Bethe's Quantum Numbers And Rigged Configurations

We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).Comment: 16 pages. Supplementary tables are included in the source file. (v2) New example at pages 8--9. (v3) Final version with minor revisio

An approximating method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems using Hamiltonian perturbation theory

In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical equation) that is derived from the theory of 1-st order partial differential equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati equation from the Hamilton-Jacobi equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained

Rigged Configurations and Kashiwara Operators

For types $A^{(1)}_n$ and $D^{(1)}_n$ we prove that the rigged configuration bijection intertwines the classical Kashiwara operators on tensor products of the arbitrary Kirillov-Reshetikhin crystals and the set of the rigged configurations.Comment: v2: 108 pages, the author's final version for publication, Proposition 33 added, Section 7.3 partially reworked; v3: published version (Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

On Gauge Symmetry Breaking via Euclidean Time Component of Gauge Fields

We study gauge theories with/without an extra dimension at finite temperature, in which there are two kinds of order parameters of gauge symmetry breaking. The one is the zero mode of the gauge field for the Euclidean time direction and the other is that for the direction of the extra dimension. We evaluate the effective potential for the zero modes in one-loop approximation and investigate the vacuum configuration in detail. Our analyses show that gauge symmetry can be broken only through the zero mode for the direction of the extra dimension and no nontrivial vacuum configuration of the zero mode for the Euclidean time direction is found.Comment: 22 pages, 6 figures, references and typos corrected, version to appear in PR

Large Gauge Hierarchy in Gauge-Higgs Unification

We study a five dimensional SU(3) nonsupersymmetric gauge theory compactified on $M^4\times S^1/Z_2$ and discuss the gauge hierarchy in the scenario of the gauge-Higgs unification. Making use of calculability of the Higgs potential and a curious feature that coefficients in the potential are given by discrete values, we find two models, in which the large gauge hierarchy is realized, that is, the weak scale is naturally obtained from an unique large scale such as a grand unified theory scale or the Planck scale. The size of the Higgs mass is also discussed in each model. One of the models we find realizes both large gauge hierarchy and consistent Higgs mass, and shows that the Higgs mass becomes heavier as the compactified scale becomes smaller.Comment: 21 pages, no figures, version to appear in PR