Dirac composite fermion theory of general Jain sequences

We reconsider the composite fermion theory of general Jain's sequences with filling factor $\nu=N/(4N\pm1)$. show that Goldman and Fradkin's proposal a Dirac leads to violation Haldane bound on coefficient static structure factor. To resolve this apparent contradiction, we add effective gapped chiral mode (or modes) which already exists in Fermi liquid state at $\nu=1/4$. interpret additional as an internal degree freedom fermion, related area-preserving deformations elementary droplet built up from electrons correlation holes. In addition providing suitable factor, our model also gives expected Wen-Zee shift Hall conductivity manifests Galilean invariance. charge density satisfies long-wavelength version Girvin-MacDonald-Platzman algebra.