Universally defined representations of Lie conformal superalgebras

TL;DR

This paper introduces a class of universally defined irreducible finite representations for conformal Lie superalgebras, providing explicit descriptions for certain superalgebras and highlighting their simplicity and computational advantages.

Contribution

It defines and characterizes universally defined representations for conformal superalgebras, including explicit examples for $W_n$ and $K_n$, and identifies their limitations.

Findings

Universal representations are determined by algebra relations and locality.

Explicit universally defined representation for $K_1$ Neveu-Schwarz algebra.

Analogues for $n eq 1$ are not universally defined.

Abstract

We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra $L$ is completely determined by commutation relations of $L$ and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras $W_n$, $n\ge 0$, with respect to a natural set of generators. We also consider the problem for superalgebras $K_n$. In particular, we find a universally defined representation for the Neveu--Schwartz conformal superalgebra $K_1$ and show that the analogues of this representation for $n\ge 2$ are not universally defined.