Branes and Quantization for an A-Model Complexification of Einstein Gravity in Almost Kahler Variables
aa r X i v : . [ m a t h - ph ] A p r Branes and Quantization for an A–ModelComplexification of Einstein Gravityin Almost K¨ahler Variables
Sergiu I. Vacaru ∗ Faculty of Mathematics, University ”Al. I. Cuza” Ia¸si , and The Fields Institute for Research in Mathematical Science222 College Street, 2d Floor, Toronto M5T 3J1, Canada
April 1, 2009
Abstract
The general relativity theory is redefined equivalently in almostK¨ahler variables: symplectic form, θ [ g ] , and canonical symplectic con-nection, b D [ g ] (distorted from the Levi–Civita connection by a tensorconstructed only from metric coefficients and their derivatives). Thefundamental geometric and physical objects are uniquely determinedin metric compatible form by a (pseudo) Riemannian metric g on amanifold V enabled with a necessary type nonholonomic 2 + 2 distri-bution. Such nonholonomic symplectic variables allow us to formulatethe problem of quantizing Einstein gravity in terms of the A–modelcomplexification of almost complex structures on V , generalizing theGukov–Witten method [1]. Quantizing ( V , θ [ g ] , b D [ g ]) , we derive aHilbert space as a space of strings with two A–branes which for theEinstein gravity theory are nonholonomic because of induced nonlin-ear connection structures. Finally, we speculate on relation of such amethod of quantization to curve flows and solitonic hierarchies definedby Einstein metrics on (pseudo) Riemannian spacetimes. Keywords: quantum gravity, Einstein gravity, nonholonomic man-ifolds, symplectic variables, nonlinear connections, strings and A–branes ∗
000 MSC: 83C45, 81S10, 53D55, 53B40, 53B35, 53D50PACS: 04.20.-q, 02.40.-k, 02.90.+g, 02.40.Yy
Contents g B cc , g B ′ ) strings . . . . . . . . . . 133.4 Quantization for Lagrangian nonholonomic branes . . . . . . 153.4.1 Properties of distinguished Hilbert spaces for nonholo-nomic A–models . . . . . . . . . . . . . . . . . . . . . 163.4.2 Topological restrictions and unitarity . . . . . . . . . . 173.4.3 A Hermitian inner product on g H of ( g B cc , g B ′ ) strings 19 In this paper we address the question of quantization of an A–model com-plexification of spacetime in general relativity following a new perspectiveon symplectic geometry, branes and strings proposed in Ref. [1] (the Gukov–Witten approach). Symplectic techniques has a long history in physics (see,2or instance, [2]) and, last two decades, has gained more and more interestin the theory of deformation quantization [3, 4, 5, 6, 7]. Recently, an ap-proach based on Fedosov quantization of Einstein gravity [8] and Lagrange–Finsler and Hamilton–Cartan spaces [9, 10] was elaborated in terms of al-most K¨ahler variables on nonholonomic manifolds (see [11, 12], for a reviewof methods applied to standard theories of physics, and [13, 10, 14], for al-ternative geometrizations of nonholonomic mechanics on manifolds and/orbundle spaces). The problem of quantization of nonlinear physical theories involves agrate amount of ambiguity because the quantum world requests a more re-fined and sophisticate description of physical systems than the classical ap-proach. Different types of quantization result in vary different mathematicalconstructions which lead to inequivalent quantizations of the same classicaltheories (the most important approaches to quantization are discussed in[1, 15, 16], see also references therein).Deformation quantization was concluded to be a systematic mathemat-ical procedure but considered that it is not a quantization in a standardmanner [1]. This is because a deformation quantization of the ring of holo-morphic functions on a complex symplectic/Poisson manifold requires notarbitrary choices but quantization does. It does not use deformations witha complex parameter, works with deformations over rings of formal powerseries and does not lead to a natural Hilbert space on which the deformedalgebra acts. Nevertheless, different methods and results obtained in defor-mation quantization play a very important role in all approaches to quanti-zation, as a consequent geometric formalism in nonlinear functional analysis.Perhaps, the most important results in deformation quantization can be re-defined in the language of other approaches to quantization which is veryuseful for developing new methods of quantization.In this paper we consider a new perspective on quantization of Einsteingravity based on two–dimensional sigma–models, following the A–modelquantization via branes proposed in [1]. Our purpose is to get closer toa systematic theory of quantum gravity in symplectic variables related to A pair ( V , N ), where V is a manifold and N is a nonintegrable distribution on V , iscalled a nonholonomic manifold. We emphasize that in this paper we shall not work withclassical physical theories on (co) tangent bundles but only apply in classical and quantumgravity certain methods formally elaborated in the geometry of Lagrange–Finsler spacesand nonholonomic manifolds. Readers may find in Appendix and Section 2 the mostimportant definitions and formulas from the geometry of nonholonomic manifolds anda subclass of such spaces defined by nonlinear connection, N–connection, structure (i.e.N–anholonomic manifolds).
3n almost K¨ahler formulation of general relativity. The novel results in thispaper are those that we propose an explicit application of the Gukov–Wittenquantization method to gravity and construct a Hilbert space for Einsteinspaces parametrizing it as the spaces of two nonholonomic A–brane strings.We relate the constructions to group symmetries of curve flows, bi–Hamilonstructures and solitonic hierarchies defined by (pseudo) Riemannian/ Ein-stein metrics.The new results have a strong relationship to our former results onnonholonomic Fedosov quantization of gravity [8] and Lagrange–Finser/Hamilton–Cartan systems [9, 10]. Geometrically, such relations follow fromthe fact that in all cases the deformation quantization of a nonholonomiccomplex manifold, constructed following the Gukov–Witten approach, pro-duces a so–called distinguished algebra (adapted to a nonlinear connectionstructure) that then acts in the quantization of a real almost K¨ahler mani-fold. It is obvious that different attempts and procedures to quantize gravitytheories are not equivalent. For such generic nonlinear quantum models, itis possible only to investigate the conditions when the variables from oneapproach can be re–defined into variables for another one. Then, a more de-tailed analysis allows us to state the conditions when physical results for onequantization are equivalent to certain ones for another quantum formalism.The paper is organized as follows: In Section 2, we provide an introduc-tion in the almost K¨ahler model of Einstein gravity. Section 3 is devotedto formulation of quantization method for the A–model with nonholonomicbranes. In section 4, an approach to Gukov–Witten quantization of thealmost K¨ahler model of Einstein gravity is developed. Finally, we presentconclusions in section 5. In Appendix, we summarize some important com-ponent formulas necessary for the almost K¨ahler formulation of gravity.Readers may consult additionally the Refs. [17, 18, 12, 11] on conventionsfor our system of denotations and reviews of the geometric formalism fornonholonomic manifolds, and various applications in standard theories ofphysics.
The standard formulation of the Einstein gravity theory is in variables( g , g ∇ ) , for g ∇ = ∇ [ g ] = { g p Γ αβγ = p Γ αβγ [ g ] } being the Levi–Cevita con-nection completely defined by a metric g = { g µν } on a spacetime manifold V and constrained to satisfy the conditions g ∇ g = 0 and g p T αβγ = 0 , where4 p T is the torsion of g ∇ . For different approaches in classical and quantumgravity, there are considered tetradic, or spinor, variables and 3+1 spacetimedecompositions (for instance, in the so–called Arnowit–Deser–Misner, ADM,formalism, Ashtekar variables and loop quantum gravity), or nonholonomic2 + 2 splittings, see a discussion and references in [16].
For any (pseudo) Riemannian metric g , we can construct an infinitenumber of linear connections g D which are metric compatible, g D g =0 , and completely defined by coefficients g = { g µν } . Of course, in general,the torsion g T = D T [ g ] of a g D is not zero. Nevertheless, we canwork equivalently both with g ∇ and any g D, because the distorsion tensor g Z = Z [ g ] from the corresponding connection deformation, g ∇ = g D + g Z, (1)(in the metric compatible cases, g Z is proportional to g T ) is also completelydefined by the metric structure g . In Appendix, we provide an explicit ex-ample of two metric compatible linear connections completely defined bythe same metric structure, see formula (A.27). Such torsions induced bynonholonomic deformations of geometric objects are not similar to thosefrom the Einstein–Cartan and/or string/gauge gravity theories, where cer-tain additional field equations (to the Einstein equations) are considered forphysical definition of torsion fields.Even the Einstein equations are usually formulated for the Ricci tensorand scalar curvature defined by data ( g , g ∇ ) , the fundamental equationsand physical objects and conservation laws can be re–written equivalentlyin terms of any data ( g , g D ) . This may result in a more sophisticate struc-ture of equations but for well defined conditions may help, for instance, inconstructing new classes of exact solutions or to define alternative methodsof quantization (like in the Ashtekar approach to gravity) [17, 11, 12, 16, 8].In order to apply the A–model quantization via branes proposed in Ref.[1], and relevant methods of deformation/geometric quantization, it is con-venient to select from the set of linear connections { g D } such a symplectic We follow our conventions from [12, 11] when ’boldfaced” symbols are used for spacesand geometric objects enabled with (or adapted to) a nonholonomic distribution/ nonlin-ear connection / frame structure; we also use left ”up” and ”low” indices as additionallabels for geometric/physical objects, for instance, in order to emphasize that g ∇ = ∇ [ g ]is defined by a metric g ; the right indices are usual abstract or coordinate tensor ones. for a general linear connection, we do not use boldface symbols if such a geometricobject is not adapted to a prescribed nonholonomic distribution g = { g µν } , being compatible to a well defined almost complex and sym-plectic structure, and for an associated complex manifold. In section 2and Appendix of Ref. [19], there are presented all details on the so–calledalmost K¨ahler model of general relativity (see also the constructions andapplications to Fedosov quantization of gravity in Refs. [8, 18, 16]). Forconvenience, we summarize in Appendix A some most important definitionsand component formulas on almost K¨ahler redefinition of gravity.Let us remember how almost K¨ahler variables can be introduced in clas-sical and quantum gravity: Having prescribed on a (pseudo) Riemannianmanifold V a generating function L ( u ) (this can be any function, for cer-tain models of analogous gravity [11, 12], considered as a formal regularpseudo–Lagrangian L ( x, y ) with nondegenerate L g ab = ∂ L∂y a ∂y b ), we con-struct a canonical almost complex structure J = L J (when J ◦ J = − I for I being the unity matrix) adapted to a canonical nonlinear connection (N–connection) structure N = L N defined as a nonholonomic distibution on T V . For simplicity, in this work we shall omit left labels like L if that willnot result in ambiguities; it should be emphasized that such constructionscan be performed for any regular L, i.e. they do not depend explicitly on L, or any local frames or coordinates .The canonical symplectic 1-form is defined θ ( X , Y ) + g ( JX , Y ) , for anyvectors X and Y on V , and g θ + L θ = { θ µν [ g ] } and g = L g . It is possibleto prove by straightforward computations that the form g θ is closed, i.e. d L θ = 0 , and that there is a canonical symplectic connection (equivalently,normal connection) θ b D = b D = { θ b Γ γαβ = b Γ γαβ } for which b D g θ = b Dg = . The variables ( g θ, b D ) define an almost K¨ahler model of general relativity,with distorsion of connection g Z → g b Z , for g ∇ = θ b D + g b Z , see formula(1). Explicit coordinate formulas for g ∇ , θ b D and g b Z are given by (A.27) We use the word ”pseudo” because a spacetime in general relativity is considered asa real four dimensional (pseudo) Riemannian spacetime manifold V of necessary smoothclass and signature ( − , + , + , +) . For a conventional 2 + 2 splitting, the local coordinates u = ( x, y ) on a open region U ⊂ V are labelled in the form u α = ( x i , y a ) , where indicesof type i, j, k, ... = 1 , a, b, c... = 3 , , for tensor like objects, will be considered withrespect to a general (non–coordinate) local basis e α = ( e i , e a ) . One says that x i and y a are respectively the conventional horizontal/ holonomic (h) and vertical / nonholonomic(v) coordinates (both types of such coordinates can be time– or space–like ones). Primedindices of type i ′ , a ′ , ... will be used for labeling coordinates with respect to a differentlocal basis e α ′ = ( e i ′ , e a ′ ) for instance, for an orthonormalized basis. For the local tangentMinkowski space, we can chose e ′ = i∂/∂u ′ , where i is the imaginary unity, i = − , and write e α ′ = ( i∂/∂u ′ , ∂/∂u ′ , ∂/∂u ′ , ∂/∂u ′ ) . d θ = 0 . In such a case, to perform a deformation,or other, quantization is a more difficult problem. Choosing nonholonomicdistributions generated by regular pseudo–Lagrangians, we can work onlywith almost K¨ahler variables which simplifies substantially the procedure ofquantization, see discussions from Refs. [8, 9, 12, 18].
In the canonical approach to the general relativity theory, one works withthe Levi–Civita connection ▽ = { p Γ αβγ } which is uniquely derived followingthe conditions p T = 0 and ▽ g = 0 . This is a linear connection but nota distinguished connection (d–connection) because ▽ does not preserve thenonholonomic splitting (see in Appendix the discussions relevant to (A.8))under parallelism. Both linear connections ▽ and b D ≡ θ b D are uniquely de-fined in metric compatible forms by the same metric structure g (A.1). Thesecond one contains nontrivial d–torsion components b T αβγ (A.26), inducedeffectively by an equivalent Lagrange metric g = L g (A.5) and adaptedboth to the N–connection L N , see (A.4) and (A.8), and almost symplectic L θ (A.17) structures L. Having chosen a canonical almost symplectic d–connection, we computethe Ricci d–tensor b R βγ (A.31) and the scalar curvature L R (A.32)). Then,we can postulate in a straightforward form the filed equations b R αβ −
12 ( L R + λ ) e αβ = 8 πG T αβ , (2)where b R α β = e αγ b R γβ , T αβ is the effective energy–momentum tensor, λ is the cosmological constant, G is the Newton constant in the units whenthe light velocity c = 1 , and the coefficients e αβ of vierbein decomposition e β = e αβ ∂/∂u α are defined by the N–coefficients of the N–elongated opera-tor of partial derivation, see (A.10). But the equations (2) for the canonical b Γ γαβ ( θ ) are not equivalent to the Einstein equations in general relativitywritten for the Levi–Civita connection p Γ γαβ ( θ ) if the tensor T αβ does notinclude contributions of p Z γαβ ( θ ) in a necessary form.Introducing the absolute antisymmetric tensor ǫ αβγδ and the effectivesource 3–form T β = T αβ ǫ αβγδ du β ∧ du γ ∧ du δ b R τγ = b R τγαβ e α ∧ e β of b Γ αβγ = p Γ αβγ − p b Z αβγ as b R τγ = p R τγ − p b Z τγ , where p R τγ = p R τγαβ e α ∧ e β isthe curvature 2–form of the Levi–Civita connection ∇ and the distorsion ofcurvature 2–form b Z τγ is defined by b Z αβγ (A.28), we derive the equations(2) (varying the action on components of e β , see details in Ref. [16]). Thegravitational field equations are represented as 3–form equations, ǫ αβγτ (cid:16) e α ∧ b R βγ + λ e α ∧ e β ∧ e γ (cid:17) = 8 πG T τ , (3)when T τ = m T τ + Z b T τ , m T τ = m T ατ ǫ αβγδ du β ∧ du γ ∧ du δ , Z T τ = (8 πG ) − b Z ατ ǫ αβγδ du β ∧ du γ ∧ du δ , where m T ατ is the matter tensor field. The above mentioned equations areequivalent to the usual Einstein equations for the Levi–Civita connection ∇ , p R αβ −
12 ( p R + λ ) e αβ = 8 πG m T αβ . For b D ≡ θ b D , the equations (3) define the so–called almost K¨ahler model ofEinstein gravity.Such formulas expressed in terms of canonical almost symplectic form θ (A.18) and normal d–connection b D ≡ θ b D (A.20) are necessary for encodingthe vacuum field equations into cohomological structure of quantum (in thesence of Fedosov quantization) almost K¨ahler models of the Einstein gravity,see [8, 18, 19, 16].We conclude that all geometric and classical physical information for anydata 1] (cid:0) g , g p Γ (cid:1) , for Einstein gravity, can be transformed equivalently intocanonical constructions with 2] ( g θ, θ b D ) , for an almost K¨ahler model ofgeneral relativity. A formal scheme for general relativity sketching a mathe-matical physics ”dictionary” between two equivalent geometric ”languages”(the Levi–Civita and almost K¨ahler ones) is presented in Figure 1. The goal of this section is to generalize the A–model approach to quan-tization [1] for the case when branes are nonholonomic and the symplecticstructure is induced by variables ( g θ, θ b D ) in Einstein gravity.8 ✣ ✜✢ (pseudo) Riemannian metric g arbitrary frames/coordinatesand/or 3+1 splitting any 2+2 nonholonomic splittingwith generating function L ( x, y ) g = g αβ e α ⊗ e β g = g αβ e α ⊗ e β , e α = ( e i , e a = dy a + N ak dx k )Levi–Civita connection g ∇ ; g ∇ g = 0 , g p T αβγ = 0 , canonical symplectic connection b D = { b Γ γαβ } ; b D g θ = b Dg = 0 , g ∇ = b D + g b Z , Levi–Civita variables:( g , g ∇ ) = ( g µν , g p Γ αβγ ) almost K¨ahler variables:( g θ, b D ) = ( θ αβ , b Γ γαβ )Classical Einstein SpacesBrane A–model quantization;Deformation quantization;Nonholonomic Ashtekar variables ❄❄ ❄❄ (cid:0)(cid:0)✠❅❅❘ ❄❄ Quantum almost K¨ahler Einstein Spaces ❅❅❅❅❅❅❅❘(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠
Figure 1:
Levi–Civita and almost K¨ahler Variables in Gravity .1 On quantization and nonholonomic branes We start with an almost K¨ahler model of a (pseudo) Riemannian mani-fold V (which is a nonholonomic manifold, or, more exactly, N–anholonomicmanifold [11, 12], see also definitions in Appendix) endowed with structures( g θ, θ b D ) , which we wish to quantize. Our goal is to develop the method ofquantization [1] and to apply it to the case of nonholonomic manifolds pro-vided with gravitational symplectic variables. We consider a complex linebundle L → V with a unitary connection of curvature R , like in geometricquantization [20, 21, 22, 23].In this work, we shall use an affine variety Y which, by definition, isa complexification of V such that: 1) it is a complex manifold with anantiholomorphic involution τ : Y → Y , when V is a component of the fixedpoint set of τ ; 2) there is a nondegenerate holomorphic 2–form Θ on Y suchthat its restriction, τ ∗ ( Θ ) , to V is just g θ ; 3) the unitary line bundle V → V can be extended to a unitary line bundle Y → Y enabled with a connectionof curvature Re ( Θ ) , when the action of τ on Y results to an action on Y , restricting to an identity on V . In brief, the approach to quantizationis based on a ”good” A–model associated with the real symplectic form Y θ = Im Θ . Such a choice determines a canonical coisotropic brane (in ourcase, in general, it is a nonholonomic manifold, because V is nonholonomic)in the A–model of Y [24, 25].Similarly to [1], we will make use of ordinary Lagrangian A–branes when V is also modeled as a Lagrangian submanifold and we can define a rank 1A–brane supported on such a (nonholonomic) manifold. We denote by g B ′ such a brane (one could be inequivalent choices for a not simply–connected).It will be written g B cc for the canonical coisotropic A–brane and g B foran A–brane of unspecified type. We can construct a quantum model (i.e. quantization of V enabled with variables ( g θ, θ b D )) postulating thatthe Hilbert space associated to V is the space g H of ( g B cc , g B ′ ) . This isrelated to the geometry of a vector bundle provided with nonlinear connec-tion structure (in different contexts such spaces and applications to modernphysics, mechanics and Finsler geometry and generalizations were studied inRefs. [11, 12, 13]) associated to the choice of A–brane g B ′ . The first explicitconstructions of such vector spaces (without N–connection structure) were For such a good A–model, the relevant correlation functions and observables arecomplex–valued rather than formal power series depending on a formal deformation pa-rameter; such series converge to complex–valued functions; following [1], this is possible ifthe superymmetric sigma–model with target Y can be twisted to give the A-model. Here,we also note that we follow our system of denotations relevant to nonholonomic manifoldsand corresponding geometric constructions. This will bediscussed in further sections.
Let us consider a complex symplectic manifold Y endowed with a nonde-generate holomorphic 2–form Θ of type (2 ,
0) splitting into respective real, J θ, and imaginary, K θ, parts, i.e. Θ = J θ + i K θ, (4)where I t Θ = i Θ , or I t J θ = − K θ and I t K θ = J θ, (5)for I being the complex structure on Y , which may be regarded as a lineartransformation of tangent vectors, I t denoting the transpose map acting on1–forms; Θ and I t Θ are regarded as maps from tangent vectors to 1–forms. In this work, we view Y as a real symplectic manifold with symplecticstructure Y θ = Im Θ = K θ and study the associated A–model as a casein [24], when such A–models are Lagrangian branes; for our purposes, itis enough to take a rank Y . Any rank 1 brane can be endowed with a unitary line bundle V witha connection for which we denote the curvature by F . If for I = Y θ − F , we have I = − , i.e. this is an integrable complex structure; we call sucha 1 brane to be an A–brane. We obey these conditions if we set F = J θ, when Y θ − F = K θ − J θ coincides with I from (5). So, we can construct In low energy limits, we proved that from string/brane and/or gauge gravity mod-els one generates various versions of (non) commutative Lagrange–Finsler spaces. Suchconstructions were considered for a long time to be very ”exotic” and far from scopes ofstandard physics theories. But some years latter, it was proven that Finsler like structurescan be modelled even as exact solutions in Einstein gravity if generic off-diagonal metricsand nonholonomic frames are introduced into consideration [17, 12]. More than that, theN–connection formalism formally developed in Finsler geometry, happen to be very use-ful for various geometric purposes and applications to modern gravity and quantum fieldtheory. even we consider such notations, we do not impose the condition that Y has a hyper–K¨ahler structure Y θ startingwith a complex symplectic manifold ( Y , Θ ) , for any choice of a unitary linebundle V enabled with a connection of curvature J θ = Re ( Θ ) . Following [1],we call this A–brane the canonical coisotropic brane and denote it as g B cc (we put a left up label g because for the almost K¨ahler models of Einsteinspaces there are induced by metric canonical decompositions of fundamentalgeometric objects related to almost complex and symplectic structures, seerespective formulas (A.1) with (A.5) and (A.6); (A.14) and (A.18)).We emphasize that the constructions leading to an A–model dependonly on Y θ and do not depend on the chosen almost complex structure;there is no need for the almost complex structure to be integrable. So, wecan always chose Y θ to be, let say a complexification, or proportional tothe gravitational symplectic 1–form g θ. Together with the almost complexstructure L J (A.14), this would make the A–model more concrete (we shallconsider details in section 4). Here we adapt some key constructions from[1] to the case of nonholonomic A–models, when Y θ = K θ for an almostcomplex structure K (in a particular case, Y θ = g θ and K = L J ) . So, in thissection, we consideer that Y is a nonholonomic complex manifold enabledwith a N–connection structure defined by a distribution of type (A.8). Forour purposes, it is enough to consider that such a distribution is related toa decomposition of tangent spaces of type T Y = T V ⊕ I ( T V ) , (6)when we chose such a K that IK = − KI implying that J = KI is also analmost complex structure.In general, J and K are not integrable but K always can be defined tosatisfy the properties that K θ is of type (1 ,
1) and IK = − KI , for anynonholonomic V , and the space of choices for K is contractable. This canbe verified for any Y of dimension 4 k, for k = 1 , , ..., as it was considered insection 2.1 of [1]. For nonholonomic manifolds related to general relativitythis is encoded not just in properties of compact form symplectic groups, Sp (2 k ) , acting on C k , and their complexification, Sp (2 k ) C . The nonholo-nomic 2 + 2 decomposition (similarly, we can consider n + n ), results notin groups and Lie algebras acting on some real/complex vector spaces, butinto distinguished similar geometric objects, adapted to the N–connectionstructure. In brief, they are called d–algebras, d–groups and d–vectors. Thegeometry of d–groups and d–spinors, and their applications in physics andnoncommutative geometry, is considered in details in Refs. [33, 34, 35], seealso Part III of [11], and references therein (we refer readers to those works).12ere, we note that for our constructions in quantum gravity, it is enoughto take Y of dimension 8 , for k = 2 , when V is enabled with a nonholo-nomic splitting 2 + 2 and the d–group d Sp (4) is modelled as d Sp (4) = Sp (2) ⊕ Sp (2), which is adapted to both type decompositions (A.8) and(6). We can now write a sigma–model action using an associated metric K g = − K θ K , when J = KI will be used for quantization. The first nonhlo-nomic sigma– and (super) string models where considered in works [27, 28]for the so–called Finsler–Lagrange (super) strings and (super) spaces, butin those works V = E , for E being a vector (supervector bundle). In thissection, we work with complex geometric structures on T Y = h Y ⊕ v Y , (7)when such a nonholonomic splitting is induces both by (6) and (A.8), i.e. Y is also enabled with N–connection structure and its symplectic and com-plex forms are related to the corresponding symplectic and almost complexstructures on V (in particular, those for the almost K¨ahler model of Einsteingravity). ( g B cc , g B ′ ) strings We can consider the space of ( g B cc , g B cc ) strings in a nonholonomicA–model as the space of operators that can be inserted in the A–modelon a boundary of a nonholonomic string world–sheet Σ , with a splitting T Σ = h Σ ⊕ v Σ , similarly to (A.8), that ends on the brane g B cc . In general,the constructions should be performed for a sigma–model with nonholo-nomic target Y , bosonic fields U and N–adapted fermionic d–fields − ψ = (cid:0) h − ψ, v − ψ (cid:1) , for left–moving d–spinors, and + ψ = (cid:0) h + ψ, v + ψ (cid:1) , where the h –and v –components are defined with respect to splitting (7).An example of local d–operator f ( U ) is that corresponding to a complex–valued function f : Y → C . This d–operator inserted at an interior point of Σ is invariant under supersymmetry transforms (on nonholonomic super-symmetic spaces, see [28]) of the A–model, δ U = (1 − i K ) ( + ψ ) + (1 + i K ) ( − ψ ) , (8)if f is constant. The N–connection structure results in similar transforms ofthe h - and v –projections of the fermionic d–fields. Boundary d–operatorsmust be invariant under transforms (8) of the A–model. In sigma models,one works with boundary (d-) operators, rather than bulk (d-) operators,13hen the boundary conditions are nonholonomic ones obeyed by fermionic(d-) fields, ( K g − F ) ( + ψ ) = ( K g + F ) ( − ψ ) , where, for K g = − K θ K and F = J θ, we have ( K g − F ) − ( K g + F ) = J = KI . The boundary conditions (8) can be written in equivalent form δ U = ((1 − i K ) J + (1 + i K )) ( − ψ ) , = (1 + i I )(1 + i J ) ( − ψ ) . This implies a topological symmetry of the nonholonomic A–model, δ , U = 0 and δ , U = ρ, (9) δρ = 0 , where ρ = (1 + i I )(1 + i J ) ( − ψ ) and decomposition δ U = δ , U + δ , U fordecompositions of the two parts of respective types (1,0) and (0,1) with re-spect to the complex structure I . The N–splitting (7) results in ρ = ( h ρ, v ρ ) , induced by h– , v–splitting − ψ = (cid:0) h − ψ, v − ψ (cid:1) . It follows from (9) that the topological supercharges of the nonholonomicA–model corresponds to the ∂ operator of Y . We wrote ”supercharges”because one of them is for the h–decomposition and the second one is for thev–decomposition. The observables of the nonholonomic A–model correspondadditively to the graded d–vector space H ,⋆∂ ( Y ) , where Y is viewed bothas a complex manifold with complex structure I and as a nonholonomicmanifold enabled with N–connection structure. We shall work with theghost number zero part of the ring of observables which corresponds to theset of holomorphic functions on Y . For instance, all boundary observables of the nonholonomic A–model canbe constructed from U and ρ = ( h ρ, v ρ ) . Let us fix the local complex coordi-nates on Y to correspond to complex fields ˜U α ( τ, σ ) = ( ˜ X i ( τ, σ ) , ˜ Y a ( τ, σ )) , for string parameters ( τ, σ ) , see also begining of section 4 on coordinateparametrizations on Y . This allows us to construct general d–operators: • of q –th order in ρ, having d–charge q under the ghost number sym-metry of the A–model, ρ α ρ α ...ρ α q f α α ...α q ( U , U ) , which is an d–operator as a (0 , q )–form on Y ; • of p –th order in h ρ, having h–charge p under the ghost number symme-try of the A–model, ρ i ρ i ...ρ i p f i i ...i p ( X , X ) , which is an h–operatoras a (0 , p )–form on Y ; 14 of s -th order in v ρ, having v–charge s under the ghost number symme-try of the A–model, ρ a ρ a ...ρ a s f a a ...a s ( Y , Y ) , which is an v–operatoras a (0 , s )–form on Y . For a series of consequent h- and v–operators, it is important to considerthe order of such operators.We conclude that the constructions in this subsection determine the d–algebra A = ( h A , v A ) of ( g B cc , g B cc ) strings. Our first purpose, in this section, is to find something (other than itself)that the d–algebra A = ( h A , v A ) can act on. The simplest constructionis to introduce a second nonholonomic A–brane g B ′ , which allows us todefine a natural action of A on the space of ( g B cc , g B ′ ) strings. In thispaper, we consider g B ′ to be a Lagrangian A–brane of rank 1, enabledwith a nonholonomic distribution, i.e. g B ′ is supported on a Lagrangiannonholonomic submanifold V also endowed with a flat line bundle V ′ . Weassume that J θ is nondegenerate when restricted to V and consider ( V , J θ )as a symplectic manifold to be quantized. For a given V, it is convenient tofurther constrain K such that T V is J –invariant, when the values I , K and J = KI obey the algebra of quaternions.The quantization of ( g B cc , g B ′ ) strings leads to quantization of the sym-plectic manifold ( V , J θ ) , and in a particular case of the almost K¨ahler modelof Einstein gravity. We do not present a proof of this result because it issimilar to that presented in section 2.3 of [1] by using holonomic manifoldsand strings. For nonholonomic constructions, we have only a formal h– andv–component dubbing of geometric objects because of the N–connectionstructure.There are also necessary some additional constructions with the action ofa string ending on a nonholonomic brane with a Chan–Paton connection A , which is given by a boundary term R ∂ Σ A µ d U µ . For ( g B cc , g B ′ ) strings, wecan take that the Chan–Paton bundle V ′ , with a connection A ′ , of the brane g B ′ is flat but the Chan–Paton bundle of g B cc is the unitary line bundle V , with connection A of curvature J θ. Next step, we consider a line bundle The proposed interpretation of V ′ is oversimplified because of relation of branes toK–theory, possible contributions of the the so-called B–field, disc instanton effects etc, seediscussion in Ref. [1]. For the quantum gravity model to be elaborated in this paper,this is the simplest approach. Here we also note that our system of denotations is quitedifferent from that by Gukov and Witten because we work with nonholonomic spaces anddistinguished geometric objects. V = V ⊗ V ′ over V which is a unitary bundle with a connection B = A − A ′ of curvature J θ. This corresponds, with a corresponding approximation, toa classical action for the zero modes Z dτ (cid:0) A µ − A ′ µ (cid:1) du µ dτ ≈ Z dτ B µ du µ dτ . To quantize the zero modes with this action, and related ”quantum” correc-tions, is a quantization of V with prequantum line bundle V . We have not yet provided a solution of the problem of quantization forthe almost K¨ahler model of Einstein gravity. It was only solved the secondaim to understand that if the A–model of a nonholonomic Y exists, thenthe space of ( g B cc , g B ′ ) strings can be modelled as a result of quantizing V with prequantum line bundle g V . It is still difficult to describe such spacesexplicitly, in a general case, but the constructions became well defined for( g θ, θ b D ) on V . Using such information from the geometry of almost K¨ahlerspaces, induced by a generating function with nonholonomic 2+2 splitting,we can establish certain important general properties of the A–model tolearn general properties of quantization.
There are two interpretations of the space of ( g B cc , g B ′ ) strings: Thefirst one is to say that there is a Hilbert space g H in the space of such stringsin the nonholonomic A–model of Y θ = K θ. The second one is to consider aHilbert space g e H of such strings in the B–model of complex structure J . Fora compact V , or for wave functions required to vanish sufficiently rapidly atinfinity, both spaces H and e H are the same: they can be described as thespace of zero energy states of the sigma–model with target Y . The model iscompactified on an interval with boundary conditions at the ends determinedby nonholonomic g B cc and g B ′ . It should be noted here that both g H and g e H are Z –graded by the ”ghost number” but differently, such grading beingconjugate, because the A—model is of type K θ and the B–model is of type Y θ. We can use a sigma–model of target Y when the boundary conditions areset by two branes of type ( A , B , A ) . In this case, the model has an SU (2)group of so–called R–symmetries, called SU (2) R , with two ghost numbersymmetries being conjugate but different U (1) subgroups of SU (2) R . Fornonholonomic models, we have to dub the groups and subgroups corre-spondingly for the h– and v–subspaces. In the simplest case, we can per-16orm quantization for trivial grading, i.e. when g V is very “ample“ as a linebundle in complex structure J . For the B–model, we can chose any K¨ahler metric, for instance, to rescalethe metric of Y as to have a valid sigma–model perturbation theory. In sucha limit, it is possible to describe g e H by a ∂ cohomology, g e H = ⊕ dim C V j =0 H j ( V , K / ⊗ g V ) , with a similar decomposition for g H ∼ = g e H , i.e. g H = ⊕ dim C V j =0 H j ( V , K / ⊗ g V ) . (10)The value K / is the square root (this is a rough approximation) of thecanonical line bundle K on V . This is because, in general, V may not bea spin manifold. Such nonholonomic configurations related to gerbes arediscussed in [36, 37]. In relation to K –theory, details are given in [1] (non-holonomic configurations existing in our constructions do not change thoseconclusions). Here, we note that for very ample g V the cohomology van-ishes except for j = 0 and its Z –grading is trivial. This is one problem.The second limitation is that g H is described as a vector space which doesnot lead to a natural description as a Hilbert space with a hermitian innerproduct. This description of g H has certain resemblance to constructionsin geometric quantization because the above cohomology defines quantiza-tion with a complex polarization. Nevertheless, this paper is based on theGukov–Witten approach to quantization and does not provide a variant ofgeometric quantization, see in [1] why this is not geometric quantization. There is a topological obstruction to having a
Spin c structure and thereare further obstructions to having a flat Spin c structure on V . Such non-holonomic spinor constructions were analyzed firstly in relation to definitionof Finsler–Lagrange spinors [33], further developments are outlined in Refs.[34, 35]. The problem of definition of spinors and Dirac operators for non-holonomic manifolds can be solved by the same N–connection methods withthat difference that instead of vector/tangent bundles we have to work withmanifolds enabled with nonholonomic distributions.In general,
Spin c structures on V are classified topologically: we have tochose a way of lifting the second Stieffel–Whitney class w ( V ) ∈ H ( V , Z )to an integral cohomology class ζ ∈ H ( V , Z ) . In their turn, flat
Spin c structures are parametrized and classified by a choice of a lift ζ as a torsion17lement of H ( V , Z ) . We emphasize that a sympletic manifold that doesnot admit a flat
Spin c structure cannot be quantized in the sense [1]. Per-haps, the cohomological analysis used in Fedosov quantization for almostK¨ahler models of gravity [8, 9, 18], can be re–defined for the Gukov–Wittenapproach. In this work, we shall consider such gravitational fields and theirquantization when the flat Spin c structure exists and they are distinguishedinto h– and v–components adapted to N–connection structures defined bycertain generating functions.Our next purpose is to define a Hermitian inner product on g H (10).For our further application in quantum gravity, we chose a nonholonomicA–model as a twisted version of a standard physical model, unitary, definedalso as a supersymmetric field theory. Such a theory has an antilinear CPTsymmetry, in our approach denoted Ξ . This operator maps any ( B , B )string into a ( B , B ) which also defines an antilinear map from ( g B cc , g B ′ )strings into ( g B ′ , g B cc ) strings, but this is not a symmetry of an A–model.In explicit form, the definition of an A–model depends on a choice of adifferential Q as a complex linear combination of supercharges. For non-holonomic configurations, we work with couples of h– and v–supercharges Q =( h Q, v Q ) . The maps with CPT symmetry transform Q into its Hermi-tian adjoint Q + =( h Q + , v Q + ) being the differential of a complex conjugatenonholonomic A–model.Let us suppose that the nonholonomic complex manifold Y admits aninvolution τ, i.e. a N–adapted diffeomorphism obeying τ = 1 , with the(odd) property: τ ∗ ( K θ ) = − K θ. (11)Such an operator can be always introduced on Y by construction. This τ maps a nonholonomic A–model into a conjugate nonholonomic A–model and Ξ τ = τ Ξ is a N–adapted antilinear map from the nonholonomic A–modelto itself. This is a general property which holds both for holonomic andnonholonomic geometrical models of certain underlying physical theory; onemay be a twisting of the underlying model and in such a case the structure K can be chosen to be integrable.We consider explicitly the antiholomorphic involution τ for a nonholo-nomic complex symplectic manifold defined by data ( Y , I , Θ ) , with a liftto V , where Θ = J θ + i K θ, I = J θ − K θ, for J θ being the curvature of theChan–Paton bundle of the ( τ –invariant) brane g B cc , when τ ∗ ( J θ ) = J θ, τ ∗ ( I ) = − I , τ ∗ ( Θ ) = Θ . Using the topological inner product ( , ) as the pairing between (in general,nonholonomic) ( B , B ) strings and ( B , B ) , we can introduce the inner18roduct on g H as < ψ, ψ ′ > = < Ξ τ ψ, ψ ′ > . So, we conclude that if B and B are τ –invariant nonholonomic A–branes,we can use Ξ τ to define a Hermitian inner product on the (already Hilbert)space g H of ( B , B ) strings. g H of ( g B cc , g B ′ ) strings We suppose that the nonholonomic spacetime manifold V is τ –invariantand that there is a lift of τ to act on the Chan–Paton line bundle V ′ on V . In such cases, the corresponding nonholonomic Lagrangian A–brane g B ′ isalso τ –invariant. This allows us to construct a Hermitian form, and corre-sponding inner product < , >, on the space g H of ( g B cc , g B ′ ) strings if τ maps V to itself.Nearly the classical limit, the norm of a state ψ ∈ g H is approximated < ψ, ψ > = R V ψ ( τ u ) ψ ( u ) δu ; this form is positive–definite only if τ actstrivially on V . In general, < ψ, ψ > is nondegenerate but not necessarilypositive definite. These are some consequences from nondegeneracy of topo-logical inner product ( , ) and the property that Ξ τ = 1 . We can chose any ψ such that ( ψ , ψ ) = 0 and set ψ = Ξ τ ψ which results in < ψ , ψ > = 0 , i.e. nondegenerate property, but this dos not constrain that < ψ, ψ > > ψ ∈ g H . For the inverse construction, we consider V tobe a component of the fixed point set of map τ. Because of property (11),we get that V is Lagrangian for K θ and that J θ is nondegenerate whenrestricted to V . Such properties are automatically satisfied for the almostK¨ahler model of Einstein gravity with splitting (7). The map τ acts as 1and -1 on summands T V and I T V and J θ is the sum of nondegenerate2–forms on respective spaces.There is a construction leading to the classical limit [1], even in our casewe have certain additional nonholonomic distributions. For this, we can takethe space of ( g B cc , g B cc ) strings and perform the deformation quantizationof the complex nonholonomic symplectic manifold Y (such constructions arepresented in detail for almost complex models of gravity [8, 9, 18]). We getan associative d–algebra g A . Then we chose an antiholomorphic involution τ, with a lift to the line bundle Y → Y , and a component V of the fixedpoint set supporting a τ –invariant nonholonomic A–brane g B ′ . Now, wecan say that g A acts on g H defined as the space of ( g B cc , g B ′ ) strings.To consider the action Ξ τ we model such a map acts on a function on Y , defining a ( g B cc , g B cc ) string as the composition of τ with complexconjugation. For an operator O f : g H → g H associated to a function f, we19efine the Hermitian adjoint of O f to be associated with the function τ ( f ) . Working with a real function f when restricted to V and if τ leaves V fixedpointwise, we get τ ( f ) = f with a Hermitian O f . Finally, in this section, we conclude that the Gukov–Witten method[1] really allows us to construct a physical viable Hilbert space for quantumalmost K¨ahler models of Einstein gravity related to the Fedosov quantizationof a corresponding complex nonholonomic symplectic manifold Y . In this section, we provide explicit constructions for quantum physicalstates of the almost K¨ahler model of Einstein gravity.
The local coordinates on a nonholonomic complex manifold Y are de-noted u e α = ( u α , iu ` α ) , for i = − , where u α = ( x i , y a ) are local coordinateson V , and ` u α = u ` α = (` x i = x ` ı , ` y a = y ` a ) are real local (pseudo) Euclidean co-ordinates of a conventional ”left–primed” nonholonomic manifold `V . We em-phasize that ”nonprimed” indices can not be contracted with primed indices,because they label objects on different spaces. For (non)holonomic Einsteinspaces, the coordinate indices will run values i, j, ... = 1 , a, b, ... = 3 , ı, ` j, ... = 1 ,
2; ` a, ` b, ... = 3 , . We can also treat ˜ u α = u e α = (˜ x j = x j + i ` x j , ˜ y a = y a + i ` y a ) as complex coordinates on Y , when, for instance, ˜ u = u + i ` u . In brief, such coordinates are labelled respectively ˜ u = (˜ x, ˜ y ) , u = ( x, y ) and` u = (` x, ` y ) . In general, such real manifolds V , `V and complex manifold Y are en-abled with respective N–connection structures, see formula (A.9), N = { N ai ( u ) } , `N = { ` N ai = N ` a ` ı (` u ) } and ˜N = { ˜ N ai = N ˜ a ˜ ı (˜ u ) } = { N ai (˜ u )+ i ` N ai (˜ u ) } . For `V and Y , the corresponding N–adapted bases (A.10) are `e ν = (cid:18) `e i = ∂∂ ` x i − ` N ai (` u ) ∂∂ ` y a , ` e a = ∂∂ ` y a (cid:19) = e ` ν = (cid:18) e ` ı = ∂∂x ` ı − N ` a ` ı (` u ) ∂∂y ` a , e ` a = ∂∂y ` a (cid:19) and ˜e ν = (cid:18) ˜e i = ∂∂ ˜ x i − ˜ N ai (˜ u ) ∂∂ ˜ y a , ˜ e a = ∂∂ ˜ y a (cid:19) `e µ = (cid:16) ` e i = d ` x i , `e a = d ` y a + ` N ai (` u ) d ` x i (cid:17) = e ` µ = (cid:16) e ` ı = dx ` ı , e ` a = dy ` a + N ` a ` ı (` u ) dx ` ı (cid:17) and ˜e µ = (cid:16) ˜ e i = d ˜ x i , ˜e a = d ˜ y a + ˜ N ai (˜ u ) d ˜ x i (cid:17) . We can also use parametrizations ˜e ν = e ˜ ν = (cid:18) e ν i e ` ν (cid:19) ∈ T Y and ˜e µ = e ˜ µ = ( e µ , − i `e µ ) ∈ T ∗ Y . The above presented formulas for N–adapted (co) bases are derived followingsplitting (6) and (7).The almost complex structure K on `V is defined similarly to (A.14) K = K αβ ` e α ⊗ ` e β = K αβ ∂∂ ` u α ⊗ d ` u β = K α ′ β ′ `e α ′ ⊗ `e β ′ = − ` e i ⊗ ` e i + `e i ⊗ `e i . The almost symplectic structure on a manifold V is defined by J θ = g θ = θ = L θ, see formula (A.18). A similar construction can be defined on `V as K θ = `g ` θ = ` θ = ` L ` θ ` θ = ` L ` θ = 12 ` L ` θ ij (` u )` e i ∧ ` e j + 12 ` L ` θ ab (` u ) `e a ∧ `e b = ` g ij (` x, ` y ) h d ` y i + ` N ik (` x, ` y ) d ` x k i ∧ d ` x j . Here, it should be noted that, in general, we can consider different geomet-ric objects like the generation function L ( x, y ) , and metric g ij ( x, y ) on V and, respectively, ` L (` x, ` y ) and ` g ij (` x, ` y ) on `V , (as some particular cases, wecan take two different exact solutions of classical Einstein equations). Butthe constructions from this section hold true also if we dub identically thegeometric constructions on V and `V . A canonical holomorphic 2–form Θ of type (2 ,
0) on Y , see (4), inducedfrom the almost K¨ahler model of Einstein gravity, is computed Θ = Θ ˜ µ ˜ ν e ˜ µ ∧ e ˜ ν , for Θ ˜ µ ˜ ν = ˜Θ µν = (cid:18) J θ µν (˜ u ) 00 − i K θ µν (˜ u ) (cid:19) and general complex coordi-nates ˜ u µ . This form is holomorphic by construction, ∂ Θ = , for ∂ definedby complex adjoints of ˜ u µ . An explicit construction of a Hilber space g H with a Hermitian in-ner product on ( g B cc , g B ′ ) strings for the almost K¨ahler model of Einsteingravity is possible if we prescribe in the theory certain generic groups of sym-metries. There are proofs that metric structures on a (pseudo) Riemannianmanifold [38, 39] can be decomposed into solitonic data with correspond-ing hierarchies of nonlinear waves. Such constructions hold true for moregeneral types of Finsler–Lagrange–Hamilton geometries and/or their Ricciflows [40, 41] and related to the geometry of curve flows adapted to a N–connection structure on a (pseudo) Riemannian (in general, nonholonomic)manifold V . Our idea is to consider in quantum gravity models the samesymmetries as for the ”solitonic” encoding of classical gravitational interac-tions.A well known class of Riemannian manifolds for which the frame cur-vature matrix constant consists of the symmetric spaces M = G/H forcompact semisimple Lie groups G ⊃ H. A complete classification and sum-mary of main results on such spaces are given in Refs. [42, 43]. The classof nonholonomic manifolds enabled with N–connection structure are charac-terized by conventional nonholonomic splitting of dimensions. For explicitconstructions, we suppose that the ”horizontal” distribution is a symmetricspace hV = hG/SO ( n ) with the isotropy subgroup hH = SO ( n ) ⊃ O ( n )and the typical fiber space is a symmetric space F = vG/SO ( m ) with theisotropy subgroup vH = SO ( m ) ⊃ O ( m ) . This means that hG = SO ( n + 1)and vG = SO ( m + 1) which is enough for a study of real holonomic andnonholonomic manifolds and geometric mechanics models. A non–stretching curve γ ( τ, l ) on V , where τ is a real parameter and l is the arclengthof the curve on V , is defined with such evolution d–vector Y = γ τ and tangent d–vector X = γ l that g ( X , X ) =1 . Such a curve γ ( τ, l ) swept out a two–dimensional surface in T γ ( τ, l ) V ⊂ T V . we can consider hG = SU ( n ) and vG = SU ( m ) for geometric models with spinor andgauge fields and in quantum gravity M = G/H arecovariantly constant and G –invariant resulting in constant curvature ma-trices. Such constructions are related to the formalism of bi–Hamiltonianoperators, originally investigated for symmetric spaces with M = G/SO ( n )with H = SO ( n ) ⊃ O ( n −
1) and when G = SO ( n + 1) , SU ( n ) , see [44] andreferences in [40, 41].For nonholonomic manifolds, our aim was to solder in a canonic way(like in the N–connection geometry) the horizontal and vertical symmetricRiemannian spaces of dimension n and m with a (total) symmetric Rieman-nian space V of dimension n + m, when V = G/SO ( n + m ) with the isotropygroup H = SO ( n + m ) ⊃ O ( n + m ) and G = SO ( n + m + 1) . There arenatural settings to Klein geometry of the just mentioned horizontal, verticaland total symmetric Riemannian spaces: The metric tensor hg = { ˚ g ij } on h V is defined by the Cartan–Killing inner product < · , · > h on T x hG ≃ h g restricted to the Lie algebra quotient spaces h p = h g /h h , with T x hH ≃ h h , where h g = h h ⊕ h p is stated such that there is an involutive automorphismof hG under hH is fixed, i.e. [ h h ,h p ] ⊆ h p and [ h p ,h p ] ⊆ h h . We can alsodefine the group spaces and related inner products and Lie algebras,for vg = { ˚ h ab } , < · , · > v , T y vG ≃ v g , v p = v g /v h , with T y vH ≃ v h ,v g = v h ⊕ v p , where [ v h ,v p ] ⊆ v p , [ v p ,v p ] ⊆ v h ; (12)for g = { ˚ g αβ } , < · , · > g , T ( x,y ) G ≃ g , p = g / h , with T ( x,y ) H ≃ h , g = h ⊕ p , where [ h , p ] ⊆ p , [ p , p ] ⊆ h . Any metric structure with constant coefficients on V = G/SO ( n + m ) canbe parametrized in the form˚ g = ˚ g αβ du α ⊗ du β , where u α are local coordinates and˚ g αβ = " ˚ g ij + ˚ N ai ˚ N bj ˚ h ab ˚ N ej ˚ h ae ˚ N ei ˚ h be ˚ h ab . (13)The constant (trivial) N–connection coefficients in (13) are computed ˚ N ej =˚ h eb ˚ g jb for any given sets ˚ h eb and ˚ g jb , i.e. from the inverse metrics coefficientsdefined respectively on hG = SO ( n + 1) and by off–blocks ( n × n )– and( m × m )–terms of the metric ˚ g αβ . This way, we can define an equivalent23–metric structure of type (A.1) ˚g = ˚ g ij e i ⊗ e j + ˚ h ab ˚e a ⊗ ˚e b , (14) e i = dx i , ˚e a = dy a + ˚ N ai dx i defining a trivial ( n + m )–splitting ˚g =˚ g ⊕ ˚N ˚ h because all nonholonomy co-efficients ˚ W γαβ and N–connection curvature coefficients ˚Ω aij are zero.It is possible to consider any covariant coordinate transforms of (14) pre-serving the ( n + m )–splitting resulting in w γαβ = 0 , see (A.12) and Ω aij = 0(A.13). Such trivial parametrizations define algebraic classifications of sym-metric Riemannian spaces of dimension n + m with constant matrix curva-ture admitting splitting (by certain algebraic constraints) into symmetricRiemannian subspaces of dimension n and m, also both with constant ma-trix curvature. This way, we get the simplest example of nonholonomicRiemannian space of type ˚V = [ hG = SO ( n + 1) , vG = SO ( m + 1) , ˚ N ei ]possessing a Lie d–algebra symmetry so ˚ N ( n + m ) + so ( n ) ⊕ so ( m ) . We can generalize the constructions if we consider nonholonomic distri-butions on V = G/SO ( n + m ) defined locally by arbitrary N–connectioncoefficients N ai ( x, y ) , with nonvanishing w γαβ and Ω aij but with constant d–metric coefficients when ˚g = g = ˚ g ij e i ⊗ e j + ˚ h ab e a ⊗ e b , (15) e i = dx i , e a = dy a + N ai ( x, y ) dx i . This metric is equivalent to a d–metric g α ′ β ′ = [ g i ′ j ′ , h a ′ b ′ ] (A.1) with con-stant coefficients induced by the corresponding Lie d–algebra structure so ˚ N ( n + m ) . Such spaces transform into nonholonomic manifolds ˚V N =[ hG = SO ( n + 1) , vG = SO ( m + 1) , N ei ] with nontrivial N–connection cur-vature and induced d–torsion coefficients of the d–connection (A.19). Onehas zero curvature for this d–connection (in general, such spaces are curvedones with generic off–diagonal metric (15) and nonzero curvature tensor forthe Levi–Civita connection). So, such nonholonomic manifolds posses thesame group and algebraic structures of couples of symmetric Riemannianspaces of dimension n and m but nonholonomically soldered to the sym-metric Riemannian space of dimension n + m. With respect to N–adaptedorthonormal bases, with distinguished h– and v–subspaces, we obtain thesame inner products and group and Lie algebra spaces as in (12).The bi–Hamiltonian and solitonic constructions are based on an extrinsicapproach soldering the Riemannian symmetric–space geometry to the Kleingeometry [44]. For the N–anholonomic spaces of dimension n + m, with a24onstant d–curvature, similar constructions hold true but we have to adaptthem to the N–connection structure. In Ref. [39], we proved that any(pseudo) Riemannian metric g on V defines a set of metric compatible d–connections of type e Γ γ ′ α ′ β ′ = (cid:16)b L i ′ j ′ k ′ = 0 , b L a ′ b ′ k ′ = b L a ′ b ′ k ′ = const, b C i ′ j ′ c ′ = 0 , b C a ′ b ′ c ′ = 0 (cid:17) (16)with respect to N–adapted frames (A.10) and (A.11) for any N = { N a ′ i ′ ( x, y ) } being a nontrivial solution of the system of equations2 b L a ′ b ′ k ′ = ∂N a ′ k ′ ∂y b ′ − ˚ h a ′ c ′ ˚ h d ′ b ′ ∂N d ′ k ′ ∂y c ′ (17)for any nondegenerate constant–coefficients symmetric matrix ˚ h d ′ b ′ and itsinverse ˚ h a ′ c ′ . Here, we emphasize that the coefficients p Γ γ ′ α ′ β ′ of the corre-sponding to g Levi–Civita connection g ∇ are not constant with respect toN–adapted frames.By straightforward computations, we get that the curvature d–tensor ofa d–connection e Γ γ ′ α ′ β ′ (16) defined by a metric g has constant coefficients e R α ′ β ′ γ ′ δ ′ = ( e R i ′ h ′ j ′ k ′ = 0 , e R a ′ b ′ j ′ k ′ = b L c ′ b ′ j ′ b L a ′ c ′ k ′ − b L c ′ b ′ k ′ b L a ′ c ′ j ′ = cons, e P i ′ h ′ j ′ a ′ = 0 , e P c ′ b ′ j ′ a ′ = 0 , e S i ′ j ′ b ′ c ′ = 0 , e S a ′ b ′ d ′ c ′ = 0)with respect to N–adapted frames e α ′ = [ e i ′ , e a ′ ] and e α ′ = [ e i ′ , e a ′ ] when N d ′ k ′ are subjected to the conditions (17). Using a deformation relation oftype (A.27), we can compute the corresponding Ricci tensor p R αβγδ forthe Levi–Civita connection g ∇ , which is a general one with ’non-constant’coefficients with respect to any local frames.A d–connection e Γ γ ′ α ′ β ′ (16) has constant scalar curvature, ∼ ←→ R + g α ′ β ′ e R α ′ β ′ = ˚ g i ′ j ′ e R i ′ j ′ + ˚ h a ′ b ′ e S a ′ b ′ = ∼ −→ R + ∼ ←− S = const. Nevertheless, the scalar curvature ∇ R of g ∇ , for the same metric, is notconstant.The constructions with different types of metric compatible connectionsgenerated by a metric structure are summarized in Table 1.We conclude that the algebraic structure of nonholonomic spaces en-abled with N–connection structure is defined by a conventional splittingof dimensions with certain holonomic and nonholonomic variables (defin-ing a distribution of horizontal and vertical subspaces). Such subspaces are25able 1: Some metric connections generated by a d–metric g = { g αβ } Geometric Levi–Civita normal d–connection constant coefficientsobjects for: connection d–connectionCo-frames e β = A ββ ( u ) du β e α = [ e i = dx i , e α ′ = [ e i ′ = dx i ′ , e a = dy a − N aj dx j ] e a ′ = dy a ′ − N a ′ j ′ dx j ′ ]Metric decomp. g αβ = A αα A ββ g αβ g αβ = [ g ij , h ab ] , g α ′ β ′ = [ ˚ g i ′ j ′ , ˚ h a ′ b ′ ] , g = g ij e i ⊗ e j g = ˚ g i ′ j ′ e i ′ ⊗ e j ′ + h ab e a ⊗ e b +˚ h a ′ b ′ e a ′ ⊗ e b ′ g i ′ j ′ = A ii ′ A jj ′ g ij ,h a ′ b ′ = A aa ′ A bb ′ h ab Connections p Γ γαβ p Γ γαβ = b Γ γαβ + p Z γαβ e Γ γ ′ α ′ β ′ = ( b L i ′ j ′ k ′ = 0 , and distorsions b L a ′ b ′ k ′ = b L a ′ b ′ k ′ = const., b C i ′ j ′ c ′ = 0 , b C a ′ b ′ c ′ = 0)Riemannian p R αβγδ b R αβγδ e R α ′ β ′ γ ′ δ ′ = (0 , e R a ′ b ′ j ′ k ′ (d–)tensors = const., , , , modelled locally as Riemannian symmetric manifolds and their propertiesare exhausted by the geometry of distinguished Lie groups G = GO ( n ) ⊕ GO ( m ) and G = SU ( n ) ⊕ SU ( m ) and the geometry of N–connections on aconventional vector bundle with base manifold M, dim M = n, and typicalfiber F, dim F = m. For constructions related to Einstein gravity, we haveto consider n = 2 and m = 2 . This can be formulated equivalently in termsof geometric objects on couples of Klein spaces. The bi–Hamiltonian andrelated solitonic (of type mKdV and SG) hierarchies are generated naturallyby wave map equations and recursion operators associated to the horizontaland vertical flows of curves on such spaces [38, 39, 40, 41].The approach allowed us to elaborate a ”solitonic” formalism when thegeometry of (semi) Riemannian / Einstein manifolds is encoded into non-holonomic hierarchies of bi–Hamiltonian structures and related solitonicequations derived for curve flows on spaces with conventional splitting ofdimensions. The same distinguished group (d–group) formalism may be ap-plied for quantum string models and nonholonomic A–branes for the almostK¨ahler model of Einstein gravity.
Let us consider a metric field g which is a solution of usual Einsteinequations (3). For different purposes, we can work equivalently with any26inear connection g ∇ = { g p Γ αβγ } , b D = { b Γ γαβ } , or e D = { e Γ γαβ } , for g and/or g θ ( X , Y ) + g ( JX , Y ) . For any g θ = ( h θ, v θ ) , we can associatetwo symplectic forms related formally to a Chern–Simons theory with acompact gauge d–group G = GO (2) ⊕ GO (2) , or G = SU (2) ⊕ SU (2) (forsimplicity, hereafter we shall consider the case of unitary d–groups, onefor the h–part, h G = SU (2) , and another for the v–part, v G = SU (2)) . We chose two two–manifolds without boundary, denoted h C and v C andconsider G V = ( h V, v V ) to be defined by a couple of moduli spaces ofhomomorphisms (up to conjugation) from π ( h C ) into h G and, respectively, π ( v C ) into v G, of given topological types. We can consider the same localcoordinate parametrization for G V and open regions of a nonholonomicspacetime V . The further geometric constructions are related to a symplecticd–structure on the infinite–dimensional linear d–spaces G A = ( h A , v A )as all couples of d–connections on a distinguished G –bundle E → C , for C = h C ⊕ v C. We fix such a parametrization of coefficients of a gravitation symplecticd–form g θ = ( h θ, v θ ) which in a point of V is proportional to the coefficients h ∗ θ = 14 π Z h C T r δ h A ∧ δ h A and v ∗ θ = 14 π Z v C T r δ v A ∧ δ v A, (18)when, in brief, in ”boldfaced” form, g ∗ θ = π R C Tr δ A ∧ δ A . The tracesymbol
T r is considered respectively for the h – and v –forms, as invariantones on the Lie algebras h g , of h G, and v g , of v G, in our case, in the2–dimensional representation. Such h ∗ θ, or v ∗ θ, are normalized to ∗ θ/ π be-ing the image in de Rham cohomology of a generator of H ( h V, Z ) ∼ = Z , or H ( v V, Z ) ∼ = Z . Here, it should be emphasized that such local identificationsof the gravitational almost K¨ahler symplectic structures with couples of sym-plectic structure of respective Chern–Simons theories (with G V modellingthe classical phase d–spaces for such models) do not impose elaborationof classical and/or gravitational models with structures d–groups of type G . We only fixed an explicit common parametrization for a ”background”curve flow network the chosen method of quantization and generating grav-itational solitonic hierarchies, like in [39]. Real classical/quantum Einsteingravitational interactions can be generated by deformations of connections g ∇ = g D + g Z (1), where g D is any necessary type connection, for instance,parametrized as a gauge one in a Chern–Simons d–model, A =( h A, v A )with coefficients determined by a d–connection (A.22), or any its nonholo-nomic transform, but the related g Z is such a distorsion tensor which non-holonomically deforms g D into g ∇ defined by an Einstein solution, in the27lassical limit. Any such schemes with equivalent geometric objects definedby a d–metric g but for suitable nonholonomic and topologic structures cor-respond to a N–connection splitting and adapted frames as it is describedin Figure 2. ✤✣ ✜✢ Levi–Civita variables:( g , g ∇ ) = ( g µν , g p Γ αβγ ) L ( x, y ) nonholonomic N–adaptedframe transforms almost K¨ahler variables:( g θ, b D , L b N ) = ( θ αβ , b Γ γαβ ) constant coefficient variables:( ◦ g αβ , ◦ D , N )metricity: b Dg = 0 , g p Γ αβγ = b Γ αβγ + g p Z αβγ metricity: ◦ D ◦ g = 0 , g p Γ αβγ = ◦ Γ αβγ + ◦ p Z αβγ Classical Einstein Spaces ❄❄ (cid:0)(cid:0)✠❅❅❘ ❄
Quantum almost K¨ahler Einstein Spaces ❅❅❅❅❅❅❅❘(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠
Figure 2:
The Levi–Civita, normal and constant coefficients con-nections in Einstein gravity
Next we introduce a distinguished line bundle (line d–bundle). Fixingtwo integers b k = ( h b k, v b k ) , we can quantize our nonholonomic model G V asin [1] but using a symplectic d–form ∗ θ = ( h b k h ∗ θ, v b k v ∗ θ ) with a prequantum This can be constructed as in usual Chern–Simons theory [45, 46] but using d–groups. ∗ V = h b k ∗ V ⊕ v b k ∗ V . Taking G Y to be the distinguishedmoduli space of homomorphysms (preserving the splitting by a prescribedN–connection structure) from π ( C ) to G C = h G C ⊕ v G C , for C = h C ⊕ v C and h G C and v G C being the respective complexifications of the respective h–and v–groups (up to conjugation), we define G Y as a natural noholonomiccomplex manifold.We denote by r = ( h r, v r ) certain finite–dimensional representationsof complex Lie d–group h G C ⊕ v G C and consider two oriented closedloops s = ( h s, v s ) on h C ⊕ v C and the holonomies of respective twoflat connections (defining a d–connection) around s , denoted respectively Hol ( s ) = Hol ( h s ) ⊕ Hol ( v s ) . This way, we define H r ( s ) = T r h r Hol ( h s ) + T r v r Hol ( v s ) (19)is a holomorphic function on G Y . For a gauge g C –valued d–connection A =( h A, v A ) , the function (19)can be written using an oriented exponential product P on both h – and v –subspaces, H r ( s ) = T r P exp (cid:18) − I h s h A (cid:19) + T r P exp (cid:18) − I v s v A (cid:19) . The restrictions of such holomorphic functions on G Y define a dense setof functions on G V associated to V . Using nonholonomic transforms pre-serving the N–connection structure, and corresponding deformations of d–connections, we can relate such distinguished group constructions to thosewith a gravitationally induced symplectic form on Y . The g C –valued d–connection A also generates a nondegenerate holomor-phic distinguished 2–form ∗ Θ = (cid:0) h ∗ Θ , v ∗ Θ (cid:1) ; we use the complexified for-mulas (19). We can consider G Y as a complex symplectic manifold enabledby symplectic d–form G Θ = (cid:16) h b k h ∗ Θ , v b k v ∗ Θ (cid:17) , with a restriction of Θ to G V coinciding with ∗ θ. This also allows us to construct a nonholonomicA–model of G Y with symplectic structure GY θ = Im G Θ like we have doneat the beginning of section 3.2. Such an A–model dubs the constructionsfrom [1] (see there sections 1.3 and 2.3) and also can be endowed with acomplete hyper–K¨ahler metric (consisting from h– and v–parts) extendingits structure as a complex symplex manifold. It is a ”very good” A–model, The end of this section, we shall discuss how we can nonholonomically deform themanifold G Y and its fundamental geometric symplectic structures into a manifold Y withinduced gravitational symplectic variables. C = h C ⊕ v C and definecomplex structures on G Y in a natural way (requiring no structures on h C and v C except corresponding orientations).There is also a natural antiholomorphic involution G τ : G Y → G Y as acomplex conjugation of al monodromies (preserving h– and v–components),were G V is the component of the fixed point set of G τ (i.e. is is the locusin G Y of all monodromie with values in G = SU (2) ⊕ SU (2)) . Having introduces two branes in the A–model of G Y , we can performthe Gukov–Witten quantization of G V . The first brane is a distinguishedone consisting from two canonical isotropic branes, G B cc = ( h B cc , v B cc ) , with curvature form Re G Θ and support by all G Y . We wont to quantizethe symplectic d–form ∗ θ = ( h b k h ∗ θ, v b k v ∗ θ ) which is the restriction of Re G Θ to G V . The second brane G B ′ is defined as a unique one up to anisomorphism preserving h– and v–splitting when G V is a simply–connectedspin manifolds; this is also a rank 1 A–brane supported on G V . The N–adapted diffeomorphisms of C = h C ⊕ v C that are continuouslyconnected to the identity on h –part and identity on v –part act triviallyboth on the A–model of G Y and on G Y . Such diffeomorphisms do notpreserve hyper–K¨ahler metrics on G Y , but this is not a problem becausethe A–model observables do not depend on a fixed hyper–K¨ahler metric,see more details in [10], on nonholonomic manifolds and Hamilton–Cartanspaces and their deformation quantization. Following [1], we can considerthe Teichmuller space T of C when any point ζ ∈ T determines (in a uniqueway, up to isotopy) a complex structure on C and (as a result) a hyper–K¨ahler polarization of (cid:0) G Y , G V (cid:1) . The interesting thing is that the space G ζ H of (cid:0) G B cc , G B ′ (cid:1) strings constructed with such a polarization does notdepend on ζ. This follows from the fact that the A–model is invariant undera local change in the hyper–K¨ahler poarlization and we can also define G ζ H as a typical fiber of a flat d–vector bundle G H over T . In general, it is a difficult task to compute exactly the (Hilbert) space G H for certain general nonholonomic manifolds (this would imply the indextheorem for a family of nonholonomic Dirac operators, see [36, 37, 11]). Nev-ertheless, in the case relevant to Einstein gravity and solitonic hierarchies,i.e. for G = SU (2) ⊕ SU (2) , one holds a standard (N–adapted) algebro–geometric description of a physical Hilbert space of a nonholonomic Chern–Simons theory at levels b k = ( h b k, v b k ) , when G H = H ( G V , h b k ∗ V ⊕ v b k ∗ V ) . Here one should be noted that in the nonholonomic A–model the (classical)commutative algebra of holonomy functions H r ( s ) (19) is nonholonomi-cally deformed to a noncommutative d–algebra A = ( h A , v A ) , i.e. the30pace of (cid:0) G B cc , G B cc (cid:1) strings, which acts on G H . When we work with anonholonomic Chern–Simon gauge theory, the quantization transforms Will-son loops on a distingushed C = h C ⊕ v C into operators that acts on thequantum Hilbert space.The Gukov–Witten quantization method is very powerful because it al-lows us to consider nonholonomic deformations of geometric classical andquantum structures and the very same d–algebra A = ( h A , v A ) acts on thespaces of any nonholonomic strings for any other choice of nonholonomic A–brane B . Let us explain these new applications to gravity and nonholonomicgeometries which were not considered in [1]:The coefficients of the gauge g C –valued d–connection A =( h A, v A ) usedfor constructing our nonholonomic Chern–Simons theory can be identifiedwith the coefficients of a d–connection of type b Γ ij = b L ijk e k + b C ijk e k (A.22),but with constant d–connection coefficients, i.e. ˜Γ γ ′ α ′ β ′ (16), when via cor-responding distorsion tensor, see Figure 2, ˜Γ ij = ˜ L ijk e k with h A ijk = ˜ L ijk and v A = 0 (but this may be for an explicit local parametrization, in general v A is not zero). Further nonholonomic transforms from ˜Γ to b Γ change thenonholonomic structure of geometric objects but does not affect the definedabove hyper–K¨ahler polarization. This means that the former g C –valuedd–connection structure A =( h A, v A ) relevant to a chosen parametriza-tion of curve flows, for spaces (cid:0) G Y , G V (cid:1) , deforms nonholonomically, butequivalently, into a couple of nonholonomic manifolds ( g Y , g V ) if the met-ric structure of d–group G = SU (2) ⊕ SU (2) maps nonholonomically intoa d–metric g (A.1). For such constructions, the corresponding (classical)Levi–Civita connection g p Γ is constrained to be a solution of the Einsteinequations (3). Using vierbein transforms of type (A.6) relating ( L g , L N ) , see(A.5) and (A.7), to an ”Einstein solution” ( g , N ) , and similar transformsto ( ˚g , N ) (15), we nonholonomically deform the space of (cid:0) G B cc , G B cc (cid:1) strings into that of ( g B cc , g B cc ) strings. This obviously result in equiva-lent transforms of G H = H ( G V , h b k ∗ V ⊕ v b k ∗ V ) into the Hilber space g H = ⊕ dim C V j =0 H j ( V , K / ⊗ g V ) (10) which is a good approximation forboth holonomic and nonholonomic quantum Einstein spaces.Finally, we conclude that crucial for such a quantization with nonholo-nomic A–branes and strings relevant to Einstein gravity are the construc-tions when we define the almost K¨ahler variables in general relativity, inSection 2, and coordinate parametrizations for almost K¨ahler gravitationalA–models, in Section 4.1. The associated constructions with a nonholo-nomic Chern–Simons theory for a gauge d–group G = SU (2) ⊕ SU (2) arealso important because they allow us to apply both a computation tech-31iques formally elaborated for topological gauge models and relate the con-structions to further developments for quantum curve flows, nonholonomicbi–Hamilton structures and derived solitonic hierarchies. In the present paper we have applied the Gukov–Witten formalism [1] toquantize the almost K¨ahler model of Einstein gravity. We have used someour former results on deformation and loop quantization of gravity follow-ing ideas and methods from the geometry of nonholonomic manifolds and(non) commutative spaces enabled with nonholonomic distributions and as-sociated nonlinear connections structures [8, 9, 16, 18]. It was shown thatthe approach with nonholonomic A–branes endowed with geometric struc-tures induced by (pseudo) Riemannian metrics serves to quantize standardgravity theories and redefine previous geometric results on the languageof string theories and branes subjected to different types of nonholonomicconstrains.The A–model approach to quantization [1, 24, 25, 46] seems to be efficientfor elaborating quantum versions of (non) commutative gauge models ofgravity [29, 32], nonholonomic Clifford–Lie algebroid systems [35, 34], gerbesand Clifford modules [36, 37] etc when a synthesis with the methods ofgeometric [20, 21, 22, 23] and deformation [3, 4, 5] quantization is considered.Further perspectives are related to nonholonomic Ricci flows and almostK¨ahler models of spaces with symmetric and nonsymmetric metrics [19, 40].It was shown that deformation quantization of the relativistic particlesgives the same results as the canonical quantization and path integral meth-ods and the direction was developed for systems with second class con-straints. On such results, we cite a series of works on commutative andnoncommutative physical models of particles and strings [47, 48, 49, 50, 51,52, 53, 6, 7] and emphasize that the Stratonovich–Weyl quantizer, Weylcorrespondence, Moyal product and the Wigner function were obtained forall the analyzed systems which allows a straightforward generalization tononholonomic spaces and related models of gravity, gauge and spinor in-teractions and strings [8, 9, 29, 32, 27, 28]. Introducing almost K¨ahlervariables for gravity theories, such constructions and generalizations canbe deformed nonholonomically to relate (for certain well defined limits)the Gukov–Witten quantization to deformation and geometric quantization, and other more general nonholonomic geometric and physical models, for instance,various types Finsler–Lagrange and Hamilton–Cartan spaces etc Acknowledgements
The work was partially performed during a visitat Fields Institute, Toronto.
A Almost K¨ahler Variables in Component Form
We parametrize a general (pseudo) Riemannian metric g on a spacetime V in the form: g = g i ′ j ′ ( u ) e i ′ ⊗ e j ′ + h a ′ b ′ ( u ) e a ′ ⊗ e b ′ , (A.1) e a ′ = e a ′ − N a ′ i ′ ( u ) e i ′ , where the vierbein coefficients e α ′ α of the dual basis e α ′ = ( e i ′ , e a ′ ) = e α ′ α ( u ) du α , (A.2)define a formal 2 + 2 splitting.Let us consider any generating function L ( u ) = L ( x i , y a ) on V (it isa formal pseudo–Lagrangian if an effective continuous mechanical modelof general relativity is elaborated, see Refs. [12, 11]) with nondegenerateHessian L h ab = 12 ∂ L∂y a ∂y b , (A.3)when det | L h ab | 6 = 0 . We define L N ai = ∂G a ∂y i , (A.4) G a = 14 L h a i (cid:18) ∂ L∂y i ∂x k y k − ∂L∂x i (cid:19) , L h ab is inverse to L h ab and respective contractions of h – and v –indices, i, j, ... and a, b..., are performed following the rule: we can write,for instance, an up v –index a as a = 2 + i and contract it with a low index i = 1 , . Briefly, we shall write y i instead of y i , or y a . The values (A.3)and (A.4) allow us to consider L g = L g ij dx i ⊗ dx j + L h ab L e a ⊗ L e b , (A.5) L e a = dy a + L N ai dx i , L g ij = L h i j . A metric g (A.1) with coefficients g α ′ β ′ = [ g i ′ j ′ , h a ′ b ′ ] computed withrespect to a dual basis e α ′ = ( e i ′ , e a ′ ) can be related to the metric L g αβ =[ L g ij , L h ab ] (A.5) with coefficients defined with respect to a N–adapted dualbasis L e α = ( dx i , L e a ) if there are satisfied the conditions g α ′ β ′ e α ′ α e β ′ β = L g αβ . (A.6)Considering any given values g α ′ β ′ and L g αβ , we have to solve a systemof quadratic algebraic equations with unknown variables e α ′ α , see details inRef. [16]. Usually, for given values [ g i ′ j ′ , h a ′ b ′ , N a ′ i ′ ] and [ L g ij , L h ab , L N ai ] , we can write N a ′ i ′ = e ii ′ e a ′ a L N ai (A.7)for e ii ′ being inverse to e i ′ i . A nonlinear connection (N–connection) structure N on V can be intro-duced as a nonholonomic distribution (a Whitney sum) T V = h V ⊕ v V (A.8)into conventional horizontal (h) and vertical (v) subspaces. In local form, aN–connection is given by its coefficients N ai ( u ) , when N = N ai ( u ) dx i ⊗ ∂∂y a . (A.9)A N–connection on V n + n induces a (N–adapted) frame (vielbein) struc-ture e ν = (cid:18) e i = ∂∂x i − N ai ( u ) ∂∂y a , e a = ∂∂y a (cid:19) , (A.10)and a dual frame (coframe) structure e µ = (cid:0) e i = dx i , e a = dy a + N ai ( u ) dx i (cid:1) . (A.11)35he vielbeins (A.11) satisfy the nonholonomy relations[ e α , e β ] = e α e β − e β e α = w γαβ e γ (A.12)with (antisymmetric) nontrivial anholonomy coefficients w bia = ∂ a N bi and w aji = Ω aij , where Ω aij = e j ( N ai ) − e i (cid:0) N aj (cid:1) (A.13)are the coefficients of N–connection curvature (defined as the Neijenhuistensor on V n + n ) . The particular holonomic/ integrable case is selected bythe integrability conditions w γαβ = 0 . A N–anholonomic manifold is a (nonholonomic) manifold enabled withN–connection structure (A.8). The geometric properties of a N–anholonomicmanifold are distinguished by some N–adapted bases (A.10) and (A.11). Ageometric object is N–adapted (equivalently, distinguished), i.e. a d–object,if it can be defined by components adapted to the splitting (A.8) (one usesterms d–vector, d–form, d–tensor). For instance, a d–vector X = X α e α = X i e i + X a e a and a one d–form e X (dual to X ) is e X = X α e α = X i e i + X a e a . We introduce a linear operator J acting on vectors on V following for-mulas J ( e i ) = − e i and J ( e i ) = e i , where and J ◦ J = − I , for I beingthe unity matrix, and construct a tensor field on V , J = J αβ e α ⊗ e β = J αβ ∂∂u α ⊗ du β (A.14)= J α ′ β ′ e α ′ ⊗ e β ′ = − e i ⊗ e i + e i ⊗ e i = − ∂∂y i ⊗ dx i + (cid:18) ∂∂x i − L N ji ∂∂y j (cid:19) ⊗ (cid:16) dy i + L N ik dx k (cid:17) , defining globally an almost complex structure on V completely determinedby a fixed L ( x, y ) . In this work we consider only structures J = L J inducedby a L N ji , i.e. one should be written L J , but, for simplicity, we shall omitleft label L, because the constructions hold true for any regular generatingfunction L ( x, y ) . Using vielbeins e αα and their duals e αα , defined by e α ′ α solving (A.6), we can compute the coefficients of tensor J with respect toany local basis e α and e α on V , J αβ = e αα J αβ e ββ . In general, we can define we use boldface symbols for spaces (and geometric objects on such spaces) enabledwith N–connection structure We can redefine equivalently the geometric constructions for arbitrary frame andcoordinate systems; the N–adapted constructions allow us to preserve the conventionalh– and v–splitting.
36n almost complex structure J for an arbitrary N–connection N , stating anonholonomic 2 + 2 splitting, by using N–adapted bases (A.10) and (A.11).The Neijenhuis tensor field for any almost complex structure J definedby a N–connection (equivalently, the curvature of N–connection) is J Ω ( X , Y ) + − [ X , Y ] + [ JX , JY ] − J [ JX , Y ] − J [ X , JY ] , (A.15)for any d–vectors X and Y . With respect to N–adapted bases (A.10) and(A.11), a subset of the coefficients of the Neijenhuis tensor defines the N–connection curvature,Ω aij = ∂N ai ∂x j − ∂N aj ∂x i + N bi ∂N aj ∂y b − N bj ∂N ai ∂y b . (A.16)A N–anholonomic manifold V is integrable if Ω aij = 0 . We get a complexstructure if and only if both the h– and v–distributions are integrable, i.e.if and only if Ω aij = 0 and ∂N aj ∂y i − ∂N ai ∂y j = 0 . One calls an almost symplectic structure on a manifold V a nondegen-erate 2–form θ = 12 θ αβ ( u ) e α ∧ e β = 12 θ ij ( u ) e i ∧ e j + 12 θ ab ( u ) e a ∧ e b . An almost Hermitian model of a (pseudo) Riemannian space V equippedwith a N–connection structure N is defined by a triple H = ( V , θ, J ) , where θ ( X , Y ) + g ( JX , Y ) for any g (A.1). A space H is almost K¨ahler,denoted K , if and only if dθ = 0 . For g = L g (A.5) and structures L N (A.4) and J canonically definedby L, we define L θ ( X , Y ) + L g ( JX , Y ) for any d–vectors X and Y . Inlocal N–adapted form form, we have L θ = 12 L θ αβ ( u ) e α ∧ e β = 12 L θ αβ ( u ) du α ∧ du β (A.17)= L g ij ( x, y ) e i ∧ dx j = L g ij ( x, y )( dy i + L N ik dx k ) ∧ dx j . Let us consider the form L ω = ∂L∂y i dx i . A straightforward computationshows that L θ = d L ω, which means that d L θ = dd L ω = 0 , i.e. thecanonical effective Lagrange structures g = L g , L N and J induce analmost K¨ahler geometry. We can express the 2–form (A.17) as θ = L θ = 12 L θ ij ( u ) e i ∧ e j + 12 L θ ab ( u ) e a ∧ e b (A.18)= g ij ( x, y ) h dy i + N ik ( x, y ) dx k i ∧ dx j , L θ ab = L θ i j are equal respectively to the coeffi-cients L θ ij . It should be noted that for a general 2–form θ constructed forany metric g and almost complex J structures on V one holds dθ = 0 . Butfor any 2 + 2 splitting induced by an effective Lagrange generating function,we have d L θ = 0 . We have also d θ = 0 for any set of 2–form coefficients θ α ′ β ′ e α ′ α e β ′ β = L θ α ′ β ′ (such a 2–form θ will be called to be a canonical one).We conclude that having chosen a regular generating function L ( x, y ) ona (pseudo) Riemannian spacetime V , we can always model this spacetimeequivalently as an almost K¨ahler manifold.A distinguished connection (in brief, d–connection) on a spacetime V , D = ( hD ; vD ) = { Γ αβγ = ( L ijk , v L abk ; C ijc , v C abc ) } , (A.19)is a linear connection which preserves under parallel transports the distribu-tion (A.8). In explicit form, the coefficients Γ αβγ are computed with respectto a N–adapted basis (A.10) and (A.11). A d–connection D is metric com-patible with a d–metric g if D X g = 0 for any d–vector field X . If an almost symplectic structure θ is considered on a N–anholonomicmanifold, an almost symplectic d–connection θ D on V is defined by theconditions that it is N–adapted, i.e. it is a d–connection, and θ D X θ = 0 , forany d–vector X . From the set of metric and/or almost symplectic compatibled–connections on a (pseudo) Riemannian manifold V , we can select thosewhich are completely defined by a metric g = L g (A.5) and an effectiveLagrange structure L ( x, y ) :There is a unique normal d–connection b D = n h b D = ( b D k , v b D k = b D k ); v b D = ( b D c , v b D c = b D c ) o (A.20)= { b Γ αβγ = ( b L ijk , v b L i j k = b L ijk ; b C ijc = v b C i j c , v b C abc = b C abc ) } , which is metric compatible, b D k L g ij = 0 and b D c L g ij = 0 , and completelydefined by a couple of h– and v–components b D α = ( b D k , b D c ) , with N–adapted coefficients b Γ αβγ = ( b L ijk , v b C abc ) , where b L ijk = 12 L g ih (cid:0) e k L g jh + e j L g hk − e h L g jk (cid:1) , (A.21) b C ijk = 12 L g ih (cid:18) ∂ L g jh ∂y k + ∂ L g hk ∂y j − ∂ L g jk ∂y h (cid:19) . In general, we can omit label L and work with arbitrary g α ′ β ′ and b Γ α ′ β ′ γ ′ withthe coefficients recomputed by frame transforms (A.2).38ntroducing the normal d–connection 1–form b Γ ij = b L ijk e k + b C ijk e k , (A.22)we prove that the Cartan structure equations are satisfied, de k − e j ∧ b Γ kj = − b T i , d e k − e j ∧ b Γ kj = − v b T i , (A.23)and d b Γ ij − b Γ hj ∧ b Γ ih = − b R ij . (A.24)The h– and v–components of the torsion 2–form b T α = (cid:16) b T i , v b T i (cid:17) = b T ατβ e τ ∧ e β from (A.23) are computed b T i = b C ijk e j ∧ e k , v b T i = 12 L Ω ikj e k ∧ e j + ( ∂ L N ik ∂y j − b L ikj ) e k ∧ e j , (A.25)where L Ω ikj are coefficients of the curvature of the canonical N–connection N ik defined by formulas similar to (A.16). The formulas (A.25) parametrizethe h– and v–components of torsion b T αβγ in the form b T ijk = 0 , b T ijc = b C ijc , b T aij = L Ω aij , b T aib = e b (cid:0) L N ai (cid:1) − b L abi , b T abc = 0 . (A.26)It should be noted that b T vanishes on h- and v–subspaces, i.e. b T ijk = 0 and b T abc = 0 , but certain nontrivial h–v–components induced by the nonholo-nomic structure are defined canonically by g = L g (A.5) and L. Similar formulas holds true, for instance, for the Levi–Civita linear con-nection ▽ = { p Γ αβγ } which is uniquely defined by a metric structure byconditions p T = 0 and ▽ g = 0 . It should be noted that this connectionis not adapted to the distribution (A.8) because it does not preserve underparallelism the h- and v–distribution. Any geometric construction for thecanonical d–connection b D can be re–defined by the Levi–Civita connectionby using the formula p Γ γαβ = b Γ γαβ + p Z γαβ , (A.27)where the both connections p Γ γαβ and b Γ γαβ and the distorsion tensor p Z γαβ with N–adapted coefficients where p Z ajk = − C ijb g ik h ab −
12 Ω ajk , p Z ibk = 12 Ω cjk h cb g ji − Ξ ihjk C jhb , p Z abk = + Ξ abcd ◦ L cbk , p Z ikb = 12 Ω ajk h cb g ji + Ξ ihjk C jhb , p Z ijk = 0 , (A.28) p Z ajb = − − Ξ adcb ◦ L cdj , p Z abc = 0 , p Z iab = − g ij (cid:2) ◦ L caj h cb + ◦ L cbj h ca (cid:3) , ihjk = ( δ ij δ hk − g jk g ih ) , ± Ξ abcd = ( δ ac δ bd + h cd h ab ) and ◦ L caj = L caj − e a ( N cj ) . If we work with nonholonomic constraints on the dynamics/ geometry ofgravity fields, it is more convenient to use a N–adapted approach. For otherpurposes, it is preferred to use only the Levi–Civita connection.We compute also the curvature 2–form from (A.24), b R τγ = b R τγαβ e α ∧ e β (A.29)= 12 b R ijkh e k ∧ e h + b P ijka e k ∧ e a + 12 b S ijcd e c ∧ e d , where the nontrivial N–adapted coefficients of curvature b R αβγτ of b D are b R ihjk = e k b L ihj − e j b L ihk + b L mhj b L imk − b L mhk b L imj − b C iha L Ω akj (A.30) b P ijka = e a b L ijk − b D k b C ija , b S abcd = e d b C abc − e c b C abd + b C ebc b C aed − b C ebd b C aec . Contracting the first and forth indices b R βγ = b R αβγα , we get the N–adaptedcoefficients for the Ricci tensor b R βγ = (cid:16) b R ij , b R ia , b R ai , b R ab (cid:17) . (A.31)The scalar curvature L R = b R of b D is L R = L g βγ b R βγ = g β ′ γ ′ b R β ′ γ ′ . 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