TITLE: What's the right way to think about "anomalies" in 3d TQFTs?
QUESTION [17 upvotes]: 3d TQFTs constructed from modular tensor categories don't in general give an honest 3d TQFT, instead they have an "anomaly."  My vague understanding from Kevin Walker's talks and from skimming Freed-Hopkins-Lurie-Teleman is that what's really going on is that its a 4d TQFT that's almost boring on the 4d part and that's what the "anomaly" means.  Does anyone know how to make this more precise?

REPLY [8 votes]: Here is my understanding from physics point of view.
A quantum field theory is anomalous if it lacks of a UV completion. In other words there is no lattice theory in the same dimension, whose continuum limit produces the quantum field theory.
However, even an anomalous quantum field theory can be produced at the boundary of a topological ordered state on lattice in one higher dimension. 
In other words, for a given quantum field theory, there is always a gapped lattice theory in one higher dimension, whose continuum limit at the boundary produces the quantum field theory. Thus 
the types of anomalies in quantum field theories = types of topological orders in one higher dimension
This leads to a classification of gravitional anomalies in terms of topological orders in one higher dimension.
Also
the types of anomalies in topological quantum field theories （ie gapped quantum field thoeries）= types of topological orders with gappable boundary in one higher dimension （ie topological quantum field theories in one higher dimension that have state sum realization ）
I have two papers discussing those points in details:
arXiv:1303.1803
    Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders (for gauge anomaly and group cohomology)
    Xiao-Gang Wen 
arXiv:1405.5858
    Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions (for gravitational anomaly and n-category, as well as the notion of H-type and L-type anomalies)
    Liang Kong, Xiao-Gang Wen 
Kapustin also has a paper
arXiv:1404.3230
    Anomalies of discrete symmetries in various dimensions and group cohomology (for gauge anomalies)
    Anton Kapustin, Ryan Thorngren