arXiv subjects
Publications and source records attributed to Satoshi Watamura.
We give a construction of the monopole bundles over fuzzy complex projective spaces as projective modules. The corresponding Chern classes are calculated. They reduce to the monopole charges in the N -> infinity limit, where N labels the representation of the fuzzy algebra.
We construct the boundary state for the D-brane in the SU(2) group manifold directly in terms of the group variables. We propose a matching condition for the left- and the right-moving sectors including the zero modes that describes a D-brane of the Neumann-type. The free field realization of the WZW model is used to obtain the boundary state subject to the matching condition. We show that the resulting state coincides with Cardy's state. The structure of the BRST cohomology is realized by imposing the invariance of the state under the Weyl group of the current algebra.
The differential algebra on the fuzzy sphere is constructed by applying Connes' scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left $U(N+1)$ transformation of the field, where a field is a bimodule over the quantized algebra $\CA_N$. The interaction with a complex scalar field is also given.
We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes' spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived.
We give a derivation of the Dirac operator on the noncommutative $2$-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.
With the aim to construct a dynamical model with quantum group symmetry, the $q$-deformed Schrödinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space is investigated. After reviewing the differential calculus on the $q$-Euclidian space, the $q$-analog of the creation-annihilation operator is constructed. It is shown that it produces systematically all eigenfunctions of the Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the $q$-analysis. We represent the Schrödinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions. The problem of the involution corresponding to the reality condition is discussed.
We consider the $q$-deformed Schrödinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space. The creation and annihilation operator are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the $q$-series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the $q$-analysis. We represent the Schrödinger equation by the $q$-difference equation and solve it by using $q$-polynomials and $q$-exponential functions.
We investigate the $q$-deformation of the BRST algebra, the algebra of the ghost, matter and gauge fields on one spacetime point using the result of the bicovariant differential calculus. There are two nilpotent operations in the algebra, the BRST transformation $\brs$ and the derivative $d$. We show that one can define the covariant commutation relations among the fields and their derivatives consistently with these two operation as well as the $*$-operation, the antimultiplicative inner involution.