Demazure Characters and Affine Fusion Rules

The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmann's derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result, however. Only a modified version of the Littlewood-Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of affine $A_r$ algebras. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl group must be implicated in any combinatorial rule for affine fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice.Comment: 23 pages, TeX, uses harvma

On Fusion Algebras and Modular Matrices

We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We also find new, systematic sources of zeros in the modular matrix $S$. In addition, we obtain a formula relating the entries of $S$ at fixed points, to entries of $S$ at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.Comment: 28 pages, plain Te

On Robin boundary conditions and the Morse potential in quantum mechanics

The physical origin is investigated of Robin boundary conditions for wave functions at an infinite reflecting wall. We consider both Schr\"odinger and phase-space quantum mechanics (a.k.a. deformation quantization), for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wave functions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Dirichlet boundary conditions (standard walls) are obtained unless a mass-dependent fine tuning (to a reflection resonance) is imposed. This phenomenon was already observed for piece-wise flat potentials, so it is not removed by smoothing. We argue that it explains why standard quantum walls are standard. Next we consider the Wigner functions (the symbols of both diagonal and off-diagonal density operator elements) of phase-space quantum mechanics. Taking the (fine-tuned) limit, we show that our Wigner functions do reduce to the expected ones on the half-line. This confirms that the Wigner transform should indeed be unmodified for this contact interaction.Comment: 19 page

An imaging and spectroscopic study of the planetary nebulae in NGC 5128 (Centaurus A): Planetary nebulae catalogues

Imaging and spectroscopic observations of planetary nebulae (PNe) in the nearest large elliptical galaxy NGC 5128 (Centaurus A), were obtained to find more PNe and measure their radial velocities. NTT imaging was obtained in 15 fields in NGC 5128 over an area of about 1 square degree with EMMI using [O III] and off-band filters. Newly detected sources, combined with literature PNe, were used as input for VLT FLAMES multi-fibre spectroscopy in MEDUSA mode. Spectra of the 4600-5100A region were analysed and velocities measured based on emission lines of [O III]4959,5007A and often H-beta. The chief results are catalogues of 1118 PN candidates and 1267 spectroscopically confirmed PNe in NGC 5128. The catalogue of PN candidates contains 1060 PNe discovered with EMMI imaging and 58 from literature surveys. The spectroscopic PN catalogue has FLAMES radial velocity and emission line measurements for 1135 PNe, of which 486 are new. Another 132 PN radial velocities are available from the literature. For 629 PNe observed with FLAMES, H-beta was measured in addition to [O III]. Nine targets show double-lined or more complex profiles, and their possible origin is discussed. FLAMES spectra of 48 globular clusters were also targetted: 11 had emission lines detected (two with multiple components), but only 3 are PNe likely to belong to the host globular. The total of 1267 confirmed PNe in NGC 5128 with radial velocity measurements (1135 with small velocity errors) is the largest collection of individual kinematic probes in an early-type galaxy. This PN dataset, as well as the catalogue of PN candidates, are valuable resources for detailed investigation of the stellar population of NGC 5128. [Abridged]Comment: 19 pages, 11 figures. Accepted for publication in Astronomy and Astrophysics. Tables 7 - 11 available in electronic form at CDS. Replaced with a few typos fixe