Matrix Model and Elliptic Curve

TL;DR

This paper explores the reduced matrix model of IKKT type with non-zero fermions, proposing our universe is rational rather than continuous, and connects solutions to elliptic curves, non-commutative geometry, and cosmological implications.

Contribution

It introduces a novel approach linking matrix models to elliptic curves and suggests a rational number structure for the universe, with new interpretations of gravity and the cosmological constant.

Findings

Reduced matrix model solutions relate to elliptic curves.

Normalization expressed via Weierstrass and Dedekind functions.

Proposes a non-commutative geometric quantization framework.

Abstract

Solution to the reduced matrix model of IKKT type is studied with non-zero fermion fields. A suggestion is made that our universe is made of rational numbers rather than being a continuum. To substantiate this proposal, the reduced Yang-Mills equation is written in the form of an elliptic curve. The normalization of the solution can be expressed in terms of the Weierstrass function generically or in terms of the Dedekind function in the case of 3-brane. A way to define the gravitational field in the matrix model is proposed with some new interpretation of the cosmological constant. The (first) quantization of the system is done within the framework of non-commutative geometry.