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


*

*What is the definition of gravitational anomaly here?

*How are they related?
 A: 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 gapped 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.
