arXiv research

Study invariant operations on Fedosov manifolds.

Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.

Obstructions found for closed Fedosov star products on symplectic and Kähler manifolds.

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 cohomological vector field $Q$ constructed on $L[1]\oplus L/A$ by the Fedosov iteration method arises as a byproduct of the Poincaré--Birkhoff--Witt map establ…

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 flatte…

We study obstructions to the existence of closed Fedosov's star products on a given Kähler manifold. In our previous paper, we proved that the Levi-Civita connection of a Kähler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer) Fedosov's star product only if it is a zero of the Cahen-Gutt moment…

Study symplectic scalar curvature on supermanifolds.

Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.

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 …

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.

The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.

We introduce the notion of a conformally Fedosov structure and construct an associated Cartan connection. When an appropriate curvature vanishes, this allows us to construct a family of natural differential complexes akin to the BGG complexes from parabolic geometry.

Quantizes functions on Kähler manifolds without formal deformation.

Study quantization schemes on Kähler manifolds linking star products and BV quantizations.

Using Fedosov's approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kaehler-Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variabl…

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannia…

The paper proves curvature identities for symplectic connections.

In this paper we study geometry of symmetric torsion-free connections which preserve a given symplectic form

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 classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transi…

A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.

We show that if a compact Kaehler manifold $M$ of non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cah…

We consider the space of germs of Fedosov structures at a point, together with the group of origin-preserving diffeomorphisms acting on it. We calculate dimensions of moduli spaces of $k$-jets of generic structures and construct Poincaré series. It is shown to be a rational function.

We study symplectic manifolds $(M^{2l},ω)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative…

We introduce, for every $\mathbb{Z}$-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of $\mathbb{Z}$-graded manifolds and we extend the Emmrich--Wein…

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…

Deform moment map on symplectic connections using star product algebras.

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 …

We study the relations between the projective and the almost conformally symplectic structures on a smooth even dimensional manifold. We describe these relations by a single almost conformally symplectic connection with totally trace--free torsion sharing the geodesics (up to parametrization) with the projective class.…

Let $(M,ω)$ be a symplectic manifold and $F$ be a Finsler structure on $M$. In the present paper we define a lift of the symplectic two-form $ω$ on the manifold $TM\backslash 0$, and find the conditions that the Chern connection of the Finsler structure $F$ preserves this lift of $ω$. In this situation if $M$ admits a …

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class o…

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…

Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a represent…

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using …

The main result of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (non necessarily pseudo-Riemannian) in terms of the existence of certain connection. The result generalizes the well-known result of Ambrose and Singer for Riemannian homogeneous spaces, as…

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of…

Quantizes Kähler manifolds using sheaves and differential operators.

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original p…

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 wit…

Let $(\mathcal{M}, Q)$ be a dg manifold. The space of vector fields with shifted degrees $(\mathcal{X}(\mathcal{M})[-1], L_Q)$ is a Lie algebra object in the homology category $\mathrm{H}((C^{\infty}_{\mathcal{M}},Q)\mathrm{-}\mathbf{mod})$ of dg modules over $(\mathcal{M},Q)$, the Atiyah class $α_{\mathcal{M}}$ being …

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-…

In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces which originally had been introduced by Chen. We explain the notion of a vector orbibundle and characterize the good sections of a reduced vector orbibundle as the smooth stratified sections. In the second part of…

The paper quantizes Kähler manifolds using differential operators.

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…

Let $Ω_0$ be a polygon in $\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $Ω_\e$ is a family of surfaces with $\calC^\infty$ boundary which converges to $Ω_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov …

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…

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) -- for example the solution operator of …

We associate to a parametrized family $f$ of nonlinear Fredholm maps possessing a trivial branch of zeroes an {\it index of bifurcation} $β(f)$ which provides an algebraic measure for the number of bifurcation points from the trivial branch. The index $β(f)$ is derived from the index bundle of the linearization of the …