# Gravitational anomalies and topological order

I wonder the relation of gravitational anomaly and topological order. Specifically:

1. What is the definition of gravitational anomaly here?

2. How are they related?

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–  Qmechanic Dec 10 '14 at 6:46

Here is a more detailed discussion. A low energy effective theory (which can be gapped or gapless) described by an effective action $S_\text{eff}$ is anomalous if such a low energy effective theory cannot be realized by any well defined local bosonic quantum model in the same dimension. A low energy effective theory is anomaly-free if it can be realized by a well defined local bosonic quantum model in the same dimension. (This is a UV completion.)
However, different gapped low energy effective theories may correspond to the same type of gravitational anomaly. To address this issue, we can introduce an equivelance relation: $S^{T_1}_\text{eff}$ and $S^{T_2}_\text{eff}$ are equivalent if there exist anomaly-free low energy effective theories $S^{C_1}_\text{eff}$ and $S^{C_2}_\text{eff}$ such that the combined effective theories $S^{T_1}_\text{eff}+S^{C_1}_\text{eff}$ and $S^{T_2}_\text{eff}+S^{C_2}_\text{eff}$ can deform into each other without encounter phase transitions. This leads to a notion of types of gravitational anomalies, which are defined as the equivalent classes gapped low energy effective theories under the above equivalence relation.