In the article Topological Field Theories in 2 dimension, Constantin Teleman has the following commentary " By constrast, in higher dimension, there seem to be no interesting theories: all examples are built from characteristic classes." There is any material where I can find these examples?
2 Answers
In high dimensions you always have available variants of DijkgraafWitten theory (references at the bottom of the page). In dimension $n$ this is a TFT constructed from a finite group $G$ and a characteristic class $\alpha \in H^n(BG, U(1))$ (where $U(1)$ has the discrete topology). On a closed oriented $n$manifold $M$ it returns the number
$$Z_{G, \alpha}(M) = \int_{f \in [M, BG]} \int_M f^{\ast}(\alpha)$$
where
 $[M, BG]$ denotes the groupoid of principal $G$bundles, or equivalently maps $f : M \to BG$,
 integrating a function $f : \pi_0(X) \to \mathbb{C}$ on a groupoid $X$ means taking the sum $\sum_{x \in \pi_0(X)} \frac{f(x)}{\pi_1(X, x)}$, and
 integrating $f^{\ast}(\alpha) \in H^n(M, U(1))$ over $M$ means pairing it with the fundamental class.
When $\alpha = 0$ and $M$ is connected this evaluates to $\frac{\text{Hom}(\pi_1(M), G)}{G}$.
DijkgraafWitten theory should in fact be a fully extended TFT, and for a variation $BG$ can be replaced by a "$\pi$finite space" (a space with finitely many nonzero homotopy groups, each of which is finite). I think this generalization is briefly discussed in some papers of Freed which should be listed in the references above.
If Teleman means to claim that these are the only highdimensional examples I'm not sure I agree; it depends on what target categories he has in mind. There should be $n$dimensional examples coming from factorization homology of $E_n$algebras (see, for example, Scheimbauer), although these have the drawback that in codimension $0$ they don't return a number. There should also be $n$dimensional "classical" examples coming from iterated spans (see, for example, Haugseng).

$\begingroup$ Ah, I see that Teleman has in mind $\text{Vect}$ as the target category. $\endgroup$ Sep 19, 2014 at 19:46

$\begingroup$ If you take $E_n$ of linear categories, you should still get numbers, right? $\endgroup$ May 4, 2016 at 7:40
In three dimensions there are ChernSimons theories which are indeed built from characteristic classes. As a physicist I find them extremely interesting. In four dimensions, we have DijkgraafWitten TQFT and also the CraneYetter TQFT, which are constructed using the algebraic data of a premodular ribbon fusion category.