be formed around the $2$-dimensional topological defect which is represented by the integer-valued field defined on the dual lattice $ \Sigma = \frac{1}{2\pi}^*([d
Z^{\prime}]_{{\rm mod} 2\pi} - d Z^{\prime})$. (Here we used the notations of differential forms on the lattice. For a definition of those notations see, for example,  [@forms]. Lattice field $Z^\prime$ is defined in Eg. (\[Z1\_\]).) Therefore, $\Sigma$ can be treated as the worldsheet of a [*quantum*]{} $Z$-string [@Chernodub_Nambu]. Then, the worldlines of quantum Nambu monopoles appear as the boundary of the $Z$-string worldsheet: $ j_Z =
\delta \Sigma $.

For historical reasons in lattice simulations we fix unitary gauge $\Phi_2 = 0$; $\Phi_1
\in {\cal R}$; $\Phi_1 \ge 0$ (instead of the usual $\Phi_1 = 0$; $\Phi_2 \in {\cal R}$), and the lattice Electroweak theory becomes a lattice $U(1)$ gauge theory with the $U(1)$ gauge field $$A_{xy}  =  A^{\mu}_{x} \;
 = \,[Z^{\prime}  + 2\theta_{xy}]  \,{\rm mod}
 \,2\pi, \label{A}$$ (The usual lattice Electromagnetic field is related to $A$ as $ A_{\rm EM}  = A -
Z^{\prime} + 2 \,{\rm sin}^2\, \theta_W Z^{\prime}$.) One may try to extract monopole trajectories directly from $A$. The monopole current is given by $$j_{A} = \frac{1}{2\pi} {}^*d([d A]{\rm mod}2\pi)
\label{Am}$$ Both $j_Z$, and 