Lie superalgebras of string theories

TL;DR

This paper classifies simple complex Lie superalgebras related to string theories, highlighting their structures, central extensions, and cocycles, and explores their role in superconformal string models.

Contribution

It introduces four series and four exceptional simple stringy superalgebras, detailing their central extensions and cocycles, and connects them to superconformal string theories.

Findings

13 distinguished superalgebras with nontrivial central extensions

Exactly 16 superizations of key physical equations

Identification of a unique Kac-Moody superalgebra with nonsymmetrizable Cartan matrix

Abstract

We define and describe simple complex Lie superalgbras of vector fields on "supercircles" - simple stringy superalgebras. There are four series of such algebras and four exceptional stringy superalgebras. The 13 of the simple stringy Lie superalgebras are distinguished: only they have nontrivial central extensions; since two of the distinguish algebras have 3 nontrivial central extensions each, there are exactly 16 superizations of the Liouville action, Schroedinger equation, KdV hierarchy, etc. We also present the three nontrivial cocycles on the N=4 extended Neveu-Schwarz and Ramond superalgebras in terms of primary fields and describe the "classical" stringy superalgebras close to the simple ones. One of these stringy superalgebras is a Kac-Moody superalgebra G(A) with a nonsymmetrizable Cartan matrix A. Unlike the Kac-Moody superalgebras of polynomial growth with symmetrizable Cartan matrix, it can not be interpreted as a central extension of a twisted loop algebra.The stringy superalgebras are often referred to as superconformal ones. We discuss how superconformal stringy superalgebras really are.