Matrix realizations of exceptional superconformal algebras

TL;DR

This paper constructs matrix realizations of certain superconformal algebras, including exceptional ones, over Weyl algebras, and identifies limitations for larger cases.

Contribution

It provides explicit matrix realizations of specific superconformal algebras and establishes non-existence results for larger N.

Findings

Realizations of $K(2)$, $\, ext{and}\,\,\,\,\, ext{hat}K'(4)$, and $CK_6$ as matrix subalgebras.

No matrix realization exists for $K(2N)$ when N ≥ 4.

Matrix realizations are over Weyl algebras of size $2^N imes 2^N$ for N=1,2,3.

Abstract

We give a general construction of realizations of the contact superconformal algebras $K(2)$ and $\hat{K}'(4)$, and the exceptional superconformal algebra $CK_6$ as subsuperalgebras of matrices over a Weyl algebra of size $2^N\times 2^N$, where $N = 1, 2$ and $3$. We show that there is no such a realization for $K(2N)$, if $N\geq 4$.