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

Abstract

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