Related papers: Generalized Berezin-Toeplitz quantization of Kaehl…

We propose generalized quantization axioms for Nambu-Poisson manifolds, which allow for a geometric interpretation of n-Lie algebras and their enveloping algebras. We illustrate these axioms by describing extensions of Berezin-Toeplitz…

We prove sharp remainder bounds for the Berezin-Toeplitz quantization and present applications to semiclassical quantum measurements.

This is a survey on our recent works which reveal new relationships among deformation quantization, geometric quantization, Berezin-Toeplitz quantization and BV quantization on K\"ahler manifolds.

We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show…

We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable Z_2 -graded Hilbert space of…

We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…

Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries…

We extend the homological method of quantization of generalized Drinfeld--Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.

We axiomatize path integral quantization of symplectic manifolds. We prove that this path integral formulation of quantization is equivalent to an abstract operator formulation, ie. abstract coherent state (or Berezin) quantization. We use…

If f is a smooth function on a Hodge manifold, we construct a canonical sequence of real algebraic functions that converge to f in the smooth topology. The definition of of the approximants is inspired by Berezin-Toeplitz quantization. The…

An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well…

On compact foliated manifolds, we extend the theorem on the existence and uniqueness of solutions to generalized Kazdan-Warner equations. We provide examples of PDEs that we solve, including the transverse Hitchin equation for a diagonal…

We explore the possibility of extending the well-known Berezin-Toeplitz quantization to reproducing kernel spaces of vector-valued functions. In physical terms, this can be interpreted as accommodating the internal degrees of freedom of the…

Generalized coherent states provide a means of connecting square integrable representations of a semi-simple Lie group with the symplectic geometry of some of its homogeneous spaces. In the first part of the present work this point of view…

We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the K\"ahler case, state spaces arise as spaces of…

It is shown that the theory of spherical Harish-Chandra modules naturally provides the algebras of covariant, contravariant and mixed symbols on generalized flag manifolds. The general proof of the correspondence principle for all these…

The purpose of this paper is to apply the framework of non- commutative differential geometry to quantum deformations of a class of Kahler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm…

I repeat my definition for quantization of a vector bundle. For the case of Toeplitz and geometric quantization of a compact Kaehler Manifold, I give a construction for quantizing any smooth vector bundle which depends functorially on a…

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct…

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…