# Linking phase of flux lines and excitation energy of monopole

I am reading this paper and on the left-hand side of pp.10 it states the following relation between linking phase and excitation energy of monopole:

Now the $\theta = \pi$ term in the bulk implies that when two closed bulk $2\pi$ flux lines link there is a phase of $(-1)$. This linking phase ensures that when a single $2\pi$ flux line is cut open to produce a strength-1 monopole it costs infinite energy unless it binds to $\pm\frac{1}{2}$ electric charge. The binding to the electric charge removes the linking phase ambiguity of an open flux tube and enables the resulting $(\pm\frac{1}{2},1)$ dyon to have finite energy, exactly consistent with the Witten effect.

I have quite a few questions regarding this statement.

(1) what is the linking phase of two flux lines? and what is the linking phase ambiguity of an open flux tube?

(2) how to calculate the linking phase from the $\theta$ term and in particular get the $(-1)$ phase out of $\theta = \pi$?

(3) why does the $(-1)$ linking phase ensure the energy cost of producing strength-1 monopole via cutting flux line open is infinite?

(4) why does the binding to the electric charge removes the linking phase ambiguity and enables the resulting $(\pm\frac{1}{2},1)$ dyon to have finite energy?