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 FreedHopkinsLurieTeleman 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?
4 Answers
You can find some of the details in a draft version of my TQFT notes on my web page. Here's a summary.
Given an ncategory with strong duality (by which I mean, more or less, pivotal if $n=2$ and a higher dimensional version of pivotal for $n > 2$; this is stronger than what Lurie means by an ncategory "with duals"), there's a standard procedure to construct an n+1dimensional TQFT. This procedure works for free for the 0through ndimensional parts of the (extended) n+1dimensional TQFT. In these dimensions, we need no additional assumptions on the ncat, and there is no need to choose a decomposition of the manifolds, so there are no combinatorial topology moves to check. The construction is manifestly invariant.
To construct the top, n+1dimensional part of the TQFT, we need to make some finiteness assumptions on the ncategory. (This corresponds to the topdimensional part of what Lurie et al mean by "fully dualizable".) If the ncategory satisfies these assumption, then we get, for each handle decomposition of the n+1manifold, a state sum type expression for the path integral of the n+1manifold. It is not hard to show that this state sum is invariant under handle slides and handle cancelation, so we get a welldefined invariant of n+1manifolds that interacts with the rest of the TQFT (via gluing formulas) in the correct way.
(Small technicality: the path integral construction depends on a choice of element in the Hilbert space of the nsphere, corresponding to the path integral of the n+1ball. Multiplying this choice by $\lambda$ changes the path integral by $\lambda^\chi$, where $\chi$ is the Euler characteristic of the n+1manifold.)
A modular tensor category is a 3category with strong duality and the right sort of finiteness properties, so we can apply the above construction to get a 3+1dimensional TQFT.
In dimension 3 the vector space we construct is an oldfashioned skein module: finite linear combinations of ribbon graphs in M^3 modulo local relations. (Actually, the dual of this vector space.) If M is closed this in 1dimensional. More generally, if M has boundary then it has the same dimension as the WittenReshetikhinTuraev vector space associated to the boundary of M.
In dimension 4, the type of state sum we get depends on the type of handle decomposition. For a general handle decomposition we get the CraneYetter state sum. For 2handles attached to a 4ball along a framed link L we get the ReshetikhinTuraev surgery formula for L. For a 4dimensional neighborhood of a 2complex we get the Turaev "shadow" state sum. For a closed 4manifold we find that the path integral is equal $a^\chi b^\sigma$, where $\chi$ is the Euler characteristic and $\sigma$ is the signature of the 4manifold. By choosing $\lambda$ above appropriately we can make a=1. b is related to the central charge of the MTC (or to the value of the RT surgery formula on framing +1 unknot). For a 4manifold with boundary we find that the state sum computes the WittenReshetikhinTuraev invariant of the boundary of the 4manifold.
In dimension 2, the 1category associated to a closed surface is a full matrix category; i.e. it is Morita trivial. For a surface with boundary k circles the category is Morita equivalent to k copies of the MTC thought of as a 1category.
In all of the above cases, we find that the TQFT invariant of $Z(X)$, where dim(X) = 2 or 3 or 4, depends strongly on the boundary of $X$ but only weakly (i.e. only up to bordism) in the interior of $X$. So we can define a new 2+1 dimensional TQFT $Z'$ via the formula
$$Z'(Y) := Z(\mathrm{boundary}^{1}(Y))$$
This TQFT has an anomaly, since we need to enhance Y with enough extra structure to pick out an inverseboundary, up to bordism.

$\begingroup$ I should add that the above in no way contradicts Chris's answer. $\endgroup$ Commented Oct 13, 2009 at 5:31

$\begingroup$ I'm somewhat confused on one point, I thought that TQFTs coming from TuraevViro (in other words when the MTC is the Drinfel'd center of a semisimple spherical cateogry) didn't have an anomaly. But by your description they still see the 4d structure because their total quantum dimension isn't 1. $\endgroup$ Commented Oct 13, 2009 at 6:20

1$\begingroup$ I should have said central charge rather than total qdim (and now I've edited away the error). The total qdim plays a prominent role in the state sum, but not in the final answer for a closed 4manifold. $\endgroup$ Commented Oct 13, 2009 at 6:54

$\begingroup$ There are some recent preprints by Daniel Freed which expand on this way of looking at anomalies: arxiv.org/abs/1404.7224 and arxiv.org/abs/1406.7278 $\endgroup$ Commented Oct 23, 2014 at 1:36
There are several different kinds of TQFTs that can be defined. For example there are oriented theories, unoriented theories, framed theories, and many more. The 3D TQFTs coming from Modular Tensor Categories don't define oriented TQFTs, hence they have an "anomaly". One way to make this precise is to realize that they do give a TQFT, just not one based on oriented manifolds, but manifolds equipped with an additional structure.
This additional structure goes under various names, e.g. "p1structure", "rigging" (there is a weaker version known as a "2framing"). There are also several ways to define this structure. Some of these approaches do not yield equivalent structures, and it depends on your MTC as to which one will work (although there is a universal choice).
One common version of this structure on a 3manifold is an equivalence class of choices 4manifolds which bound it. The equivalence relation is that these 4manifolds are taken up to bordism (hence you really only have the signature). This gives you a central extension of the bordism category, and (some of) the MTC TQFTs are defined on this central extension. But it also means that there is a way to interpret such TQFTs as assigning data to 4manifolds, but just in a (mostly) trivial way.
Now the 4manifold structure is fine for ChernSimmons theory, but my understanding is that it is not the universal structure (which has to do with the determinant bundle on the moduli spaces of surface, see Segal's "Definition of Conformal Field Theory" manuscript).
My own understanding of anomalies in TQFTs:
In cases that I have seen, "anomaly" in general refers to central extensions and line bundles. Physicists have long thought of these issues in a very explicit way, in terms of integrals to compute Feynman diagrams. In many quantum field theories, the Feynman diagram integrals do not converge even though the quantum field theories are healthy just from counting degrees of terms in the Lagrangian (renormalizable). A modern explanation of this (that I do not know firsthand, but have been told) is that nonconvergence comes from trying to compute something that is not a number, but rather a value in a nontrivial line bundle. "Anomaly cancellation" means that you find two things to compute that both have anomalies, but whose product does not because the line bundle is trivial.
A 3D TQFT is, among other things, a functor from the category of cobordisms between surfaces to the category of vector spaces. This category has central extensions, in the same sense that groups have central extensions. Its universal central extension over ℚ is given by tensoring the torsor of framings with ℚ and it is onedimensional. (Here "torsor" just means an affine space over a group. Framings up to homotopy are a torsor over the abelian group of the homotopy space $[M,SO(3)]$.) So, if you want a central extension of cobordisms before making your functor, you want framings of 3manifolds. The only point of 2framings is to cancel the 2torsion part of the torsor of framings by hand; it has always seemed better to me to tensor with ℚ instead.
This central extension appears naturally in the ChernSimons QFT, a.k.a., ReshetikhinTuraev invariants of 3manifolds. And the TuraevViro invariant is a famous example of anomaly cancellation. If the input category to that invariant is modular (it need only be a spherical fusion category in general), then it is two factors of ReshetikhinTuraev with opposite phase.
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) XiaoGang Wen
arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions (for gravitational anomaly and ncategory, as well as the notion of Htype and Ltype anomalies) Liang Kong, XiaoGang 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

3$\begingroup$ Prof. Wen, I feel confused about the meaning of
'lattice model'
here: e.g., we say FQHE states can be realized on 'lattice' just as solid state experimentalist did, so it should be called anomaly free from your definition. However, from Kevin's answer, a ChernSimons theory is anomalous because it has nontrivial central charge, or related to language of lattice, it can not be realized as TuraevViro/LevinWen TQFT. In that sense, the 'mathematical lattice' seems to imply a Local Commuting Projector Hamiltonian as well, however, a 'physical lattice' does not requires that. $\endgroup$ Commented Sep 24, 2015 at 16:16