# 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?

-
–  Qmechanic Dec 10 '14 at 6:46

Let us define gravitational anomaly as low energy effective theory with no UV completion.

Then gravitational anomalies are classified by topological orders in one higher dimension. In other words, all gravitational anomalies can be realized by the boundary of topological orders in one higher dimension. My paper with Liang Kong addresses this issue. http://arxiv.org/abs/1405.5858

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.

Conjecture: The types of gravitational anomalies are classified by topological orders in one higher dimension.

-
Hi Prof Wen. Can you explain by an example? Let's say we have a 3D conventional superconductor, which is described by BF model with coefficient k=2. Then it has 8 GSD on a 3 torus, which is an indication that it is topologically ordered. Then does the surface of such conventional superconductors have gravitatinal anomalies? Thank you. –  hehuan0430 Feb 24 at 17:46