Sp(4,H) / Z2 Pair Universe in E6 Matrix Models

We construct interacting Sp(4,H)/Z_2 pair matrix models inside the compact E6matrix models. Generally, models based on the compact E6 seem to always includedoubly the degrees of freedom that we need physically. In this paper, wepropose one solution to this problem. A basic idea is that: `we regard thateach point of space-time corresponds to the center of projection of twofundamental figures (i.e. two internal structures), and assume that theprojection of these fundamental figures from each point possesses onetransformation group as a whole.' Namely, we put emphasis on the `analogy' withthe projective geometry. Given that the whole symmetry is compact E6 * Gauge,the space $(\Vec{\mathfrak{J}_H} \oplus i \Vec{H^3}) \times \Vec{\mathcal{G}}$is promising as two subspaces seen from such a viewpoint. When this situationis seen from the standpoint of Klein's Erlangen Program, each fundamentalfigure should also have an independent transformation group. The symmetrycorresponding to this is Sp(4,H)/Z_2 * Gauge. This is ensured by theintroduction of the Yokota mapping Y. As a consequence, we result in thepicture of interacting pair universes which are being pi/2[rad]-phase-shifted.This picture is applicable to all the models based on the compact E6, which maybe not only matrix models but also field theories. An interactingbi-Chern-Simons model is provided when this result is applied to our previousmatrix model. This paper is one answer of the author to the doubling problemwhich has been left in the previous paper.Comment: 42 pages, 9 figures, LaTeX 2