To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of the cotangent bundle to M to the formal neighborhood of the diagonal of the product M x M~, where M~ is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we "dequantize" Fedosov's quantization.
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 non regular Poisson structures, presymplectic manifolds, complex analytic symplectic manifolds, etc.
Let $(M,ω)$ be a symplectic manifold, $\mathcal{D}\subset TM$ a real polarization on $M$ and $\wp$ a leaf of $\mathcal{D}$. We construct a Fedosov-type star-product $\ast_L$ on $M$ such that $C^\infty (\wp)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\infty (M)[[h]], \ast_L)$. This generalizes the results of 0708.2626.
B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally,…
A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.
We study various aspects of Fedosov star-products on symplectic manifolds. By introducing the notion of "quantum exponential maps", we give a criterion characterizing Fedosov connections. As a consequence, 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, one obtains 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 submanifol…
The description of all deformation quantizations with separation of variables on a Kaehler manifold obtained in our earlier paper is used to identify the Fedosov star-product of Wick type constructed by M. Bordemann and S. Waldmann. This star-product is shown to be the one with separation of variables which corresponds to the trivial deformation of the Kaehler form. To this end a formal Fock bundle on a Kaehler manifold is introduced and an associative multiplication on its sections is defined.
We study quantization schemes on a Kähler manifold and relate several interesting structures. We first construct Fedosov's star products on a Kähler manifold $X$ as quantizations of Kapranov's $L_\infty$-algebra structure. Then we investigate the Batalin-Vilkovisky (BV) quantizations associated to these star products. A remarkable feature is that they are all one-loop exact, meaning that the Feynman weights associated to graphs with two or more loops all vanish. This leads to a succinct cochain level formula in de Rham cohomology for the algebraic index.
We review our construction of star-products on Poisson manifolds and discuss some examples. In particular, we work out the relation with Fedosov's original construction in the symplectic case.
Fedosov's simple geometrical construction for deformation quantization of symplectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does not have to be of Weyl/symmetric or Wick/normal type. (3) The initial geometric structures are allowed to depend on Planck's constant.
In certain neighborhood $U$ of an arbitrary point of a symplectic manifold $M$ we construct a Fedosov-type star-product $\ast_L$ such that for an arbitrary leaf $\wp$ of a given polarization $\mathcal{D}\subset TM$ the algebra $C^\infty (\wp \cap U)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\infty (U)[[h]], \ast_L)$. With certain additional assumptions on $M$, $\ast_L$ becomes a so-called star-product with separation of variables.
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 $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal M}$ 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 ${{\mathcal T}^*_ρ}{\ma…
Let $A$ be a star product on a symplectic manifold $(M,ω_0)$, $\frac{1}{t}[ω]$ its Fedosov class, where $ω$ is a deformation of $ω_0$. We prove that for a complex polarization of $ω$ there exists a commutative subalgebra, $O$, in $A$ that is isomorphic to the algebra of functions constant along the polarization. Let $F(A)$ consists of elements of $A$ whose commutator with $O$ belongs to $O$. Then, $F(A)$ is a Lie algebra which is an $O$-extension of the Lie algebra of derivations of $O$. We prove a formula which relates the class of this extension, the Fedosov class, and the Chern class of $P$…
We give a classification of polarized deformation quantizations on a symplectic manifold with a (complex) polarization. Also, we establish a formula which relates the characteristic class of a polarized deformation quantization to its Fedosov class and the Chern class of the polarization.
A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newland…
Various types of Lagrange and Finsler geometries and the Einstein gravity theory, and modifications, can be modelled by nonholonomic distributions on tangent bundles/ manifolds when the fundamental geometric objects are adapted to nonlinear connection structures. We can convert such geometries and physical theories into almost Kahler/ Poisson structures on (co)tangent bundles. This allows us to apply the Fedosov quantization formalism to almost symplectic connections induced by Lagrange-Finsler and/or Einstein fundamental geometric objects. There are constructed respective nonholonomic version…
This is an expository note on Fedosov's construction of deformation quantization. Given a symplectic manifold and a connection on it, we show how to calculate the star-product step by step. We draw simple diagrams to solve the recursive equations for the Fedosov connection and for flat sections of the Weyl algebra bundle corresponding to functions. We also reflect on the differences of symplectic and Riemannian geometries.
A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove that the classical moment map could be quantized if there exists an equivariant connection. Along the way we touch upon the general question of quantization with values in a bundle of algebras. We consider Fedosov's construction of deformation quantization in general. In the App…
We construct a canonical deformation quantization for symplectic supermanifolds. This gives a novel proof of the super-analogue of Fedosov quantization. Our proof uses the formalism of Gelfand-Kazhdan descent, whose foundations we establish in the super-symplectic setting.
We extend Fedosov deformation quantization to general contact manifolds. Unlike the case of symplectic manifolds, not every classical observable on a contact manifold is generally quantized. On examination of possible obstructions to quantization, we obtain some new invariants of contact manifolds.
Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product $\star$ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a Kähler manifold $X$ equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined $(C^\infty_X[[\hbar]], \star)$. This sheaf can be viewed as the $\hbar$-expansion of $L^{\otimes k}$ as $k \to \…
Consider a group $Γ$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,ω)$. This canonically induces an action of $Γ$ by symplectomorphisms on $M$. We examine the reverse problem of extending group actions by symplectomorphisms to the deformation quantization. To do this we first define a notion of extension that does not impose restrictions on the Fedosov connection realizing $\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition of pull-back with certain inner automorphisms of sections of the Weyl bundle. We refo…
In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universa…
Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV i…
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.
We show that for every vector bundle E over any given symplectic manifold M there exists an explicitly given super-Poisson bracket on the space of sections of the dual Grassmann bundle associated to E built out of symplectic structure of M, a fibre metric on E, and a connection in E compatible with the given fibre metric. Moreover, we construct a deformation quantization for this space of sections by means of a Fedosov type procedure.
We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kaehler manifold which respects the almost-Kaehler structure to construct a Fedosov-type deformation quantization on this manifold.
We present explicit formulas for deformation quantization on the co-adjoint orbits of the real diamond Lie group. From this we obtain quantum half-plans, quantum hyperbolic cylinders, quantum hyperbolic paraboloids via Fedosov deformation quantization and finally, the corresponding unitary representations of this group.
We demonstrate the main idea of constructing irreducible unitary representations of Lie groups by using Fedosov deformation quantization in the concrete case of the group Aff(R) of affine transformations of the real straight line. By an exact computation of the star-product and the operator $\hat{\ell}_Z$, we show that the resulting representations exhausted all the irreducible representations of this groups.
Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures. For deformations of these structures we prove the classification theorems and a general a general index theorem.
This paper is part II of a series of papers on the deformation quantization on the cotangent bundle of an arbitrary manifold $Q$. For certain homogeneous star products of Weyl ordered type (which we have obtained from a Fedosov type procedure in part I, see q-alg/9707030) we construct differential operator representations via the formal GNS construction (see q-alg/9607019). The positive linear functional is integration over $Q$ with respect to some fixed density and is shown to yield a reasonable version of the Schrödinger representation where a Weyl ordering prescription is incorporated. Furt…
In this paper we introduce two classes of Poisson brackets on algebras (or on sheaves of algebras). We call them locally free and nonsingular Poisson brackets. Using the Fedosov's method we prove that any locally free nonsingular Poisson bracket can be quantized. In particular, it follows from this that all Poisson brackets on an arbitrary field of characteristic zero can be quantized. The well known theorem about the quantization of nondegenerate Poisson brackets on smooth manifolds follows from the main result of this paper as well.
In this paper we define Courant algebroids in a purely algebraic way and study their deformation theory by using two different but equivalent graded Poisson algebras of degree -2. First steps towards a quantization of Courant algebroids are proposed by employing a Fedosov like deformation quantization.
Let $(X,ω)$ be a symplectic orbifold which is locally like the quotient of a $\mathbb{Z}_2$ action on $\reals^n$. Let $A^{((\hbar))}_X$ be a deformation quantization of $X$ constructed via the standard Fedosov method with characteristic class being $ω$. In this paper, we construct a universal deformation of the algebra $A^{((\hbar))}_X$ parametrized by codimension 2 components of the associated inertia orbifold $\widetilde{X}$. This partially confirms a conjecture of Dolgushev and Etingof in the case of $\mathbb{Z}_2$ orbifolds. To do so, we generalize the interpretation of Moyal star-product …
We provide a general study for triangular dynamical r-matrices using Poisson geometry. We show that a triangular dynamical r-matrix always gives rise to a regular Poisson manifold. Using the Fedosov method, we prove that non-degenerate (i.e., the corresponding Poisson manifolds are symplectic) triangular dynamical r-matrices (over $ \frakh^* $ and valued in $\wedge^{2}\frakg$) are quantizable, and the quantization is classified by the relative Lie algebra cohomology $H^{2}(\frakg, \frakh)[[\hbar ]]$. We also generalize this quantization method to splittable triangular dynamical r-matrices, whi…
On every split supermanifold equipped with the Rothstein even super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fibre metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order supercommutator proportional to the Rothstein superbracket. Moreover, we discuss two constructions related to the above result, namely the quantized BRST-coh…
In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence of Kähler metrics of constant scalar curvature. In the third part of the paper we outline recent results of both authors, and in particular show that the constant scalar curvature Kähler metric problem and the study of deformation quantization meet at the notion of trace (density) for star product. We formulate a cohomology formula for the inv…
We consider a class of \textit{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 \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely conn…
We present a method of quantizing analytic spaces $X$ immersed in an arbitrary smooth ambient manifold $M$. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold $M$. Using a super-manifold framework we modify the Fedosov construction in a way such that the $\star$-product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on $M$. This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on $(\mc{O…
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…
In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E^*, where E -> M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concr…
We start with a short exposition of developments in physics and mathematics that preceded, formed the basis for, or accompanied, the birth of deformation quantization in the seventies. We indicate how the latter is at least a viable alternative, autonomous and conceptually more satisfactory, to conventional quantum mechanics and mention related questions, including covariance and star representations of Lie groups. We sketch Fedosov's geometric presentation, based on ideas coming from index theorems, which provided a beautiful frame for developing existence and classification of star-products …
