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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 ?

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The question seems vague. Can you be more specific? –  Qiaochu Yuan May 24 '13 at 20:43
@Qiaochu I had noticed TQFT can help make certain group character calculations really short by using the topology of surfaces. However, TQFT seem to be more about studying G-bundles on these surfaces (or even more general objects) in terms of various group invariants. –  john mangual May 26 '13 at 16:32
@ john mangual, Qiaochu: hopefully you will find these References very helpful - Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory, arxiv.org/abs/1404.7854 and Twisted Gauge Theory Model of Topological Phases in Three Dimensions arxiv.org/abs/1409.3216 –  Idear Mar 17 at 0:26

1 Answer 1

up vote 13 down vote accepted

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...

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The self-promotion is fine, since it's dead-on. I didn't know you could get Hurwitz theory from Dijkgraaf-Witten theory as in your paper. –  john mangual May 26 '13 at 16:37
I got caught on this topic trying to count the number of solutions to $\phi(k) = \# \{(g,h): ghg^{-1}h^{-1} = k \}$ This seems to be a kind of discretized character variety. –  john mangual May 26 '13 at 16:38
Right. Hurwitz numbers (and variants) are computing the volume of these discretized character varieties. The TQFT interpretation of this is analogous to Witten's computation of the volume of character varieties of compact groups in "On Quantum Gauge Theories in 2 Dimensions"(1991). –  Sam Gunningham May 26 '13 at 18:46
By the way, I think this is a great question, and I am very interested to find out about other aspects of character theory that can be simplified using TQFT. –  Sam Gunningham May 26 '13 at 18:54
You might find these notes interesting. They are from a lecture series given by David Ben-Zvi a few years ago. math.utexas.edu/~benzvi/GRASP/lectures/NWTFT/nwtft.pdf –  Sam Gunningham May 26 '13 at 18:58

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