Sp(4,H)/Z_2 Pair Universe in E6 Matrix Models

TL;DR

This paper constructs a novel pair matrix model within the compact E6 framework, proposing a solution to the doubling problem and introducing an interacting pair universe concept based on projective geometry and symmetry considerations.

Contribution

It introduces an Sp(4,H)/Z_2 pair matrix model inside E6 models, addressing the doubling problem and proposing a universe interaction mechanism via phase-shifted pairs.

Findings

Constructed an Sp(4,H)/Z_2 pair matrix model within E6.

Proposed a solution to the doubling problem in E6-based models.

Presented an interacting bi-Chern-Simons model as an application.

Abstract

We construct interacting Sp(4,H)/Z_2 pair matrix models inside the compact E6 matrix models. Generally, models based on the compact E6 seem to always include doubly the degrees of freedom that we need physically. In this paper, we propose one solution to this problem. A basic idea is that: `we regard that each point of space-time corresponds to the center of projection of two fundamental figures (i.e. two internal structures), and assume that the projection of these fundamental figures from each point possesses one transformation group as a whole.' Namely, we put emphasis on the `analogy' with the 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 situation is seen from the standpoint of Klein's Erlangen Program, each fundamental figure should also have an independent transformation group. The symmetry corresponding to this is Sp(4,H)/Z_2 * Gauge. This is ensured by the introduction of the Yokota mapping Y. As a consequence, we result in the picture 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 may be not only matrix models but also field theories. An interacting bi-Chern-Simons model is provided when this result is applied to our previous matrix model. This paper is one answer of the author to the doubling problem which has been left in the previous paper.