lagrange fedosov nonholonomic manifolds we outline a unified approach to geometrization of lagrange mechanics finsler geometry and geometric methods of constructing exact solutions with generic off diagonal terms and nonholonomic variables in gravity theories such geometries with induced almost symplectic structure are modeled on nonholonomic manifolds provided with nonintegrable distributions defining nonlinear connections we introduce the concept of lagrange fedosov spaces and fedosov nonholonomic manifolds provided with almost symplectic connection adapted to the nonlinear connection structure we investigate the main properties of generalized fedosov nonholonomic manifolds and analyze exact solutions defining almost symplectic einstein spaces

the atiyah bott patodi method in deformation quantization we give a new proof of the index theorem for deformation quantization following the atiyah bott patodi scheme the main ingredient consists of tamarkin s theorem on topological invariants of symplectic connections which is a symplectic analog of the well known theorems for riemannian connections

controllability and observability of linear retarded systems in banach spaces abstract controllability and observability are studied for linear retarded systems in a banach space x both the x exact and the x approximate controllability are considered and a number of necessary and sufficient conditions in terms of the coefficient operators and or the fundamental solution are presented various corresponding concepts for observability are introduced and the duality between controllability and observability is clarified an applicable condition for the x approximate controllability and the x observability is established under the assumptions that the control is a finite dimensional one and the system of generalized eigenspaces is projectively complete the condition is stated in terms of eigenvectors and controllers

local well posedness for membranes in the light cone gauge in this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge a hamiltonian reduction gives a system with one constraint the area preserving constraint the hamiltonian evolution equations corresponding to this system however fail to be hyperbolic making use of the area preserving constraint an equivalent system of evolution equations is found which is hyperbolic and has a well posed initial value problem we are thus able to solve the initial value problem for the hamiltonian evolution equations by means of this equivalent system we furthermore obtain a blowup criterion for the membrane evolution equations and show making use of the constraint that one may achieve improved regularity estimates

lowast sdym fields and heavenly spaces i lowast sdym equations as an integrable system it is shown that the self dual yang mills sdym equations for the bracket lie algebra on a heavenly space can be reduced to one equation the master equation two hierarchies of conservation laws for this equation are constructed then the twistor transform and a solution to the riemann hilbert problem are given

a formal model of berezin toeplitz quantization we give a new construction of symbols of the differential operators on the sections of a quantum line bundle l over a kahler manifold m using the natural contravariant connection on l these symbols are the functions on the tangent bundle tm polynomial on fibres for high tensor powers of l the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on tm corresponding to some pseudo kahler structure on tm surprisingly this star product is intimately related to the formal symplectic groupoid with separation of variables over m we extend the star product on tm to generalized functions supported on the zero section of tm the resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the bergman projection operator using this idempotent we define an algebra of toeplitz elements and show that it is naturally isomorphic to the algebra of berezin toeplitz deformation quantization on m

fedosov and riemannian supermanifolds generalizations of symplectic and metric structures for supermanifolds are analyzed two types of structures are possible according to the even odd character of the corresponding quadratic tensors in the even case one has a very rich set of geometric structures even symplectic supermanifolds or equivalently supermanifolds with nondegenerate poisson structures even fedosov supermanifolds and even riemannian supermanifolds the existence of relations among those structures is analyzed in some detail in the odd case we show that odd riemannian and fedosov supermanifolds are characterized by a scalar curvature tensor however odd riemannian supermanifolds can only have a constant curvature supersymmetric extensions of anti de sitter spaces are considered

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

on gammelgaard s formula for a star product with separation of variables we show that gammelgaard s formula expressing a star product with separation of variables on a pseudo kahler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal fock space we prove this inversion formula directly and thus offer an alternative approach to gammelgaard s formula which gives more insight into the question why the directed graphs in his formula have no cycles

formality for lie algebroids using dolgushev s generalization of fedosov s method for deformation quantization we give a positive answer to a question of p xu can one prove a formality theorem for lie algebroids as a direct application of this result we obtain that any triangular lie bialgebroid is quantizable

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

l2 regularity for parabolic partial integrodifferential equations with delay in the highest order derivatives abstract a partial integrodifferential equation is studied in which the derivatives of highest order also contain a discrete and a distributed delay by means of abstract regularity results global existence and uniqueness of a strict solution are obtained moreover a characterization of the infinitesimal generator of the solution operator is given

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

gauge theories in noncommutative homogeneous kahler manifolds we construct a gauge theory on a noncommutative homogeneous kahler manifold where we employ the deformation quantization with separation of variables for kahler manifolds formulated by karabegov a key point in this construction is to obtaining vector fields which act as inner derivations for the deformation quantization we give an explicit construction of this gauge theory on noncommutative cpn and noncommutative chn

deformations of momentum maps and g systems in this note we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a lie group action on rd this family of deformations is parametrized by the formal g systems introduced in b dherin and i mencattini g systems and deformations of g actions on rd j math phys 55 011702 2014 see also b dherin and i mencattini quantization of volume preserving actions on rd e print arxiv 1202 0886 and allows us to obtain classical invariant hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal g systems

fedosov deformation quantization as a brst theory the relationship is established between the fedosov deformation quantization of a general symplectic manifold and the bfv brst quantization of constrained dynamical systems the original symplectic manifold is presented as a second class constrained surface in the fibre bundle which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure the second class system is converted into the first class one by continuation of the constraints into the extended manifold being a direct sum of and the tangent bundle t this extended manifold is equipped with a nontrivial poisson bracket which naturally involves two basic ingredients of fedosov geometry the symplectic structure and the symplectic connection the constructed first class constrained theory being equivalent to the original symplectic manifold is quantized through the bfv brst procedure the existence theorem is proven for the quantum brst charge and the quantum brst invariant observables the adjoint action of the quantum brst charge is identified with the abelian fedosov connection while any observable being proven to be a unique brst invariant continuation for the values defined in the original symplectic manifold is identified with the fedosov flat section of the weyl bundle the fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum brst invariant observables

deformation quantization of almost kahler models and lagrange finsler spaces finsler and lagrange spaces can be equivalently represented as almost kahler manifolds endowed with a metric compatible canonical distinguished connection structure generalizing the levi civita connection the goal of this paper is to perform a natural fedosov type deformation quantization of such geometries all constructions are canonically derived for regular lagrangians and or fundamental finsler functions on tangent bundles

formal series of generalized functions and their application to deformation quantization foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed

non abelian conversion and quantization of nonscalar second class constraints we propose a general method for the deformation quantization of any second class constrained system on a symplectic manifold the constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a nontrivial vector bundle over the phase space manifold the covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the constraint bundle which becomes a key ingredient of the conversion procedure for the nonscalar constraints unlike in the case of scalar second class constraints no abelian conversion is possible in general within the brst framework a systematic procedure is worked out for converting nonscalar second class constraints into non abelian first class ones the brst extended system is quantized yielding an explicitly covariant quantization of the original system an important feature of second class systems with nonscalar constraints is that the appropriately g

on higher order relations in fedosov supermanifolds higher order relations existing in normal coordinates between affine extensions of the symplectic curvature tensor and basic objects for any fedosov supermanifolds are derived representation of these relations in general coordinates is discussed

fedosov dg manifolds associated with lie pairs given any pair l a of lie algebroids we construct a differential graded manifold l 1 oplus l a q which we call fedosov dg manifold we prove that the homological vector field q constructed on l 1 oplus l a by the fedosov iteration method arises as a byproduct of the poincare birkhoff witt map established in 18 finally using the homological perturbation lemma we establish a quasi isomorphism of dolgushev fedosov type the differential graded algebras of functions on the dg manifolds a 1 d a and l 1 oplus l a q are homotopy equivalent

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

homogeneous fedosov star products on cotangent bundles i weyl and standard ordering with differential operator representation in this paper we construct homogeneous star products of weyl type on every cotangent bundle t q by means of the fedosov procedure using a symplectic torsion free connection on t q homogeneous of degree zero with respect to the liouville vector field by a fibrewise equivalence transformation we construct a homogeneous fedosov star product of standard ordered type equivalent to the homogeneous fedosov star product of weyl type representations for both star product algebras by differential operators on functions on q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex valued functions on t q polynomial in the momenta where an arbitrary fixed torsion free connection 0 on q is used motivated by the flat case t n another homogeneous star product of weyl type corresponding to the weyl ordering prescription is constructed the example of the cotangent bundle of an arbitrary lie group is explicitly computed and the star product given by gutt is rederived in our approach

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

a formality theorem for hochschild chains abstract we prove tsygan s formality conjecture for hochschild chains of the algebra of functions on an arbitrary smooth manifold m using the fedosov resolutions proposed in math qa 0307212 and the formality quasi isomorphism for hochschild chains of r y 1 y d proposed in paper math qa 0010321 by shoikhet this result allows us to describe traces on the quantum algebra of functions on an arbitrary poisson manifold

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

fedosov quantization on symplectic ringed spaces we expose the basics of the fedosov quantization procedure placed in the general framework of symplectic ringed spaces this framework also includes some poisson manifolds with nonregular poisson structures presymplectic manifolds complex analytic symplectic manifolds etc

formal gns construction and states in deformation quantization in this paper we develop a method of constructing hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well known gns construction for complex c algebras in this approach the corresponding positive linear functionals states take their values not in the field of complex numbers but in a suitable extension field of the field of formal complex laurent series in the formal parameter by using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations such as the bargmann and schrodinger representation come out correctly both formally and in a suitable convergence scheme for certain hamiltonian functions contained in the gel fand ideal of the positive functional a formal solution to the time dependent schrodinger equation is shown to exist moreover we show that for every kahler manifold equipped with the fedosov star product of wick type all the classical delta functionals are positive and give rise to some formal bargmann representation of the deformed algebra

almost tangent and cotangent structures in the large we examine some global properties of integrable almost tangent and cotangent manifolds in particular we extend several results which essentially characterize tangent and cotangent bundles as respectively regular almost tangent and cotangent structures

approximate controllability of linear retarded systems in hilbert spaces the object of this paper is to construct some results on the control problems for retarded functional differential equations of parabolic type with unbounded principal operators in hilbert spaces in order to establish relations between controllability of the given equation and observability of the adjoint system we investigate the equivalent relation for the completeness of generalised eigenspaces of the infinitesimal generators finally when the control space is a finite dimensional space a necessary and sufficient condition for the approximate controllability of retarded equations is given by the so called rank condition

a universal formula for deformation quantization on kahler manifolds abstract we give an explicit local formula for any formal deformation quantization with separation of variables on a kahler manifold the formula is given in terms of differential operators parametrized by acyclic combinatorial graphs

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

some geometric structures associated with a k symplectic manifold a canonical connection is attached to any k symplectic manifold we study the properties of this connection and its geometric applications to k symplectic manifolds in particular we prove that under some natural assumptions any k symplectic manifold admits an ehresmann connection discuss some corollaries of this result and find vanishing theorems for characteristic classes on a k symplectic manifold

multicomponent wentzel kramers brillouin approximation on arbitrary symplectic manifolds a star product approach it is known that in the wentzel kramers brillouin approximation of multicomponent systems like the dirac equation or born oppenheimer approximation an additional phase appears apart from the berry phase so far this phase was only examined in special cases or under certain restrictive assumptions namely that the eigenspaces of the matrix or endomorphism valued symbol of the hamiltonian form trivial bundles we give a completely global derivation of this phase which does not depend on any choice of local trivializing sections this is achieved using a star product approach to quantization furthermore we give a systematic and global approach to a reduction of the problem to a problem defined completely on the different polarizations finally we discuss to what extent it is actually possible to reduce the problem to a really scalar one and make some comments on obstructions to the existence of global quasiclassical states

approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions in this paper we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable the results are obtained using new properties of symmetric operators which are proved in this paper compact semigroups the schauder fixed point theorem and or the contraction mapping principle

geometric quantization of vector bundles and the correspondence with deformation quantization i repeat my definition for quantization of a vector bundle for the cases of the toeplitz and geometric quantizations of a compact kahler manifold i give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle

the characteristic classes of morita equivalent star products on symplectic manifolds in this paper we give a complete characterization of morita equivalent star products on symplectic manifolds in terms of their characteristic classes two star products and on m are morita equivalent if and only if there exists a symplectomorphism colon m m such that the relative class t is 2 i integral for star products on cotangent bundles we show that this integrality condition is related to dirac s quantization condition for magnetic charges

the canonical connection of a bi lagrangian manifold we prove that the canonical connection of a bi lagrangian manifold introduced by hess is a levi civita connection showing that a bi lagrangian manifold i e a symplectic manifold endowed with two transversal lagrangian foliations is endowed with a canonical semi riemannian metric

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

explicit formulas for noncommutative deformations of cpn and chn we give explicit expressions of a deformation quantization with separation of variables for cpn and chn this quantization method is one of the ways to perform a deformation quantization of kahler manifolds which is introduced by karabegov star products are obtained as explicit formulas in all order in the noncommutative parameter we also give the fock representations of the noncommutative cpn and chn

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

cotangent bundle quantization entangling of metric and magnetic field for manifolds of noncompact type endowed with an affine connection for example the levi civita connection and a closed 2 form magnetic field we define a hilbert algebra structure in the space and construct an irreducible representation of this algebra in this algebra is automatically extended to polynomial in momenta functions and distributions under some natural conditions this algebra is unique the non commutative product over is given by an explicit integral formula this product is exact not formal and is expressed in invariant geometrical terms our analysis reveals that this product has a front which is described in terms of geodesic triangles in the quantization of functions induces a family of symplectic reflections in and generates a magneto geodesic connection on this symplectic connection entangles on the phase space level the original affine structure on and the magnetic field in the classical approximation the 2 part of the quantum product contains the ricci curvature of and a magneto geodesic coupling tensor

samples of noncommutative products in certain differential equations a set of associative noncommutative products is considered in different differential equations of the ordinary and partial types a method of separation of variables is considered for a large set of those systems the products involved include for example some products and some products based on nijenhuis tensors which are embedded in the differential equations of the laplace poisson lax and schr dinger styles a comment on the products of reshetikhin jambor sykora type is also given in relation to products of vey type

weyl underhill emmrich quantization and the stratonovich weyl quantizer weyl underhill emmrich wue quantization and its generalization are considered it is shown that an axiomatic definition of the stratonovich weyl quantizer leads to severe difficulties quantization on the cylinder within the wue formalism is discussed

compatible symplectic connections on a cotangent bundle and the fedosov quantization a global construction of a family of symplectic connections on a cotangent bundle compatible with some linear symmetric connection on the base space is proposed examples of the compatible symplectic connections are given a detailed analysis of an abelian connection and of flat sections for this kind of symplectic connection in the fedosov algorithm are presented some properties of the product determined by the compatible symplectic connection are shown

deformation quantization of the n tuple point contrary to the classical methods of quantum mechanics the deformation quantization can be carried out on phase spaces which are not even topological manifolds in particular the moyal star product gives rise to a canonical functor f from the category of affine analytic spaces to the category of associative in general non commutative algebras curiously if x is the n tuple point xn 0 then f x is the algebra of n n matrices

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived