# Wick type deformation quantization of contact metric manifolds

Boris M. Elfimov  Physics Faculty, Tomsk State University, Lenin ave. 36,
Tomsk 634050, Russia<EMAIL_ADDRESS>and  Alexey A. Sharapov  Physics
Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia
<EMAIL_ADDRESS>
###### Abstract.

We construct a Wick-type deformation quantization of contact metric manifolds.
The construction is fully canonical and involves no arbitrary choice. Unlike
the case of symplectic or Poisson manifolds, not every classical observable on
a general contact metric manifold can be promoted to a quantum one due to
possible obstructions to quantization. We prove, however, that all these
obstructions disappear for Sasakian manifolds.

###### Key words and phrases:

Contact metric manifolds, deformation quantization, Sasakian geometry, Wick
quantization

###### 2010 Mathematics Subject Classification:

Primary 53D55; Secondary 53D35

The work of B.M.E. was supported by the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS”. The work of A.A.Sh. was supported
by the FAPESP Grant 2022/13596-8.

##  1\. Introduction

In the recent paper [8], we formulated the problem of deformation quantization
for contact manifolds and solved it using a modification of Fedosov’s method
[9]. The main novelty compared to the case of symplectic or Poisson manifolds
is that not every classical observable – a Reeb-invariant function on a
contact manifold – admits quantization. As a result, the algebra of quantum
observables may well be smaller than the algebra of classical ones. The study
of obstructions to quantization yielded some new invariants of contact
manifolds.

One can regard the above quantization as a contact counterpart of the Weyl-
type deformation quantization of symplectic manifolds. This is the most
natural quantization scheme if no extra geometric structure is assumed on
symplectic or contact manifolds. In many interesting situations, however,
symplectic manifolds come equipped with (integrable) Lagrangian polarizations,
which are important, for example, for their physical interpretation as phase
spaces of mechanical systems. An invariant Lagrangian polarization is also an
integral part of the geometric quantization method. In the most favorable
case, there is a pair of transverse Lagrangian polarizations, which may be
complex and mutually conjugate. The last situation is best exemplified by
(almost) Kähler manifolds with their holomorphic and antiholomorphic
polarizations. By definition, a Wick quantization of a Kähler manifold is a
quantization adapted to the underlying holomorphic structure, see Sec. 5.
There is a large body of literature devoted to various aspects of and
approaches to the problem, starting with the pioneering works of Berezin [1,
2]. In Fedosov’s approach, the Wick deformation quantization of Kähler
manifolds was in-depth studied in [4, 20, 6, 11, 13].

The odd-dimensional counterpart of almost Kähler geometry is known as contact
metric geometry. This can be defined as the geometry of contact manifolds
endowed with compatible Riemannian metrics. Incorporating a metric structure
in the fabric of contact geometry allows one to polarize a (maximally
nonintegrable) contact distribution into holomorphic and antiholomorphic
parts. In this paper, we use this polarization to formulate a Wick-type
deformation quantization of contact metric manifolds. As in [8], our
construction largely relies on Fedosov’s method. The rich geometry of contact
metric manifolds allows for several types of integrability conditions to be
imposed. These lead to three special contact metric geometries: K-contact, CR,
and Sasakian; the last one is the intersection of the first two. All these
integrability conditions play an important part in the Wick deformation
quantization developed below. In particular, we show that no obstructions
arise for the Wick deformation quantization of Sasakian manifolds and the
property of being K-contact ensures the absence of the first obstruction to
quantization.

To conclude this introduction, we list the basic differences between the Weyl-
and Wick-type deformation quantizations of contact manifolds. First, the
contact metric manifolds enjoy a natural contact connection, making the Wick
quantization completely canonical. The absence of a canonical connection on
general contact manifolds introduces an inevitable arbitrariness in the Weyl
quantization. Second, the obstructions to the Weyl quantization of classical
observables start from the second order in the deformation parameter, while in
the Wick quantization, they arise already in the first order. Third, the
differential operators that control the leading obstructions to quantization
are completely different: they are third-order for Weyl quantization and
first-order for Wick.

##  2\. Contact metric geometry

We follow the definitions and conventions of the classical papers [16, 17,
18]. They differ slightly from those adopted in more recent expositions of
contact metric geometry [3, 5]. The difference is explained in [15, Thm. 7.5].

One can regard contact manifolds as odd-dimensional analogues of symplectic
ones. Recall that a $1$-form $\lambda$ on a ($2m+1$)-dimensional manifold $M$
is called contact if

(2.1) $\lambda\wedge(d\lambda)^{m}\neq 0$

at each point of $M$. The pair $(M,\lambda)$ is called a contact manifold.
Geometrically, Eq. (2.1) means that the contact distribution
$\mathcal{C}=\ker\lambda$ on $M$ is ‘maximally nonintegrable’. As a
consequence, each contact manifold carries the canonical volume form
$v=\lambda\wedge(d\lambda)^{m}$ and an exact presymplectic structure defined
by the $2$-form $\omega=d\lambda$ of rank $2m$. The kernel of the
presymplectic form $\omega$, called characteristic distribution, is generated
by the Reeb vector field $\xi$ which is chosen to satisfy

(2.2) $i_{\xi}\omega=0\,,\qquad i_{\xi}\lambda=1\,.$

This gives the canonical splitting $TM=\mathcal{C}\oplus L_{\xi}$, where
$L_{\xi}$ is the trivial line bundle generated by $\xi$.

It is known that every contact manifold admits a Riemannian metric $g$ and a
tensor field $\phi$ of type $(1,1)$ that satisfy the following relations:

(2.3)
$\begin{array}[]{l}\lambda(X)=g(X,\xi)\,,\qquad\omega(X,Y)=g(X,\phi(Y))\,,\\\\[8.53581pt]
g(\phi(X),\phi(Y))=g(X,Y)-\lambda(X)\lambda(Y)\,,\qquad\phi^{2}=-I+\xi\otimes\lambda\,,\\\\[8.53581pt]
\phi(\xi)=0\,,\qquad\lambda\circ\phi=0\,.\end{array}$

Here, $X$ and $Y$ are arbitrary vector fields on $M$ and we treat the
$(1,1)$-tensor field111In [16], the tensor $\phi$ was dubbed a fundamental
singular collineation, but this name did not stick. $\phi$ as defining an
endomorphism of the tangent bundle $TM$. One usually calls $g$ an associated
metric. It is known that the associated metric is far from being unique and
not all of Rels. (2.3) are independent: the last two, for example, follow from
the others. We refer to the data $(\phi,\xi,\lambda,g)$ as a contact metric
structure and call $M$, endowed with such a structure, a contact metric
manifold.

Every contact metric manifold carries a canonical Jacobi structure [12, 14].
This is given by the Reeb vector field $\xi$ and the bivector $\pi$ defined by
the rule

(2.4) $(\alpha\wedge\beta)(\pi)=\bar{g}(\alpha,\phi^{t}(\beta))\,.$

Here, $\alpha$ and $\beta$ are arbitrary $1$-forms; $\bar{g}$ and $\phi^{t}$
stand for the inverse metric and the transpose of the endomorphism $\phi$. One
can see that

(2.5) $[\pi,\pi]_{SN}=2\pi\wedge\xi\,,\qquad[\xi,\pi]_{SN}=0\,.$

The subscript SN refers to the Schouten–Nijenhuis bracket on poly-vector
fields. Alternatively, using the aforementioned volume form $v$, one can
define the same bivector $\pi$ as dual to the form
$\lambda\wedge\omega^{m-1}$. This shows that the Jacobi structure $(\xi,\pi)$
is completely determined by the contact structure alone [18, Sec. 6]. Given a
Jacobi structure, one can make the space of smooth functions on $M$ into a Lie
algebra for the following bracket:

(2.6) $\\{f,g\\}=(df\wedge dg)(\pi)+fdg(\xi)-gdf(\xi)\,.$

This is called the Jacobi bracket of functions $f,g\in C^{\infty}(M)$. It
should be noted that (2.6) is not a Poisson bracket as it fails to satisfy the
Leibniz identity:

$\\{f,gh\\}-\\{f,g\\}h-g\\{f,h\\}=ghdf(\xi)\,.$

Notice, however, that the Jacobi bracket gives rise to a Poisson algebra
structure on the commutative subalgebra of $\xi$-invariant functions
$C^{\infty}_{\xi}(M)\subset C^{\infty}(M)$.

By a deformation quantization of a contact manifold $(M,\lambda)$ we mean a
deformation quantization of the Poisson algebra
$\big{(}C^{\infty}_{\xi}(M),\\{\,\cdot\,,\,\cdot\,\\}\big{)}$. A precise
definition will be given in the next section.

In order to develop the differential geometry of contact metric manifolds one
usually introduces one or another adapted connection. By definition, a contact
metric connection $\nabla$ is an affine connection on $M$ that parallelizes
the contact metric structure:

(2.7) $\nabla\lambda=0\,,\quad\nabla\xi=0,\quad\nabla\omega=0\,,\quad\nabla
g=0\,,\quad\nabla\phi=0\,.$

The definition is somewhat redundant since the first three relations follow
from the last two [19, Lem. 4.1]. It is known that a contact metric connection
always exists and can be constructed as follows [17, Thm. 11]. Let
$\nabla^{g}$ denote the Levi-Civita connection that is compatible with the
metric $g$. In each local coordinate system, define a $(1,2)$-tensor $S$ by
the expression

(2.8)
$S^{i}_{jk}=-\frac{1}{2}\phi^{i}_{l}\nabla^{g}_{k}\phi^{l}_{j}-\frac{1}{2}\lambda_{j}\nabla^{g}_{k}\xi^{i}+\xi^{i}\nabla^{g}_{k}\lambda_{j}\,.$

Using (2.3) one can readily verify that the new affine connection
$\nabla=\nabla^{g}+S$ meets all conditions (2.7). As usual, the commutator of
covariant derivatives evaluated on a vector field $v$,

(2.9)
$[\nabla{}_{k},\nabla{}_{l}]v^{i}=R^{i}_{jkl}v^{j}+T^{j}_{kl}\nabla_{j}v^{i}\,,$

gives the curvature $R^{i}_{jkl}$ and the torsion tensor
$T^{i}_{jk}=S^{i}_{jk}-S^{i}_{kj}$. The latter is always nonzero. Indeed, it
follows from the first identity in (2.7) that

(2.10) $\omega_{ij}=\lambda_{k}T_{ji}^{k}\,.$

The tensor $S$ is known as the potential of a metric connection
$\nabla=\nabla^{g}+S$. The following easily verified identity shows that the
torsion of a metric connection contains the same information as its potential:

(2.11)
$S_{ij}^{k}=\frac{1}{2}T^{k}_{ij}+\frac{1}{2}(g_{im}T^{m}_{jn}+g_{jm}T^{m}_{in})g^{nk}\,.$

Let us also introduce the covariant torsion and curvature tensors

(2.12) $T_{kij}=\omega_{kn}T^{n}_{ij}\,,\qquad
R_{klij}=\omega_{kn}R^{n}_{lij}\,.$

They enjoy the following symmetry properties:

(2.13) $\qquad R_{klij}=R_{lkij}=-R_{klji}\,,\qquad
T_{kij}+T_{ijk}+T_{jki}=0\,.$

Let $\hat{\phi}$ denote the restriction of the endomorphism
$\phi:TM\rightarrow TM$ onto the contact distribution $\mathcal{C}\subset TM$.
Then $\hat{\phi}^{2}=-I$; and hence, $\phi$ induces an almost complex
structure on $\mathcal{C}$. Therefore, the complexification of the vector
bundle $\mathcal{C}$ splits into the two transverse and mutually conjugate
subbundles composed of $\pm i$-eigenspaces of the operator $\hat{\phi}$:

(2.14)
$\begin{array}[]{c}\mathcal{C}_{\mathbb{C}}=\mathcal{P}\oplus\bar{\mathcal{P}}\,,\qquad\mathcal{P}\cap\bar{\mathcal{P}}=0\,,\\\\[8.53581pt]
\mathcal{P}=\\{X-i\hat{\phi}X\;|\;\forall
X\in\mathcal{C}\\}\,,\qquad\bar{\mathcal{P}}=\\{X+i\hat{\phi}X\;|\;\forall
X\in\mathcal{C}\\}\,.\end{array}$

This splitting equips $M$ with an almost CR structure of hyper surface type,
which can be viewed as an odd-dimensional counterpart of the almost complex
structure222The abbreviation CR means either ‘Cauchy–Riemann’ or ‘Complex-
Real’. . Furthermore, the presence of the positive definite Levi form

(2.15) $L_{\lambda}(X,Y)=\omega(\phi(X),Y)=g(X,Y)\qquad\forall
X,Y\in\mathcal{C}\,,$

allows one to identify this almost CR structure as strongly pseudoconvex [7].
An almost CR structure is called integrable (or just CR structure) if the
holomorphic subbundle $\mathcal{P}$ is involutive, i.e.,
$[\mathcal{P},\mathcal{P}]\subset\mathcal{P}$. The last condition amounts to
the vanishing of the $(1,2)$-tensor field

(2.16)
$Q_{ij}^{k}=\nabla^{g}_{j}\phi_{i}^{k}+\xi^{k}\phi_{i}^{n}\nabla^{g}_{j}\lambda_{n}+\phi^{k}_{n}\nabla^{g}_{j}\xi^{n}\lambda_{i}\,.$

This result is due to Tanno [21].

For the purposes of deformation quantization, we also introduce the dual
vector bundles $\mathcal{C}^{\ast}$ and $\mathcal{P}^{\ast}$. The sections of
the former are naturally identified with the $1$-forms on $M$ that annihilate
the Reeb vector field $\xi$. To characterise the latter it is convenient to
equip the complexified cotangent bundle $T^{\ast}_{\mathbb{C}}M$ with the
Hermitian form

(2.17)
$h(\alpha,\beta)=\bar{g}(\alpha,\beta+i\phi^{t}(\beta))=i\pi(\alpha,\beta)+\bar{g}(\alpha,\beta)\,,$

where $\alpha$ and $\beta$ are arbitrary $1$-forms. Let $\hat{h}$ denote the
restriction of $h$ onto $\mathcal{C}^{\ast}$. Then
$\mathcal{P}^{\ast}\subset\mathcal{C}^{\ast}_{\mathbb{C}}$ coincides with the
right kernel of $\hat{h}$, that is, $\hat{h}(-,\mathcal{P}^{\ast})=0$. In view
of Hermiticity the antiholomorphic distribution $\bar{\mathcal{P}}^{\ast}$
generates then the left kernel of $\hat{h}$. The contact metric connection
(2.7) clearly respects $h$. We will refer to the Hermitian form (2.17) as the
Wick tensor.

Finally, a contact metric structure $(\phi,\xi,\lambda,g,)$ is said to be
$K$-contact if $\xi$ is a Killing vector of the associated metric $g$, i.e.,
$\mathcal{L}_{\xi}g=0$. If, in addition, the corresponding CR structure is
integrable ($Q=0$), then the contact metric manifold is called Sasakian. It is
known [18, Thm. 11] that the property of being Sasakian is equivalent to the
identity

(2.18) $2\nabla^{g}_{k}\omega_{ij}=\lambda_{i}g_{kj}-\lambda_{j}g_{ki}\,.$

In dimension $3$, the holomorphic distribution $\mathcal{P}$, being one-
dimensional, is automatically integrable. Therefore every $3$-dimensional
K-contact manifold is Sasakian.

##  3\. Wick deformation quantization of contact metric manifolds

As mentioned in the previous section, by the deformation quantization of a
contact metric manifold $M$ we understand the quantization of the Poisson
algebra $(C^{\infty}_{\xi}(M),\\{\,\cdot\,,\,\cdot\,\\})$ associated with the
canonical Jacobi structure on $M$. It is remarkable that the Jacobi bracket
$\\{\,\cdot\,,\,\cdot\,\\}$ is determined by the contact structure alone and
in no way involves the associated metric. The general definition of the Weyl
deformation quantization of contact manifolds was formulated in our recent
paper [8]. The presence of the canonical Hermitian form (2.17) on $M$ suggests
the following definition of Wick deformation quantization.

######  Definition 3.1.

Let $\nu$ be a formal deformation parameter and $C_{\mathbb{C}}^{\infty}(M)$
be the commutative algebra of smooth complex-valued functions on $M$. By a
Wick deformation quantization of a contact metric manifold
$(M;\phi,\xi,\lambda,g)$ we understand a pair $(\ast,\Delta)$, where $\ast$
and $\Delta$ are, respectively, bilinear and linear operators on
$C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$ of the form

$a\ast
b=ab+\frac{\nu}{2}h(da,db)+\sum_{k=2}^{\infty}\nu^{k}C_{k}(a,b)\,,\qquad\Delta
a=\xi a+\sum_{k=1}^{\infty}\nu^{k}\Delta_{k}a\,.$

It is also assumed that

  1. (1)

$C_{k}$ and $\Delta_{k}$ are differential operators on $C^{\infty}(M)$, which
are extended to $C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$ by
$\mathbb{C}[\\![\nu]\\!]$-linearity;

  2. (2)

the elements of the subspace $\ker\Delta$ form an associative algebra w.r.t.
the $\ast$-product, that is,

$\Delta(a\ast b)=0\quad\mbox{and}\quad(a\ast b)\ast c=a\ast(b\ast c)$

whenever $\Delta a=\Delta b=\Delta c=0$.

The pair $(\ker\Delta,\ast)$ is called the algebra of quantum observables.

An important property of the operators $\ast$ and $\Delta$ is their locality,
i.e., the possibility of restriction on functions defined in any open domain
$U\subset M$. By abuse of language, we will refer to this as the restriction
of the operators $\ast$ and $\Delta$ onto the domain $U$.

In order to construct the Wick deformation quantization above we will follow
Fedosov’s approach [9]. Let $U$ be a coordinate chart on $M$ with coordinates
$\\{x^{i}\\}$ and denote by $\\{y^{i}\\}$ the fibre coordinates on the tangent
bundle $TU$ w.r.t. the natural frame $\\{\partial/\partial x^{i}\\}$. For each
point $x\in M$ define the formal Wick algebra $W_{x}$ as the space spanned by
the formal power series with complex coefficients

(3.1) $a(\nu,y)=\sum_{k,l=0}^{\infty}\nu^{k}a_{k,i_{1}\cdots
i_{l}}y^{i_{1}}\cdots y^{i_{l}}$

endowed with the $\circ$-product

(3.2) $(a\circ
b)(y)=\exp\left(\frac{\nu}{2}h^{ij}(x)\frac{\partial^{2}}{\partial
y^{i}\partial z^{j}}\right)a(y)b(z)\Big{|}_{z=y}\,,$

$h^{ij}(x)$ being the components of the Wick tensor at $x$. The
$\circ$-product is known to be associative, but not commutative. Taking the
union of algebras $W_{x}$, we obtain the bundle of formal Wick algebras $W$,
whose space of sections will be denoted by $\mathcal{W}$. Prescribing the
formal variables $y$’s and $\nu$ the degrees $1$ and $2$, respectively, makes
$\mathcal{W}$ into an $\mathbb{N}$-graded algebra, that is,
$\mathcal{W}=\bigoplus_{n\geq 0}\mathcal{W}_{n}$. We refer to this grading as
the $\nu$-degree. Associated with the $\nu$-degree is the descending
filtration

${F}^{k}\mathcal{W}\supset{F}^{k+1}\mathcal{W}\,,\qquad{F}^{k}\mathcal{W}=\bigoplus_{n\geq
k}\mathcal{W}_{n}\,,\qquad k=0,1,\ldots\,,$

which defines topology and convergence in the space of formal power series.
Sometimes we will use another natural $\mathbb{N}$-grading on the vector space
${\mathcal{W}}$, which is given by the polynomial degree in $y$’s. We refer to
it as $y$-degree and write ${\mathcal{W}}=\bigoplus_{n\geq
0}{\mathcal{W}}_{(n)}$. Unlike the $\nu$-degree, the $y$-degree does not
respect the $\circ$-product.

The most interesting to us will be the subalgebra
$\mathcal{W}^{\xi}\subset\mathcal{W}$ of $\xi$-transverse sections of $W$.
This is defined as

(3.3) $\mathcal{W}^{\xi}\ni
a\quad\Longleftrightarrow\quad\xi^{i}\frac{\partial a}{\partial y^{i}}=0\,.$

Tensoring the algebra $\mathcal{W}$ with the exterior algebra
$\Lambda(M)=\bigoplus_{n\geq 0}\Lambda_{n}(M)$ of differential forms on $M$,
we obtain the bigraded algebra $\mathcal{W}\otimes\Lambda$, the second grading
being the form degree. We will use the same symbol $\circ$ for multiplication
in the tensor product algebra. Let us also introduce the commutator of two
elements of $\mathcal{W}\otimes\Lambda$ by

(3.4) $[a,b]=a\circ b-(-1)^{nm}b\circ a$

for all $a\in\mathcal{W}\otimes\Lambda_{n}$ and
$b\in\mathcal{W}\otimes\Lambda_{m}$. The commutator makes the space
$\mathcal{W}\otimes\Lambda$ into a graded Lie algebra w.r.t. the form degree.
It is easy to see that the centre of this Lie algebra is given by
$\Lambda(M)[\\![\nu]\\!]$. The algebra $\mathcal{W}\otimes\Lambda$ is endowed
with the differential $\delta$ of bidegree $(-1,1)$:

(3.5) $\delta=dx^{i}\wedge\frac{\partial}{\partial
y^{i}}\,,\qquad\delta^{2}=0\,.$

It is clear that

(3.6) $\delta(a\circ b)=\delta a\circ b+(-1)^{n}a\circ\delta b\qquad\forall
a\in\mathcal{W}\otimes\Lambda_{n}\,,\quad\forall
b\in\mathcal{W}\otimes\Lambda\,.$

Upon restriction onto the differential subalgebra
$\mathcal{W}^{\xi}\otimes\Lambda$ one can write the differential $\delta$ as

(3.7) $\delta a=\frac{i}{\nu}[\omega_{ij}y^{i}dx^{j},a]\qquad\forall
a\in\mathcal{W}^{\xi}\otimes\Lambda\,.$

In order to describe the cohomology of the differential $\delta$ it is
convenient to split $\Lambda(M)$ into the direct sum

(3.8)
$\Lambda=\Lambda^{\xi}\oplus\Lambda^{\lambda}\,,\qquad\Lambda^{\xi}=\\{a\in\Lambda\,|\,i_{\xi}a=0\\}\,,\qquad\Lambda^{\lambda}=\\{\lambda
a\,|\,\forall a\in\Lambda\\}\,.$

Since the differential $\delta$ leaves invariant the subspaces $\Lambda^{\xi}$
and $\Lambda^{\lambda}$, the groups of $\delta$-cohomology

$H_{m,n}(\mathcal{W}\otimes\Lambda)=\mathrm{ker}(\delta:\mathcal{W}_{m}\otimes\Lambda_{n}\rightarrow\mathcal{W}_{m-1}\otimes\Lambda_{n+1})/\delta(\mathcal{W}_{m+1}\otimes\Lambda_{n-1})$

split as
$H_{m,n}(\mathcal{W}\otimes\Lambda)=H_{m,n}(\mathcal{W}\otimes\Lambda^{\xi})\oplus
H_{m,n}(\mathcal{W}\otimes\Lambda^{\lambda})$.

######  Proposition 3.2.

With the above notation

$\begin{array}[]{l}H_{0,0}(\mathcal{W}^{\xi}\otimes\Lambda)\simeq
H_{0,0}(\mathcal{W}^{\xi}\otimes\Lambda^{\xi})\simeq
C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]\,,\\\\[8.53581pt]
H_{0,1}(\mathcal{W}^{\xi}\otimes\Lambda)\simeq
H_{0,1}(\mathcal{W}^{\xi}\otimes\Lambda^{\lambda})\simeq
C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]\,,\\\\[8.53581pt]
H_{m,n}(\mathcal{W}^{\xi}\otimes\Lambda)=0\,,\quad\forall
m>0\quad\mbox{or}\quad n>1\,.\end{array}$

For the sake of completeness we reproduce the proof of this statement from
[8].

###### Proof.

The $(1,1)$-tensor $P^{i}_{j}=\delta^{i}_{j}-\xi^{i}\lambda_{j}$ defines an
orthogonal projection in the tangent bundle $TM$:

$P^{2}=P\,,\qquad\mathrm{Tr}P=2m\,,$
$P^{i}_{j}=\pi^{ik}\omega_{kj}\,,\quad\lambda_{i}P^{i}_{j}=0\,,\quad
P^{i}_{j}\xi^{j}=0\,,\quad
P^{k}_{i}\omega_{kj}=\omega_{ij}\,,\quad\pi^{ik}P^{j}_{k}=\pi^{ij}\,.$

Define the operator
$\delta^{\ast}:\mathcal{W}^{\xi}_{m}\otimes\Lambda_{n}\rightarrow\mathcal{W}^{\xi}_{m+1}\otimes\Lambda_{n-1}$
as

$\delta^{\ast}=y^{i}P_{i}^{j}i_{\frac{\partial}{\partial x^{j}}}\,.$

It is easy to see that $(\delta^{\ast})^{2}=0$, $\delta^{\ast}\lambda=0$, and

(3.9) $(\delta\delta^{\ast}+\delta^{\ast}\delta)a=(m+n)a-\lambda
i_{\xi}a\qquad\forall a\in\mathcal{W}^{\xi}_{m}\otimes\Lambda_{n}\,.$

Notice that $\delta^{\ast}$ is not a derivation of the $\circ$-product, but it
maps $\xi$-transverse forms of ${\mathcal{W}}^{\xi}\otimes\Lambda$ to
$\xi$-transverse. Now define the operator

(3.10)
$\delta^{-1}:\mathcal{W}^{\xi}_{m}\otimes\Lambda_{n}\rightarrow\mathcal{W}^{\xi}_{m+1}\otimes\Lambda_{n-1}$

by the relations

(3.11)
$\delta^{-1}a^{\prime}=\left\\{\begin{array}[]{cc}\frac{1}{m+n}\delta^{\ast}a^{\prime}\,,&m+n>0\,;\\\
0\,,&m=n=0\,;\end{array}\right.\qquad\delta^{-1}a^{\prime\prime}=\left\\{\begin{array}[]{cc}\frac{1}{m+n-1}\delta^{\ast}a^{\prime\prime}\,,&m+n>1\,;\\\
0\,,&m+n=1\,;\end{array}\right.$

for all $a^{\prime}\in\mathcal{W}_{(m)}\otimes\Lambda_{n}^{\xi}$ and
$a^{\prime\prime}\in\mathcal{W}_{(m)}\otimes\Lambda^{\lambda}_{n}$. Writing
$a=a^{\prime}+a^{\prime\prime}$, one can check that

(3.12) $(\delta\delta^{-1}+\delta^{-1}\delta)a=a-\Pi a\,,$

where the operator $\Pi$ acts as

$\Pi
a=a(x,0,0,\nu)+\lambda(i_{\xi}a)(x,0,0,\nu)\qquad\forall\,a=a(x,y,dx,\nu)\in\mathcal{W}^{\xi}\otimes\Lambda\,.$

Clearly, $\Pi^{2}=\Pi$. Eq. (3.12) implies that all nontrivial
$\delta$-cocycles are nested in the space $\mathrm{ker}(1-\Pi)$ spanned by the
sums $a+\lambda b$ with $a,b\in C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$. On
the other hand, the elements of the form $a+\lambda b$ are obviously
nontrivial cocycles and the proposition follows. ∎

Every contact metric connection $\nabla$ on the tangent bundle $TM$ induces a
linear connection on the associated bundle of formal Wick algebras $W$. The
latter, in turn, gives rise to an exterior covariant derivative on
$\mathcal{W}^{\xi}\otimes\Lambda$. In each coordinate chart, it is defined as

(3.13) $d_{\nabla}a=dx^{i}\wedge\nabla_{i}a=dx^{i}\wedge\left(\frac{\partial
a}{\partial x^{i}}-y^{j}\Gamma_{ji}^{k}\frac{\partial a}{\partial
y^{k}}\right)\qquad\forall a\in{\mathcal{W}}^{\xi}\otimes\Lambda\,,$

$\Gamma_{ij}^{k}=\Gamma_{ij}^{k}(x)$ being the coefficients of the contact
metric connection. Since $\nabla$ respects the Wick tensor $h$, the graded
Leibniz rule holds

(3.14) $d_{\nabla}(a\circ b)=d_{\nabla}a\circ b+(-1)^{n}a\circ
d_{\nabla}b\qquad\forall a\in{\mathcal{W}}\otimes\Lambda_{n}\,,\quad\forall
b\in{\mathcal{W}}\otimes\Lambda\,.$

Define the elements

(3.15) $T=\frac{1}{2}T_{kij}y^{k}dx^{i}\wedge
dx^{j}\in{\mathcal{W}}^{\xi}_{1}\otimes\Lambda_{2}\quad\mbox{and}\quad
R=\frac{1}{4}R_{ijkl}y^{i}y^{j}dx^{k}\wedge
dx^{l}\in{\mathcal{W}}^{\xi}_{2}\otimes\Lambda_{2}$

associated with the torsion and curvature of the contact metric connection
(2.12). With the help of Rels. (2.13) and the identity

(3.16) $g^{in}R_{nkl}^{j}+g^{jn}R_{nkl}^{i}=0$

one can find

(3.17) $(d_{\nabla}\delta+\delta d_{\nabla})a=-\frac{i}{\nu}[T,a]\,,\qquad
d_{\nabla}^{2}a=\frac{i}{\nu}[R,a]\qquad\forall
a\in{\mathcal{W}}^{\xi}\otimes\Lambda\,.$

The Jacobi identity for the double commutators of the operators $\delta$ and
$d_{\nabla}$ implies the following Bianchi identities:

(3.18) $\delta T=0\,,\qquad\delta R=d_{\nabla}T\,,\qquad d_{\nabla}R=0\,.$

Following Fedosov, we consider more general connections in the Wick bundle
$W$, which are defined by exterior covariant derivatives of the form

(3.19) $Da=d_{\nabla}a-\delta
a+\frac{i}{\nu}[r,a]=d_{\nabla}a+\frac{i}{\nu}[r-\omega_{ij}y^{i}dx^{j},a]$

for some $r\in F^{2}{\mathcal{W}}^{\xi}\otimes\Lambda_{1}$. A straightforward
calculation yields

(3.20) $D^{2}a=\frac{i}{\nu}[\Omega,a]\,,$

where

(3.21) $\Omega=\omega-\delta r+T+R+d_{\nabla}r+\frac{i}{\nu}r\circ
r\in{\mathcal{W}}^{\xi}\otimes\Lambda_{2}\,,\qquad\omega=\frac{1}{2}\omega_{ij}dx^{i}\wedge
dx^{j}\,.$

The two-form $\Omega$ is called the Weyl curvature of the Fedosov connection
$D$. According to the Bianchi identity $D\Omega=0$. A connection $D$ is called
abelian if $D^{2}=0$. This implies that the curvature $\Omega$ of an abelian
connection is a closed two-form of $\Lambda_{2}(M)[\\![\nu]\\!]$.

######  Theorem 3.3.

For a given contact metric connection $\nabla$ there exists a unique abelian
connection (3.19) with Weyl curvature $\Omega=\omega$ and $\delta^{-1}r=0\,.$

The theorem states the existence of a unique solution to the equations

(3.22) $\delta r=T+R+d_{\nabla}r+\frac{i}{\nu}r\circ
r\,,\qquad\delta^{-1}r=0\,.$

The proof of this fact essentially repeats that of [8], and we omit it here.
In practical terms, the desired solution is obtained by solving the equation

(3.23) $r=\delta^{-1}(T+R)+\delta^{-1}\left(d_{\nabla}r+\frac{i}{\nu}r\circ
r\right)\,,$

which is an obvious consequence of (3.22) and (3.12). Since
$\delta^{-1}(R+T)\in\mathcal{W}^{\xi}_{2}\otimes\Lambda_{2}$ and the operator
$\delta^{-1}$ increases the $\nu$-degree by one unit, we can solve Eq. (3.23)
by iteration. It also follows from (3.23) that $\delta^{-1}r=0$. Iterating Eq.
(3.23) yields

(3.24)
$r=\frac{1}{3}y^{i}y^{m}P^{j}_{m}T_{ijk}P^{k}_{n}dx^{n}+\frac{1}{2}y^{i}y^{j}T_{ijk}\xi^{k}\lambda+\cdots\,,$

where dots stand for elements of $F^{3}{\mathcal{W}}^{\xi}\otimes\Lambda_{1}$.
The higher-degree terms involve the curvature tensor.

Associated with an abelian connection $D$ is the algebra of flat sections
${\mathcal{W}}_{D}$. By definition,

${\mathcal{W}}_{D}=\\{a\in{\mathcal{W}}^{\xi}\;|\;Da=0\\}\,.$

Since $D$ differentiates the $\circ$-product, the flat sections form a
subalgebra in ${\mathcal{W}}^{\xi}$. The algebra ${\mathcal{W}}_{D}$ is the
central object of the Fedosov quantization. To clarify the structure of the
space ${\mathcal{W}}_{D}$ we need the following lemma of [8].

######  Lemma 3.4.

For any $a\in\mathcal{W}^{\xi}$ the equation

(3.25) $b=a+\delta^{-1}(D+\delta)b$

has a unique solution for $b\in\mathcal{W}^{\xi}$. Furthermore, $\Pi b=\Pi a$.

The statement follows from the observation that the linear operator
$\delta^{-1}(D+\delta)$ raises the $\nu$-degree by one unit. Therefore one can
solve Eq. (3.25) by iterations to get a unique solution for $b$. Since the
image of $\delta^{-1}$ depends on positive powers of $y$’s, applying the
projection $\Pi$ to both sides of (3.25) gives $\Pi b=\Pi a$. The assignment
$a\mapsto b$ defines an injective $\mathbb{C}[\\![\nu]\\!]$-linear map

(3.26)
$Q:C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]\rightarrow\mathcal{W}^{\xi}\,,$

called a quantum lift. Using (3.24) and iterating (3.25) for $a\in
C^{\infty}_{\mathbb{C}}(M)[\\![\nu]\\!]$, one can find that up to the second
order in $\nu$-degree

(3.27) $Qa=\displaystyle
a+y^{j}P^{i}_{j}\nabla_{i}a+\frac{1}{2}y^{n}y^{m}P^{i}_{n}P^{j}_{m}\nabla_{i}\nabla_{j}a+\frac{1}{6}y^{i}y^{n}P^{j}_{n}T_{ijk}\pi^{km}\nabla_{m}a+\cdots$

Again, the higher-order terms involve the curvature tensor and its covariant
derivatives.

The composition of maps

(3.28) $\Delta=i_{\xi}\Pi DQ$

defines an operator $\Delta:C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]\rightarrow
C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$. Using Eqs. (3.24) and (3.27), one
can find that

(3.29)
$\Delta=\xi-\frac{\nu}{8}T_{ijk}\xi^{k}H^{ij|nm}\nabla_{n}\nabla_{m}+\frac{\nu}{24}T_{ijk}T_{nmp}\xi^{p}H^{ij|nm}\pi^{kl}\nabla_{l}+\mathcal{O}(\nu^{2})\,,$

where

(3.30)
$\begin{array}[]{c}H^{ij|nm}=\pi^{in}g^{jm}+\pi^{jm}g^{in}+\pi^{jn}g^{im}+\pi^{im}g^{jn}\,,\\\\[8.53581pt]
H^{ij|nm}=H^{ji|nm}=H^{ij|mn}=-H^{nm|ij}\,.\end{array}$

The next theorem is completely analogous to Theorem 4.4 of Ref. [8].

######  Theorem 3.5.

Each element $a\in{\mathcal{W}}_{D}$ is completely determined by its
projection $\Pi a\in C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$ through the
formula

(3.31) $a=Q\Pi a\,.$

An element $a_{0}\in C_{\mathbb{C}}^{\infty}(M)[\\![\nu]\\!]$ is the
projection of a flat section if and only if $\Delta a_{0}=0$.

As an immediate corollary of this theorem, we conclude that
${\mathcal{W}}_{D}\simeq\ker\Delta$. The quantum lift (3.26) allows one to
pull back the $\circ$-product from ${\mathcal{W}}^{\xi}$ to
$C_{\mathbb{C}}^{\infty}[\\![\nu]\\!]$. This results in the following
$\ast$-product:

(3.32) $a\ast b=\Pi(Qa\circ Qb)\qquad\forall a,b\in
C^{\infty}_{\mathbb{C}}(M)[\\![\nu]\\!]\,.$

Unlike the $\circ$-product, the $\ast$-product is not associative. Notice,
however, that it becomes associative upon restriction to the subspace
$\ker\Delta$. By construction, the associative algebra $(\ker\Delta,\ast)$ is
isomorphic to the algebra of flat sections $({\mathcal{W}}_{D},\circ)$.

Now, it is straightforward to check that the linear and bilinear operators
(3.28) and (3.32) do satisfy all the properties assumed by Definition 3.1.
Hence, they define a Wick deformation quantization on a contact metric
manifold $(M;\phi,\xi,\lambda,g)$. Notice that $1\in{\mathcal{W}}_{D}$ and the
algebra of quantum observables $(\ker\Delta,\ast)$ is unital.

##  4\. Quantum observables

In this section, we will examine more closely the space of quantum observables
$\ker\Delta$. In particular, we will consider the implications of a contact
metric manifold being K-contact.

To study the solution space of the equation $\Delta a=0$, we expand the
operator $\Delta$ and an element $a\in
C^{\infty}_{\mathbb{C}}(M)[\\![\nu]\\!]$ in powers of $\nu$:

(4.1) $\Delta=\xi+\nu\Delta_{1}+\nu^{2}\Delta_{2}+\cdots\,,\qquad a=a_{0}+\nu
a_{1}+\nu^{2}a_{2}+\cdots\,.$

On substituting these back into the equation, we get an infinite sequence of
relations, which start as

(4.2) $\xi a_{0}=0\,,\qquad\Delta_{1}a_{0}+\xi a_{1}=0\,,\quad\ldots$

The first equation says that $a_{0}$ is a $\xi$-invariant function on $M$. As
mentioned in Sec. 2, the $\xi$-invariant functions form a closed Poisson
algebra $C^{\infty}_{\xi}(M)$ w.r.t. the Jacobi bracket (2.6). It is natural
to call $\big{(}C^{\infty}_{\xi}(M),\\{\cdot\,,\,\cdot\\}\big{)}$ the algebra
of classical observables. We thus conclude that the $\nu$-expansion of any
quantum observable starts from a classical one. The higher powers of $\nu$ are
‘quantum corrections’ we need to add to a classical observable $a_{0}$ to
promote it at the quantum level. The corrections are to be found successively
from the chain of equations (4.2). The solvability of these equations is not
ensured in advance. It depends on a particular contact metric structure and a
classical observable $a_{0}$ to be quantized. Therefore, in contrast to the
deformation quantization of Poisson manifolds, not every classical observable
on a contact manifold admits quantization. We say that a classical observable
$a_{0}\in C^{\infty}_{\xi}(M)$ is quantizable if system (4.2) admits a
solution. Otherwise, we speak about obstructions to quantization. The first
obstruction comes with the second equation in (4.2). It says that the function
$-\Delta_{1}a_{0}$, defined by a classical observable $a_{0}$, must be a
gradient of some other function $a_{1}\in C^{\infty}(M)$ along the Reeb vector
field $\xi$. Recall that $\Delta_{1}$ is a second-order differential operator
defined by Eq. (3.29). After lengthy, albeit straightforward calculations, one
can bring this operator into the form

(4.3)
$\Delta_{1}=-\frac{1}{4}(\mathcal{L}_{\xi}g^{ij})\nabla_{i}\nabla_{j}+\frac{1}{12}(\mathcal{L}_{\xi}g^{ij})(\nabla^{g}_{i}\omega_{jk})\pi^{kn}\nabla_{n}\,.$

The following proposition is now obvious.

######  Proposition 4.1.

On every K-contact manifold, $\Delta_{1}=0$.

Thus, every classical observable on a K-contact manifold can be promoted to a
quantum one up to the first order in $\nu$ (e.g. setting $a_{1}=0$). Using the
physics language, we can say that every classical observable on a K-contact
manifold is quantizable in semi-classical approximation.

In the general case, the operator $\Delta_{1}$ is different from zero, so the
equation $\xi a_{1}+\Delta_{1}a_{0}=0$ may present a real obstruction to
quantization. To better understand this obstruction it is useful to rewrite
the operator (4.3) in terms of the metric connection:

(4.4)
$\Delta_{1}=-\frac{1}{4}(\mathcal{L}_{\xi}g^{ij})\nabla^{g}_{i}\nabla^{g}_{j}-\frac{1}{4}(\mathcal{L}_{\xi}g^{ij})S_{ij}^{k}\nabla^{g}_{k}+\frac{1}{12}(\mathcal{L}_{\xi}g^{ij})(\nabla^{g}_{i}\omega_{jk})\pi^{kn}\nabla^{g}_{n}\,.$

(The operator is supposed to act on scalar functions.) Next, one can find that

(4.5)
$(\mathcal{L}_{\xi}g^{ij})S_{ij}^{k}\nabla^{g}_{k}=\frac{1}{2}(\mathcal{L}_{\xi}g^{ij})(\nabla^{g}_{i}\omega_{jk})\pi^{kn}\nabla^{g}_{n}+\frac{1}{2}(\mathcal{L}_{\xi}g^{ij})(\mathcal{L}_{\xi}g_{ij})\nabla_{\xi}^{g}\,.$

The standard formulas for the commutator of the Lie and covariant derivatives,

(4.6)
$[\mathcal{L}_{\xi},\nabla_{i}]f=0\,,\qquad[\mathcal{L}_{\xi},\nabla_{i}^{g}]\alpha_{j}=-({}^{g}\\!R^{k}_{jni}\xi^{n}+\nabla^{g}_{i}\nabla^{g}_{j}\xi^{k})\alpha_{k}\,,$

allows us to write

(4.7)
$(\mathcal{L}_{\xi}g^{ij})\nabla^{g}_{i}\nabla^{g}_{j}=[\mathcal{L}_{\xi},\mathbb{L}]-g^{ij}[\mathcal{L}_{\xi},\nabla^{g}_{i}]\nabla^{g}_{j}=[\mathcal{L}_{\xi},\mathbb{L}]+({}^{g}\\!R^{k}_{nlm}g^{nm}\xi^{l}+g^{nm}\nabla_{n}^{g}\nabla_{m}^{g}\xi^{k})\nabla_{k}^{g}\,,$

where ${}^{g}\\!R_{nlm}^{k}$ is the Riemann curvature tensor and

(4.8) $\mathbb{L}=g^{nm}\nabla_{n}^{g}\nabla_{m}^{g}$

is the Laplace operator on scalar functions. Combining all the parts together,
we finally get

(4.9)
$-4\Delta_{1}=[\mathcal{L}_{\xi},\mathbb{L}]+\frac{1}{2}(\mathcal{L}_{\xi}g_{ij})(\mathcal{L}_{\xi}g^{ij})\nabla_{\xi}^{g}+\nabla_{\zeta}\,.$

Here we introduced the vector field

(4.10)
$\zeta=\Big{(}{}^{g}\\!R^{i}{}_{nkm}g^{nm}\xi^{k}+g^{nm}\nabla^{g}_{n}\nabla^{g}_{m}\xi^{i}+\frac{1}{6}(\mathcal{L}_{\xi}g^{nm})(\nabla^{g}_{n}\omega_{mk})\pi^{ki}\Big{)}\frac{\partial}{\partial
x^{i}}\,.$

The commutator in (4.9) suggests to look for the first quantum correction in
the form

(4.11) $a_{1}=\frac{1}{4}\mathbb{L}a_{0}+\frac{1}{4}a_{1}^{\prime}\,,$

where $a^{\prime}_{1}$ satisfies the first-order differential equation

(4.12) $\xi a_{1}^{\prime}=\zeta a_{0}\,.$

Below, we state some necessary conditions for this equation to have a
solution. Recall that integral curves of the Reeb vector field $\xi$ are
called characteristics of a contact metric manifold $M$. The function $a_{0}$,
being $\xi$-invariant, must be constant on each characteristic. The most
interesting are closed characteristics, that is, periodic orbits of the Reeb
flow333The famous Weinstein’s conjecture states that every compact manifold
$M$ enjoys at least one closed characteristic, see [3, Ch. 3.4].. Let us
multiply both sides of (4.12) by $\lambda$ and then integrate over a closed
characteristic $\gamma$. As the left-hand side vanishes for an obvious reason,
we get the condition

(4.13) $\Psi_{\gamma}[a_{0}]:=\int_{\gamma}\lambda\zeta a_{0}=0\,.$

We call $\Psi_{\gamma}[a_{0}]$ the characteristic of a classical observable
$a_{0}$ (associated with a closed characteristic $\gamma\subset M$). A nonzero
number $\Psi_{\gamma}[a_{0}]$ represents thus the first obstruction to the
quantization of $a_{0}\in C^{\infty}_{\xi}(M)$.

Suppose now that the contact metric manifold $M$ is compact. Then we can endow
the space of smooth functions $C^{\infty}(M)$ with a positive-definite inner
product

(4.14) $(a,b)=\int_{M}v{a}b\,.$

Here $v$ is the canonical volume form on $M$. Let $f^{\prime}$ denote the
derivative of a smooth function $f\in C^{\infty}(\mathbb{R})$. Multiplying
both sides of Eq. (4.12) by $f^{\prime}(a_{0})$ and using the $\xi$-invariance
of $a_{0}$, we get

(4.15) $\xi(f^{\prime}(a_{0})a_{1})=\zeta f(a_{0})\,.$

Since the volume form is $\xi$-invariant, integrating the last equality over
$M$ yields

(4.16) $\Phi_{f}[a_{0}]:=\int_{M}v\zeta f(a_{0})=0\,.$

We call this functional the $f$-character of a classical observable $a_{0}$.
Again, a nonzero value of $\Phi_{f}[a_{0}]$ implies an obstruction to
quantization. Define the character of a contact metric manifold $M$ by the
formula

(4.17) $\chi:=\mathrm{div}_{v}\zeta=\nabla_{i}^{g}\zeta^{i}\,.$

The equality takes place because the canonical and Riemannian volume forms on
$M$ coincide up to a constant factor [15, Thm. 6]. Integrating by parts in
(4.16), we get

(4.18) $\Phi_{f}[a_{0}]=-(\chi,f(a_{0}))\,.$

Eqs. (4.16, 4.18) say that any smooth function of a quantizable classical
observable $a_{0}$ must be orthogonal to the character of $M$. It would be
interesting to clarify the geometric properties of the vector field $\zeta$
and implications of the conditions $\zeta=0$ or $\chi=0$.

##  5\. Sasakian manifolds and Wick property

Now is the time to explain the adjective ‘Wick’ in the name of the star
product above. First, we recall the definition of the Wick deformation
quantization on complex manifolds. Let $M$ be a complex manifold endowed with
a $\ast$-product, that is, a formal associative deformation of the pointwise
multiplication of complex-valued functions on $M$. Since the deformation is
assumed to be local, one can restrict the $\ast$-product to any open domain
$U\subset M$. Then the $\ast$-product is said to be of Wick type444Another
commonly used term is a deformation quantization with separation of variables
[10]. if for any holomorphic function $a$ and any antiholomorphic function $b$
on $U$ the following equalities hold:

(5.1) $c\ast a=c\cdot a\,,\qquad b\ast c=b\cdot c\,.$

Here $c$ is an arbitrary smooth function on $U$. Below, we will establish
similar properties for the $\ast$-product (3.32) on Sasakian manifolds. This
result is quite expected, given the analogy between Kähler and Sasakian
geometries. Less obvious is the fact that any classical observable on a
Sasakian manifold is quantizable.

Let us start with some auxiliary constructions and statements. First, we
define the following two subspaces in the $\circ$-product algebra
${\mathcal{W}}^{\xi}\otimes\Lambda$:

(5.2)
$\begin{array}[]{l}\mathrm{L}=\big{\\{}a\in{\mathcal{W}}^{\xi}\otimes\Lambda\quad\big{|}\quad(b\circ
a)|_{y=0}=0\,,\quad\forall
b\in{\mathcal{W}}^{\xi}\otimes\Lambda\big{\\}}\,,\\\\[8.53581pt]
\mathrm{R}=\big{\\{}a\in{\mathcal{W}}^{\xi}\otimes\Lambda\quad\big{|}\quad(a\circ
b)|_{y=0}=0\,,\quad\forall
b\in{\mathcal{W}}^{\xi}\otimes\Lambda\big{\\}}\,.\end{array}$

Clearly, $\mathrm{L}$ is a left ideal and $\mathrm{R}$ is a right ideal in
${\mathcal{W}}^{\xi}\otimes\Lambda$. Then the intersection
$\mathrm{A}=\mathrm{L}\cap\mathrm{R}$ is a subalgebra in
${\mathcal{W}}^{\xi}\otimes\Lambda$, while the subspaces $\mathrm{L}$ and
$\mathrm{R}$ are bi-modules over $\mathrm{A}$:

(5.3)
$\mathrm{A}\circ\mathrm{A}\subset\mathrm{A}\,,\qquad\mathrm{A}\circ\mathrm{L}\circ\mathrm{A}\subset\mathrm{L}\,,\qquad\mathrm{A}\circ\mathrm{R}\circ\mathrm{A}\subset\mathrm{R}\,.$

######  Lemma 5.1.

The subspaces $\mathrm{L}$, $\mathrm{R}$, and $\mathrm{A}$ are invariant under
the action of the operators $d_{\nabla}$ and $\delta^{-1}$.

###### Proof.

Let $a\in\mathrm{L}$ and $b\in{\mathcal{W}}^{\xi}\otimes\Lambda_{n}$. Using
the definition of the covariant differential (3.13) and the Leibniz rule
(3.14), we can write

(5.4) $\begin{array}[]{c}(b\circ d_{\nabla}a)|_{y=0}=(-1)^{n}d_{\nabla}(b\circ
a)|_{y=0}-(-1)^{n} (d_{\nabla}b\circ a)|_{y=0}\\\\[8.53581pt]
=(-1)^{n}\left.\Big{(}d(b\circ a)-dx^{i}\wedge
y^{j}\Gamma_{ji}^{k}\frac{\partial}{\partial y^{k}}(b\circ
a)\Big{)}\right|_{y=0}=(-1)^{n}d\Big{(}(b\circ
a)|_{y=0}\Big{)}=0\,.\end{array}$

In order to prove the invariance of $\mathrm{L}$ under $\delta^{-1}$ it is
enough to show that $y^{i}\cdot a\in L$ for all $a\in L$. We start with the
obvious identity

(5.5) $y^{i}\cdot a=y^{i}\circ a+X^{i}a\,,$

where

(5.6) $X^{i}=-\frac{\nu}{2}h^{ij}\frac{\partial}{\partial y^{j}}\,.$

Note that the operators $X^{i}$ differentiate the $\circ$-product. Therefore,
we may proceed as follows:

(5.7) $\begin{array}[]{rcl}b\circ(y^{i}\cdot a)|_{y=0}&=&(b\circ y^{i})\circ
a|_{y=0}+b\circ X^{i}a|_{y=0}\\\\[8.53581pt] &=&X^{i}(b\circ
a)|_{y=0}-(X^{i}b)\circ a|_{y=0}\\\\[8.53581pt] &=&y^{i}\cdot(b\circ
a)|_{y=0}-(y^{i}\circ b)\circ a|_{y=0}=0\,.\end{array}$

This proves555In fact, we proved more. Since
${\mathcal{W}}^{\xi}\otimes\Lambda$, viewed as graded commutative algebra
w.r.t. the dot product, is generated by the $y$’s, $b\cdot a\in\mathrm{L}$ for
all $a\in\mathrm{L}$ and $b\in{\mathcal{W}}^{\xi}\otimes\Lambda$. that
$\mathrm{L}$ is invariant under the action of $\delta^{-1}$. The invariance of
$\mathrm{R}$ is proved in the same way. Taken together, this means the
invariance of the algebra $\mathrm{A}=\mathrm{L}\cap\mathrm{R}$. ∎

######  Lemma 5.2.

On every contact metric manifold a contact metric curvature $R\in\mathrm{A}$.

###### Proof.

Using Rels. (2.3), (2.4), and (2.17), one can easily verify that

(5.8) $\omega_{ij}h^{in}h^{jm}=0\,,\qquad h^{ni}h^{mj}\omega_{ij}=0\,.$

The form $h$ being Hermitian, the equalities are obtained from each other by
complex conjugation. By the definition of a contact metric connection,

(5.9) $0=[\nabla_{k},\nabla_{l}]h^{ij}=R^{i}_{nkl}h^{nj}+R^{j}_{nkl}h^{in}\,.$

As a consequence,

(5.10)
$\omega_{pi}(R^{i}_{nkl}h^{nj}+R^{j}_{nkl}h^{in})h^{pm}=R_{pnkl}h^{pm}h^{nj}+R^{j}_{nkl}(\omega_{pi}h^{pm}h^{in})=R_{pnkl}h^{pm}h^{nj}=0\,.$

On the other hand,

(5.11) $(R\circ
b)|_{y=0}=\frac{1}{4}\left(\frac{\nu}{2}\right)^{2}R_{ijkl}dx^{k}\wedge
dx^{l}h^{in}h^{jm}\left.\frac{\partial^{2}b}{\partial y^{n}\partial
y^{m}}\right|_{y=0}=0\,.$

This means $R\in\mathrm{R}$. Similarly, one can see that $R\in\mathrm{L}$; and
hence, $R\in\mathrm{A}=\mathrm{L}\cap\mathrm{R}$. ∎

######  Lemma 5.3.

On every Sasakian manifold $\delta^{-1}T\in\mathrm{A}$.

###### Proof.

Starting from definition (2.8), we first compute the covariant potential
tensor

(5.12)
$\begin{array}[]{rcl}S_{ijk}:=\omega_{in}S^{n}_{jk}&=&\displaystyle-\frac{1}{2}\omega_{in}\phi^{n}_{m}\nabla^{g}_{k}\phi^{m}_{j}-\frac{1}{2}\omega_{in}\lambda_{j}\nabla_{k}^{g}\xi^{n}\\\\[8.53581pt]
&=&\displaystyle\frac{1}{2}(g_{in}-\lambda_{i}\lambda_{n})\nabla^{g}_{k}\phi^{n}_{j}+\frac{1}{2}g_{nm}\phi^{m}_{i}\lambda_{j}\nabla^{g}_{k}\xi^{n}\\\\[8.53581pt]
&=&\displaystyle\frac{1}{2}\nabla^{g}_{k}\omega_{ij}-\frac{1}{2}\lambda_{i}\lambda_{n}\nabla^{g}_{k}\phi^{n}_{j}+\frac{1}{2}\phi^{n}_{i}\lambda_{j}\nabla_{k}^{g}\lambda_{n}\\\\[8.53581pt]
&=&\displaystyle\frac{1}{2}\nabla^{g}_{k}\omega_{ij}+\frac{1}{2}\lambda_{i}\phi^{n}_{j}\nabla^{g}_{k}\lambda_{n}+\frac{1}{2}\lambda_{j}\phi^{n}_{i}\nabla_{k}^{g}\lambda_{n}\,.\end{array}$

Here we used the algebraic identities
$\omega_{ik}\phi^{k}_{j}=-g_{ij}+\lambda_{i}\lambda_{j}$ and
$\phi_{i}^{j}\lambda_{j}=0$. Since every Sasakian manifold is automatically
K-contact,

(5.13) $\nabla^{g}_{i}\lambda_{j}=\frac{1}{2}\omega_{ij}$

and we can write

(5.14)
$\begin{array}[]{rcl}S_{ijk}&=&\displaystyle\frac{1}{2}\nabla^{g}_{k}\omega_{ij}+\frac{1}{4}\lambda_{i}\phi^{n}_{j}\omega_{kn}+\frac{1}{4}\lambda_{j}\phi^{n}_{i}\omega_{kn}\\\\[8.53581pt]
&=&\displaystyle\frac{1}{2}\nabla^{g}_{k}\omega_{ij}+\frac{1}{4}(\lambda_{i}\lambda_{j}\lambda_{k}-\lambda_{i}g_{jk}-\lambda_{j}g_{ik})\,.\end{array}$

Combining the last formula with the identity

(5.15)
$\nabla^{g}_{i}\omega_{jk}+\nabla^{g}_{j}\omega_{ki}+\nabla^{g}_{k}\omega_{ij}=0\,,$

we get the following expression for the covariant torsion tensor on
$K$-contact manifolds:

(5.16)
$T_{ijk}=S_{ijk}-S_{ikj}=\frac{1}{2}\nabla_{i}^{g}\omega_{kj}-\frac{1}{4}(\lambda_{j}g_{ki}-\lambda_{k}g_{ji})\,.$

Using the defining property of Sasakian manifolds (2.18), we can simplify the
last expression to

(5.17) $T_{ijk}=\frac{1}{2}(\lambda_{k}g_{ij}-\lambda_{j}g_{ik})\,.$

Hence,

(5.18) $T=\frac{1}{2}y^{i}T_{ijk}dx^{j}\wedge
dx^{k}=\frac{1}{2}y^{i}g_{ij}dx^{j}\wedge\lambda\quad\mbox{and}\quad\delta^{-1}T=\frac{1}{4}y^{i}y^{j}(g_{ij}-\lambda_{i}\lambda_{j})\lambda\,.$

Notice that

(5.19)
$g_{ij}h^{in}h^{jm}=h^{ni}h^{mj}g_{ij}=0\,,\qquad\lambda_{i}\lambda_{j}h^{in}h^{jm}=h^{ni}h^{mj}\lambda_{i}\lambda_{j}=\xi^{n}\xi^{m}\,.$

Now we are ready to show that $\delta^{-1}T\in\mathrm{R}$. Indeed,

(5.20) $(\delta^{-1}T\circ
b)|_{y=0}=\frac{1}{4}\left(\frac{\nu}{2}\right)^{2}\lambda(g_{ij}-\lambda_{i}\lambda_{j})h^{in}h^{jm}\left.\frac{\partial^{2}b}{\partial
y^{n}\partial y^{m}}\right|_{y=0}=0$

for all $b\in{\mathcal{W}}^{\xi}\otimes\Lambda$. In the same way, one can see
that $\delta^{-1}T\in\mathrm{L}$. Therefore, $\delta^{-1}T\in\mathrm{A}$.

∎

As mentioned in Sec. 2, every Sasakian manifold carries a CR structure. This
is given by the integrable (anti)holomorphic distribution (2.14). Let $U$ be
an open domain in $M$. A complex-valued function $a\in
C_{\mathbb{C}}^{\infty}(U)$ is said to be CR-holomorphic (resp. CR-
antiholomorphic) if it is annihilated by all vector fields belonging to the
antiholomorphic distribution $\bar{\mathcal{P}}|_{U}$ (resp.
$\mathcal{P}|_{U}$). By $\mathbb{C}[\\![\nu]\\!]$-linearity, these definition
extends to the space $C_{\mathbb{C}}^{\infty}(U)[\\![\nu]\\!]$. Of particular
interest to us will be CR-(anti)holomorphic functions that are
$\xi$-invariant. It is easy to see that any $\xi$-invariant CR-holomorphic
(resp. -antiholomorphic) on $U$ function $a$ satisfies the equations

(5.21) $h^{ji}\frac{\partial a}{\partial
x^{i}}\Big{|}_{U}=0\qquad\Big{(}\mbox{resp.}\quad\frac{\partial a}{\partial
x^{i}}h^{ij}\Big{|}_{U}=0\Big{)}\,.$

######  Theorem 5.4.

Let $U$ be an open domain in a Sasakian manifold $M$. Consider the restriction
of $\ast$ and $\Delta$ on $U\subset M$. Then any $\xi$-invariant CR-
holomorphic (resp. -antiholomorphic) function $a$ on $U$ is a quantum
observable, i.e., $\Delta a=0$. Moreover, the Wick property holds

(5.22) $c\ast a=c\cdot a\qquad(\mbox{resp.}\quad a\ast c=a\cdot c)$

for any quantum observable $c$ on $U$.

###### Proof.

The quantum lift (3.26) of a CR-holomorphic function $a$ is determined by the
Fedosov connection (3.19), where $r$ is found recursively from Eq. (3.23). By
Lemmas 5.1 – 5.3, $\delta^{-1}(T+R)\in\mathrm{A}$. Since $d_{\nabla}$ and
$\delta^{-1}$ leave invariant the $\circ$-subalgebra $\mathrm{A}$, iteration
of Eq. (3.23) gives a unique solution $r$ belonging to $\mathrm{A}$. Let us
expand both $r$ and $Qa$ according to the $\nu$-degree:

(5.23) $r=r_{2}+r_{3}+r_{4}+\cdots\,,\qquad Qa=a+a_{1}+a_{2}+\cdots\,.$

The expansion for $Qa$ follows from the recurrence relations

(5.24) $a_{1}=\delta^{-1}d_{\nabla}a\,,\qquad
a_{n+1}=\delta^{-1}\Big{(}d_{\nabla}a_{n}+\frac{i}{\nu}\sum_{k=1}^{n}[r_{n-k+2},a_{k}]\Big{)}\,,\quad
n=1,2,\ldots$

We claim that $a_{1}\in\mathrm{L}$. Indeed, $a_{1}=y^{i}\nabla_{i}a$ and

(5.25) $(c\circ a_{1})|_{y=0}=\frac{\nu}{2}\frac{\partial c}{\partial
y^{j}}h^{ji}\frac{\partial a}{\partial x^{i}}\Big{|}_{y=0}=0\qquad\forall
c\in{\mathcal{W}}^{\xi}\otimes\Lambda\,,$

as $a$ is $\xi$-invariant and CR-holomorphic on $U$. Since
$r_{k}\in\mathrm{A}$ and the $\mathrm{A}$-bimodule $\mathrm{L}$ is invariant
under the action of $d_{\nabla}$ and $\delta^{-1}$, Eq. (5.24) implies that
all $a_{k}\in\mathrm{L}$.

By Theorem 3.5, $Qa$ is a flat section of ${\mathcal{W}}_{D}$ iff

(5.26) $\Delta a=i_{\xi}\Pi DQa=(D_{\xi}Qa)|_{y=0}=0\,.$

Explicitly,

(5.27) $\Delta a=\xi a-\sum_{k=1}^{\infty}\xi^{i}\frac{\partial
a_{k}}{\partial
y^{i}}\Big{|}_{y=0}+i_{\xi}\sum_{k=1}^{\infty}\Big{(}d_{\nabla}a_{k}+\frac{i}{\nu}[r,a_{k}]\Big{)}\Big{|}_{y=0}=0\,.$

Here the first summand vanishes due to the $\xi$-invariance of $a$, the first
sum is zero as all $a_{k}$’s are $\xi$-transversal, and the second sum goes to
zero as an element of $\mathrm{L}$.

It remains to prove the Wick property (5.22). Let $c$ be an arbitrary quantum
observable, then

(5.28) $c\ast a=\Pi(Qc\circ Qa)=c\cdot a+\sum_{k=1}^{\infty}(Qc\circ
a_{k})|_{y=0}=c\cdot a$

because all $a_{k}\in\mathrm{L}$. The proof for CR-antiholomorphic functions
is analogous.

∎

It follows from the theorem above that for every CR-holomorphic function $a$
and every CR-antiholomorphic function $b$ on $U\subset M$, the product $b\ast
a=b\cdot a$ is a quantum observable, that is, $\Delta(ab)=0$. On the other
hand, the products of $\xi$-invariant CR-holomorphic and -antiholomorphic
functions generate the whole space of $\xi$-invariant functions on $U$. This
means that the operator $\Delta$, being local, must be of the form
$\Delta=B\xi$ for some formal differential operator $B=1+\mathcal{O}(\nu)$.
Thus, we arrive at

######  Corollary 5.5.

On every Sasakian manifold $M$, all classical observables – the elements of
$C^{\infty}_{\xi}(M)$ – are quantizable without any quantum corrections.

This remarkable property distinguishes Sasakian manifolds from other contact
metric manifolds and conforms nicely to their intuitive understanding as odd-
dimensional analogues of Kähler manifolds.

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