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

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show 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 group cohomology ${\text{cH}}^d(G,\mathbb{R}\mathrm{/}\mathbb{Z})$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by ${\text{cH}}^d(G,\mathbb{R}\mathrm{/}\mathbb{Z})$, where $G\mathrm{/}{G}_g=G_s$. 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, where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).