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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?

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In high dimensions you always have available variants of Dijkgraaf-Witten 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 $\frac{|\text{Hom}(\pi_1(M), G)|}{|G|}$.

Dijkgraaf-Witten 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 high-dimensional 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).

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  • $\begingroup$ Ah, I see that Teleman has in mind $\text{Vect}$ as the target category. $\endgroup$ – Qiaochu Yuan Sep 19 '14 at 19:46
  • $\begingroup$ If you take $E_n$ of linear categories, you should still get numbers, right? $\endgroup$ – Manuel Bärenz May 4 '16 at 7:40
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In three dimensions there are Chern-Simons theories which are indeed built from characteristic classes. As a physicist I find them extremely interesting. In four dimensions, we have Dijkgraaf-Witten TQFT and also the Crane-Yetter TQFT, which are constructed using the algebraic data of a premodular ribbon fusion category.

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