High dimensional topological field theory

Question

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?

Answer 1

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).

Answer 2

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.