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I am studying the lattice gauge theory coupled with matter (Higgs) fields in the 1979 Fradkin-Shenker paper (https://inspirehep.net/record/132906). Taking the matter fields $\sigma$ and the link fields $U$ to be $\mathbb{Z}_2$ variables, the action is $$ S = \beta \sum_{\text{links}}\sigma U \sigma + K\sum_{\square} UUUU. $$

In the limit of frozen gauge fields ($K\rightarrow\infty$), the action reduces to $$ S = \beta \sum_{\text{links}}\sigma U \sigma, $$ but Fradkin and Shenker write (general case on page 3683, $\mathbb{Z}_2$ case on page 3686)

Their action in an axial gauge $U_{\mu}(r) = I$ (identity of $G$) ($\mu=1$, for instance) is ... $$S = \beta \sum_{\text{links}} \sigma(r)\sigma(r+\mu)$$

I am confused about the axial gauge. Why can $U$, which is a $\mathbb{Z}_2$ variable, just be set to the group identity, $1$? Is the model changed at all when you choose this gauge?

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