In the continuum limit of (2+1)-dimensional compact $U(1)$ gauge field, the instantons are input by hand in terms of nonconservation of magnetic flux $\int b$: \begin{eqnarray} \int dxdy [b(x,y,t+\epsilon)-b(x,y,t-\epsilon)]=2\pi, \end{eqnarray} by insertion of a monopole operator $\Psi^\dagger$ at time $t$.
My question is how to understand it on the lattice compact $U(1)$ gauge field with the spatial lattice as a torus, i.e. a periodic boundary condition? In many existing papers, the sentence like “a flux 2π passes through a single plaquette of the surface. Now let this flux 2π slowly expand to form a smooth flux distribution over some portion of the surface” takes place, but I have difficulty in understanding it on lattice with periodic boundary condition. Could someone explain this instanton event in more details, such as by an explicit gauge field construction on each link?
