Anomalies and symmetry fractionalization in reflection-symmetric topological order

One of the central ideas regarding anomalies in topological phases of matter is that they imply the existence of higher-dimensional physics, with an anomaly in a D-dimensional theory typically being canceled by a bulk (D+1)-dimensional symmetry-protected topological phase (SPT). We demonstrate that for some topological phases with reflection symmetry, anomalies may actually be canceled by a D-dimensional SPT, provided that it comes embedded in an otherwise trivial (D+1)-dimensional bulk. We illustrate this for the example of ${\mathbb{Z}}_N$ topological order enriched with reflection symmetry in (2+1)D, and along the way establish a classification of anomalous reflection symmetry fractionalization patterns. In particular, we show that anomalies occur if and only if both electric and magnetic quasiparticle excitations possess nontrivial fractional reflection quantum numbers.

Figure 1

Illustration of the geometry we consider and the gauge fields present in our theory, viewed at a single time slice. The $a$ and $b$ cochains represent electric and magnetic quasiparticle strings, and live on the surface $X$, although our theory only involves their restriction to the mirror axis $\partial X$. The ${\mathcal{G}}_{\alpha }$ represent the electric and magnetic ${\mathbb{Z}}_2^{\mathcal{P}}$ gauge fields, and are initially also defined on $\partial X$. Deciding whether or not the ${\mathcal{G}}_{\alpha }$ also must extend down into the mirror plane $\mathrm{\Sigma }$ (as shown in the figure) amounts to determining whether or not the surface theory is anomalous.

Figure 2

Illustration of Poincaré duality for a region of a two-dimensional triangulated manifold $X$, shown in black. The Poincaré dual of $X$ is shown in pink.

Figure 3

Geometric meaning of the term $a\wedge db$. Since $a$ ($db$) is a 1-cochain (2-cochain), it is represented by a codimension 1 (2) submanifold in (2+1)D space-time. The intersection of these submanifolds (marked by the x) is a codimension 3 submanifold, and is represented by the cup product $a\wedge db$.

Figure 4

What fractionalization looks like from a geometric perspective. Here the fiber $N$ describes the topological phase, and the base space $G$ describes the symmetry group. $s$ is a section (also known as projective representation) which lifts $G$ up into the fibers. The fractionalization factor set ${\mathfrak{w}}_{\alpha }$ measures the failure of $s$ to be linear.

Figure 5

Illustration of the folding trick. We fold the surface along the mirror axis $\partial X$, so that the system becomes a bilayer system with edge $\partial X$ on which reflection acts as a layer-exchange symmetry.