The 1D Ising model at criticality is given by the Hamiltonian $H=-\mathcal{N} \sum_i (\sigma^x_i \sigma^x_{i+1} + \sigma_i^z)$ in terms of Pauli operators and a normalization $\mathcal{N}$. In CFT language, the operator content of this theory is supposed to be described by the primary fields usually called $\mathbb{1}, \epsilon,\psi$ and $\bar{\psi}$ with scaling dimensions $\Delta_\mathbb{1}=0$, $\Delta_\epsilon = 1$ and $\Delta_\psi =\Delta_\bar{\psi}=1/2$. In addition, plenty of papers (such as this one) mention the fields $\sigma$ and $\mu$ with $\Delta_\sigma = \Delta_\mu = 1/8$.
Question: What physical (local) operators/observables do these fields correspond to? My understanding so far:
The identity field $\mathbb{1}$ seems to be equivalent to a local identity operator.
Is the "energy" $\epsilon$ at site $i$ simply $\sigma^x_i \sigma^x_{i+1} + \sigma_i^z$? The Ising CFT Wikipedia article also mentions a field called $\epsilon^\prime$ with $\Delta_{\epsilon^\prime}=4$, whose meaning is unexplained.
The fields $\psi$ and $\bar{\psi}$ seem to correspond to the Majorana modes $c_{2i-1}$ and $c_{2i}$ after Jordan-Wignerizing the Hamiltonian.
The "order" $\sigma$ appears to correspond to the Pauli operator $\sigma^x_ i$ at site $i$.
I have no idea what the "disorder" $\mu$ corresponds to, but it seems to be an order parameter on the dual lattice.
The only proper textbook I have currently available (Christe/Henkel's "Introduction to Conformal Invariance and Its Application to Critical Phenomena") talks about these fields but doesn't explain $\sigma$ and $\mu$, only mentioning that they can't be written locally in terms of $\epsilon$ and the $\psi,\bar{\psi}$.