arXiv research

Classifies star products on Lie algebroid duals and extends to projectable quantizations.

Study star products on Poisson manifolds compatible with reduction.

Deform moment map on symplectic connections using star product algebras.

We derive a closed formula for a star-product on complex projective space and on the domain $SU(n+1)/S(U(1)\times U(n))$ using a completely elementary construction: Starting from the standard star-product of Wick type on $C^{n+1} \setminus \{ 0 \}$ and performing a quantum analogue of Marsden-Weinstein reduction, we ca…

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection introduced by the first author. We establish a necess…

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

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…

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

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…

Constructs non-unital monoidal category of contact manifolds and Legendrian correspondence calculus

We study quantum moment maps of $G$-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a $G$-invariant star product is differentiable. This property gives us a new method for the class…

New star-product defined on Poisson manifolds using Toeplitz operators.

I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products …

It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetri…

We etablish a necessary and sufficient condition under which there exists a tangential and well graded star product, differential or not, on the dual g^* of a nilpotent Lie algebra g. We also give enlightening examples with explicit computations.

Researchers create a star product on a Grassmannian with separation of variables.

We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.

Study provides explicit formula for complex 2D Kähler manifold quantization.

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…

In the present paper we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a divergence-free Poisson bivector field on R^d, the Kontsevich star-product with the harmonic angle function is cyclic. We also prove a globalization of this theorem in the case of arbitrary P…

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 paper proves new inequalities in hyperbolic space using Euclidean methods.

For arbitrary compact quantizable Kaehler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their semi-classical behaviour (their asymptotic expansion) due to Bordemann, Meinrenken and Schlichenmaier are used in an essential man…

For a real symmetric domain $G_{\mathbb R}/K_{\mathbb R}$, with complexification $G_{\mathbb C}/K_{\mathbb C}$, we introduce the concept of "star-restriction" (a real analogue of the "star-products" for quantization of Kähler manifolds) and give a geometric construction of the $G_{\mathbb R}$-invariant differential ope…

Quantizes symplectic manifolds with toric singularities using Toeplitz operators.

In this note, we will show one example of hamiltonian Lie algebra action which has no invariant star product.

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation par…

Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $π$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $Γ(π)\to\Bbbk$ that take…

We consider formal deformations of the Poisson algebra of functions (with singularities) on $T^*M$ which are Laurent polynomials of fibers. Tn the case: $\dim M=1$ ($M=S^1, {\bf R}$), there exists a non-trivial $\star$-product on this algebra non-equivalent to the standard Moyal product.

Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.

We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known whether holomorphic star products exist on complex analytic manifolds. The main p…

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.

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…

Noncommutative geometry connects higher order connections to quantization.

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …

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.

Develops methods for structured variational inference with star-structured models.

Paper studies Hessian quotient equations in warped product manifolds.

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].

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…

Paper proves Koszul duality for weighted A-infinity algebras.

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

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-…

If M is a smooth compact oriented Riemannian manifold of dimension n=4k+2, with or without boundary, and F is a vector bundle on M with an inner product and a flat connection, we construct a modification of the Hodge star operator on the parabolic cohomology H^{2k+1}_{par}(M;F). The operator gives a canonical complex s…

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner p…

Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.

We consider the warped product manifold, $\mathbb{R}_+ \times_{\bf{Id}} M^n$, with Riemannian metric $γ\equiv \mathrm{d} r^2 \oplus r^2 σ$, where $(M^n, σ)$ is a smooth closed Riemannian $n$-manifold. We investigate what sufficient curvature condition is required of $σ$ to ensure that a solution to the inverse mean cur…