Multiderivations of Coxeter arrangements

Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $\alpha_H \in V^*$ such that $H = {\rm ker}(\alpha_H).$ For each nonnegative integer $m$, define the derivation module \sD^{(m)}({\cal A}) = \{\theta \in {\rm Der}_S | \theta(\alpha_H) \in S \alpha^m_H\}. The module is known to be a free $S$-module of rank $\ell$ by K. Saito (1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$. The main result of this paper is that this is the case for all $m$. Moreover we explicitly construct a basis for \sD^{(m)} (\cal A). Their degrees are all equal to $mh/2$ (when $m$ is even) or are equal to $((m-1)h/2) + m_i (1 \leq i \leq \ell)$ (when $m$ is odd). Here $m_1 \leq ... \leq m_{\ell}$ are the exponents of $G$ and $h= m_{\ell} + 1$ is the Coxeter number. The construction heavily uses the primitive derivation $D$ which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of $G$.) Some new results concerning the primitive derivation $D$ are obtained in the course of proof of the main result.Comment: dedication and a footnote (thanking a grant) adde

Two Dimensional Black Hole Evapolation in the Light-Cone Gauge

Quantization of the pure $1+1$ dimensional dilaton gravity is examined in the light-cone gauge. It is found that the total action including ghosts generates a $c=0$ free conformal field theory without modification of the classical action, which is required in the conformal gauge. We also study semiclassical equations of the dilaton gravity coupled to $N$ scalar fields. It is shown that the black hole singularity is not removed even for $N<24$ in the light-cone gauge. This indicates that the semiclassical analysis breaks down for small $N$.Comment: 10 pages, KANAZAWA-92-1

Exchange and spin Jahn-Teller distortions for a triangular cluster of spin-1/2

We study the effects of magnetoelastic coupling on the degenerate ground state of the spin-1/2 antiferromagnetic Heisenberg model for the regular triangular spin cluster. Static displacement of spins spontaneously lifts the degeneracy of the ground state through the distance dependence of exchange coupling, i.e., a spin Jahn-Teller mechanism takes place. On the other hand, dynamical displacement does not lift the degeneracy, though the cluster distorts spontaneously. The energy decrease obtained by dynamical theory is twice as large as that obtained by static theory because of quantum fluctuation.Comment: 4 pages, 1 figure. Accepted by JPSJ. Clarified some setences. Corrected typo

Softly Broken Supersymmetric Gauge-Higgs-Yukawa Theories as Renormalizable Composite Models

We examine the (softly broken) supersymmetric gauge-Higgs-Yukawa theories satisfying the compositeness conditions at a certain scale. In these theories the Higgs superfields can be regarded as the chiral composite fields. It is found that there are the fundamental theories, which contain the dimension 5 interactions and the hard SUSY breaking in perturbation, turn to be renormalizable and also softly broken theories in the nonperturbative framework. The soft SUSY breaking parameters as well as the Yukawa coupling in the corresponding gauge-Higgs-Yukawa theories are restricted by the renormalization group invariant relations.Comment: 14 pages, latex, 2 figure

Simple-root bases for Shi arrangements

In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases

Equivalent models for gauged WZW theory

We modify the $SL(2,{\bf R})/U(1)$ WZW theory, which was shown to describe strings in a 2D black hole, to be invariant under chiral $U(1)$ gauge symmetry by introducing a Steukelberg field. We impose several interesting gauge conditions for the chiral $U(1)$ symmetry. In a paticular gauge the theory is found to be reduced to the Liouville theory coupled to the $c=1$ matter perturbed by the so-called black hole mass operator. Also we discuss the physical states in the models briefly.Comment: 13pages, KANAZAWA 93-0

Higgs and Top quark coupled with a conformal gauge sector

We propose a dynamical scenario beyond the standard model, in which the radiative correction to the Higgs mass parameter is suppressed due to a large anomalous dimension induced through a conformal invariant coupling with an extra gauge sector. Then the anomalous dimension also suppresses the Yukawa couplings of the Higgs field. However, the large top Yukawa coupling can be generated effectively through mixing among top quarks and the fermions of the conformal gauge sector. This scenario is found to predict a fairly heavy Higgs mass of about 500 GeV. We present an explicit model and show consistency with the Electro-Weak precision measurements of the S and T parameters as well as the Z boson decay width.Comment: 12 pages, 3 Postscript figures, uses RevTex.sty; corrected typos and reference

Democratic mass matrices induced by strong gauge dynamics and large mixing angles for leptons

We consider dynamical realization of the democratic type Yukawa coupling matrices as the Pendelton-Ross infrared fixed points.?Such fixed points of the Yukawa couplings become possible by introducing many Higgs fields, which are made superheavy but one massless mode. Explicitly, we consider a strongly coupled GUT based on $SU(5) \times SU(5)$, where rapid convergence to the infrared fixed point generates sufficiently large mass hierarchy for quarks and leptons. Especially, it is found that the remarkable difference between mixing angles in the quark and lepton sectors may be explained as a simple dynamical consequence. We also discuss a possible scenario leading to the realistic mass spectra and mixing angles for quarks and leptons. In this scheme, the Yukawa couplings not only for top but also for bottom appear close to their quasi-fixed points at low energy and, therefore, $\tan \beta$ should be large.Comment: 25 pages, 5 figure