density: $\mathcal{L}=\mathcal{L}_b+\mathcal{L}_f-(A+a)_\mu j^\mu_b-(-a_\mu) j^\mu_f$, where $\mathcal{L}_{b,f}$ are the Lagrangian densities and $j_{b,f}$ are the number currents of $b$ and $f$ respectively.

Composite Fermi liquid states of the 1/2-filled Landau level can be obtained by having the bosonic partons, which are at $1/2$-filling with respect to the external magnetic flux, form an incompressible abelian (Laughlin-like) fractional quantum Hall state, $\mathcal{L}_b = -\frac{2i}{4\pi}a_bda_b$, where $a_b$ is a vector field whose flux is the boson current: $j_b = \frac{da_b}{2\pi}$. This mean-field ansatz is equivalent to the “flux-attachment" picture typically utilized in composite fermion descriptions of quantum Hall states, as each boson in the $\nu=1/2$ FQH state is accompanied by $4\pi$ flux of $a$ due to their quantum Hall response, which the composite fermions $f = b^\dagger c$ see has $-4\pi$ flux. The fermionic partons, $f$, then see zero average magnetic flux, and may form a simple Fermi-surface (this is the usual HLR theory, which is maximally particle-hole symmetry violating). Alternatively, we may access the PHS composite Dirac state by putting $f$ into a band-structure consisting of two Dirac cones, with flavor index $i=1,2$: $$\begin{aligned}
\mathcal{L}_f = \sum_{i=1,2}\psi^\dagger_{a,i}{\left(}i{\partial}_\tau-\mu+v{\mathbf{p}}\times {\boldsymbol{\sigma}}_{ab}-m_i\sigma^z_{ab}{\right)}\psi_{b,i}
\label{eq:Lf}\end{aligned}$$ a PHS low-energy theory is obtained by keeping one flavor of massless Dirac fermions 