Quantum Mechanics on Manifolds

In this paper, we investigate the relationships between quantum mechanics and the theory of partial differential equations. We closely follow the De Broglie and Schrödinger picture. Namely, we consider the well-known wave-particle duality as a relation between solutions of partial differential equations, describing waves, and singularities of solutions, that is particles. Our analysis of these relations shows that the necessary ingredients of any quantum mechanical picture are two connections. The first one is a connection in the tangent bundle of the configuration manifold and the second one is a connection in the trivial linear bundle.We also consider mechanical systems equipped with an inner structure and show that quantization of these systems requires a linear connection in the corresponding vector bundle.These are gravity and electromagnetic fields, or Yang–Mills fields if the configuration space is the Minkowski space. In the case of general mechanical systems, they should be considered as natural generalizations of these fields.Explicit formulas for quantizations of some mechanical systems and the corresponding star-products are given.     

Yang–Mills Theory over Compact Symplectic Manifolds

In this paper, Yang–Mills theory over a compactKähler manifold is naturally extended to a compactsymplectic manifold. The relation betweenthe Yang–Mills equation and symplecticstructure is explicitly clarified, and the moduli spaceof Yang–Mills connections over a compactsymplectic manifold is constructed. Furthermore, theabsolute minima of the Yang–Mills functional arecharacterized, and finite dimensionality ofthe moduli space of the minimizers of the Yang–Millsfunctional is shown.     

Random Holonomy for Yang–Mills Fields: Long-Time Asymptotics

We study weak and strong convergence of the stochastic parallel transport for time t on Euclidean space. We show that the asymptotic behavior can be controlled by the Yang–Mills action and the Yang–Mills equations. For open paths we show that under appropriate curvature conditions there exits a gauge in which the stochastic parallel transport converges almost surely. For closed paths we show that there exists a gauge invariant notion of a weak limit of the random holonomy and we give conditions that insure the existence of such a limit. Finally, we study the asymptotic behavior of the average of the random holonomy in the case of t'Hooft's 1-instanton.     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

The ‘Three-Line’ Theorem for the Vinogradov C-Spectral Sequence of the Yang–Mills Equations

The Vinogradov C-spectral sequence for the Yang–Mills equations is considered and the three-line theorem for the term E₁ of the C-spectral sequence is proved: E₁ ^p,q = 0 if p > 0 and q < n – 2, where n is the dimension of spacetime.     

First-Order Equations of Motion in the Supersymmetric Yang–Mills Theory with a Scalar Multiplet

We propose a system of first-order equations of motion all solutions of which are solutions of a system of second-order equations of motion for the supersymmetric Yang–Mills theory with a scalar multiplet. We find N = 1 transformations under which the systems of first- and second-order equations of motion are invariant.     

n-Dimensional submanifolds with an (n−1)-dimensional asymptotic distribution and with 1-asymptotic lines in a space form Rm(k)

In this paper we prove a theorem about the reduction of the codimension of n-dimensional submanifolds of a space form R^m(k), with an (n-1)-dimensional asymptotic distribution and with 1-asymptotic lines (i.e. asymptotic curves for which the (1+1)-dimensional osculating spaces are subspaces of the tangent spaces). We also give a result for (n–1)-asymptotic lines in connection with Enneper's formula. Finally we construct an example.Dedicated to Prof.dr.J.Bilo on his 65th birthday     

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.     

Peculiarities of Decomposition of the SU(N) Yang–Mills Field

Several aspects of extension of the Faddeev–Niemi decomposition for the SU(2) Yang–Mills field to the case of the SU(N) gauge group are discussed. Bibliography: 4 titles.     

Geometric evolution equations in critical dimensions

We make a qualitative comparison of phenomena occurring in two different geometric flows: the harmonic map heat flow in two space dimensions and the Yang–Mills heat flow in four space dimensions. Our results are a regularity result for the degree-2 equivariant harmonic map flow, and a blow-up result for an equivariant Yang–Mills-like flow. The results show that qualitatively differing behaviours observed in the two flows can be attributed to the degree of the equivariance.     

Analytic cycles,Bott–Chern forms,and singular sets for the Yang–Mills flow on Kähler manifolds

It is shown that the singular set for the Yang–Mills flow on unstable holomorphic vector bundles over compact Kähler manifolds is completely determined by the Harder–Narasimhan–Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a singular Bott–Chern formula relating the second Chern form of a smooth metric on the bundle to the Chern current of an admissible metric on the associated graded sheaf. This is used to show that the multiplicities of the top dimensional bubbling locus defined via the Yang–Mills density agree with the corresponding multiplicities for the Harder–Narasimhan–Seshadri filtration. The set theoretic equality of singular sets is a consequence.     

Geometry of Solutions of the N=2 SYM Theory in Harmonic Superspace

The classical equations of motion of the D=4, N=2 supersymmetric Yang–Mills (SYM) theory for Minkowski and Euclidean spaces are analyzed in harmonic superspace. We study dual superfield representations of equations and subsidiary conditions corresponding to classical SYM solutions with different symmetries. In particular, alternative superfield constructions of self-dual and static solutions are described in the framework of the harmonic approach.     

Perturbation of Solutions with Moving Singularities

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.     

An elliptic theory of indicial weights and applications to non-linear geometry problems

Given an elliptic operator P on a non-compact manifold (with proper asymptotic conditions), there is a discrete set of numbers called indicial roots. It's known that P is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial. We show that an elliptic theory exists even when the weight is indicial. We also discuss some simple applications to Yang–Mills theory and minimal surfaces.     

Some regularizations of the temporal gauge and propagator of the Yang-Mills field

The paper continues earlier work of Slavnov and the author. Propagators of the Yang—Mills field are found by means of the Faddeev—Popov transition from the Coulomb to the temporal gauge with the use of two types of regularization. A comparison with already known results is made. Further possibilities are discussed.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp. 408–417, March, 1993.     

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

Limit holonomy and extension properties of Sobolev and Yang-Mills bundles

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally L^m/2-integrable curvature and trivial limit holonomy extends through singularity of codimension 2. Additionally, if the connection satisfies the Yang-Mills-Higgs equation, the extension also satisfies the equation.     

On the minimizers of the Ginzburg-Landau energy for high kappa: the axially symmetric case

The Ginzburg-Landau theory of superconductivity is examined in the case of a special geometry of the sample, the infinite cylinder. We restrict to axially symmetric solutions and consider models with and without vortices. First putting the Ginzburg-Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg-Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg-Landau energy is analyzed and different convergence results are obtained. Our main result states that, when κ is large, the minimum of the energy is reached when there are about κ vortices at the center of the cylinder. Numerical computations illustrate the various behaviours.