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We show, using the methods of geometric algebra, that Runge-Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, RL vector is a consequence of geometric symmetry of $\mathbb{R}^4\times \mathbb{R}^4$ phase space., Comment: Corrected typographical error. The final version is published in J. Phys. A

We introduce the basic equivariant quantity $Q$ in the gauge theory on the noncommutative descrete $Z_{2}$ space, which plays an important role for the equivariant dimensional reduction. If the gauge configuration of the ground state on the extra dimensional space is described by the equivariant $Q$, then the extra dimensional space is invisible. Especially, using the equivariance principle, we show that the Yang-Mills theory on $R^{2}\times Z_{2}$ space is equivalent to the Yang-Mills-Higgs model on $R^{2}$ space. It can be said that this model is the simplest model of this type., Comment: Some discussion added

We show that vortices of Yang-Mills-Higgs model in $R^{2}$ space can be regarded as instantons of Yang-Mills model in $R^{2}\times Z_{2}$ space. For this, we construct the noncommutative $Z_{2}$ space by explicitly fixing the $Z_{2}$ coordinates and then show, by using the $Z_{2}$ coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation., Comment: 19 pages, various arguments are added, the exposition is improved

There seems to be close relationship between the moduli space of vortices and the moduli space of instantons, which is not yet clearly understood from a standpoint of the field theory. We clarify the reasons why many similarities are found in the methods for constructing the moduli of instanton and vortex, viewed in the light of the notion of the self-duality. We show that the non-Abelian vortex is nothing but the instanton in $R^{2} \times Z_{2}$ from a viewpoint of the noncommutative differential geometry and the gauge theory in discrete space. The action for pure Yang-Mills theory in $R^{2} \times Z_{2}$ is equivalent to that for Yang-Mills-Higgs theory in $R^{2} $., Comment: 19 pages, various arguments are added, the exposition is improved

We point out that the similarities in N = 1 supersymmetric SO, SP gauge theories can be explained by using the trick of extrapolating the groups to the negative dimensions. One of the advantages of this trick is that anomaly matching is automatically satisfied., Comment: 6 pages, LaTeX, no figure, Manuscript has been substantially modified because the duality we proposed before as an application of our method was wrong. The title itself has been changed accordingly

We calculate the $F/D$ ratios of spin-nonflip baryon vertex for an arbitrary number of color degrees of freedom $N_c$ both in the non-relativistic quark model with the $SU(6)$ spin-flavor symmetry and in the chiral soliton model with $SU(3)$ flavor symmetry. We find that the spin-nonflip $F/D$ ratio tends to $-1$ in the limit of $N_c \to \infty$. We show that this leading value $F/D= -1$ of spin-nonflip baryon vertex in the $1/N_c$ expansion corresponds to the isoscalar dominance while the well known leading value $F/D=1/3$ of the spin-flip vertex corresponds to the isovector dominance. We discuss origins of the dominance of isovector in spin-flip and isoscalar in spin-nonflip baryon vertices, referred to as the $I=J$ rule. \par In terms of the matrix elements of the operator which transform as the generator $\lambda^8$ of the $SU(3)$ symmetry we derive the model independent isoscalar formula for baryon vertices and apply this to the mass formula and the isoscalar part of the baryon magnetic moments. The same Okubo-Gell-Mann mass relation and its refined relation among the octet baryons as the one for the case $N_c=3$ is derived model independently for arbitrary color degrees of freedom $N_c$. Contrary to $\lambda^8$, Comment: 25pages, LaTeX file, with uuencoded 3 postscript figures

We calculate the $F/D$ ratios of spin 1/2 baryon vertex for both the non-relativistic quark model and the chiral soliton model with arbitrary number of color degrees of freedom $N_c$ and examine the results in terms of the consistency condition approach for the baryon vertices recently developed by Dashen, Jenkins and Manohar from the viewpoint of QCD. We show that the $1/N_c$ corrections have two different origins, i.e. one is from the baryon states or baryon wave functions and the other from the vertex operators. Although in the limit of $N_c \to \infty$ the $F/D$ tends to 1/3 in all models, the $1/N_c$ expansion of $F/D$ ratio does not converge for $N_c=3$ in the chiral soliton model in contrast to the non-relativistic quark model., Comment: 14 pages, Latex, 1 figure