Fractionalizing global symmetry on looplike topological excitations

Symmetry fractionalization (SF) on topological excitations is one of the most remarkable quantum phenomena in orders with symmetry, i.e., symmetry-enriched phases. While much progress has been theoretically and experimentally made 2D, understanding SF 3D far from complete. A long-standing challenge to understand looplike which are spatially extended objects. In this work, we construct a powerful topological-field-theoretical framework approach for orders, leads systematic characterization classification SF. For systems Abelian gauge groups ($G_g$) symmetry ($G_s$), successfully establish equivalence classes that lead finite number patterns SF, although there notoriously infinite Lagrangian-descriptions system. We compute topologically distinct types fractional charges carried by particles. Then, each type, statistical phases braiding processes among loop external fluxes. As result, able unambiguously list all physical observables pattern present detailed calculations many concrete examples. an example, find untwisted $\mathbb{Z}_2\times \mathbb{Z}_2$ order $\mathbb{Z}_2$ classified $ (\mathbb{Z}_2)^6\oplus (\mathbb{Z}_2)^2\oplus (\mathbb{Z}_2)^2$. If twisted, reduces $(\mathbb{Z}_2)^6$ particle always carry integer charge. Pure algebraic formalism given by: \bigoplus_{\nu_i} \mathcal H^4 ( G_g\leftthreetimes_{\nu_i}G_s, U (1)) /\Gamma_\omega ({\nu_i}) \,$. Several future directions proposed.