Degenerate link variable configuration in $Z_2$ lattice gauge theory (Wen's QFT book)

Question

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. $$ s_{ij}^{(m,n)}=f_x^m(i,j)f_y^n(i,j)s^0_{ij}. $$

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?

Answer 1

The statement "the four-fold degenerate configuration are not globally gauge equivalent" means that these four configurations belong to four different equivalence classes. To begin with, we know that the operator $$ U(C)=s_{i j} s_{j k} \cdots s_{l i} $$ is gauge invariant (under $\mathbb{Z}_2$ gauge $\tilde{s}_{i j}=W_i s_{i j} W_j^{-1}$ ). But if we chose C to be a topological untrivial loop on the torus, that is "going around the torus" as you mentioned, we will find that the operator $U(C)$ is different under different configuration (chosing different m and n). That means these four configurations are not equivalent under gauge transformation, belong to four different equivalence classes.