A Criterion for the Algebraic Density Property of Affine $S{L}_2$-Manifolds

Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an ${\mathfrak{s}\mathfrak{l}}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of ${\mathrm{S}\mathrm{L}}_2$-varieties arising in physics: Classical Calogero–Moser spaces, Calogero–Moser spaces with "inner degrees of freedom’’ and smooth cyclic quiver varieties.