Classification of simple quasifinite modules for contact superconformal algebras with $N\ne4$
Yan-an Cai, Rencai Lü
TL;DR
The work classifies simple jet modules for the contact superconformal algebras $\mathcal{K}(N;\epsilon)$ with $N\neq 4$ and, via their universal central extension $\widehat{\mathcal{K}}(N;\epsilon)$, all simple quasifinite modules, establishing Martínez-Zelmanov's conjecture in this setting. A uniform approach based on the weighting functor and the $A$-cover yields tensor-type jet modules $\mathcal{F}_a(V,c)$ from finite-dimensional $\mathbb{C}z\oplus\mathfrak{so}_N$-modules, and it is shown that every simple jet module is a quotient of some $\mathcal{F}_a(V,c)$. The main result classifies simple quasifinite modules as either highest-weight, lowest-weight, or simple quotients of $\mathcal{F}_a(V,c)$; the paper also provides explicit full descriptions of simple bounded modules for $N=2$ and $N=3$, including detailed sector-by-sector analyses and submodule structures. Additionally, equivalences of bounded categories under algebra isomorphisms between different sectors are established, clarifying how sector choices affect the representation theory. The results furnish a complete, uniform framework for simple quasifinite representations of large-rank superconformal algebras in the $N\neq 4$ regime, with concrete realizations in low-rank cases that connect to prior works on the Virasoro and super-Virasoro families.
Abstract
In this paper, we classify all simple jet modules for contact superconformal algebras $\mathcal{K}(N;ε)$ with $N\neq4$. Then all simple quasifinite modules for $\widehat{\mathcal{K}}(N;ε)$ ($N\neq4$), the universal central extension of $\mathcal{K}(N;ε)$, are classified. Our results show that Matínez-Zelmanov's conjecture in \cite{MZe1} holds for $\mathcal{K}(N;ε)$ ($N\ne4$).