The n-Category Café

Scott wrote:

Well Khovanov homology is known to give trivial (or more precisely uninteresting) invariants for surfaces in 4-space.

Maybe that’s true for closed surfaces… is that what you mean? But Khovanov homology definitely does say very interesting things about 2d discs embedded in 4d space. So I’m afraid nonexperts will get the wrong impression from what you wrote here!

From the Prehistory:

One exciting aspect of Khovanov’s homology theory is that it breathes new life into Crane and Frenkel’s dream of understanding the special features of smooth 4-dimensional topology in a purely combinatorial way, using categorification. For example, Rasmussen has used Khovanov homology to give a purely combinatorial proof of the Milnor conjecture—a famous problem in topology that had been solved earlier in the 1990’s using ideas from quantum field theory, namely Donaldson theory. And as the topologist Gompf later pointed out, Rasmussen’s work can also be used to prove the existence of an exotic $\mathbb{R}^4$.

In outline, the argument goes as follows. A knot in $\mathbb{R}^3$ is said to be smoothly slice if it bounds a smoothly embedded disc in $\mathbb{R}^4$. It is said to be topologically slice if it bounds a topologically embedded disc in $\mathbb{R}^4$ and this embedding extends to a topological embedding of some thickening of the disc. Gompf had shown that if there is a knot that is topologically but not smoothly slice, there must be an exotic $\mathbb{R}^4$. However, Rasmussen’s work can be used to find such a knot!

Before this, all proofs of the existence of exotic $\mathbb{R}^4$’s had involved ideas from quantum field theory: either Donaldson theory or its modern formulation, Seiberg–Witten theory. This suggests a purely combinatorial approach to Seiberg–Witten theory is within reach. Indeed, Ozsváth and Szabó have already introduced a knot homology theory called ‘Heegaard Floer homology’ which has a conjectured relationship to Seiberg-Witten theory. Now that there is a completely combinatorial description of Heegaard–Floer homology, one cannot help but be optimistic that some version of Crane and Frenkel’s dream will become a reality.

I think Quinn would say the better analogy is “Requiring a map of topological spaces to be a homeomorphism enormously restricts it. To get more power, we need to consider the larger class of continuous maps.”

I think Quinn’s point is that requiring that TQFTs satisfy a large number of higher order decomposition axioms means there will be fewer examples of TQFTs. Weakening the axioms means there will be more examples or TQFTs, some of which might be more powerful in the sense that they distinguish/detect the objects we are interested in.

(I’m not necessarily endorsing Quinn’s view. Personally I’m quite fond of the higher-order TQFT axioms.)

I think Jacob’s viewpoints 1-3 are very closely connected, and I don’t find Quinn’s position about extended TFT reasonable in a long-term sense even from POV 1, whether or not one can point yet to concrete statements about 3-manifold topology coming say from extended TFT.

The way I understand the cobordism hypothesis is as an attempt to capture something close to the richness of structure in a physicist’s TFT (specifically in a topologically twisted supersymmetric QFT). The assignments of traditional TFT capture a mere shadow of this structure, as is already evident in the 2d case (as Kontsevich discovered in homological mirror symmetry). I (learning from Hopkins and Freed) think of what one assigns to a point as a version of the physicists’ action - or more precisely of the action integrated over all fields on a small disc. The cobordism hypothesis teaches us how to perform this integration on larger manifolds - and in particular gives us criteria to tell if this integration is possible. In order to do so one needs higher category theory. Thus points 2,3 are very closely tied - the higher category is a powerful way to make sense of the local structure of path integrals in the topologically twisted SUSY setting.

As to 1, one of the reasons we should think hard about the structures physicists have in place (and thus model them as we do eg with extended TFT) is that they are potentially much more powerful than what we have available. The classic paradigm for this is dualities – physicists can identify apparently very different looking theories, drawing powerful conclusions. (My favorite of these being S-duality). By working with extended TFTs one can hope to get much closer to understanding these dualities, by separating the complexity of the theory and of the topological spaces on which we study the theory.

Witten once described what mathematicians tend to do with ideas trickling in from string theory as “in vitro QFT” - we take the beautiful rich structures of the physics, cut off a piece, let it shrivel and die in the lab, and see what conclusions we can draw. The Seiberg-Witten revolution in topology (among many) teaches us that these structures are much richer and smarter than the ones we used before – in particular in vivo Seiberg-Witten theory is much richer than what mathematicians usually mean. So I don’t think it’s the case that the theories that have given rise to important applications in topology began life as nonextended theories – they may have first made their appearance within mainstream math that way, but underlying them is something much more extensive and living. I think of extended TFT as an attempt to study in vivo QFT, and (to the extent that this language succeeds) I expect it will make an impact in topology as well.

a much more ambitious theory of nontopological QFTs.

What can we say about cobordisms equipped with (suitably well behaved) maps to a fixed smooth/topological space $X$?

Here is a simple low-dimensional observation, that smells a bit like it might be the indication of something. Probably of something obvious, but I feel like mentioning it anyway.

for $X$ a smooth manifold one can define a smooth category (i.e. 1-category valued stack on $CartSp$ or the like) $P_1(X)$ of smooth paths in $X$, whose morphisms are (certain classes of) certain smooth maps $[0,t] \to X$ for $t \in \mathbb{R}_+$. This is a bit like $Bord_1(X)$, but without the symmetric monoidal structure.

Smooth functors $tra : P_1(X) \to Vect$ correspond to smooth vector bundles with connection on $X$.

Now, if this vector bundle is finite rank, then there is a unique extension of this data to a symmetric monoidal morphism $Bord_1(X) \to Vect$, where the value assigned to a circle in $X$ is the trace of the value of $tra$ on the path obtained by cutting open the circle to obtain a path.

Apologies for this lengthy description of the obvious. The simple observation I am meaning to get at is that this looks like the beginning of a free/forgetful adjunction

$$\array{ \underline{P_1(X) \to U(Vect_{\otimes})} \\ Bord_1(X) \to Vect_{\otimes} } \,,$$

where $U$ is the forgetful functor from symmetric monoidal (smooth) categories to bare (smooth) categories.

It kind of makes me want to think of $Bord_1(X)$ as $F(P_1(X))$, the “free smooth symmetric monoidal category with duals” on $P_1(X)$.

For $X$ the point we have essentially $P_1({pt}) = {pt}$ (not quite, possibly, depending on details that i want to be glossing over for the time being). So this kind of perspective might connect to the bare cobordism hypothesis, which, too, one would still hope to be the restriction to the point of a genuine free/forgetful adjunction.