Galliano Valent

New geometry from heterotic supersymmetry

The authors show that the bosonic sector of a (4,0) supersymmetric sigma -model is a quasi Ricci-flat manifold with a closed torsion and vanishing scalar curvature; in the torsionless limit they recover the hyper-Kahler (4,4) models. Using the harmonic superspace and the self-duality equations they derive the (4,0) generalizations of the metrics of Taub-NUT and Eguchi-Hanson.

One loop renormalizability of the Poisson-Lie sigma models

Abstract We present the proof of the one loop renormalizability in the strict field theoretic sense of the Poisson–Lie σ-models. The result is valid for any Drinfeld double and it relies solely on the Poisson–Lie structure encoded in the target manifold.

Integrability Versus Separability for the Multi-Centre Metrics

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-Stäckel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors.

Some cubic birth and death processes and their related orthogonal polynomials

AbstractThe orthogonal polynomials with recurrence relationn(λ,n +μn-z)Fn(z) = μn+1Fn+1(z)+λn-1Fn-1(z) with two kinds of cubic transition rates λn and μn, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.

Integrable geodesic flows and multi-centre versus Bianchi A metrics

It is shown that most, but not all, of the four-dimensional metrics in the multi-centre family with integrable geodesic flows may be recognized as belonging to spatially homogeneous Bianchi type A metrics. We show that any diagonal bi-axial Bianchi II metric has an integrable geodesic flow, and that the simplest metric in this family displays a finite-dimensional W-algebra for its observables. Our analysis also sheds light on a non-diagonal Bianchi VII0 metric which seems to be new. We conclude by showing that the elliptic coordinates advocated in the literature do not separate the Hamilton–Jacobi equation for the tri-axial Bianchi IX metric.

Indeterminate Moment Problems and a Conjecture on the Growth of the Entire Functions in the Nevanlinna Parametrization

Combining previous results on a quartic indeterminate Stieltjes moment problem and new results for a cubic one, we present a conjecture on the order, type, and Phragmen-Lindelof indicator for a large class of indeterminate Stieltjes moment problems.

Neumann-like integrable models

Abstract A countable class of integrable dynamical systems, with four-dimensional phase space and conserved quantities in involution ( H n , I n ) are exhibited. For n = 1 we recover Neumann system on T ∗ S 2 . All these systems are also integrable at the quantum level.

Bianchi type II, III and V diagonal Einstein metrics re-visited

We present, for both Minkowskian and Euclidean signatures, short derivations of the diagonal Einstein metrics for Bianchi type II, III and V. For the first two cases we show the integrability of the geodesic flow while for the third case a somewhat unusual bifurcation phenomenon takes place: for Minkowskian signature elliptic functions are essential in the metric while for Euclidean signature only elementary functions appear.

Screened Coulomb scattering versus Thomson scattering

We examine a simplified approach to the classical scattering of a particle by a neutral atom, first for a Thomson-like model and then for different screened Coulomb potentials. As an application we discuss in detail why, in the experiments of Geiger and Marsden on the scattering of α particles by matter, screening effects are negligible. On the way we encounter several unusual phenomena in classical scattering theory.

Further results on non-diagonal: Bianchi type III vacuum metrics

We present the derivation, for these vacuum metrics, of the Painlevé VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows to give the precise connection with some old results due to Kinnersley. The hyperkähler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperkähler metrics.