link $\ell$, note that $S$ is conserved around a vertex, and one can introduce a height field $h(i)$ on the same sites where $\sigma$ lives, such that $h$ increases by $\pi$ in crossing an arrow pointing up from the right. This discrete height renormalizes at long distances to a Gaussian free field, a conformal field theory with central charge $c=1$, and via this mapping one can compute dimensions of operators. Noting that $ \sigma_i\sigma_j = \prod_{\ell\in \Gamma_{ij}} -ie^{i\pi S_\ell /2 }
\propto e^{i h(i)/2 }e^{-i h(j)/2 } $, $\sigma$ is found to have scaling dimension $3/8$. Similarly, noting that the two point function of $\tau$ in eq.  corresponds to inserting two vortices where the height field has discontinuity of $\pm 4\pi$, $\tau$ has dimension $2/3$. Identifying $\phi\equiv h/2$, one has the effective theory $$\begin{aligned}
\mathscr{L}
 &= \frac{g}{4\pi} (\nabla \phi)^2 \, , \quad g=\frac{4}{3}\, .\end{aligned}$$ The appearance of half integer electric charges follows also naturally by considering the torus partition function. Indeed on $4L\times 4L'$ lattices, periodic boundary conditions for the $\sigma$’s select only even frustrations for the height field as it winds around a cycle, resulting in half integer electric charges and even magnetic charges. Microscopically, $\sigma$ is a Hermitian 