Refining the Elliptic Genus

We show how special forms of an $N=2$ Landau-Ginzburg potential directly imply the presence of an $N=2$ super-$W$ algebra. If the Landau-Ginzburg model has a super-$W$ algebra, we show how the elliptic genus can be refined so as to give much more complete information about the structure of the model. We study the super-$W_3$ model in some detail, and present some results and conjectures about more general models.