From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a topological generalization of representation theory.

Dijkgraaf and Witten write down a few interesting formulas in Section 6:

$\displaystyle Z(S^3) = \frac{1}{|G|} $

$\displaystyle Z(S^2 \times S^1) = 1$

$\displaystyle Z(\mathbb{R}P^3) = \frac{1}{2}(1 + (-1)) = 0$

$\displaystyle Z( M ) = \frac{1}{|G|} \sum_{\gamma \\, : \\, \pi_1(M)\to G} e^{2\pi i S}$

$\displaystyle Z(S^1 \times S^1 \times S^1) = \frac{1}{|G|}\sum_{[g,h]=[h,k]=[k,g]=1} W(g,h,k) = \sum_{g \in C} r(N_g; c_g)$, the sum over certain conjugacy classes.

$\displaystyle Z(S^3/\mathbb{Z}_n) = \langle \varnothing |(TST)^n|\varnothing \rangle = \left\{ \begin{array}{ll}\frac{1}{2}(1 + (-1)^{n/2}) & n \text{ even } \\\\ \frac{1}{2} & n \text{ odd } \end{array} \right.$

A few years later, there seem to have been written Freed-Hopkins-Lurie-Teleman seem mainly concerned with the categorical structure of this story.

About a year ago, I wrote a short group theory note generalizing the 5/8 bound using a tiny bit of TQFT (itself based on an MO question).

I wonder, what these TQFT's can say about surfaces (or higher dim topological spaces) or groups aand their representations ?

G-bundleson these surfaces (or even more general objects) in terms of various group invariants. $\endgroup$ – john mangual May 26 '13 at 16:32