Singular projective varieties and quantization

By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantization are explained. In particular, the notions of projective varieties, embeddings, singularities, and quotients appearing in geometric invariant theory are recalled.Comment: 21 pages, 3 figure

Higher-Dimensional Integrable Systems from Multilinear Evolution Equations

A multilinear M-dimensional generalization of Lax pairs is introduced and its explicit form is given for the recently discovered class of time-harmonic, integrable, hypersurface motions.Comment: 5 page

Integrable field theories from Poisson algebras

New integrable 1 + 1 dimensional classical field theories are found that include infinite dimensional analogues of N-particle Toda-and Calogero-Moser systems, as well as non-relativistic theories with an interaction that is polynomial in the first (spatial) derivative of the field. The existence, as well as the involutivity, of an infinite set of independent conserved quantities follows most easily from a 2 + 1 dimensional Lax-pair which uses as its underlying infinite dimensional Lie algebra a Poisson algebra of functions in two variables

Curved Space (Matrix) Membranes

Hamiltonian formulations of M-branes moving in curved backgrounds are given

The Weyl bundle as a differentiable manifold

Construction of an infinite dimensional differentiable manifold ${\mathbb R}^{\infty}$ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra and a Weyl algebra bundle are presented. Continuity of the $\circ$-product in the Tichonov topology is proved. Construction of the $*$-product of the Fedosov type in terms of theory of connection in a fibre bundle is explained.Comment: 31 pages; revised version - some typoes have been eliminated, notation has been simplifie