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

摘要

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, v_d), where G is an extension of G_s by G_g and v_d an element in group cohomology H~d(G,R/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 H~d(G,B./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 + 1 dimensions [(2+ 1)D], 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 + 1)D].