Goldfish Geodesics and Hamiltonian Reduction of Matrix Dynamics

We describe the Hamiltonian reduction of a time-dependent real-symmetric N×N matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars–Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.      

Central extensions and semi-infinite wedge representations of Krichever-Novikov algebras for more than two points

For the generalized Krichever-Novikov algebras of meromorphic vector fields and their induced modules of weight a different basis is given. With respect to this basis the module structure is generalized graded. Local central extensions of these algebras and their representations on the space of semi-infinite wedge product of forms of weight are studied. In this generalization, one again obtains c = -2(6² -6 +1) as the value for the central charge.     

Krichever-Novikov algebras for more than two points: Explicit generators

For the Krichever-Novikov algebras and modules of meromorphic vector fields and forms with more than two poles on higher genus Riemann surfaces, explicit expressions of the generators are given. These are expressions in terms of theta functions and prime forms for g1, expressions in terms of the Weierstraß function for g=1 and rational expressions for g=0.     

Krichever-Novikov algebras for more than two points

Krichever-Novikov algebras of meromorphic vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight is studied.     

Apparent singularities of Fuchsian equations and the Painlevé property for Garnier systems

We study movable singularities of Garnier systems using the connection of the latter with Schlesinger isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.     

A class of solutions of the dispersionless KP equation

We construct a class of (complex-valued) solutions to the dispersionless KP equation using a meromorphic function on a plane algebraic curve as a variable dependent on suitable Abelian integrals.     

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given.The research of Victor Katsnelson was supported by the Minerva Foundation.     

Hypergeometric systems of Gelfand-type equations and series in orthogonal polynomials

We describe multivariable hypergeometric series in orthogonal polynomials. These series are solutions of special systems of Gelfand-type equations. The difference andq-analogs are also given.     

Chiral fermions on a compact Riemann surface: A sheaf cohomology analysis

We prove that the correlation functions of a system of chiral fermions on a compact Riemann surface are determined by postulating their behaviour at coincident points and a principle of maximal analyticity. The proof proceeds by a reformulation as a problem of sheaf cohomology. Wick's theorem and the Fay identities are rigorous consequences of our analysis.     

Symmetry of Lagrangians of nonholonomic systems

In this Letter, a new symmetry of nonholonomic systems is put forward. Firstly, the definition and the criterion of the symmetry are given. Secondly, the condition under which the symmetry can lead to a conserved quantity is presented. Finally, two examples are shown to illustrate the application of the results.     

Sinc-collocation method for solving the Blasius equation

Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.     

Star products for SL(2)

The deformation program (the use of star products in harmonic analysis) leads to the definition of an adapted Fourier transform, unitary transformation between spaces of square integrable functions of the group G and on the dual of its Lie algebra, describing the unitary dual of G and its Plancherel transform. This Letter is an application of this program to the universal covering group of SL(2).     

Normal Bundles to Laufer Rational Curves in Local Calabi–Yau Threefolds

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank two holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.     

Extended non-abelian gauge symmetries in the classical WZNW model

Higher spin extensions of the non-Abelian gauge symmetries for the classical WZNW model are considered. Both linear and nonlinear realizations of the extended affine Kac-Moody algebra are obtained. It is a characteristic property of the WZNW model that it admits a higher spin linear realization of the extended affine Kac-Moody algebra which is equivalent to the corresponding higher spin nonlinear realization of the same algebra. However, in both cases, the higher spin Noether currents do not span an invariant space with respect to their generating transformations. Here, the current space is extended to an invariant space which allows us to gauge the symmetry.Supported by Bulgarian Foundation on Fundamental Research under contract Ph-318/93-95.     

Self-similar potentials and theq-oscillator algebra at roots of unity

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schrödinger equation provide bases of representations of theq-deformed Heisenberg-Weyl algebra. When the parameterq is a root of unity, the functional form of the potentials can be found explicitly. The generalq ³ = 1 and the particularq ⁴ = 1 potentials are given by the equi-anharmonic and (pseudo) lemniscatic Weierstrass functions, respectively.     

Coulomb gas representation of the SU(2) WZW correlators at higher genera

We extend a previous analysis to the case with insertion points. The result allows us to express the correlation functions of the SU(2) WZW conformal field theory on Riemann surfaces of genus > 1 by finite-dimensional integrals.     

Invariant fourth root Finsler metrics on the Grassmannian manifolds

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds SO(p+q)/SO(p)×SO(q)$SO(p+q)/SO\left(p\right)\times SO\left(q\right)$. By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2m$2m$-th root Finsler metrics are also given.     

The Riemann–Roch theorem and zero-energy solutions of the Dirac equation on the Riemann sphere

In this paper, we revisit the connection between the Riemann–Roch theorem and the zero-energy solutions of the two-dimensional Dirac equation in the presence of a delta-function-like magnetic field. Our main result is the resolution of a paradox—the fact that the Riemann–Roch theorem correctly predicts the number of zero-energy solutions of the Dirac equation despite counting what seem to be functions of the wrong type.     

The dirac operator on the fuzzy sphere

We introduce the fuzzy analog of spinor bundles over the sphere on which the noncommutative analog of the Dirac operator acts. We construct the complete set of eigenstates including zero modes. In the commutative limit, we recover known results.Supported by the Ministry of Science and Research, Austria. Part of Project No. P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in Österreich.