Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces

Deformation quantization of almost Kähler modelsand Lagrange-Finsler spaces

Sergiu I. Vacarua�

The Fields Institute for Research in Mathematical Science, 222 College Street, SecondFloor, Toronto, Ontario M5T 3J1, Canada

�Received 12 July 2007; accepted 13 November 2007; published online 12 December 2007�

Finsler and Lagrange spaces can be equivalently represented as almost Kählermanifolds endowed with a metric compatible canonical distinguished connectionstructure generalizing the Levi-Civita connection. The goal of this paper is toperform a natural Fedosov-type deformation quantization of such geometries. Allconstructions are canonically derived for regular Lagrangians and/or fundamentalFinsler functions on tangent bundles. © 2007 American Institute ofPhysics. �DOI: 10.1063/1.2821249�

I. INTRODUCTION

An almost Kähler manifold �space� is a Riemannian manifold K= �M2n ,g� of even dimension2n and metric g together with a compatible almost complex structure J such that the symplecticform ��g�J · , · � is closed. Therefore, a space K defines a symplectic geometry with preferredmetric and almost complex structures. In a more general case of almost Hermitian manifolds H,the form � is not integrable �not closed�, see details in Refs. 1 and 2.

Let us consider a smooth n-dimensional real manifold M, its tangent bundle TM= �TM ,� ,M� with surjective projection � :TM→M, total space TM, and base M, and denote

TM˜�TM \ �0�, where 0 means the image of the null cross section of �.3 A Finsler space Fn

= �M ,F� consists of a Finsler metric �fundamental function� F�x ,y� defined as a real valued

function F :TM→R with the properties that the restriction of F to TM˜ is a function �1� positive,�2� of class C� and F is only continuous on �0�, �3� positively homogeneous of degree 1 withrespect to yi, i.e., F�x ,�y�= ��F�x ,y�, ��R, and �4� the Hessian Fgij = �1 /2��2F /�yi�yj, defined

on TM˜, is positive definite. There is a classical result by Matsumoto4 that a Finsler geometry Fn

can be modeled as an almost Kähler space FK= �TM , F��, where F�� Fg�J · , · �. for J adapted to acanonical Finsler nonlinear connection structure.

A Lagrange space Ln= �M ,L� is defined by a Lagrange fundamental function L�x ,y�, i.e., aregular real function L :TM→R, for which the Hessian

Lgij = �1/2��2L/�yi�yj �1�

is not degenerate. The concept was introduced by Kern5 and developed as a model of geometricmechanics by using methods of Finsler geometry in a number of works of Miron’s school onLagrange-Finsler geometry and generalizations.6–12 For researches interested in applications tomodern physics, we note that in our approach the review in Ref. 13 is a reference for everythingconcerning the geometry of nonholonomic manifolds and generalized Lagrange and Finsler spacesand applications to standard theories of gravity and gauge fields. It should be emphasized here thata Lagrange space Ln is a Finsler space Fn if and only if its fundamental function L is positive andtwo homogeneous with respect to variables yi, i.e., L=F2. For simplicity, in this paper we shall

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work in the bulk with Lagrange spaces, considering the Finsler ones to consist of a more particu-lar, homogeneous subclass.

This paper is motivated by the results of Oproiu14–16 who proved that any Lagrange space canbe realized as an almost Kähler model LK= �TM , L��, where L�� Lg�J · , · � for J adapted to acanonical nonlinear connection structure. This allows us to elaborate a Fedosov quantizationscheme17–20 for Lagrange and Finsler spaces. We shall develop for nonholonomic tangent bundlesthe approach from Ref. 21 in order to prove that any Lagrange and/or Finsler space can bequantized by geometric deformation methods. The geometric constructions will be performed forcanonical nonlinear and linear connections in Lagrange and Finsler geometry, see similar results inRef. 22 for Lagrange-Fedosov spaces and Fedosov nonholonomic manifolds provided with almostsymplectic connection adapted to the nonlinear connection structure.

For simplicity, we shall work on tangent bundles even when the results have a natural exten-sion for N-anholonomic manifolds, i.e., manifolds provided with nonintegrable distributions de-fining nonlinear connection structures. Such generalizations of Lagrange-Finsler methods to geo-metric and �in our cases� nonholonomic deformations will be used for elaborating certain modelsof quantum gravity for lifts of the Einstein gravity on the tangent bundle23 and almost Kählermodels of the Einstein gravity constructed on nonholonomic semi-Riemannian manifolds.24

The work is organized as follows: In Sec. II we establish our notations and remember thebasic definitions and results on the Lagrange and Finsler geometry, canonical nonlinear and dis-tinguished connections, and metric structures. We show how such geometric models on tangentbundles can be reformulated equivalently as almost Kähler nonholonomic structures. Section III isdevoted to basic constructions in nonholonomic deformation and quantization. We consider starproducts for symplectic manifolds provided with nonlinear connection structure and define ca-nonical distinguished connections adapted to the nonlinear connection and related almost complexstructure. There the Fedosov operators for Lagrange-Finsler spaces are defined. This allows us, inSec. IV, to formulate and sketch the proofs of Fedosov’s theorems for such nonholonomic tangentbundles and provide a deformation quantization of Lagrange spaces. Finally, we compute theimportant coefficient c0 of zero degree of cohomology classes of quantized Lagrange spaces.

A. Conventions

We shall use “left-up” and “left-low” labels like LD and Lg in order to emphasize that thegeometric object g is defined canonically by a regular Lagrange function L. A tensor analysis inLagrange-Finsler spaces requires a more sophisticated system of notations, see details in Ref. 13.Moreover, we use Einstein’s summation convention in local expressions. The system of notationsis a general one considered in a series of our works on nonholonomic Einstein spaces, generalizedRicci-Lagrange flows, and nonholonomic deformation quantization.

II. ALMOST KÄHLER LAGRANGE STRUCTURES

In this section, we outline briefly the almost Kähler model of Lagrange and Finslerspaces.4,6,7,13–16

A nonlinear connection �N connection� N on a tangent bundle TM can be defined by aWhitney sum �nonholonomic distribution�

TTM = hTM � vTM , �2�

given locally by a set of coefficients Nia�x ,y� defined with respect to a coordinate basis ��

=� /�u�= ��i=� /�xi ,�a=� /�ya� and its dual du�= �dxj ,dyb�. In a particular case, we get linearconnections for Ni

a=�iba �x�yb. The curvature of N connection is defined as the Neijenhuis tensor,

�ija =

�Nia

�xj −�Nj

a

�xi + Nib�Nj

a

�yb − Njb�Ni

a

�yb .

Let L�x ,y� be a regular Lagrangian with nondegenerate Lgij �1� and action integral

123509-2 Sergiu I. Vacaru J. Math. Phys. 48, 123509 �2007�


S�� = �0

1

L�x��,y��d

for yk��=dxk�� /d for x�� parametrizing smooth curves on a manifold M with � �0,1�. Wecan formulate certain very important results on geometrization of Lagrange mechanics:25

• The Euler-Lagrange equations �d /d��L /�yi�−�L /�xi=0 are equivalent to the “nonlineargeodesic” �equivalently, semispray� equations

d2xk

d2 + 2Gk�x,y� = 0,

where

Gk =1

4gkjyi �2L

�yj�xi −�L

�xjdefines the canonical N connection �for Lagrange spaces�

LNja =

�Ga�x,y��yj . �3�

• The regular Lagrangian L�x ,y� defines a canonical �Sasaki-type� metric structure on TM,

Lg = Lgij�x,y�ei� ej + Lgij�x,y�ei

� e j , �4�

where the preferred frame structure �defined linearly by LNja� is e= �ei ,ea� for

ei =�

�xi − LNia�u�

�

�ya and ea =�

�ya , �5�

and the dual frame �coframe� structure is e�= �ei ,ea�, where

ei = dxi and ea = dya + LNia�u�dxi, �6�

satisfying nontrivial nonholonomy relations

�e�,e�� = e�e� − e�e� = W�� e�, �7�

with �antisymmetric� nontrivial anholonomy coefficients Wiab =�aNi

b and Wjia =�ij

a .26

• We get a Riemann-Cartan canonical model RCL on TM of Lagrange space Ln if we choose the

canonical metrical distinguished connection D= �hD ,vD�= �Lijk , Cjc

i � �in brief, d connection,which is a linear connection preserving under parallelism the splitting �2��,

�ij = �i

j�e� = Lijke

k + Cjci ec,

for Lijk= La

bk , Cjci = Cbc

a in �ab= �a

b�e�= Labke

k+ Cbca ec, and

Lijk = 1

2gih�ekgjh + e jgkh − ehgjk�, Cabc = 1

2gae�ebgec + ecgeb − eegbc� , �8�

which are just the generalized Christoffel indices.27 We note that RCL contains a nonholo-nomically induced torsion structure defined by 2-forms

�i = Cijce

i ∧ ec and �a = − 12�ij

a ei ∧ ej + �ebNia − La

bi�ei ∧ eb �9�

computed from Cartan’s structure equations

123509-3 Deformation quantization and Lagrange-Finsler spaces J. Math. Phys. 48, 123509 �2007�


dei − ek ∧ �ik = − �i, dea − eb ∧ �a

b = − �a,

�10�d�i

j − �kj ∧ �i

k = − � ji ,

in which the curvature 2-form is denoted by � ji, see explicit formulas for coefficients in Refs.

6, 7, and 13 and formula �22�.• In principle, we can work also with the torsionless Levi-Civita connection L�, constructed for

the same metric �4� which with respect to N-adapted bases �5� and �6� is given by the samecoefficients �8� but subjected to the condition that they must solve the structure equations�10� with ��i= ��a=0 and � j

i � �� ji. This provides a Riemann-type geometrization RL on

TM of Lagrange space Ln. We note that � does not preserve under parallelism theN-connection splitting �2�, i.e., it is not adapted to the N-connection structure defined canoni-cally by a regular Lagrangian L. From a formal point of view, we can work equivalently with

both types of connections because D and L� are uniquely defined by the same data �Lg andLNj

a�, i.e., by the same fundamental function L, in metric compatible forms, D Lg=hD Lg=vD Lg=0 and L� Lg=0. Even D has a nonzero torsion; it is induced canonically by thesame Lg and LNj

a.

The canonical N connection LNja �3� induces an almost Kähler structure defined canonically by

a regular L�x ,y� �Refs. 14–16� �in this paper, we use the constructions from Refs. 6, 7, and 13�.We introduce an almost complex structure for Ln as a linear operator J acting on the vectors onTM the following formulas:

J�ei� = − ei and J�ei� = ei,

where the superposition J �J=−I, for I being the unity matrix. The operator J reduces to acomplex structure J if and only if the distribution �2� is integrable.

A regular Lagrangian L�x ,y� induces a canonical 1-form

L =1

2

�L

�yiei

and metric Lg �4� induces a canonical 2-form

L� = Lgij�x,y�ei ∧ ej �11�

associating to J the formulas L��X ,Y�� Lg�JX ,Y� for any vectors X and Y on TM decomposedwith respect to an N-adapted basis �5�.

We can prove the following results:

�1� A regular L defines on TM an almost Hermitian �symplectic� structure L� for which dL = L�.

�2� The canonical N connection LNja �3� and its curvature have the properties

�ijk

Lgl�iL� jk

l � = 0, Lgij�k − Lgik�j = 0, ekLgij − ej

Lgik = 0,

where �ijk� means symmetrization of indices and

Lgij�k = ekLgij − LBik

s Lgsj − LBjks Lgis

for LBiks =ei

LNks, which means that the almost Hermitian model of a Lagrange space Ln is an almost

Kähler manifold with dL�=0. We conclude that the triad K2n= �TM , Lg ,J� defines an almostKähler space �see details in Refs. 14–16�.

Proofs of properties �1� and �2� follow from the computation



dL� =1

6 ��ijk�

LgisL� jk

s ei ∧ ej ∧ ek +1

2�Lgij�k − Lgik�j�e

i ∧ ej ∧ ek +1

2�ek

Lgij − eiLgkj�e

k ∧ ei ∧ ej .

The next step is to define the concept of almost Kähler d connection �D, which is compatibleboth with the almost Kähler �L� ,J� and N-connection structures LN and satisfies the conditions

�DXLg = 0 and �DXJ = 0

for any vector X=Xiei+Xaea. By a straightforward computation, we prove �see details in Refs. 6and 7� the following.

Theorem 1: The canonical d-connection ��= �La

bk , Cbca � with coefficients (8) defines also a

(unique) canonical almost Kähler d connection �D= D for which, with respect to N-adapted

frames (5) and (6), the coefficients Tjki =0, torsion vanishes on hTM, and Tbc

a =0, torsion vanishes

on vTM, but there are cross nonzero coefficients of type �9�, Tjci = Ci

jc, Tija =�ij

a , and Tiba =ebNi

a

− Lbia .There are two important particular cases: If L=F2, for a Finsler space, we get an almost

Kähler model of Finsler space,4 when �D= D transforms into the so-called Cartan-Finslerconnection.28 We get a Kählerian model of a Lagrange or Finsler space if the respective almostcomplex structure J is integrable.

III. NONHOLONOMIC DEFORMATIONS AND QUANTIZATION

The geometry of Lagrange-Fedosov manifolds was investigated in Ref. 22. The aim of thissection is to provide a nonholonomic modification of Fedosov’s constructions in order to performa geometric quantization of Lagrange �in particular, Finsler� spaces provided with canonical metricand nonlinear and linear connection structures defined by respective fundamental Lagrange �Fin-sler� functions, see next section. We shall use the approach to Fedosov quantization of geometrieswith arbitrary metric compatible affine connections on almost Kähler manifolds and related sym-plectic structures for a manifold M elaborated in Ref. 21. We shall redefine the constructions fromM and TM, respectively, on TM and TTM endowed with canonical N connection, metric, sym-plectic, and almost Kähler structures uniquely defined by fundamental Lagrange �Finsler� func-tions.

A. Star products for symplectic manifolds

Let us denote by C��V��v�� the spaces of formal series in variable v with coefficients fromC��V� on a Poisson manifold �V , �· , · ��. Following Refs. 29–31, a deformation quantization is anassociative algebra structure on C��V��v�� with a v-linear and v-adically continuous star product

1f � 2f = �r=0

�

rC�1f , 2f�vr, �12�

where rC ,r�0, are bilinear operators on C��V� with 0C�1f , 2f�= 1f 2f and 1C�1f , 2f�− 1C�2f , 1f�= i�1f , 2f�, with i being the complex unity. Following conventions from Ref. 13, we use “up” and“low” left labels which are convenient to be introduced on Finsler-like spaces in order to notcreate confusions with a number of “horizontal” and “vertical” indices and labels which must bedistinguished if the manifolds are provided with N-connection structure. We note that, in ourfurther constructions, the manifold V will be a tangent bundle, V=TM, or a nonhlonomic manifoldV=V �for instance, a Riemann-Cartan manifold� provided with a nonholonomic distribution de-fining an N connection.

If all operators rC ,r�0 are bidifferential, a corresponding star product � is called differential.We can define different star products on a �V , �· , · ��. Two differential star products � and �� areequivalent if there is an isomorphism of algebras A : �C��V��v�� , � �→ �C��V��v�� ,��, whereA=�r�1

�rAvr, for 0A being the identity operator and rA being differential operators on C��V�.



For a particular case of Poisson manifolds, when �V ,�� is a symplectic manifold, each differ-ential star product � on V belongs to its characteristic class cl�� 1 / iv��+H2�V ,C��v��, whereC is the field of complex numbers, and �1 / iv��+H2�V ,C��v�� is an affine vector space, seedetails in Refs. 19, 32, and 33. For symplectic structures, the equivalence classes of differentialstar products on �V ,�� can be bijectively parametrized by the elements of the corresponding affinevector space using the map �→cl��.

The bibliography on existence proofs, methods, and descriptions of equivalent classes for thefirst examples of star products �Moyal-Weyl and Wick star products, asymptotic expansions witha numerical parameter, Planck constant, �→0, Berezin-Toeplitz deformation quantization� is out-lined in Refs. 21 and 34–38. A very important conclusion following from the above-mentionedworks is that a natural deformation quantization can be constructed on an arbitrary compact almostKähler manifold. The question of existence of deformation quantization and corresponding geo-metric formalism on general Poisson manifolds were solved in Kontsevich’s work.39,40 In thiswork, we are interested in describing this deformation quantization for Lagrange-Finsler spacesdefined by corresponding nonholonomic structures on arbitrary almost Kähler spaces.

B. Canonical d connections and complex structures

The deformation quantization using Fedosov’s machinery can be also constructed for a naturalclass of affine connections, in general, considered by Yano.41 For Lagrange-Finsler spaces andtheir almost Kähler models, we work with the canonical d connection �8�: the constructions will beredefined with respect to N-adapted bases in a form when the results from Ref. 21 will hold truefor nontrivial nonholonomic structures.

Let K2n= �TM˜ , Lg ,J� define an almost Kähler model of Lagrange space with canonical d

connection �D= D. For a chart U�TM, we set the local coordinates u�= �xi ,ya� and parametrizeJ�ei�=−ei and J�ei�=ei for e�= �ei ,ea� and denote

J�e�� = J��e��, or J�ei� = Ji

i�ei� and J�ea� = Jaa�ea�. �13�

We also write

�� = ��e�,e��, or �ij = ��ei,e j� and �ab = ��ea,eb� , �14�

corresponding to metric �4� with

Lg�� = Lg�e�,e��, or Lgij = Lg�ei,e j� and Lgab = Lg�ea,eb� . �15�

Using the inverse matrices corresponding to those considered above, we can write

J�� = Lg�� = Lg��

or in horizontal and vertical �in brief, h and v� components,

Jii� = Lgij�

ji� = Lgi�j�� j�i and Jaa� = Lgab�ba� = Lga�b��b�a.

We can define J��

as the inverse to J��. In our further considerations, we shall omit decompo-

sitions into h and v components if it would be possible to write formulas in a more compactifiedform with Greek indices not creating ambiguities in distinguishing the nonholonomicN-connection structure.

The Nijenhuis tensor � for the complex structure J on TM provided with N connection N isdefined in the form

��X,Y� � �JX,JY� − J�JX,Y� − J�X,Y� − �X,Y� ,

where X and Y are vector fields on TM.42



Let D= �� be any metric compatible d connection on TM �it is any affine connection

preserving the splitting �2� and satisfying D�g�=0 for a given metric structure g.� A componentcalculus with respect to N-adapted bases �5� and �6� for ��e� ,e��=��

� e� results in

�� = 4T��

� , �16�

where T�� is the torsion of ��

� .43 For the case of Lagrange spaces with canonical d-connection D�8�, the nontrivial components T��

� are given by nonzero components of 2-forms �9� being definedcompletely by Lgij and LNi

a, which in turn are generated by a regular L�x ,y�.Proposition 1: Any metric compatible d connection D with the torsion given by formula (16)

respects the symplectic form ��X ,Y��g�JX ,Y� and therefore the complex structure J.Proof: It consists in a straightforward verification of conditions DXg=0 and DXJ=0 which is

an N-adapted calculus with respect to �5� and �6�, see similar computations proving Proposition2.1 in Ref. 21. In our case, we work not with affine metric compatible connections but withrespective d connections. �

As a consequence of Theorem 1 and Proposition 1, we have the following.

Corollary 1: The unique metric compatible canonical d connection ��= �La

bk , Cbca � (8), with

torsion components Tjki =0, Tbc

a =0, Tjki = Ci

jc, Tija =�ij

a , and Tiba =ebNi

a− Labi computed with respect

to N-adapted bases (5) and (6), satisfies the formula �� =4T��

� and respects the canonicalsymplectic structure L� (11) constructed for the canonical N connection LNj

a (3) and metric Lg (4).In a particular case, for L=F2, similar results hold true for the Cartan connection on Finslerspaces and respective canonical Finsler symplectic, metric, and N-connection structures.

We conclude that by geometrizing a Lagrange or Finsler space in terms of geometric objectsof a nonholonomic almost Kähler manifold, we can perform directly a natural deformation quan-tization by adapting the constructions from Ref. 21 to the canonical N connection and for respec-tive canonical geometrical objects �such as metric, symplectic form, d connection, induced byfundamental Lagrange or Finsler functions�.

Our further considerations will perform a generalization to the nonholonomic tangent bundlesTM of the results and methods elaborated in the above-mentioned reference for usual manifolds Mendowed with metric compatible affine connections and respective almost Kähler structures. Inother turns, we shall emphasize the constructions only for the case of canonical Lagrange orFinsler geometric objects. For simplicity, we shall omit details for local computations and proofsif they will be certain “N-adapted” Lagrange-Finsler analogs of formulas and results obtained inRefs. 21 and 41 or Refs. 17–20.

C. Fedosov operators for Lagrange-Finsler spaces

In this section, we modify Fedosov’s constructions to provide all data necessary for deforma-tion quantization of almost Kähler models of Lagrange-Finsler spaces.

On TM endowed with canonical Lagrange structures, we introduce the tensor L�� L��

− iLg��, see related formulas �4�, �11�, �14�, and �15�. The local coordinates on TM are param-etrized in the form u= �u�� and the local coordinates on TuTM are labeled �u ,z�= �u� ,z��, where z�

are the second order fiber coordinates. We shall work with the formal series

�17�

to emphasize what we perform our constructions for spaces provided with N-connection structure�.We use the formal Wick product on Wu for two elements a and b defined by formal series of type�17�,



a � b�z� � expiv2

L�� 2

�z��z�1�� a�z�b�z�1��z=z�1�

. �18�

It is possible to construct a nonholonomic bundle W=�uWu of formal Wick algebras definedas a union of algebras Wu distinguished by the N-connection structure, see Refs. 44 and 45 onsuch “d-algebras,” for instance, on gauge and spinor field geometries adapted to N-connectionstructures. The fiber product �18� can be trivially extended to the space of W-valued N-adapteddifferential forms LW � � by means of the usual exterior product of the scalar forms �, where LWdenotes the sheaf of smooth sections of W �we put the left label L in order to emphasize that theconstructions are adapted to the canonical N-connection structure induced by a regular Lagrang-ian�. There is a standard grading on �, denoted dega. It is possible to introduce gradings degv,degs, dega on LW � � defined on homogeneous elements v, z�, e� as follows: degv�v�=1,degs�z��=1, dega�e��=1, and all other gradings of the elements v, z�, e� are set to zero. In thiscase, the product � from �18� on LW � � is bigraded, we write with respect to the grading deg=2 degv+degs and the grading dega, see also conventions from Ref. 21.

The canonical d connection LD= �L�� can be extended to an operator on LW � � following

the formula

LD�a � �� e��a� − u�L�� ze��a�� e� ∧ �� + a � d� , �19�

where e� and e� are defined, respectively, by formulas �5� and �6� and ze� is a similar to e� onN-anholonomic fibers of TTM, depending on z variables �for holonomic second order fibers, wecan take ze�=� /�z��. For second order mechanical or Finsler models, ze� can be constructedcanonically from higher order Lagrangians and respective semi-spray configurations and Nconnections.8–12 In superstring theory and nonholonomic �super�gravity and higher order spinorstructures, such effective higher order N connections have to be defined from �super�vielbeinconfigurations.46,47 It should be noted that the operator �19� can be similarly defined for arbitrarymetric compatible d connection D= ��

� � and arbitrary N-connection structures on TTM, but forpurposes of this paper, we consider only the case of geometric objects induced canonically by a

fundamental function L or F. Using formulas �18� and �19�, we can show that LD is an N-adapteddega-graded derivation of the distinguished algebra �LW � � , � �, in brief called d-algebra.

Now, we can introduce on LW � � the Fedosov operators L� and L�−1 �we put additional leftlabels in order to emphasize that in this work they are completely generated by a regular Lagrangeor Finsler canonical structure on TM�:

L��a� = e� ∧ ze��a� ,

L�−1�a� = �i/�p + q��z�e��a� if p + q � 0

0 if p = q = 0,�

where a� LW � � is homogeneous with respect to the gradings degs and dega with degs�a�= p anddega�a�=q. We get the formula

a = �L�L�−1 + L�−1L� + ��a� ,

where a��a� is the projection on the �degs ,dega�-bihomogeneous part of a of degree zero;degs�a�=dega�a�=0. We can verify that L� is also a dega-graded derivation of the d-algebra�LW � � , � �.

The Lagrange canonical d connection LD �8� on TM induces respective torsion and curvatureon LW � �,



T �z�

2L��T��

�u�e� ∧ e� �20�

and

R �z�z�

4L��R

��u�e� ∧ e�, �21�

where the torsion T�� 9� has nontrivial components Tjk

i = Cijc, Tij

a =�ija , and Tib

a = ebNi

a− Labi and the

curvature R�� has nontrivial components

Rihjk = ekL

ihj − e jL

ihk + Lm

hjLimk − Lm

hkLimj − Ci

ha�akj ,

�22�Pi

jka = eaLijk − DkC

ija, Sa

bcd = edCabc − ecC

abd + Ce

bcCa

ed − CebdCa

ec,

all computed in a form when the structure equations �10� are solved.Using the formulas �17� and �18� and the identity

L��R�� = L��R

��, �23�

we prove the formulas

�LD, L�� =i

vadWick�T� and LD2 = −

i

vadWick�R� , �24�

where �·,·� is the dega-graded commutator of endomorphisms of LW � � and adWick is defined viathe dega-graded commutator in �LW � � , � �.48

IV. FEDOSOV QUANTIZATION OF LAGRANGE SPACES

We generalize the standard statements of Fedosov’s theory for the case of Lagrange-Finslerspaces provided with canonical N-connection and d-connection structures. The class c0 of thedeformation quantization of Lagrange geometry is calculated.

A. Fedosov’s theorems for Lagrange-Finsler spaces

Using the formalism of Fedosov operators on Lagrange spaces, we formulate and sketch theproof of two theorems generalizing similar ones to the case of N-connection and metric compat-ible d-connection geometries on TM, induced by fundamental Lagrange �Finsler� functions.

Let us denote the deg-homogeneous component of degree k of an element a� LW � � by a�k�.

Theorem 2: For any regular Lagrangian L on TM˜, there is a flat canonical Fedosov dconnection

LD � − L� + LD −i

vadWick�r�

satisfying the condition LD2=0, where the unique element r� LW � �, dega�r�=1, L�−1r=0,solves the equation

L�r = T + R + LDr −i

vr � r

and this element can be computed recursively with respect to the total degree deg as follows:

r�0� = r�1� = 0, r�2� = L�−1T ,



r�3� = L�−1R + LDr�2� −i

vr�2� � r�2� ,

r�k+3� = L�−1LDr�k+2� −i

v�l=0

k

r�l+2� � r�l+2�, k � 1.

Proof: We sketch the idea of the proof which is similar to the standard Fedosov constructions butN adapted. By induction, we use the identities

L�T = 0 and L�R = LDT .

In Ref. 21, these identities were proved for arbitrary affine connections with torsion and almostKähler structures on M. In our case, we work with a more particular class of geometric objects,induced canonically from Ln, on TM. In another turn, the constructions are generalized to non-holonomic bundles. �

We note that the canonical Fedosov d connection LD is a dega-graded derivation of the algebra

�LW � � , � �. This means that LWD�ker�LD�� LW is an N-adapted subalgebra of �LW , � �.The next theorem gives a rule on how to define and compute the star product induced by a

regular Lagrangian.Theorem 3: A star product on the canonical almost Kähler model of Lagrange (Finsler)

spaces K2n= �TM˜ , Lg ,J� is defined on C��TM˜��v�� by the formula

1f � 2f � ��1f�� 2f�� ,

where the projection �: LWD→C��TM˜��v�� onto the part of degs degree zero is a bijection and

the inverse map : C��TM˜��v��→ LWD can be calculated recursively with respect to the total

degree deg,

�f��0� = f

and, for k�0,

�f��k+1� = L�−1LD�f��k� −i

v�l=0

k

adWick�r�l+2��f��k−l�� .

Proof: We note that the connection LD and its almost Kähler version defined by Proposition 1 andTheorems 1 and 2 in the case of almost Kähler manifolds is a special N-adapted case of the starproduct constructed in Ref. 49. �

The statements of the above presented Fedosov theorems generalized for Lagrange-Finsler

spaces can be extended for arbitrary metric compatible d connections on TM˜. For Finsler spaces,we can use the so-called Miron procedure of computing all metric compatible d connections for agiven metric g, see Refs. 6 and 7 �from a formal point of view, we shall have the same formulaswithout “hats” and L labels but with arbitrary d torsions and corresponding curvatures�. It shouldbe noted that there is also the so-called Kawaguchi metrization procedure, which allows to workwith metric noncompatible d connections, described in detail in the above-mentioned Miron andAnastasiei monographs. In Ref. 50, such constructions were elaborated for nonholonomic mani-folds with the aim of applying Finsler methods in modern gravity theories.

B. Cohomology classes of quantized Lagrange spaces

It follows from the results obtained in Ref. 38 that the characteristic class of the star productfrom Ref. 49 is �1 / iv��− �1 /2i��, where � is the canonical class for an underlying Kählermanifold. This canonical class can be defined for any almost complex manifold. In this section, we



calculate the crucial part of the characteristic class cl of the star product � which we haveconstructed in Theorem 3 for an almost Kähler model of Lagrange �Finsler� spaces, i.e., we shallcompute the coefficient c0 at the zeroth degree of v. Only the coefficient c0�� of the classcl��= �1 / iv��L��+c0��+¯ can not be recovered from Deligne’s intrinsic class.32 Here we alsonote that the cohomology class of the formal Kähler form parametrizing a quantization withseparation of variables on a Kähler manifold differs from the characteristic class of this quantiza-tion only in the coefficient c0 as it is proved in Ref. 38.

Let us recall the rigorous definition of the class c0 of a star product �12� �see details, forinstance, in Ref. 20� adapting the constructions for tangent bundles provided with theN-connection structure. One denotes by f� the corresponding Hamiltonian vector field correspond-ing to a function f �C��TM� on a symplectic tangent bundle �TM ,�� and considers the antisym-metric part

− C�1f , 2f� �12 �C�1f , 2f� − C�2f , 1f��

of bilinear operator C�1f , 2f�. A star product �12� is normalized if

1C�1f , 2f� =i

2�1f , 2f� ,

where �·,·� is the Poisson bracket. For normalized �, the bilinear operator 2−C is a de Rham–

Chevalley 2-cocycle. In this case, there is a unique closed 2-form L�, induced by a regularLagrangian L, such that

2C�1f , 2f� = 12

L�� f1�, f2�� 25�

for all 2f , 2f �C��TM�. The class c0 of a normalized star product � is stated as the equivalenceclass c0�� L��.

A fiberwise equivalence operator on LW can be defined by the formula

LG � exp�− vL�� ,

where

L� = 18

Lg��ze�ze� + ze�

ze��

for nonholonomic configurations on the second order fibers on TTM, or

L� =1

4Lg�� 2

�z��z�

if we elaborate a model with trivial N connection for the second order fibers on TTM. We cancheck directly the formulas

�LD, L�� = �LD, LG� = 0 and �L�, L�� = �L�, LG� = 0, �26�

which allows us to define a fiberwise star product on LW,

a��b � LG�LG−1a � LG−1b� ,

which is the Weyl star product

a��b�z� = exp iv4

L��e�ze� − ze�e�� = exp iv

2L��e�

�

�z� −�

�z�e�for holonomic 2d order fibers

The next step is to push forward the Fedosov d connection LD from Theorem 2 using theformulas in �26�. We get a new canonical d-connection operator,



LD� = LGLDLG−1 = L� + LD −i

vadWeyl�r�� ,

where r�=LGr and adWeyl is calculated with respect to the ��-commutator.For symplectic manifolds, it is well known that each star product is equivalent to a normalized

one. The class c0�� of a star product � is defined as the cohomology class c0�� of an equivalentnormalized star product ��. We have first to construct an equivalent normalized star product inorder to calculate the class c0�� for � from Theorem 3. This procedure is described in detail inSec. IV of Ref. 21 for arbitrary affine metric compatible connection on a manifold M. In our case,those formulas have to be redefined with respect to N-adapted bases and canonical d-connection,N-connection, and metric structures. For simplicity, we omit in this work such tedious but trivialgeneralizations but present only the most important formulas and definitions.

A straightforward computation of 2C from �25�, using statements of Theorem 2, results in aproof of the following lemma.

Lemma 1: The unique 2-form L� can be expressed in the form

L� = −i

8J

��R��e� ∧ e� − iL� ,

where the exact N-adapted 1-form L�=dL� for

L� = 16J

��T��e�,

with nontrivial components of curvature and torsion defined by the canonical d-connection com-puted following formulas (9) and (22).

For trivial N-connection structures and arbitrary metric and metric compatible affine connec-tions, Lemma 1 is equivalent to statements of Lemma 4.1 from Ref. 21, in our case, redefined forRiemann-Cartan geometries modeled on TM. We reformulated the results in a form when gener-alizations for arbitrary metric g compatible d-connection and N-connection structures, D and N onTM, can be performed following the formal rule of omitting hats and L-labels.

Let us recall the definition of the canonical class � of an almost complex manifold �M ,J� andredefine it for NTTM =hTM � vTM �2� stating an N-connection structure N. The distinguishedcomplexification of such second order tangent bundles is introduced in the form

TC�NTTM� = TC�hTM� � TC�vTM� .

For such nonholonomic bundles, the class N� is the first Chern class of the distributionsTC��NTTM�=TC��hTM� � TC��vTM� of couples of vectors of type �1,0� both for the h and v parts.Our aim is to calculate the canonical class L� �we put the label L for the constructions canonicallydefined by a regular Lagrangian L� for the almost Kähler model of a Lagrange space Ln. We take

the canonical d-connection LD that was used for constructing � and consider h and v projections,

h� = 12 �idh − iJh� and v� = 1

2 �idv − iJv� ,

where Idh and Idv are the respective identity operators and Jh and Jv are defined by the formulasin �13�, which are projection operators onto corresponding �1,0� subspaces. It follows from �23�that Tr R=Tr��

��=0, see �10�. The matrix �h� ,v��R�h� ,v��T, where �¯�T means transposi-

tion, is the curvature matrix of the N-adapted restriction of the connection LD to TC��NTTM�. Now,we can compute the Chern-Weyl form

L� = − i Tr��h�,v��R�h�,v��T� = − i Tr��h�,v��R� = −1

4J

��R��e� ∧ e�

to be closed. By definition, the canonical class is L�� L��. The proof of the following theoremfollows from Lemma 1 and the above presented considerations.



Theorem 4: The zero-degree cohomology coefficient c0�� for the almost Kähler model ofLagrange space is computed,

c0�� = − �1/2i�L� ,

where the value L� is canonically defined by a regular Lagrangian L�u�.Finally we note that the formula from this theorem can be directly applied for the Cartan

connection in Finsler geometry with L=F2. In our partner works,23,24 we consider its extensions,respectively, for generalized Lagrange spaces or canonical nonholonomic lifts of semi-Riemanianmetrics on TM and nonholonomic deformations of the Einstein gravity into almost Kähler modelson nonholonomic manifolds.

ACKNOWLEDGMENTS

The author is grateful to Professor Vasile Oproiu for very important discussions and referenceson almost Kähler models of Lagrange spaces. He also thanks the referee for hard work and veryuseful suggestions. This work belongs to a research program for a visitor at Fields Institute.

1 S. Goldberg, Proc. Am. Math. Soc. 21, 96 �1969�.2 A. Gray, Tohoku Math. J. 28, 233 �1976�.3 For coordinates u�= �xi ,ya� on TM, when indices i , j , . . . ,a , . . . ,b , . . . run values 1 ,2 , . . . ,n, we get also coordinates on

TM˜ if not all fiber coordinates ya vanish; in brief, we shall write u= �x ,y�.4 M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces �Kaisisha, Shigaken, Japan, 1986�.5 J. Kern, Arch. Math. 25, 438 �1974�.6 R. Miron and M. Anastasiei, Vector Bundles and Lagrange Spaces with Applications to Relativity �Geometry Balkan,Bukharest, 1997�.

7 R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications �Kluwer Academic, Dordrecht,1994�, FTPH No. 59.

8 R. Miron and Gh. Atanasiu, Seminarul de Mecanica �unpublished�, Vol. 40, p. 1; Revue Roumaine de MathematiquesPures et Appliquee, 1996, Vol. 41, pp. 3, 4, 205, 237, and 251.

9 R. Miron and Gh. Atanasiu, Lagrange Geometry, Finsler Spaces and Nois Applied in Biology and Physics, Mathematicaland Computer Modelling Vol. 20 �Pergamon, Oxford, 1994�, p. 41.

10 R. Miron, The Geometry of Higher-Order Lagrange Spaces, Application to Mechanics and Physics �Kluwer Academic,Boston, 1997�, FTPH No. 82.

11 R. Miron, The Geometry of Higher-Order Finsler Spaces �Hadronic, Palm Harbor, 1998�.12 I. Bucataru and R. Miron, e-print arXiv:0705.3689.13 S. Vacaru, e-print arXiv:0707.1524.14 V. Oproiu, Rendiconti Seminario Facoltà Scienze Università Cagliari 55, 1 �1985�.15 V. Oproiu, An. St. Univ. “Al. I. Cuza” Iaşi, Romania, XXXIII, s. Ia. f. 1 �1987�.16 V. Oproiu, Math. J. Toyama Univ. 22, 1 �1999�.17 B. Fedosov, Funkc. Anal. Priloz. 25, 1984 �1990�.18 B. Fedosov, J. Diff. Geom. 40, 213 �1994�.19 B. Fedosov, Deformation Quantization and Index Theory �Akademie, Berlin, 1996�.20 I. Gelfand, V. Retakh, and M. Shubin, Adv. Math. 136, 104 �1998�.21 A. Karabegov and M. Schlichenmaier, Lett. Math. Phys. 57, 135 �2001�.22 F. Etayo, R. Santamaría, and S. Vacaru, J. Math. Phys. 46, 032901 �2005�.23 S. Vacaru, e-print arXiv:0707.1526.24 S. Vacaru, e-print arXiv:0707.1667.25 Proofs consist of straightforward computations.26 For simplicity, in this work, we shall omit left labels L in formulas if it will not result in ambiguities; we shall use

boldface indices for spaces and objects provided or adapted to an N-connection structure.27 We contract horizontal and vertical indices following the rule i=1 is a=n+1; i=2 is a=n+2; . . .; i=n is a=n+n�.28 E. Cartan, Les Espaces de Finsler �Hermann, Paris, 1935�.29 F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Lett. Math. Phys. 1, 521 �1977�.30 F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Ann. Phys. �N.Y.� 111, 61 �1978�.31 F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Ann. Phys. �N.Y.� 111, 111 �1978�.32 P. Deligne and P. Lecompte, Selecta Math., New Ser. 1, 667 �1995�.33 R. Nest and B. Tsygan, Adv. Math. 11, 223 �1995�.34 F. Berezin, Math. USSR, Izv. 8, 1109 �1974�.35 M. Bordermann, E. Meinrenken, and M. Schlichenmaier, Commun. Math. Phys. 165, 281 �1994�.36 V. Dolgushev, S. Lyakhovich, and A. Sharapov, Nucl. Phys. B 606, 647 �2001�.37 A. Karabegov, Commun. Math. Phys. 180, 745 �1996�.38 A. Karabegov, Lett. Math. Phys. 43, 347 �1998�.39 M. Kontsevich, Lett. Math. Phys. 66, 157 �2003�.



40 M. Kontsevich, Lett. Math. Phys. 48, 35 �1999�.41 K. Yano, Differential Geometry on Complex and Almost Complex Spaces �Pergamon, New York, 1965�.42 We chose a definition of this tensor as in Ref. 6 which is with minus sign compared to the definition used in Ref. 21; we

also use a different letter for this tensor, like for the N-connection curvature, because in our case, the symbol N is usedfor N connections.

43 This formula is a nonholonomic analog, for our conventions, with inverse sign, of formula �2.9� from Ref. 21.44 S. Vacaru, J. Math. Phys. 46, 042503 �2005�.45 S. Vacaru, J. Math. Phys. 47, 093504 �2006�.46 S. Vacaru, Nucl. Phys. B 434, 590 �1997�.47 S. Vacaru, J. High Energy Phys. 09, 011 �1998�.48 It should be noted that formulas �20�–�24� can be written for any metric g and metric compatible d connection D, Dg

=0, on TM, provided with arbitrary N connection N �we have to omit hats and labels L�. It is a more sophisticatedproblem to define such constructions for Finsler geometries with the so-called Chern connection which are metricnoncompatible �Ref. 51�. For applications in standard models of physics, we chose the variants of Lagrange-Finslerspaces defined by metric compatible d connections, see discussion in Ref. 13.

49 M. Bordemann and St. Waldmann, Lett. Math. Phys. 41, 243 �1997�.50 S. Vacaru, P. Stavrinos, E. Gaburov, and D. Gonţa, Clifford and Riemann-Finsler Structures in Geometric Mechanics and

Gravity, Selected Works, Differential Geometry—Dynamical Systems Monograph 7 �Geometry Balkan Press, Bucharest,2006� �www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023�.

51 D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics Vol.200 �Springer-Verlag, Berlin, 2000�.



Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces

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