I am reading this paper on toric code for my research and has some questions about it.
$\sigma_j^Z$ appears in the equation for $A_S$ (last but five line) and in the last but four line, it’s defined to be $\sigma^Z$ acts on the copy of $\mathbb{C}^2$ at edge $j$. What does this mean? How does it act on the edge?
Furthermore, why can we view it as “operator that acts as $\sigma^Z$ ... and identity everywhere else”?
I am really confused about the definition of $A_S$ and $B_P$, and would appreciate if someone could tell me why they are very useful (I know they are, but I don’t have much physics background unfortunately).
Thank you!