Anomaly indicators for time-reversal symmetric topological orders

Abstract: Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensions without breaking time reversal symmetry; instead, they can only be realized on the surface of certain three-dimensional systems. We propose two quantities, which we call {\it anomaly indicators}, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, $\eta_2$, applies to bosonic systems while the second indicator, $\eta_f$, applies to fermionic systems in the DIII class. We conjecture that $\eta_2$, together with a previously known indicator $\eta_1$, can detect the two known $\mathbb Z_2$ anomalies in the bosonic case, while $\eta_f$ can detect the $\mathbb Z_{16}$ anomaly in the fermionic case.