arXiv subjects
Publications and source records attributed to P. Mathieu.
Fermionic-type character formulae are presented for charged irreduciblemodules of the graded parafermionic conformal field theory associated to the coset $osp(1,2)_k/u(1)$. This is obtained by counting the weakly ordered `partitions' subject to the graded $Z_k$ exclusion principle. The bosonic form of the characters is also presented.
This paper carries on the investigation of the non-unitary su(2)_{-1/2} WZW model. An essential tool in our first work on this topic was a free-field representation, based on a c=-2 ηξghost system, and a Lorentzian boson. It turns out that there are several ``versions'' of the ηξsystem, allowing different su(2)_{-1/2} theories. This is explored here in details. In more technical terms, we consider extensions (in the c=-2 language) from the small to the large algebra representation and, in a further step, to the full symplectic fermion theory. In each case, the results are expressed in terms of su(2)_{-1/2} representations. At the first new layer (large algebra), continuous representations appear which are interpreted in terms of relaxed modules. At the second step (symplectic formulation), we recover a logarithmic theory with its characteristic signature, the occurrence of indecomposable representations. To determine whether any of these three versions of the su(2)_{-1/2} WZW is well-defined, one conventionally requires the construction of a modular invariant. This issue, however, is plagued with various difficulties, as we discuss.
We study models of continuous time, symmetric, $\Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0, and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.
This work initiates the study of {\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly.
A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.
We first review the construction of the supersymmetric extension of the (quantum) Calogero-Moser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models ({\it s} for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the 4N (N being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of 2N commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying 2N-2 extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered.
We report Small-Angle Neutron Scattering (SANS) measurements on the vortex lattice in a PbIn polycrystal in the presence of an applied current. Using the rocking curves as a probe of the distribution of current in the sample, we observe that vortex pinning is due to the surface roughness. This leads to a surface current that persists in the flux flow region. We show the influence of surface treatments on the distribution of this current.
The peak effect has been investigated in clean Nb crystals with artificially corrugated surfaces by measuring the linear surface impedance in the 1kHz-1MHz frequency range. From a two-mode analysis of the complex spectra, we establish that vortex dynamics is governed by surface pinning and deduce the associated vortex slippage length. We demonstrate experimentally and theoretically that the peak effect is related to a transition from collective to individual surface pinning. A proper account of the peak-effect anomalies implies softening of the shear rigidity by disorder-induced lattice deformations. This leads to a vortex crystal-glass transition induced by surface defects.
We review some contributions on fusion rules that were inspired by the work of Sharp, in particular, the generating-function method for tensor-product coefficients that he developed with Patera. We also review the Kac-Walton formula, the concepts of threshold level, fusion elementary couplings, fusion generating functions and fusion bases. We try to keep the presentation elementary and exemplify each concept with the simple $\su(2)_k$ case.
If X is a d-dimensional uniformly elliptic diffusion, with initial law nu, we show that F(X) is a Dirichlet process, whenever F satisfies an integrability condition linking its weak derivative to the coefficients of the diffusion and the initial law nu. We then show that F(X) satisfies an Ito formula, giving a construction of the stochastic integral of grad F(X) with respect to X, provided that the two first weak derivatives of F satisfy integrability conditions involving the coefficients of the diffusion and the initial law. Si X est une diffusion uniformement elliptique d-dimensionnelle, de loi initiale nu, on montre que F(X) est un processus de Dirichlet, lorsque F verifie une condition d'integrabilite qui lie ses derivees faibles aux coefficients de la diffusion et a la loi initiale nu. On montre ensuite qu'on peut ecrire une formule d'Ito pour F(X), en donnant une construction de l'integrale stochastique de grad F(X) par rapport a X. Les conditions requises sur F sont des conditions d'integrabilite liant ses derivees faibles, premiere et seconde, aux coefficients de la diffusion et a la loi initiale nu.
The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.
We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model's Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials.
In a recent paper, Matl et al present a high-frequency study of the complex resistivity of a pinned vortex lattice in YBaCuO . They focus on the inductive-to-resistive transition which is investigated as a function of temperature at a constant field $B_0=2$ T, so that the transition is associated with the vanishing of vortex pinning strength. To our view, their conclusions rely on a rather brittle experimental body and the collapse of C66 results from an involved analysis of the finite frequency corrections to $ρ(ω)$. These corrections are not necessary since the complex frequency spectrum has been previously interpreted by the two modes model, first proposed for low Tc materials. We think that it is more adequate to interpret the present data and should be at least considered.
A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for $\su(2)$, 432 for $\su(3)$ and quite surprisingly, it reduces to 36 for $\su(4)$, while it is only of order 4 for $\sp(4)$. This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For $\su(2)$ and $\su(3)$, it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For $\su(3)$, new symmetries of the fusion coefficients are found.
Two bases of states are presented for modules of the graded parafermionic conformal field theory associated to the coset $\osp(1,2)_k/\uh(1)$. The first one is formulated in terms of the two fundamental (i.e., lowest dimensional) parafermionic modes. In that basis, one can identify the completely reducible representations, i.e., those whose modules contain an infinite number of singular vectors; the explicit form of these vectors is also given. The second basis is a quasi-particle basis, determined in terms of a modified version of the $\ZZ_{2k}$ exclusion principle. A novel feature of this model is that none of its bases are fully ordered and this reflects a hidden structural $\Z_3$ exclusion principle.
A new basis of states for highest-weight modules in $\ZZ_k$ parafermionic conformal theories is displayed. It is formulated in terms of an effective exclusion principle constraining strings of $k$ fundamental parafermionic modes. The states of a module are then built by a simple filling process, with no singular-vector subtractions. That results in fermionic-sum representations of the characters, which are exactly the Lepowsky-Primc expressions. We also stress that the underlying combinatorics -- which is the one pertaining to the Andrews-Gordon identities -- has a remarkably natural parafermionic interpretation.
Some simple (namely, single-channel) correlation functions involving an arbitrary number of fields are computed by means of a direct application of the residue calculus, through partial fraction expansions. Examples are presented in minimal models and parafermionic conformal theories. A generic factorized expression is deduced for the corresponding single-channel structure constants.
Through an $\hbar$-expansion of the confined Calogero model with spin exchange interactions, we extract a generating function for the involutive conserved charges of the Frahm-Polychronakos spin chain. The resulting conservation laws possess the spin chain yangian symmetry, although they are not expressible in terms of these yangians.