!Õ R Ÿ % Ñ R Ÿ}Ö~× ºDAv$
i  Ùb
9 ¹;
R R
—VÑ R D! ¢ ÑR
¬ËÚR <! ¢ 2 b Ñ R D!Õ
R
Let
be a differentiable manifold as in Section 3. Moreover, we will assume that there is a Lie algebroid
over
with anchor map
which satisfies the following conditions:
1. If
phism
is the space of the cross sections of
such that
then there exists a Lie algebra homomor-
2. For all
where
3. For all
We will denote by
is the fibre over
we have that
such that
the Lie bracket on
Furthermore, there is a vector subbundle
of
and by
the restriction to of the anchor map
induced homomorphism between the cross sections of
4.1.
2 R *Ñ R <! —ÎD Ñ R s<! b | ~
We also will denote by
and the vector fields on
the
Lie algebroids and H-Chevalley-Eilenberg cohomology
In this section, we introduce a cohomology associated to the Lie algebroid and we study the relation between
this cohomology and the H-Chevalley-Eilenberg cohomology (for a detailed study of the cohomologies
associated to a Lie algebroid we refer to [23]).
We consider the cohomology of the Lie algebra
relative to the representation of
on
defined by
V UR i 9 ] ; R R h´ &
j<! W&
i #QR ,C! ¬ #$R i Šb
R 73
M. de León, J. C. Marrero and E. Padrón
Ü £ R ,+ V &
W
Šb . ›  R w´Œ bO› bK b ¼ › àOR àOà
-´+ V R hDW!&
V
›Ü R -+ ‰% áâ p 9±9±ã ã Q › › € R -+ -+ "
h! "! ›  › € R -
,+ + Š
\; ; g R R ã }sä R -+ V W&
W
hD! sä  € … R -+ V W&
ç
ã .sä ™ # å àOàKà # ó% æçÄè ä AVv$
¬ ç # ç ì i . ä # å àOàKà # ê ç àOàKà # W
R Rä
R R R R
. åæç†ëBì AVvw
 . ä 9 # R ç # R ì ; R # R å àOàKà # êR ä ç àOàKà # Rê ì àKàOà # R ä # R å KbObKbO # R ä R Šb
Ü £ R -
+ V i P
Z› ~
ù P Œ› ~
cb <! Œ› P P
š
¬­Q UR <! | ~
¬ R P Œ› P j<! › R ,
+ V / &
¬ R P i‡–
W
i # R € ObKbObK # R › ‰%‡ ¬ # R € KbObKbO ¬ # R › Š
£#R € KbO bObO-
#+ R › WR &
Šb W
Š ã ib ¬ R P P ¬
R
£ ‡ P iWŒdқ
Í P"!¶
¬ R P Ü £ P R DV! Ü £ R -
+ V W&
ib
¬ËR Œ› ~
sD! › UR -
+ V W&
/ &
¬ R % ¬ R P > ù P Šb
¬R
¬R c£ ~
iWdÒ
s!Ô £ R ,
+ £ W&
i ã ¬­R Ü ñ™£ ò ~
<! Ü £ R -
+ W&
W
Šb
RE P Ü £ P hDP ! Ü Ýh£ Þ Ã ~
zRE QÜ ñ™£ ò ~
hD! Ü ÝF£ Þ Ã ~
º QV <! UR º R } › R ,
+ V W
W
,! ÝF› Þ Ã ~
­ W&
ºDR . › ™ € ObKbOb™ › ,% . › ÀºD € ŠKbObKbOºD › ObK<bObO! › £ V ~
iW &ã ib
ºR
. › º › £ -
+ V ,
+ W&
&
W
Š 㠀 =
R R V º R ÒÜ £ R -
+ V ÝhÞÃ ZD! Ü Ý ÝF£ Þ ~
ib
Ã
This cohomology is denoted by
Therefore, if
vector space of the -multilinear skew-symmetric mappings
then
is the real
Im
where
by
is the linear differential operator defined
(23)
for
Next, we will obtain some relations between the cohomology
the -foliated
cohomology and the de Rham cohomology.
the canonical homomorphism between
and the
We will denote by
space of the -foliated -forms on
Using (17), (23) and the fact that
is a Lie algebra homomorphism, we deduce
Theorem 7
1. Let
modules defined by
be the homomorphism of
-
(24)
for
and
Then,
induces a homomorphism of complexes
Thus, we have the corresponding homomorphism in
cohomology
2. Let
by
be the homomorphism of
-modules defined
(25)
Then induces a homomorphism of complexes
thus we have the corresponding homomorphism in cohomology
and
Now, if
(respectively,
) is the canonical
homomorphism between the -foliated cohomology (respectively, the de Rham cohomology) and the HChevalley-Eilenberg cohomology (see Theorems 2 and 3) then, using (14), (23), (24), (25) and the fact that
is a Lie algebra homomorphism, we prove
Theorem 8 Let
given by
be the homomorphism of
-modules
(26)
for
complexes
homomorphism in cohomology
gram is commutative
74
and
Then, induces a homomorphism of
and thus we have the corresponding
Moreover, the following dia-
On the geometric prequantization of brackets
ü Ü ø £ 0 R ,+ V &
W
0 0
0 0 ºR
0 0
/R P 0 01
Ü £ Pø RE P ú ú ú úü ú4û Ü Ýh£ Þ Ã ~
ú
ù P ú ú ú
RE
Ü ñ™£ ò ~
¬R
2 2
%12 b
Š\ D!¯ D!µ
€
Ÿ
Ÿ

Ÿ b … 
R
2 2 ·D!S
2 Ú
R ! Ÿ-ž }¡ Š\ Ÿ # R Ÿ R Ÿ 2 3 ( ö Š\ Ÿ 2 ô 4 ( ö R Ÿ D!
iW Ÿ # R Ÿ Ó R Ÿ =&V # _ i
2 3( ö #$
,% ¬ # R Ÿ i #}  ¤.  2 3( ö #Qb
! # R f D! R V
2 R *# “R ”D !Õ Š 
#
D
(
2 3 }
2 3( #$
™  h% 2 3( ¼ Ÿ ÁA#Q
2 “D!Õb
  V
2 ÷D!ªb R 2 2 ÷<!·
# R Ÿ ™ # € Š#±°±
F% Ÿ 2 3( ö # € \#±°Q  W
Ð. Ÿ # €  i 2 3( ö #±°w
¬
 ' # R Ÿ Î R Ÿ P # € \#±° Šb
2 D!Õ
2 3( ö #•%6' 5™¼ 3( ö Ái#}ô õ ö # R Ÿ Î R Ÿ # _ R 2 <!(
hD! 2 U
R 2 “D!
b
U
´
h
´
2
8 7? R R ( ( i#w
#
±
#
±
°
™
$
#
‰
~
%
(
(
(
(
(
7 R€ R
2 3 î > 2 3:9 2 3:9 > 2 3 î 2`L 3 î M 3 9 N
# R € # R ° & UR # & ib
4.2.
Lie algebroids and derivatives on complex line bundles
Let
by
onto
maps of
be a complex line bundle over
the fibre over
Definition 4 A
-derivative
Denote by
and by Lin
on
the space of cross sections of
the space of the -linear
is a map
Lin
which
satisfies the following conditions:
1. If
then
2. The map
Lin
.
Lin
3. For
is
and
4. If
is an open subset of
then the map
for
is a
we have
and
given by
for
If
is a
A
-derivative
, we obtain a (Hermitian)
Definition 5 Let
The curvature of
on
is said to be
(27)
,
and
is a (Hermitian) -foliated covariant derivative on
and
-differentiable local section of
-differentiable local section of
Let be a Hermitian metric on
Hermitian if
for
-linear.
and we put
-derivative.
be a complex line bundle over
is the mapping
and
a
-derivative on
given by
(28)
for
and
75
M. de León, J. C. Marrero and E. Padrón
R hD! | ~
­
¬

&
7
V
7 # R € #±R °±
i#$
‰%Ø 7 #KR ° # R € ™#$
Šb
ì I´
W
&
­
. 7 * R
UR <! V Š™º%~vŠ0
7 # R € #R ° i#$
‰%~W .;7 € # R € # R ° 4. 3i .;7 ° # R € # R ° #Qb °
.;7 € .; 7 U
° ´ hD! Y R £ D!Õ
. 7  R R
V #
±
#
±
°
,
Ì
%
. 7 R€ R
. 7 € # R € #±R °±
Ð. 3i . 7 ƒ°} # R € #KR °w
Šb
.<7
7 .%7 › -
+ 2 ã
R V ¼
9
› R -
+ V % . ›  R ´bObKb ›àKàOà ´ R hD! ;Qb V ¨ . ›
. › % . ›€ . 3 . °› › R -
+ V ã . › % ã . ›€ . 3 ã . °› b
ã° % a
Ü £ R -
+ V W
ib
Using (8), (28), Definition 4 and the fact that
deduce that
is trilinear over
and that
Thus, we have that there exist two
, such that
is a Lie algebra homomorphism, we
-bilinear skew-symmetric mappings
(29)
and
We remark that
We will denote by
induce two cross sections of the vector bundle
the map defined by
Since
completely determines to
we will say also that
Now, we can extend by linearity the operator to the space
.
is the curvature of
.
given by
is
-multilinear and skew-symmetric
In fact, if
we define
It is clear that
and, therefore, we obtain the corresponding cohomology which will be denoted by
Moreover, using (23), (27), (28), Definition 4 and proceeding as in the proof of Theorem IV.3 in [19],
we conclude
b
2
R .;7
Ü ° UR ,+ V \
Šb
2 <! / =FR ¼?7 2 > í 7 Á  UR < ! R V …
ã
. 7 > . 7 % = R @¼ 7 > … 7 Á Šb
G . 7 > J % G .<7hJ b
2 <!( 2
R 2 “D!(
.;7
2 <!Õ
2 “ D!Õ b
. 7
Theorem 9 Let
with curvature
1.
be a complex line bundle over
Then:
Suppose that
is a
-derivative on
defines a cohomology class in
2. If
is another
-derivative on
such that
there exists a
-linear mapping
In particular,
3. If is a Hermitian metric on
then
is purely imaginary.
and
is a Hermitian
2 Prequantization
D!Õ
2 ®"!Õ
UR 2
ì ¼ Áʼ ÁA#c%an # ib
F
]
%
a
5€
2 € Ÿ
-derivative on
4.3.
be an arbitrary complex line bundle over .
-derivative on
In this section, we will assume that a
conditions:
Let
(C1) If
76
then
for all
always satisfies the following
On the geometric prequantization of brackets
(C2) If
5 €  •%]y a
# & Šb
' P
and there exists
#R Ÿ Ñ R Ÿ
for all
Note that if
for
is a
— #ìR Ÿ ,% 5 €  3( ö #•% 2 ¼ € Áʼ Ÿ ÁA#
such that
2
2 D!Õ
3( ö #•%'65™¼ 3( öKÁi#}ô õnö-
-foliated covariant derivative on
# R Ÿ Î R Ÿ # _ i
2
2
and
2
then
is the
R -derivative defined by
and
then satisfies the above conditions.
Next, we will introduce a new definition of prequantization bundle for the manifold
2 <!(
89 A:D;%z*ê > ê:­ ê: > ê
Definition 6 A complex line bundle
with
= ê [%$Ðd W
over
b
is said to be prequantization bundle if
A: W&
(30)
# ,C<! ê #w
F% 2 ì ¼ 6 Á#h.¹0$243 #Q
2
R 2 “D!Õ2 «D! b
Yí 0
º R Q ° R -
+ W&
jD! ÝF° Þ Ã ~
º
2
R 2 2 D!Õ
Ø
%
.
i
3
°
.%7 .<7 € .;7
ºDR . 7 € ‰%]an ºDR 0$v 2 . 7 ƒ°±
,%~ Yí b
RE  ° ~
ZD! Ýh° Þ Ã ~
¬
0
#R £ 2 R £ I R £ D!Õ
#R £
= R Q UR j<! V W&
V ã RE  ‰% Yí . Ý ºDR = R ‰% ã Ý © . ºDR = R Šb
5 € i  ‰%a # R £  ™ÀºDAv$
i  %aBb

5 € i  Í%~y a — # R Ÿ Z% 5 €  Š # R Ÿ “Ñ R Ÿ # R £   ™ÀºDAv$
i  ib given by
-derivative on
where is a
prequantization bundle
over
.
The manifold
(31)
is said to be quantizable if there exits a
It is clear that if is quantizable in the sense of Section 3.3 (see Definition 3) then is also quantizable
in the above sense. However, in general, the converse is not true.
Let be the -coboundary in the H-Chevalley-Eilenberg complex given by (15) and let
be the homomorphism defined by (26).
Using (28), (29), (30), (31), Definition 4 and the fact that is a Lie algebra homomorphism, we deduce
<!Û
Lemma 2 The manifold
is quantizable if and only if there exist a complex line bundle
over
and a
-derivative on
with curvature
such that
Now, if
is the homomorphism given by (16) then, proceeding as in the proof
of Theorem V.2 of [19] and using (23), Lemma 2, Theorems 5, 8 and 9 and the fact that is a Lie algebra
homomorphism, we conclude
Theorem 10 The manifold
is quantizable if and only if there exist an integral closed -form
and a cross section of the dual bundle
such that:
1. If
is the
2. If
is a point of
and
3. If
is a point of
such that
-linear map induced by
with
on
then
then
and
 then
#R £  i #R Ÿ %
77
M. de León, J. C. Marrero and E. Padrón
5. The particular cases: Jacobi and Poisson manifolds
\YZ\[
£ ~
Š ã 9 ‚;
ÝhÞ Ã Ý
Ycí :n
,%Ì ã Ý © ™ A:B
‰%]Ycd \d-:B
Š : Šb
š vb œä ß ›~&
Ýh› Þ Ã ~
ù v › ~º Ö œ › › … ~€ ~
ç ~
œ
v
›
› ­œ Ö
š
ÝhÞ Ã € … ñ 6w6
›œ … € ~
hD! Ýh› Þ Ã ~
º › = A­
™ € ObKbObK › ,% = d € ObObKbO\d › Ð. æ è › Vvw
B  € B AÍd € KbObObKDd C B ObObKb™Wd › Šb
B €
ç
º › ~
› € ~
,% Ýh› Þ Ã € ñ 6w6 ~
œ › ~
Ö œ › … € ~
…
› ÝFÞ Ã › € œ ñ ç 6±6 ~Ö œ …
…
Y¹í % ã Ý © % º ° YcWag
Šb
ã Ý =R v
ÂÝF› Þ € ñ ç ~
i = Ýh› Þ 9 Vñ ;ç ~
R & à € … 6w6 ç ã
à € … 6±6
Ýh£ Þ Ã € ñ 6w6 ~
Š Ý ô FÞ GIE HJ î Dÿ K é +\+ ¼ ¡ ÁW
ç ~
…
v
£
Ü ÝhÞ Ã € … ñ 6w6
ã Ý Àº › = A­
W
‰% º ›  € ’ Mõ L = A
’ Mõ L = A­
F%ØA G Yc = J . š [Mõ ^ L = .‚› Y=~)^ A* G Yc› A € J~È
j š D”! vw
A[]›  *^ € A]~. G [` =› J ~
Šb œ œ £ ~Ö œ Ö œ £ … € ~
i ’ Nõ L õ’ ° ’ L %aBÒbœ Ö œ …
M
õÜ £ L ~
M
º › ‰£ œ ~› ~
Ö £ œ › …~€ Š~ õM
ËL D
! Ýh› Þ £ à ~
ç ~
i ã ô
œ Ö œ …€ ’
ÝhÞ Ã € … ñ 6w6 Ý ÞOGIE H"J î ÿDK é +W+ ¼ ¡ Á
’ Mõ L aBOv$
,%~YZ\a}
Šb
[Ì%ag
’ Mõ L = \a}
,%ÌA G YZ = J \a}
Â
= œ › ~
Šb
’ Mõ P Zœ › ~
UD! œ › € ~
’ Mõ P = ‰%Ø G Yc = J
Let
be a Jacobi manifold and
the Jacobi bracket.
Suposse that
is the H-Chevalley-Eilenberg complex associated to
(6) and (15), we deduce that
. Using (2), (4),
(32)
Next, we will recall the definition of the Lichnerowicz-Jacobi cohomology (see [18] and [19]).
is said to be -differentiable if it is defined by a linear differential operator
A -cochain
of order If
is the space of -vectors on
then we can identify the space
with the space of all -differentiable -cochains
as follows: define
the monomorphism given by
(33)
Then
and
which implies that the spaces
are isomorphic. Note that (see (32) and (33))
(34)
On the other hand, using that
for
is a linear differential operator of order , we deduce that
. Thus, we have the corresponding subcomplex
of the H-Chevalley-Eilenberg complex whose cohomology
will be called the -differentiable
H-Chevalley-Eilenberg cohomology of . Moreover, we obtain that (see [18, 19])
(35)
where
(36)
This last equation defines a mapping
which is in
fact a differential operator that verifies
Therefore, we have a complex
whose cohomology will be called the Lichnerowicz-Jacobi cohomology (LJ-cohomology) of and denoted
by
(see [18, 19]).
Note that the mappings
given by (33) induce an isomorphism between the complexes
and
and
consequently the corresponding cohomologies are isomorphic. Furthermore, from (36), we obtain that
(37)
Now, if
is a Poisson manifold
, it follows that (see (36))
(38)
for
defined by
Thus, we deduce that the linear differential operator
(39)
78
On the geometric prequantization of brackets
œ £ ~
Š ’ Mõ P ib
õ’ ° P %Žanb
N
Ü õM£ P ~
\YZ\[
P
—ÓR Œ› ~
•D! œ › ~
Ö
›œ  € ~
—Ó › ~
s<! œ › ­~ W&
— R ‡,
‰%~ — ‡,
ŠO — 3 à ‡–
W
‡ Œ› ~
Šb
—R > d%~ ’ Mõ L > — R b
Œ
› ~£ Í! € ~œ Š› K ~
õMLÖ œ › … € ~
R
¯
—
~
Š
\
g
d
&
D
!
~

£
£
’
ò R ~ h"! Ü õM£ L ~
œ Ö œ …
—R Ü ñ™£ —Ú
º › > — R %ÌAVv$
› RE ÎRE Œ› ~
<! Ýh› Þ Ã ~
› ù P Z› ~
Ë D!
š
Œ › £ P Š\dg
"!¯ £ ~
£ P € ~
ŠK Mõ L P
—R P P
œ ֜ …
’ £ P iWdÒ
— R P ü SO œ T £ ~
Ö œ £ … € ~
iO ’ õML QR
ù P Q Q S S — R
£ ~
Š\dg
— P Œ› P ‰! œ › ~
/ — P i ‰% — P ™‡,
™‡ € ObObKbO\‡ › ,%ÌAVv$
› ‡ — ‡ € ŠKbObObK — ‡ › W
=&V ‡ Œ› P ‡ € ObObKb™W‡ ›  € ~
i
— R P i‡,
‰%Ì — P i‡–
iO — P i3 à ‡,
ib
satisfies the condition
This fact allows us to consider the differential complex
The cohomology of this complex is the Lichnerowicz-Poisson cohomology (LP-cohomology) associated to
and it is denoted by
(see [20] and [29]).
Next, we will obtain the relation between the -foliated cohomology of a Jacobi manifold
and the LJ-cohomology.
Denote by
the map given by (4) and (5) and let
be the homomorphism of
-modules defined by
(40)
for
In [19] (see also [18]), we prove that
(41)
This implies that the mappings
induce a homomorphism of comand, thus, a homomorphism in cohomology
plexes
(see [18, 19]). In fact, using (4), (5), (6), (33) and (40), we have that
(42)
being the map given by (16).
On the other hand, if
is the space of the -foliated -forms on
and
is the canonical homomorphism, it is clear that there exists a homomorphism of complexes
such that the following diagram is commutative
In fact, if
is the homomorphism of
-modules defined by
(43)
for
,
and
then
(44)
Thus, from (6), (33), (43) and (44), we deduce that
RE P  › P hD! Ýh› Þ Ã ~
º › >=— R P ,%ÌAVv$
› RE P i
(45)
is defined by (18).
where
Therefore, using (35) and Theorem 3, we conclude that
— R P > d%~ ’ õML > — R P Šb
Consequently, for the corresponding homomorphisms in cohomology, we obtain the following commutative diagram
79
M. de León, J. C. Marrero and E. Padrón
› P Q X X X
ø Q Q X X X X Vvw
› RE P X X X
P QOU
X
R
—
ù P õÜ SOM› L T ~
º › ü XÜ Y ÝF› Þ Ã ~
V VW
S
V
V
— RS
V V AVv$
› RE
S V V V V V
V
ñÜ ™› ò ~
— P Œ› P <! œ › ~
—  £ ~
Š\dg
"!¯ œ £ ~
ŠK ’ Mõ P i — P  £ P ŠWdÒ
<!¯ œ £ ~
ŠO ’ õMP — P > ù P F% — º › >Œ— %ÌAVv$
› RE º › >Œ— P ,%ÌAVvw
› RE P ib
Ü
In the particular case when is a Poisson manifold, using the above results, we prove that the mappings
induce two homomorphisms of complexes
which satisfy the conditions
Next, we will give necessary and sufficient conditions for a Jacobi manifold
sense of Section 3.3 and in the sense of Section 4.3.
5.1.
to be quantizable in the
Prequantization bundles in the sense of Section 3.3
0 WYcW[I ‡
P
— R  ,% ’ õML — P ™‡,
ŠKvZ‡h[
ib
v ‡  WYŒ
P
—  ,%YU. ’ Mõ P — P i‡,
ib
P Ÿ % ¢ Ÿ ›~  Y
~
j
<
!

›
—
œ
/

From (34), (35), (37), (42), (43), (44), (45) and Theorem 6, it follows
Theorem 11 Let
integral closed -form
on
v
be a Jacobi manifold. Then,
is quantizable if and only if there exist an
and a -foliated -form such that
Using Theorem 11, we deduce
0
Corollary 1 Let
closed -form on
be a Poisson manifold. Then, is quantizable if and only if there exist an integral
and a -foliated -form such that
Now, we will study the case of a transitive Jacobi manifold
We distinguish the following cases:
(
for all
 —  ‰%]YZb ).
1. Symplectic manifolds: Let
be a symplectic manifold and the Poisson bivector. Then the
mapping
is an isomorphism of
-modules and
Using
these facts and Corollary 1 we recover the result of Kostant and Souriau (see [14, 26]), that is,
is
quantizable if and only if is integral.
2. Locally conformal symplectic manifolds: If a Jacobi manifold is quantizable in the sense of Section
3.3 then it is also quantizable in the sense of Section 4.3. Using this fact and the results of [19], we
conclude that a l.c.s. manifold is quantizable if and only if it is a quantizable symplectic manifold.
uB
 %a ‡%Èu
3. Contact manifolds: Let
be a contact manifold. Then, from Theorem 11, we obtain that
quantizable (it is sufficient to take
and
).
80
is
On the geometric prequantization of brackets
bundles in the sense of Section 4.3
9 ð; ¬ ‚R Prequantization
b R % ¢ £ ¸´ R
#
2
<
³
!
£
£
£
R R ­ R5 R ËW&| ~
д V W&
R
= R  R hD!
V
= R ‡h :B
,% ‡ R5 4. R :"
‡R :ÒR & R º ,%  € ~ h
-´ + V W&
Š
/b R <! R R ~
R
R 4 € R V ºD = Ýh€ Þ Ã
vQ™º € R5 R R R
0  WYcW[I
Z
— R  F% ’ Mõ L Z ib
[ Ÿ %a  ‰%~vb

[ Ÿ %®y a — ‡ Ÿ 7% [ Ÿ  `%

‡ Ÿ  Z Ÿ ¤.vQb ‡ Ÿ v
0  \Yj
Z
—  ‰%]YU. ’ Mõ P Z ‰%]Y » [–Ycb
5.2.
Let
be the Lie algebroid associated to a Jacobi manifold
of the dual bundle
Section 2) and thus a cross section
in such a way that the corresponding
is given by
Then,
(see
can be identified with a pair
-linear map
(46)
for
Therefore, if
is the homomorphism defined by (26) then,
is the linear differential operator of order
. Using
from (10), (33) and (46), we obtain that
these facts, (34), (35), (37), (42) and Theorem 10, we deduce a result which has been proved in [19].
Theorem 12 [19] Let
be a Jacobi manifold. Then,
is quantizable if and only if there exist
an integral closed -form , a vector field and a real differentiable function such that:
1.
2. If
is a point of
3. If
is a point of
and
and
then
is a -form at
such that
and
then
From Theorem 12, it follows a result of Vaisman [28].
Corollary 2 [28] Let
integral closed -form
be a Poisson manifold. Then,
and a vector field such that
is quantizable if and only if there exist an
Finally, for the transitive Jacobi manifolds we obtain the same results that in Section 5.1 (see [19] for
more details).
Acknowledgement. This work has been partially supported through grants DGICYT (Spain) (Project
PB94-0106) and University of La Laguna (Spain). ML wishes to express his gratitude for the hospitality
offered to him in the Departamento de Matemática Fundamental (University of La Laguna) where part of
this work was conceived.
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M. de León
Instituto de Matemáticas y Fı́sica Fundamental
Consejo Superior de Investigaciones Cientı́ficas
28006 Madrid, Spain
[email protected]
J. C. Marrero, E. Padrón
Departamento de Matemática Fundamental
Universidad de La Laguna
La Laguna, Tenerife, Canary Islands, Spain
[email protected], [email protected]
83