constraints on transport coefficients, we now turn to their computation from an effective field theory.

![ [**Schematic of charge vortex duality for transport -** ]{} a) An electron (blue dot) is a source of electric field ${\mathbf{E}}$ (blue line) and $-4\pi$ flux of the emergent gauge field $b$ (circulating dashed red line indicates winding of phase for Dirac composite fermion). The Dirac composite fermion object (red dot) is a dual object that is a source of emergent electric field lines ${\mathbf{e}}$ (dashed red lines) and $4\pi$ flux of the electron phase (circulating blue line). Panel b) shows the electron phase, $\phi_e$, winding by $4\pi$ across the $y$-direction for each composite fermion propagating along $x$. Panels c,d) schematically depict the composite fermion dispersion and Fermi-surface pseudospin texture for the Dirac ($\pi$ Berry phase) and HLR ($0$ Berry phase) respectively.[]{data-label="fig:Duality"}](Duality.pdf){width="0.8\columnwidth"}

[[**Thermoelectric properties of the composite Dirac liquid –**]{}]{} The composite Dirac liquid is described by the effective field theory [@Son15]: $$\begin{aligned}
\mathcal{L}_\text{eff} =& \psi^\dagger{\left(}D_\tau-\mu-iv{\mathbf{D}}\times {\boldsymbol{\sigma}}-m\sigma^z{\right)}\psi+
\nonumber\\
&+\frac{iadA}{4\pi}-\frac{iAdA}{8\pi}+\int_{{\mathbf{r'}}} \frac{b({\mathbf{r}}) b({\mathbf{r}}')}{4\pi^2{\epsilon}_r|{\mathbf{r}}-{\mathbf{r}}'|}+\dots
\label{eq:Leff}\end{aligned}$$ where $\psi$ is the Dirac CF field, $\mu=\tau,x,y$ are (Euclidean) space-time indices, $a_\mu$ is an emergent U(1) gauge field whose magnetic field $b = \nabla\times a$ is $2\pi$ times the electron density, 