\(D\)-module representations of \({\mathcal N}=2,4,8\) superconformal algebras and their superconformal mechanics (Q2860724)

The \(\mathcal{N}\)-extended one dimensional supersymmetry algebra defined by the relations \(\{Q_i,Q_j\}=2\delta _{ij} H\), \([H,Q_i]=0\), where \(i,\, j=1,\dots,\mathcal{N}\), is the dynamical Lie superalgebra of the supersymmetric quantum mechanics. In this paper, the minimal linear supermultiplets of this superalgebra are used in order to induce \(D\)-module representations for superconformal algebras. The obtained \(D\)-module representations are defined by using supermatrices whose entries are differential operators in one variable. In the first part of the paper, the authors briefly review the classification of the finite \(\mathcal{N}=2,\, 4,\, 8\) superconformal Lie algebras, some results concerning the classification of minimal linear supermultiplets for global supersymmetry and indicate their extension to \(D\)-module representations for superconformal algebras. They continue by presenting the \(D\)-module representations for the \(\mathcal{N}=2\) superconformal algebra \(A(1,1)\) and their investigation of \(D\)-module representations for \(\mathcal{N}=4\) superconformal algebras for the full list of minimal linear \(\mathcal{N}=4\) supermultiplets. They also present for \(\mathcal{N}=8\) a \(D\)-module representation for the \(D(4,1)\) superalgebra induced by the \((8,8)\) root supermultiplet and the construction of superconformal-invariant mechanics based on \(D\)-module representations of superconformal algebras.