Anomaly Indicators for Time-Reversal Symmetric Topological Orders

Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensional systems; instead,they can only be realized on the surface of three-dimensional symmetry-protected topological phases. We propose two quantities, which we call 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, η₂, applies to bosonic systems while the second indicator, ηf, applies to fermionic systems in the DIII class. We conjecture that η₂, together with apreviously known indicator η₁, can detect the two known Z₂ anomalies in the bosonic case, while η_f can detect the Z₁₆ anomaly in the fermionic case.