Semigroup Methods In Supersymmetric Theories Of Elementary Particles

TL;DR

This paper introduces a novel mathematical framework for elementary particle theories using semigroup structures, extending supersymmetry and superstring models with noninvertible and geometric concepts.

Contribution

It develops the theory of semisupermanifolds and noninvertible supergeometries, providing new supermatrix representations and a generalized algebraic approach to supersymmetric particle models.

Findings

Introduction of semisupermanifolds and noninvertible supergeometries

Development of new supermatrix representations of semigroups

Formulation of a generalized model of elementary particles using semigroup theory

Abstract

The thesis is devoted to abstract, geometric and symmetric aspects of modern elementary particle theories. A new direction in constructing supersymmetric and superstring models based on consequent and strong consideration and inclusion of semigroups, ideals and noninvertible properties into their mathematical structure is proposed. A theory of semisupermanifolds (Chapter I) and noninvertible generalization of superconformal (Chapters II-III) and hyperbolic geometries (Chapter V) are introduced. New continuous supermatrix representations of semigroups (Chapter IV) are obtained. The carried out investigations will allow us to formulate a theoretical model of elementary particles based on supersymmetry in terms of more general categories and new structures - as a theory of abstract and supermatrix semigroups which includes previous theories as a particular invertible case in abstract sense.