Boundary G/G Theory and Topological Poisson–Lie Sigma Model

We study a boundary version of the gauged WZW model with a Poisson–Lie group G as the target. The Poisson–Lie structure of G is used to define the Wess–Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson–Lie group G ^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov–Tian–Shansky.     

Generalization of the Hellmann-Feynman theorem

The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.     

Chern-Simons states at genus one

We present a rigorous analysis of the Schrödinger picture quantization for theSU(2) Chern-Simons theory on 3-manifold torusxline, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals ofsu(2)-connections on the torus, are expressed by degree 2k theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistablesu(2)-connections into nine submanifolds and by analyzing the behavior of states at four codimension 1 strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins>1/2 level implies the Verlinde dimension formula.     

Experimental and Computational Studies of the Molybdenum‐Flanking Arene Interaction in Quadruply Bonded Dimolybdenum Complexes with Terphenyl Ligands

To clarify the nature of the Mo?C_arene interaction in terphenyl complexes with quadruple Mo?Mo bonds, ether adducts of composition [Mo₂(Ar′)(I)(O₂CR)₂(OEt₂)] have been prepared and characterized (Ar′=Ar^Xyl₂, R=Me; Ar′=ArMes₂, R=Me; Ar′=Ar^Xyl2, R=CF₃) (Mes=mesityl; Xyl=2,6‐Me₂C₆H₃, from now on xylyl) and their reactivity toward different neutral Lewis bases investigated. PMe₃, P(OMe)₃ and PiPr₃ were chosen as P‐donors and the reactivity studies complemented with the use of the C‐donors CNXyl and CN₂C₂Me₄ (1,3,4,5‐tetramethylimidazol‐2‐ylidene). New compounds of general formula [Mo₂(Ar′)(I)(O₂CR)₂( L )] were obtained, except for the imidazol‐2‐ylidene ligand that yielded a salt‐like compound of composition [Mo₂(Ar^Xyl2)(O₂CMe)₂(CN₂C₂Me₄)₂]I. The Mo?C_arene interaction in these complexes has been analyzed with the aid of X‐ray data and computational studies. This interaction compensates the coordinative and electronic unsaturation of one of the Mo atoms in the above complexes, but it seems to be weak in terms of sharing of electron density between the Mo and C_arene atoms and appears to have no appreciable effect in the length of the Mo?Mo, Mo?X, and Mo? L bonds present in these molecules.     

Distortions of π‐Coordinated Arenes with Anionic Character

A qualitative analysis of the distortions that operate on the π system of bridging arenes with anionic character is presented and substantiated by computational studies at the density functional B3LYP and CASSCF levels. The observed effects of bonding to two metal atoms and of the negative charge are an expansion of the arene ring due to the partial occupation of π* orbitals, an elongation or compression distortion accompanied by a loss of the equivalence of carbon‐carbon bonds due to a Jahn–Teller distortion of the arene dianions, and a ring puckering due to a second‐order Jahn–Teller distortion that may appear independently of the existence of the first‐order effect. The workings of the orbital mixing produced by these distortions have been revealed in a straightforward way by a pseudosymmetry analysis of the HOMOs of the distorted conformations. The systems studied include Li^I and Y^III adducts of benzene, as well as trimethylsilyl‐substituted derivatives in the former case. An analysis of the structural data of a variety of purported di‐ and tetraanionic arene ligands coordinated to transition metals in several bridging modes has reproduced the main geometrical trends found in the computational study for the benzene and trimethylsilyl‐substituted benzene dianions, allowing a classification of the variety of structural motifs found in the literature.     

Star Products and Branes in Poisson-Sigma Models

We prove that non-coisotropic branes in the Poisson-Sigma model are allowed at the quantum level. When the brane is defined by second-class constraints, the perturbative quantization of the model yields Kontsevich’s star product associated to the Dirac bracket on the brane. Finally, we present the quantization for a general brane.Research supported by grant FPU, MEC (Spain).Research supported by grant FPA2003-02948, MEC (Spain).     

Poisson Reduction and Branes in Poisson–Sigma Models

We analyse the problem of boundary conditions for the Poisson–Sigma model and extend previous results showing that non-coisotropic branes are allowed. We discuss the canonical reduction of a Poisson structure to a submanifold, leading to a Poisson algebra that generalizes Diracs construction. The phase space of the model on the strip is related to the (generalized) Dirac bracket on the branes through a dual pair structure.Mathematics Subject Classifications (2000). 81T45, 53D17, 81T30, 53D55.     

Generalized Central Limit Theorem and Renormalization Group

We introduce a simple instance of the renormalization group transformation in the Banach space of probability densities. By changing the scaling of the renormalized variables we obtain, as fixed points of the transformation, the Lévy strictly stable laws. We also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter. The physical interest of a renormalization group approach to the generalized central limit theorem is discussed.