Lattice Chern-Simons model for non Abelian fractional quantum hall effect

Using discrete homological structures and topological invariance on the lattice, we construct a gauge-invariant non-Abelian Chern-Simons theory for non Abelian FQHE. This topological theory is described by a lattice functional SCSLatt[b,b¯,φ], which involves the link variable b, its dual counterpart b¯, and the flux variable φ associated with the plaquettes of the direct lattice. Within this framework, we derive two effective lattice realisations that faithfully reproduce the Pfaffian quantum Hall state. The first realisation is based on the gauge symmetry SU(2)2×U(1)−2 and includes, in addition to the non-Abelian sector variables, an Abelian link field a¯ coupled to an external gauge field A¯ associated with the Abelian sector. The second realisation relies on the symmetry U(2)(−4,2) involving the link variable c, its dual c¯, and the flux φ(c), which collectively encode both the non-Abelian and Abelian sectors. In this setup, the external field A¯ couples to the Abelian component of c¯, given by Tr(c¯). Finally, we generalize this construction to a broader class of effective lattice models governed by the gauge group ARR=(U(2)(−2k,k)×U(1)k(kM+2))/Zk which captures the topological structure of non-Abelian Read-Rezayi states characterized by the lattice filling factor νLatt=k/(kM+2).