Integral TQFT and mapping class group actions

Gregor Massbaum gave one of the shorter contributed talks at the conference in Faro, speaking in very general terms on his work on integral TQFT.

When I say TQFT here, I really mean the Witten-Reshetkhin-Turaev 3-d TQFT associated to a quantum group and a root of unity. Since I don’t want to get into the guts of what that is, let’s just accept for a moment that a 3-manifold invariant called WRT_\zeta exists. For each root of unity $\zeta$, it associates a complex number to each oriented compact 3-manifold.

Now, as I’m sure you’re all aware, a numerical invariant of 3 manifolds is a lot weaker than a 3-d TQFT. But it’s less weak than you might think.For example, we can get a TQFT out of such an invariant as follows:

So, having a 3-d TQFT (to a topologist) means that we should associate to every surface \Sigma a vector space V(\Sigma), and to each oriented 3-manfold with boundary M and each orientation preserving isomorphism \phi:\partial M\to \Sigma, we get a vector \eta(M,\phi)\in V(\Sigma). We also need some coherences, the most important being that we want a pairing \langle\cdot,\cdot\rangle:V(\Sigma)\times V(-\Sigma)\to \mathbb{C} such that WRT(M_1\bigsqcup_{\phi_1,\phi_2} M_2)=\langle\eta(M_1,\phi_1),\eta(M_2,\phi_2)\rangle  (and a slightly more general statement when you only glue part of the boundary).

Now, starting with the invariant WRT_\zeta, our “first approximation” to its TQFT is simply the abelian group freely generated by pairs (M,\phi:\partial M\to \Sigma) up to the obvious equivalence, which we’ll call \tilde V(\Sigma). Of course, the pairing \langle\cdot,\cdot\rangle of the generators corresponding to 2 manifolds is just the WRT invariant of their union along the gluing maps.

Now, I’m sure you’ll all protest that this is terrible. This is a huge infinitely generated group, and we’ll get no new insights out of this, since we didn’t put anything interesting in.

If we want to get something more recognizable as a TQFT, we should consider the kernel of the pairing and divide out by it (since this pairing is really what we’re interested in). Call this smaller group V(\Sigma). What can we say about it?

Well, you may have noticed I’ve been working over the integers thus far, but most people would have just worked over \mathbb{C}. If we had done that, then we would be able to say right away that V(\Sigma) is finite-dimensional, provided you’re willing to accept that the WRT-invariant came from a TQFT with finite-dimensional vector spaces in the first place.

But do we really need the whole complex numbers? It turns out that WRT invariants aren’t arbitrary complex numbers, but rather are cyclotomic integers (if normalized correctly). This implies that we could take our vector spaces not over the complex numbers, but rather over the cyclotomic field, and still get finite dimensional vector spaces.

But what about using the ring of integers instead of the field? It takes a lot more work, but in fact we could use the cyclotomic integers and still get the same answer, by which I mean, V(\Sigma) will be a free module over the cyclotomic integers of the same rank as we would have gotten if we had worked over \mathbb{C}.

Why do we care? Well, the WRT invariant of any manifold can basically be entirely understood by looking at the mapping class group action on the vectors corresponding to handlebodies, using a Heegaard splitting. Thus, the mapping class group action on V(\Sigma) (which we can understand as the action by composition with the map \phi:\partial M\to\Sigma) is of great interest to anybody who likes quantum invariants of 3-manifolds.

Suddenly, now we know that this action is defined not just over \mathbb{C}, but the ring of cyclotomic integers. Apparently there are some interesting things that come out of this picture (though I can say I really understood them. Gregor went very fast at the end of his talk), but the one that seems most obvious to me seems to be largely unstudied: what happens when you reduce mod a prime?

Since the linear groups over finite fields are finite, these reductions are going to have big kernels, and any two Heegaard splittings that differ by elements of these kernels will have congruent WRT invariants. Maybe this ultimately isn’t so exciting, but it sounds rather interesting to me.