Fractionalizing global symmetry on looplike topological excitations

Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (${G}_g$) and Abelian symmetry groups (${G}_s$), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}₂\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}₂$ topological order with ${\mathbb{Z}}₂$ symmetry is classified by ${({\mathbb{Z}}₂)}⁶\ensuremath{\bigoplus}{({\mathbb{Z}}₂)}²\ensuremath{\bigoplus}{({\mathbb{Z}}₂)}²\ensuremath{\bigoplus}{({\mathbb{Z}}₂)}²$. If the topological order is twisted, the classification reduces to ${({\mathbb{Z}}₂)}⁶$ in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_i}{\mathcal{H}}⁴({G}_g{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_i}{G}_s,U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_i),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_i\ensuremath{\in}{\mathcal{H}}_non²({G}_s,{G}_g)$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_i)$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_i$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_i)$. Several future directions are presented at the end of this paper.