Moduli spaces of solutions of a noncommutative sigma model

Using a noncommutative version of the uniton theory, we study the space of those solutions of the noncommutative U(1) sigma model that are representable as finite-dimensional perturbations of the identity operator. The basic integer-valued characteristics of such solutions are their normalized energy e, canonical rank r, and minimum uniton number u, which always satisfy r ≤ e and u ≤ e. Starting with the so-called BPS solutions (u = 1), we completely describe the sets of all solutions with r = 1, 2, e − 1, e (which forces u ≤ 2) and all solutions of small energy (e ≤ 5). The obtained results reveal a simple but nontrivial structure of the moduli spaces and lead to a series of conjectures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 307–327, September, 2008.     

Noncommutative orlicz-hardy spaces

Let (Φ, Ψ) be a pair of complementary N-functions and HΦ(A)$H^{\Phi }\left(A\right)$ and HΨ(A)$H^{\Psi }\left(A\right)$ be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szegö and inner-outer type factorization theorems of Hp(A)$H^p\left(A\right)$ to this case.     

Noncommutative Geometry (An Introduction to Selected Topics)

We give an introduction to noncommutative geometry and to some of its applications. Emphasis will be on noncommutative manifolds, notably noncommutative tori and spheres.     

Noncommutative Grassmannian <Emphasis Type="Italic">U</Emphasis>(1) sigma model and a Bargmann-Fock space

We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖ _HS ² , where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a, P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 347–357, December, 2007.     

K-Theory of Noncommutative Lattices

Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their K-theory. We shall do it algebraically, by studying the algebraic K-theory of the associated algebras of continuous functions which turn out to be noncommutative approximately finite dimensional (AF)C^*. We also work out several examples.     

格点上的非交换微分运算及其应用

By introducing the noncommutative differential calculus on the function space of the infinite/finite set and construct a homotopy operator, one prove the analogue of the Poincare lemma for the difference complex. As an application of the differential calculus, a two dimensional integral model can be derived from the noncommutative differential calculus.     

On the Noncommutative Residue for Pseudodifferential Operators with log-Polyhomogeneous Symbols

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.     

Acyclic Complexes Related to Noncommutative Symmetric Functions

In this paper, we show how to endow the algebra of noncommutative symmetric functions with a natural structure of cochain complex which strongly relies on the combinatorics of ribbons, and we prove that the corresponding complexes are acyclic.     

Gauge Theory Solitons on the Noncommutative Cylinder

The solution-generating technique originally suggested for gauge theories on the noncommutative plane is generalized to the noncommutative cylinder. For this, we construct partial isometry operators and a complete set of orthogonal projection operators in the algebra _C of the cylinder, and an isomorphism between the free module _C and its direct sum _{C C} with the Fock module on the cylinder. We explicitly construct the gauge theory soliton and evaluate the spectrum of perturbations about this soliton.     

Aspects of Matrix Theory and Noncommutative Geometry

We review some basic foundations of the matrix model of M-theory. We study the important problem of compactifying the matrix theory in relation to noncommutative geometry. We show that there exist solutions of this problem other than the well-known toroidal solutions of Connes, Douglas, and Schwarz. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 181–190, November, 2005.     

Path Integrals in Noncommutative Quantum Mechanics

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.     

Grothendieck Representations of Categories and Canonical Noncommutative Topologies

From a representation of a category in terms of a class of Grothendieck categories, a canonical topologisation of the objects follows. The noncommutative topology recently introduced in noncommutative geometry is an example of this. Basic results of sheaf theory are derived in this setting.     

The Noncommutative Residue about Witten Deformation

In this paper, we compute lower dimensional volumes Vol_4~((1,1)) and Vol_6~((2,2)) about Witten deformation for 4, 6-dimensional spin manifolds with boundary respectively, and get assosiated Kastler–Kalau–Walze type theorems. We also give theoritic explaination of the gravitational action for 4, 6 dimensional manifolds with boundary by these noncommutative residues.     

The Injective Spectrum of a Noncommutative Space

For a noncommutative space X, we study Inj(X), the set of isomorphism classes of indecomposable injective X-modules. In particular, we look at how this set, suitably topologized, can be viewed as an underlying “spectrum” for X. As applications we discuss noncommutative notions of irreducibility and integrality, and a way of associating an integral subspace of X to each element of Inj(X) which behaves like a “weak point.”     

Symmetries of Field Theories on the Noncommutative Plane

We report new developments concerning the symmetry properties and their actions on special solutions allowed by certain field theory models on the noncommutative plane. In particular, we seek Galilean-invariant models. The analysis indicates that this requirement strongly restricts the admissible interactions. Moreover, if a scalar field is coupled to a gauge field, then a geometric phase emerges for vortexlike solutions transformed by Galilean boosts.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 64–73, July, 2005.     

非交换加权Lorentz空间的对偶空间

设μ是一个半有限von Neumann代数.对于0P∞,0q≤∞,定义了非交换加权Lorentz空间Λ_ω~(p,q)(μ)及其associate空间Λ_ω~(p,q)(μ)',给出了空间Λ_ω~(p,q)(μ)'和Λ_ω~(p,q)(μ)'的一些基本性质.应用这些性质,还给出了非交换加权Lorentz空间Λ_ω~p(μ),0P∞的对偶空间.