The Seiberg–Witten equations, found at the end of the 20 th century, are one of the main discoveries in the topology and geometry of four-dimensional Riemannian manifolds. They are defined in terms of a Spin^c -structure that exists on any four-dimensional Riemannian manifold. Like the Yang–Mills equations, the Seiberg–Witten equations are the limit case of a more general supersymmetric Yang–Mills equations. However, unlike the conformally invariant Yang–Mills equations, the Seiberg–Witten equations are not scale invariant. Therefore, in order to obtain “useful information” from them, one must introduce a scale parameter λ and pass to the limit as λ→∞ . This is precisely the adiabatic limit studied in this paper.

The Ginzburg–Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It turned out that these equations have interesting and non-trivial generalizations. First of all, they can be extended to arbitrary compact Riemann surfaces. Next, they can be generalized to dimension 3 as dynamical (or hyperbolic) Ginzburg–Landau equations. They also have a 4-dimensional extension provided by Seiberg–Witten equations. In this review we describe all these interesting topics together with some unsolved problems.

The paper is a survey devoted to the topological phases - one of the actively developing directions in the theory of solid states. An interpretation of topological phases in terms of the generalized cohomology theories and $K$-theory is given.

This is a review of the results on Hermitian Yang–Mills equation and their generalization called the deformed Hermitian Yang–Mills equation. Solutions of Hermitian Yang–Mills equation in complex dimension 2 are given by antiself-dual connections so this equation may be considered as a multidimensional generalization of the instanton equations. Deformed Hermitian Yang–Mills equation reduce to the Hermitian Yang–Mills equation in the large volume limit.

We review some applications of noncommutative geometry to function theory and mathematical physics. In the first case we discuss relations between the spaces of real variables and operator algebras. In the second case we deal with quantization of universal Techmüller space and quantum Hall effect.

This review is devoted to one of the most interesting and actively developing fields in condensed matter physics-theory of topological insulators. Apart from its importance for theoretical physics, this theory enjoys numerous connections with modern mathematics, in particular, with topology and homotopy theory, Clifford algebras, K-theory and non-commutative geometry. From the physical point of view topological invariance is equivalent to adiabatic stability. Topological insulators are characterized by the broad energy gap, stable under small deformations, which motivates application of topological methods. A key role in the study of topological objects in the solid state physics is played by their symmetry groups. There are three main types of symmetries-time reversion symmetry, preservation of the number of particles (charge symmetry) and PH-symmetry (particle-hole symmetry). Based on the study of symmetry groups and representation theory of Clifford algebras Kitaev proposed a classification of topological objects in solid state physics. In this review we pay special attention to the topological insulators invariant under time reversion.

Исследуются топологические диэлектрики, инвариантные относительно обращения времени. Такие системы характеризуются наличием широкой энергетической щели, устойчивой к малым деформациям. Примером может служить квантовый спиновый диэлектрик Холла. Он обладает нетривиальным топологическим $\mathbb Z_2$-инвариантом, введенным Кейном и Милом.

Представлена концепция адиабатического предела в динамических уравнениях Гинзбурга-Ландау на пространстве $\mathbb R^{1+2}$ и уравнениях Зайберга-Виттена на четырехмерных симплектических многообразиях. Показано, что уравнения Зайберга-Виттена можно рассматривать как комплексную версию уравнений Гинзбурга-Ландау.

In our course we have presented the basics of twistor theory and its applications to the solution of Yang–Mills duality equations. The first part describes the twistor correspondence between geometric objects in Minkowski space and their counterparts in twistor space.

This paper is devoted to a survey of recent results in the Kähler geometry of infinite-dimensional Kähler manifolds. Three particular classes of such manifolds are investigated: the loop spaces of compact Lie groups, Hilbert–Schmidt Grassmannians, and the universal Teichmüller space. These investigations have been prompted both by requirements in Kähler geometry itself and by connections with string theory, which are considered in the last section. Bibliography: 43 titles.

Задача квантования пространства $\Omega_d$ гладких петель, принимающих значения в $d$-мерном векторном пространстве, может решаться в рамках стандартного дираковского подхода. Однако естественная симплектическая форма на $\Omega_d$ продолжается на гильбертово пополнение пространства $\Omega_d$, совпадающее с соболевским пространством $V_d:=H_0^{1/2}(\mathbb S^1,\mathbb R^{d})$ полудифференцируемых петель со значениями в $\mathbb R^{d}$. Пространство $V_d$ рассматривается как фазовое пространство теории полудифференцируемых струн. Эту теорию удается проквантовать, пользуясь идеями из некоммутативной геометрии.

Hyperbolic Ginzburg-Landau equations arise in gauge field theory as the Euler-Lagrange equations for the (2 + 1)-dimensional Abelian Higgs model. The moduli space of their static solutions, called vortices, was described by Taubes; however, little is known about the moduli space of dynamic solutions. Manton proposed to study dynamic solutions with small kinetic energy with the help of the adiabatic limit by introducing the “slow time” on solution trajectories. In this limit the dynamic solutions converge to geodesics in the space of vortices with respect to the metric generated by the kinetic energy functional. So, the original equations reduce to Euler geodesic equations, and by solving them one can describe the behavior of slowly moving dynamic solutions. It turns out that this procedure has a 4-dimensional analog. Namely, for the Seiberg-Witten equations on 4-dimensional symplectic manifolds it is possible to introduce an analog of the adiabatic limit. In this limit, solutions of the Seiberg-Witten equations reduce to families of vortices in normal planes to pseudoholomorphic curves, which can be considered as complex analogs of geodesics parameterized by “complex time.” The study of the adiabatic limit for the equations indicated in the title is the main content of this paper.

We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,ℝ d ) of half-differentiable vector functions on the circle, and the algebra of observables A d , identified with the semi-direct product of the Heisenberg algebra of V d and the algebra Vect(S 1) of tangent vector fields on the circle.

Одной из задач некоммутативной геометрии является перевод основных понятий анализа на язык банаховых алгебр. Этот перевод осуществляется с помощью процедуры квантования. Возникающее в результате операторное исчисление называют, следуя Конну, квантовым исчислением. В работе приводится ряд утверждений из указанного исчисления, касающихся интерпретации идеалов Шаттена в терминах теории функций. Основное внимание уделяется операторам Гильберта-Шмидта.

Our goal is to present an approach to the proof of the harmonic spheres conjecture based on the adiabatic limit construction. This construction allows to associate with an arbitrary Yang–Mills G-field on the Euclidean 4-dimensional space a harmonic map of the Riemann sphere to the loop space of the group G