I'm reading through Xiao-Gang Wen's Quantum Field Theory of Many-body Systems, and I just begin reading $Z_2$ lattice gauge theory.
In page 255, the author constructed a four-fold denegerate (in the sense that they give rise to the same $Z_2$ flux $F_i$) link variable configuration with the statement that they are not gauge invariant. $$ s_{ij}^{(m,n)}=f_x^m(i,j)f_y^n(i,j)s^0_{ij} $$$$ s_{ij}^{(m,n)}=f_x^m(i,j)f_y^n(i,j)s^0_{ij}. $$
To prove the statement, the author leaved a hint (consider $U(C)=s_{ij}s_{jk}\dots s_{li}$ with $C$ going around the torus) and referred to some physical picture in Chapter 9.
Referring to Chapter 9 (page 394,395), I grasped some idea that sarting from $i$, going along $x$ direction and back to $i$, we will get a phase $e^{i\pi m}$ ($e^{i\pi n}$ for $y$ direction).
Since I just started reading Chapter 6, I can't understand some fancy words in Chapter 9, thus still not knowing what's the meaning of "the four-fold degenerate configuration are not globally gauge equivalent".
Is there anyone who has read this book willing to give me some clear picture about this statement? Thanks in advance.
