Isotropic Tori,Complex Germ and Maslov Index,Normal Forms and Quasimodes of Multidimensional Spectral Problems

More than twenty years ago V. P. Maslov posed the question under what conditions it is possible to assign to invariant isotropic lower-dimensional tori of Hamiltonian systems sequences of asymptotic eigenvalues and eigenfunctions (spectral series) of the corresponding quantum mechanical and wave operators. In the present paper this question is answered in terms of the quadratic approximation to the theory of normal forms. We also discuss the quantization conditions for isotropic tori and their relation to topological, geometric, and dynamical characteristics (Maslov indices, rotation (winding) numbers, eigenvalues of dynamical flows, etc.).     

Maslov Complex Germ Method for Systems with First-Class Constraints

We develop the semiclassical mechanics of systems with first-class constraints. A convenient quantization method is the method based on modifying the inner product used in the theory. We consider semiclassical states of the wave-packet type (with small indeterminacies in both coordinates and momenta) that appear in the theory of the Maslov complex germ at a point. We show that these states have a nonzero norm only if the classical coordinates and momenta lie on the constraint surface. The set of semiclassical states of the wave-packet type forms a (semiclassical) bundle whose base is the set of admissible classical states and whose fibers are function spaces determining the form of the wave packet. In some cases, the difference between two semiclassical states has a zero norm; it is therefore possible to introduce the gauge equivalence relation. The semiclassical gauge transformations that are automorphisms of the semiclassical bundle form a Batalin quasigroup. We also study the action of semiclassical observables and of semiclassical evolution transformations. We show that they preserve the norm and the gauge equivalence relation and that the observables coinciding on the constraint surface act on semiclassical states similarly up to the gauge invariance.     

一个Maslov指标公式及应用

本文给出辛流形（Ｍ，ω）和（Ｍ，－ω）的乘积辛流形（Ｍ×Ｍ，ω⊕（－ω））中Ｌａ－ｇｒａｎｇｅ子流形ΔＭ：＝｛（ｘ，ｘ）｜ｘ∈Ｍ）的Ｍａｓｌｏｖ指标的计算公式，并讨论它的一些应用．     

Yang-Mills模空间上的有限群作用

设Ｘ是单连通光滑闭４－流形,有限群Ｇ光滑作用于Ｘ,令ｐ：Ｅ→Ｘ是Ｘ上具有光滑Ｇ作用的ＳＵ（２）从且ｐ是一个Ｇ映射．本文研究了丛Ｅ上等模空间Ｍ＾Ｇ和商丛Ｅ’→Ｘ’上的关键空间之间的关系,并讨论了不变模空间的紧致化。     

The Hörmander and Maslov Classes and Fomenko's Conjecture

Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.     

Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold

We show that in the case of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein symplectic manifold, a certain universal 1-cohomology class, analogous to the Maslov class, can be defined. In contrast to the Maslov index, the presented class is directly related to the minimality problem for Lagrangian submanifolds if the ambient pseudo-Einstein manifold admits a Kähler-Einstein metric. We interpret the presented class geometrically as a certain obstruction to the continuation of one-dimensional supercycles from the Lagrangian submanifold to the ambient symplectic manifold.     

Harmonic Maps into the Moduli Spaces of Flat Connections

It is proved in this paper that, under reasonable assumptions, for each given harmonic map into the moduli spaces of flat connections, there exists one corresponding smooth family of Yang–Mills solutions approaching to the given harmonic map as the parameter tends to zero.     

Moduli of Quaternionic Superminimal Immersions of 2-Spheres into Quaternionic Projective Spaces

The aim of this paper is to describe the moduli spaces of degree d quaternionic superminimal maps from 2-spheres to quaternionic projective spaces HPn. We show that such moduli spaces have the structure of projectivized fibre products and are connected quasi-projective varieties of dimension 2nd + 2n + 2. This generalizes known results for spaces of harmonic 2-spheres in S⁴.     

Maslov Index and a Central Extension of the Symplectic Group

In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I ²(K) and that the induced natural homomorphism from K ₂ Sp(K)I ²(K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, -x 1, –y.     

The Maslov Complex Germ in the Case of a Degenerate Rest Point of the Hamiltonian

Mathematical Notes -     

On the Motive of Moduli Spaces of Rank Two Vector Bundles over a Curve

We study the motive of the moduli spaces of rank two vector bundles on a curve. In the smooth case we obtain the Hodge numbers, intermediate Jacobians and number of points over a finite field as corollaries. In the singular case our computations yield the Poincaré–Hodge polynomial of Seshadri's smooth model.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

Relativistically Covariant Quantum Field Theory of the Maslov Complex Germ

We consider an explicitly covariant formulation of the quantum field theory of the Maslov complex germ (semiclassical field theory) in the example of a scalar field. The main object in the theory is the “semiclassical bundle” whose base is the set of classical states and whose fibers are the spaces of states of the quantum theory in an external field. The respective semiclassical states occurring in the Maslov complex germ theory at a point and in the theory of Lagrangian manifolds with a complex germ are represented by points and surfaces in the semiclassical bundle space. We formulate semiclassical analogues of quantum field theory axioms and establish a relation between the covariant semiclassical theory and both the Hamiltonian formulation previously constructed and the axiomatic field theory constructions Schwinger sources, the Bogoliubov S-matrix, and the Lehmann-Symanzik-Zimmermann R-functions. We propose a new covariant formulation of classical field theory and a scheme of semiclassical quantization of fields that does not involve a postulated replacement of Poisson brackets with commutators.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 492–512, September, 2005.     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

On the Complex of Sobolev Spaces Associated with an Abstract Hilbert Complex

We consider the complexes of Hilbert spaces whose differentials are closed densely-defined operators. A peculiarity of these complexes is that from their differentials we can construct Laplace operators in every dimension. The Laplace operator together with a sufficiently nice measurable function enables us to define a generalized Sobolev space. There exist pairs of measurable functions allowing us to construct some canonical mappings of the corresponding Sobolev spaces. We find necessary and sufficient conditions for those mappings to be compact. In some cases for a given Hilbert complex we can construct an associated Sobolev complex. We show that the differentials of the original complex are normally solvable simultaneously with the differentials of the associated complex and that the reduced cohomologies of these complexes coincide.     

Complex Symplectic Geometry of Quasi-Fuchsian Space

We study the complex symplectic geometry of the space QF(S) of quasi-Fuchsian structures of a compact orientable surface S of genus g > 1. We prove that QF(S) is a complex symplectic manifold. The complex symplectic structure is the complexification of the Weil–Petersson symplectic structure of Teichmüller space and is described in terms which look natural from the point of view of hyperbolic geometry.     

Moduli space of potentials in the Aharonov-Bohm effect

We present a simplified derivation of the fact that the set of gauge equivalence classes or moduli space of flat connections (potentials) in the abelian Aharonov-Bohm effect, is isomorphic to the circle. The length of this circle is the absolute value of the electric charge.     

On Seshadri Constants of Canonical Bundles of Compact Complex Hyperbolic Spaces

Upper and lower bounds for the Seshadri constants of canonical bundles of compact hyperbolic spaces are given in terms of metric invariants. The lower bound is obtained by carrying out the symplectic blow-up construction for the Poincaré metric, and the upper bound is obtained by a convexity-type argument.     

On the Manifold of Almost Complex Structures

Let (M, g ₀) be a smooth closed Riemannian manifold of even dimension 2_n admitting an almost complex structure. It is shown that the space of all almost complex structures on M determining the same orientation as the one determined by a fixed almost complex structure J ₀ is a smooth locally trivial fiber bundle over the space of almost complex structures orthogonal with respect to g ₀ and determining the same orientation as J ₀.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 66–71.Original Russian Text Copyright © 2005 by N. A. Daurtseva.