${\mathbb{Z}}_k^{(r)}$-Algebras, FQH Ground States, and Invariants of Binary Forms

A prominent class of model FQH ground states is those realized as correlation functions of ${\mathbb{Z}}_k^{(r)}$-algebras. In this paper, we study the interplay between these algebras and their corresponding wavefunctions. In the hopes of realizing these wavefunctions as a unique densest zero energy state, we propose a generalization for the projection Hamiltonians. Finally, using techniques from invariants of binary forms, an ansatz for computation of correlations $\langle\psi(z_1)\cdots\psi(z_{2k})\rangle \prod_{i<j}(z_i-z_j)^{2r/k}$ is devised. We provide some evidence that, at least when $r=2$, our proposed Hamiltonian realizes ${\mathbb{Z}}_k^{(2)}$-wavefunctions as a unique ground state.