D. Arnaudon

Periodic and partially periodic representations ofSU(N) q

The Gelfand-Zetlin basis is adapted toSU(N)q forq a root of unit. Extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center. A crucial identity is derived and proved, which guarantees the periodicity of the action of the generators. Full periodicity is relaxed by stages, some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation. In the extreme case of highest weight representations. all the raising and lowering operators are nilpotent. As an alternative approach an auxiliary algebra giving all the periodic representations is presented. An explicit solution of this system forN=3, while fully equivalent to the G.-Z. basis, turns out to be much simpler.

Periodic and flat irreducible representations ofSU(3) q

We construct all the periodic irreducible representations ofU(SU(3))q forq am-root of unity. Their dimensions arek(2m)2 fork=1,...,m (onlyk=1,...,m/2 for evenm). Their interest is that they could be a tool to generalize the chiral Potts model. By truncation of these representations, we construct “flat representations” ofU(SU(3))q, in which all the multiplicities of the weights are set to 1.

Flat periodic representations ofU q (G)

We give explicit expression of flat periodic representations, when they exist, of the quantum analogues of simple Lie algebras and their affine extensions for a parameter of deformationq equal to a root of unity. By “flat periodic”, we mean that these representations have no highest weight and that all the weights have multiplicity 1.

On the vanishing of the cosmological constant in four-dimensional superstring models

Abstract In the general framework of the recently constructed fermionic superstring models, which can live in any spacetime dimension below ten and including four, we prove that the cosmological constant vanishes to all orders in the loop expansion if: (i) the free string spectrum contains at least one massless spin 3 2 particle, and (ii) a simple conjecture on the spin-structure dependence of the superghost and supermoduli contribution to the amplitudes holds. This conjecture, which arises naturally in the context of four-dimensional N = 1 superstrings, is the simplest one compatible with multiloop modular invariance.

q-Analogue of Iu (n) for q a root of unity

Abstract Periodic and partially periodic representations of U(IU(n))q are constructed for qm = 1, m being an odd or even integer. The classical non-semisimple IU (n) or U (n)⋉I 2n algebra has an abelian subalgebra of dimension 2n. Gelfand-Zetlin bases and matrix elements are generalized and adapted to this case. Our previous results for U(IU(n))q for a generic q (not a root of unity) and those for SU(N)q for qm = 1 are combined in the present study giving explicit matrix elements and eigenvalues such as the second order Casimilar operator D2 = K2 cos(2π/m)(h2n+1+ … + hnn+1 + n − 1)/cos(2π/m). This displays the role of the internal parameters (hi,n+1) in the q-analogue of the classical K2 (“mass” squared). The two translation generators (Inn+1, In+1n) become periodic.

Periodic representations of SO(5)q

Abstract We give explicit expressions for periodic irreducible representations of U q ( SO (5)) (or complex quantum De Sitter group) when the parameter of deformation q is an m th root of unity. Their maximal dimension is m 4 when m is odd, as expected, and 1 4 m 4 when m is even.

Weakly self-avoiding polymers in four dimensions. Rigorous results

Abstract The Edwards model of weakly self-avoiding polymers in four dimensions is studied by a renormalization group method. We prove the validity at any order of the behaviour of physical quantities at large size of the polymers, as given by perturbation theory. Remainders are controlled by a new argument which enlarges the use of constructive field theory methods to statistical physics models.

Minimal surfaces and string tension from non-planar loops

Abstract We suggest that one-scale non-planar loops can be useful tools to study the approach of lattice gauge theory to the continuum limit. We show how to extract the string tension with reduced systematic errors from the knowledge of the minimal surface bounded by these loops. We apply this method in a Monte Carlo calculation of the string tension of the SU (3) lattice gauge theory in several color representations on a 8 4 lattice. The results are in complete agreement up to β = 6.0 with those obtained on much larger lattices from planar loop ratios or correlations of thermal loops.

Composition of kinetic momenta: The (sl(2)) casecase

The tensor products of (restricted and unrestricted) finite dimensional irreducible representations of(sl(2)) are considered forq a root of unity. They are decomposed into direct sums of irreducible and/or indecomposable representations.The tensor products of (restricted and unrestricted) finite dimensional irreducible representations of Open image in new window (sl(2)) are considered forq a root of unity. They are decomposed into direct sums of irreducible and/or indecomposable representations.

On the Witten vertex

Abstract The three-string vertex is computed by elementary means in the formalism of Witten. It is shown that in contrast with the standard overlap of Cremmer and Gervais, it is possible to carry out an explicitly convergent computation, without Neumann functions and multi-sheeted mappings. Numerical evidence is found that the Witten vertex for physical states reduces to the Veneziano vertex.