gauge field with vector potential $a^\mu$, whose electric charges are the Dirac composite fermions. However, unlike familiar bosonic vortices of a superconductor, these Dirac vortices have fermionic exchange statistics, and form a metallic state with a Fermi-surface.

To examine the thermoelectric properties of this “vortex metal" consider, a rectangular sample with spatial dimensions $L_x\times L_y$. A temperature gradient along $x$ tends to produce a Dirac CF current $j_D$, also along $x$. Each time a CF passes through, say, $x=0$, the electrons at $(x=0,y=L_y)$ acquire a relative $4\pi$ phase shift to those at $(x=0,y=0)$, resulting in a steady time-dependent winding of the electron phase-difference across the sample in the $y$ direction: $\Delta \phi_y(t) = 4\pi j_\text{D} L_y t$ (Fig. \[fig:Duality\]b). Such a time-dependent phase winding for electrons is gauge equivalent to a constant voltage difference across the $y$-direction of the sample: $V_y =- \frac{{\partial}\Delta \phi_y}{{\partial}t}$, corresponding to an electric field $E_y = \frac{V_y}{L_y} = -4\pi j_\text{D}$. In general, a composite fermion current, ${\mathbf{j}}_\text{D}$ induces an electric field ${\mathbf{E}}$: $$\begin{aligned}
{\mathbf{j}}_\text{D} = \frac{1}{4\pi}\hat{{\epsilon}}{\mathbf{E}}
\label{eq:jD}\end{aligned}$$ where $\hat{\epsilon}$ is the unit antisymmetric tensor with components ${\epsilon}^{ii}=0$, ${\epsilon}^{xy}=-{\epsilon}^{yx}=1$. Alternatively, this relation can be directly obtained by varying Eq. (\[eq:Leff\]) with respect to ${\mathbf{a}}$ and setting 