Folding the W Algebras

In the same way the folding of the Dynkin diagram of A_{2n} (resp. A_{2n-1}) produces the B_n (resp. C_n) Dynkin diagram, the symmetry algebra W of a Toda model based on B_n (resp. C_n) can be seen as resulting from the folding of a W-algebra based on A_{2n} (resp. A_{2n-1}). More generally, W algebras related to the B-C-D algebra series can appear from W algebras related to the unitary ones. Such an approach is in particular well adapted to obtain fusion rules of W algebras based on non simply laced algebras from fusion rules corresponding to the A_n case. Anagously, super W algebras associated to orthosymplectic superalgebras are deduced from those relative to the unitary A(m,n) series.Comment: 27 pages, Late

Hyperbolic Supersymmetric Quantum Hall Effect

Developing a non-compact version of the SUSY Hopf map, we formulate the quantum Hall effect on a super-hyperboloid. Based on $OSp(1|2)$ group theoretical methods, we first analyze the one-particle Landau problem, and successively explore the many-body problem where Laughlin wavefunction, hard-core pseudo-potential Hamiltonian and topological excitations are derived. It is also shown that the fuzzy super-hyperboloid emerges in the lowest Landau level.Comment: 14 pages, two columns, no figures, published version, typos correcte

Sum rules of codon usage probabilities

In the crystal basis model of the genetic code, it is deduced that the sum of usage probabilities of the codons with C and A in the third position for the quartets and/or sextets is independent of the biological species for vertebrates. A comparison with experimental data shows that the prediction is satisfied within about 5 %.Comment: 7 page

Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

We classify all Hamiltonians with rank 1 symmetry, acting on a periodic three-state spin chain, and solvable through (generalisation of) the coordinate Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev, Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians, there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a generalization of the genus 5 special branch found by Martins, plus three new ones. We get also two 14-vertex Hamiltonians. We solve all these Hamiltonians using CBA, and provide their spectrum, eigenfunctions and Bethe equations. A special attention is made to provide the specifications of our multi-parametric Hamiltonians that give back known Hamiltonians.Comment: 30 pages; web page: http://www.coulomb.univ-montp2.fr/3Ha

Many-body spin Berry phases emerging from the π\pi-flux state: antiferromagnetic/valence-bond-solid competition

We uncover new topology-related features of the $\pi$-flux saddle-point solution of the $D$=2+1 Heisenberg antiferromagnet. We note that symmetries of the spinons sustain a built-in competition between antiferromagnetic (AF) and valence-bond-solid (VBS) orders, the two tendencies central to recent developments on quantum criticality. An effective theory containing an analogue of the Wess-Zumino-Witten term is derived, which generates quantum phases related to AF monopoles with VBS cores, and reproduces Haldane's hedgehog Berry phases. The theory readily generalizes to $\pi$-flux states for all $D$.Comment: 4 pages, revise

On Casimir's Ghost

We define on the universal enveloping superalgebra of osp(1|2n) a nonstandard adjoint action, endowing it with a module structure. This allows, in particular, to construct a bosonic operator which anticommutes with all the fermionic generators and which appears to be the square root of a certain Casimir operator.Comment: LaTeX2e, 13 pages,also available at http://lapphp0.in2p3.fr/preplapp/psth/ENSLAPP587.ps.gz ; one sentence removed and a note added. Not a major revisio

Crystalizing the genetic code

New developments are presented in the framework of the model introduced by the authors in refs. [1,2] and in which nucleotides as well as codons are classified in crystal bases of the quantum group U_q(sl(2)+sl(2)) in the limit q -> 0. An operator which gives the correspondence between the amino-acids and the codons is now obtained for any known genetic code. The free energy released by base pairing of dinucleotides as well as the relative hydrophilicity and hydrophobicity of the dinucleosides are also computed. For the vertebrate series, a universal behaviour in the ratios of codon usage frequencies is put in evidence and is shown to fit nicely in our model. Then a first attempt to represent the mutations relative to the deletion of a pyrimidine by action of a suitable crystal spinor operator is proposed. Finally recent theoretical descriptions are reviewed and compared with our model