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One may similarly expect that $\\operatorname U(1)$-valued Maxwell theory may be approximated by $\\mathbb Z_k$-valued gauge theory and that, as $k\\to\\infty$, one recovers the original Maxwell theory. However, this fails: the $k\\to\\infty$ limit of $\\mathbb Z_k$-valued gauge theory is flat Maxwell theory with no local degrees of freedom. 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One may similarly expect that $\\operatorname U(1)$-valued Maxwell theory may be approximated by $\\mathbb Z_k$-valued gauge theory and that, as $k\\to\\infty$, one recovers the original Maxwell theory. However, this fails: the $k\\to\\infty$ limit of $\\mathbb Z_k$-valued gauge theory is flat Maxwell theory with no local degrees of freedom. 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