Dijkgraaf-Witten TQFT vs. Representation Theory?

Question

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 ?

Answer 1

Strictly speaking, this answer is not about the 3d TQFT which you mention in your question, but rather a 2d version of Dijkgraaf-Witten theory (described in Section 2 of Freed-Hopkins-Lurie-Teleman).

To every finite group $G$, there is a 2d TQFT $Z_G$ which assigns to a closed orientable surface $\Sigma$ the following sum over isomorphism classes of $G$ local systems $P \to \Sigma$: $$ Z_G(\Sigma) = \sum 1/|Aut(P)| = |Hom(\pi_1(\Sigma),G)|/|G|. $$ In the framework of TQFT as a symmetric monoidal functor $Bord \to Vect$, this assigns to a circle the space of class functions on $G$ (which is a commutative Frobenius algebra under convolution). We can extend further and define $Z_G$ on a point to be the category of representations of $G$ (or alternatively, the group algebra of $G$, depending on your set-up).

Analysing this TQFT on surfaces allows you to recover interesting group-theoretic identities. For example, by cutting up the surface $\Sigma$ into pairs of pants, recovers the following formula (probably first due to Frobenius): $$ Z_G(\Sigma) = \sum_{V\in \widehat{G}} \left(\frac{\dim V}{|G|}\right)^{\chi(\Sigma)}. $$ There are similar formulas involving the other entries in the character table for $G$, by considering surfaces with boundary (which can be thought of as counting $G$ local systems on a closed surface with singularities).

These formulas were used by Hausel and Rodriguez-Villegas to compute data about the Hodge numbers of character varieties in their paper Mixed Hodge Polynomials of Character Varieties.

The recent work of Ben-Zvi and Nadler Character Theory of a Complex Group is in some sense a categorified analogue of this TQFT, but where the finite group is replaced by a complex reductive group (as explained in the introduction). In ongoing work of myself with the authors, we are trying to understand what this structure says about the cohomology of character varieties.

At the risk of over advertising my own work, let me also mention this paper: Spin Hurwitz Numbers and TQFT, which describes an analogue of the Dijkgraaf-Witten TQFT for surfaces with spin structure.

There are probably many other references which I will try to add later...