In FQH liquid, the effective Lagrangian of a $\nu=1/m$ Laughlin state is given in Xiao-Gang Wen's Book Quantum Field Theory of Many-Body Systems Chapter 7: $$ \mathcal{L} = \frac{m}{4\pi}A_\mu \partial_\nu a_\lambda \epsilon^{\mu\nu\lambda}+ \frac{e}{2\pi}a_\mu\partial_\nu a_\lambda \epsilon^{\mu\nu\lambda} $$ in which $A_\mu$ is the electromagnetic field and $a_\mu$ is an emergent Chern Simons gauge field. But the interacting 2DEG has the following action: $$ S[\psi^\dagger,\psi,A_\mu]=\int d^3x \left[\psi^\dagger(D_0-\mu)\psi-\psi^\dagger\frac{D_i^2}{2m}\psi\right] + S_I[\psi^\dagger,\psi] $$ in which $D_\mu = \partial_\mu+ieA_\mu$. So can we derive the effective theory directly from the action of interacting 2DEG?
Besides, in Naoto Nagaosa's Book Quantum Field Theory in Condensed Matter Physics Chapter 6, the author introduced the Chern-Simons gauge theory in another way: 2D fermion theory is equivalent to a 2D boson theory coupled with a Chern-Simons gauge field with an odd quantum flux $$ S[\phi^\dagger,\phi,A_\mu,a_\mu]=\int d^3x \left[\phi^\dagger(D_0-\mu)\phi-\phi^\dagger\frac{D_i^2}{2m}\phi\right] + S_I[\phi^\dagger,\phi] + \frac{i}{4\theta}\int d^3x a_\mu\partial_\nu a_\lambda\epsilon^{\mu\nu\lambda} $$ in which $D_\mu = \partial_\mu+ieA_\mu + iea_\mu$. What is the relationship between these two different Chern-Simons gauge theory?
