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We study the quantized topological terms in a weak-coupling gauge theory withgauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. Weshow that the quantized topological terms are classified by a pair $(G,\nu_d)$,where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ an element in groupcohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, theweak-coupling gauge theories with quantized topological terms describe gappedsymmetry enriched topological (SET) phases (i.e. gapped long-range entangledphases with symmetry). Thus, those SET phases are classified by$\cH^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case$G_s=G_g=Z_2$, which leads to 12 different SET phases in 2+1D, wherequasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbersand fractional statistics. If the weak-coupling gauge theories are gapless,then the different quantized topological terms may describe different gaplessphases of the gauge theories with a symmetry $G_s$, which may lead to differentfractionalizations of $G_s$ quantum numbers and different fractional statistics(if in 2+1D).

Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases