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

We study the quantized topological terms in a weak-coupling gauge theory with gauge group Gg and a global symmetry Gₛ in d space-time dimensions. We show that the quantized topological terms are classified by a pair (G,νd), where G is an extension of Gₛ by Gg and νd an element in group cohomology Hᵈ(G,R/Z). When $d=3$ and/or when Gg 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 Hᵈ(G,R/Z), where G/Gg=Gₛ. We also apply our theory to a simple case Gₛ=Gg=Z₂, which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional Gₛ=Z₂ 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 Gₛ, which may lead to different fractionalizations of Gₛ quantum numbers and different fractional statistics [if in ($2+1$)D].