How is this term an angle? This question is regarding the $\Theta_{\mu\nu}$ term given in equation (3.8) of the paper (https://arxiv.org/abs/2206.07725). The term ( I have typed it below )is defined right below in (3.9) and also in some other old papers ( which are difficult to access and don't provide much insight about this theta term ). The authors have said more things below the definition, but I am having difficulty understanding it.
Let me put my questions first before I obscure them with my (incorrect) attempt:
1) How does this object represent an angle? Is this a (not so) well known formula to give angle between two vectors/one-forms?
2) How can we see from this expression that $\Theta$ changes by $2\pi$ when "electric charge crosses the world sheet" of a monopole string?
If you have qualitative/pictorial arguments instead of mathematical justification, they are also welcome.
Note: I have not typed the explicit expression in the beginning to avoid clutter ( as one anyways needs to open the paper to see how it's used in expression (3.8)). I am typing it here:
$$ \Theta_{\mu\nu}(R-r) = 2 \pi \epsilon_{\mu\nu\lambda\sigma}u_\lambda  (u.\Delta)^{-1}\Delta^{(r)}_\sigma G(R-r) \tag{3.9}$$
Here u is an arbitrary vector that represents the direction of Dirac string ( we are free to chose this direction once and for all). $\Delta$ is discrete derivative on the lattice. $(u.\Delta)^{-1}$ is an integral operator such that $(u.\Delta)(u.\Delta)^{-1}(x) = \delta(x)$, which ( I assume ) doesn't play any role in this angle. I can provide more context if it helps.
This is what I understand ( which might be wildly incorrect ):
The directions $\mu$ and $\nu$ are the directions of magnetic and electric currents respectively. Say the monopole current is in $\tau$ direction ( moving only in Euclidean time ) and the electric current is in z direction ( so, $\mu = \tau, \nu = z$ ). So we have in this case ( upto a negative sign from orientation of axes ): $$ \Theta_{\tau z}(R-r) = 2\pi \left( u_x (u.\Delta)^{-1}\Delta^{(r)}_y G(R-r) -  u_y (u.\Delta)^{-1}\Delta^{(r)}_x G(R-r) \right) $$
But I can't see how this is an angle between (R-r) and u directions. Any help is appreciated.
 A: The term Θμν(R−r) in equation (3.8) of the paper you've linked is Berry phase angle. Berry phases are geometric phases that are associated with the adiabatic evolution of quantum systems.
In this case, the term represents an angle that arises between the magnetic and electric currents due to the presence of a monopole string (also known as a Dirac string). The vector u represents the direction of the Dirac string, and the discrete derivative Δ is a lattice version of the derivative. The term G(R−r) is the Green's function on the lattice.

*

*the Θμν term does not have a simple geometric interpretation as an
angle between two vectors or one-forms. Instead, it is a
mathematical object that arises from the adiabatic evolution of the
system in the presence of a monopole string.


*Regarding your second question, it could be due to the topological defect of monopole string in the gauge field, and the crossing of the world sheet by an electric charge corresponds to a change in the topology of the gauge field. This change in topology leads to a non-trivial Berry phase, which is quantized in units of 2π.
