Holon metal, charge-density-wave and chiral superconductor from doping fractional Chern insulator and SU(3)${}_1$ chiral spin liquid

Recent experiments have observed superconductivity proximate to the nu = -2/3 fractional quantum anomalous Hall (FQAH) insulator in twisted MoTe2. A critical open question is whether the underlying normal state is a Fermi liquid with a large Fermi surface or a strongly correlated metal with low carrier density. In this work, we develop a theory of the phases emerging from doping the $\nu =-2\mathrm{/}3$ fractional Chern insulator (FCI). We establish a duality between this problem and doping a gapped SU(3)${}_1$ chiral spin liquid (CSL).