Cuspidal modules over Superconformal algebras of rank \geq 1

According to V. Kac and J. van de Leur, the superconformal algebras are the simple $\mathbb{Z}$-graded Lie superalgebras of growth one which contains the Witt algebra. We describe an explicit classification of all cuspidal modules over the known supercuspidal algebras of rank $\ge 1$, and their central extensions. Our approach reveals some unnoticed phenomena. Indeed the central charge of cuspidal modules is trivial, except for one specific central extension of the contact algebra $\text{K}(4)$. As shown in the paper, this fact also impacts the representation theory of $\text{K}(3)$, $\text{CK}(6)$ and ${\text{K}}^{(2)}(4)$. Besides these four cases, the classification relies on general methods based on highest weight theory.