$D_\mu = {\partial}_\mu + ia_\mu$ are covariant derivatives, and we have abbreviated Chern-Simons terms ${\epsilon}^{\mu\nu\lambda}A_\mu{\partial}_\nu A_\lambda\equiv AdA$. The last term represents Coulomb interactions between electrons with dielectric constant ${\epsilon}_r$, and $(\dots)$ indicates other terms that are less important at low-energies. Particle-hole symmetry requires vanishing Dirac mass, $m=0$.

A straightforward derivation of Eq. (\[eq:Leff\]) from a parton description, as well as a computation of its transport properties obtained by a self-consistent RPA treatment of gauge fluctuations (in a similar spirit to analysis of conductivity in a U(1) spin liquid by Ioffe and Larkin [@IoffeLarkin]) is presented in Appendices \[app:Partons\],\[app:PartonResponse\]. Here, we instead obtain the same results by a simple physical picture by making an analogy to vortex motion in a superfluid. Specifically, due to the $\frac{adA}{4\pi}$ term in Eq. (\[eq:Leff\]), electrons see the Dirac composite fermions as vortices with flux $4\pi$ (in units $\hbar,c,e\equiv1$). The external magnetic field $B$ required to produce half-filling for the electrons, induces a finite density (equal to the electron density) of these vortices, (analogous to a superfluid subjected to $B>H_{c1}$). Just as for a superfluid, the fermionic vortices are non-local objects with long-range interactions which can be viewed as being mediated by a fluctuating emergent electromagnetic-like 