# Is it possible to directly derive the $K$ matrix for a topological order described by a gauge-theory Hamiltonian?

To be concrete, Let us consider a $Z_2$ gauge theory in the deconfined phase coupled to matter field, \begin{align} S_{Z_2}=\beta\sum_{\vec r\mu}\phi(\vec r)U_\mu(\vec r)\phi(\vec r+\vec e_{\mu}) + K \sum_{\vec r \mu\nu}\left[U_\mu(\vec r)U_{\nu}(\vec r +\vec e_{\mu})U_\mu(\vec r+\vec e_{\nu})U_{\nu}(\vec r )\right] \end{align} where $\vec r$ is defined on a cubic spacetime lattice, $U=\pm 1$ is a $Z_2$ gauge field and $\phi$ is a matter field. It is easy to show that this model has toric-code topological order, with $e$ and $m$ anyons with nontrivial mutual statistics.

Its effective theory is described by a Chern-Simons theory \begin{align} S_{CS}=\frac{K_{IJ}}{4\pi}\int a^I\wedge da^J + a^I J_I, \end{align} where $a^I$ are emergent gauge fields, $K=\begin{bmatrix}0&2\\2&0\end{bmatrix}$ and $J_I$ are currents of the anyonic excitations.

The way I learned this is that $S_{CS}$ is constructed by "reverse engineering" to capture the correct statistics of the anyons.

However, since both $S_{Z_2}$ and $S_{CS}$ are gauge theories, is it possible to derive $S_{CS}$ from $S_{Z_2}$ through some RG procedures?