Classification of supersymmetries

TL;DR

This paper classifies simple linearly compact Lie superalgebras, explores their role in superconformal algebras, and discusses their representation theory, including connections to modular forms and potential implications for the Standard Model.

Contribution

It extends Lie's classification problem to supermanifolds, providing a comprehensive classification of superalgebras and their representations, with applications to superconformal algebras and theoretical physics.

Findings

Classification of simple linearly compact Lie superalgebras completed.

Identification of superconformal algebra structures including affine superalgebras and super Virasoro extensions.

Development of a unified representation theory linking superalgebras to modular forms and the Standard Model.

Abstract

In the first part of my talk I will explain a solution to the extension of Lie's problem on classification of "local continuous transformation groups of a finite-dimensional manifold" to the case of supermanifolds. (More precisely, the problem is to classify simple linearly compact Lie superalgebras, i.e. toplogical Lie superalgebras whose underlying space is a topological product of finite-dimensional vector spaces). In the second part I will explain how this result is used in a classification of superconformal algebras. The list consists of affine superalgebras and certain super extensions of the Virasoro algebra. In the third part I will discuss representation theory of affine superalgebras and its relation to "almost" modular forms. Furthermore, I will explain how the quantum reduction of these representations leads to a unified representation theory of super extensions of the Virasoro algebra. In the forth part I will discuss representation theory of exceptional simple infinite-dimensional linearly compact Lie superalgebras and will speculate on its relation to the Standard Model.