The n-Category Café

Topological field theory from perfect stacks

Early on we learn about $2$-dimensional TQFT’s as functors from $2Cob$ to $Hilb$ or something of that sort. Here $2Cob$ is the category whose objects are $1$-dimensional manifolds and whose morphisms are $2$-dimensional manifolds whose boundary we think of as having an “incoming” $1$-dimensional manifold as source and similarly an “outgoing” $1$-dimensional manifold as target. Of course, $Hilb$ could have been replaced with a variety of different categories.

Eventually we grow up and have to learn new things. So today we are going to change our notion of $2Cob$: let $2Cob$ denote the $\infty$-category whose objects are compact oriented $1$-manifolds and whose morphisms $2Cob(C_1,C_2)$ consist of the classifying spaces of oriented topological surfaces with fixed incoming and outgoing components $C_1$, $C_2$. This is given symmetric monoidal structure by disjoint union of $1$-manifolds.

Then of course we need to replace $Hilb$ by a symmetric monoidal $\infty$-category $\mathcal{C}$.

Now if we know what these symmetric monoidal $\infty$-categories are then we are ready to define the notion of topological field theory. Since all the hard work was pushed into understanding $2Cob$, then the definition is obvious:

Definition

A $2$-dimensional topological field theory is a symmetric monoidal functor $F: 2Cob \rightarrow \mathcal{C}$.

So that was the background that we need. Maybe in a few years we will say “Early on we learn about $2$-dimensional TFT’s as symmetric monoidal functors from $2Cob$ to $\mathcal{C}$” and everyone will know that we mean $\infty$-categories.

Next we define the values of our TFT’s for the present example. We first fix a derived ring $k$.

Definition

Let $Cat_\infty^{ex}$ denote the $(\infty,2)$-category of stable, cocomplete $k$-linear $\infty$-categories, with $1$-morphisms consisting of continuous exact functors.

So when I hit ‘$(\infty,2)$-category’ something funny happens and the rest of the sentence comes out as “…of blah-blah-blah-categories, with $1$-morphisms consisting of continuous exact functors.” This shouldn’t really happen since stable, cocomplete and $k$-linear are friendly enough concepts, but it definitely tells me I should slow down and learn to love $(\infty,2)$-categories.

I am going to again put off explanations for a later date at the $n$Lab.

Thinking about the symmetric monoidal structure for a moment gets us to the first of our two results which are approximately of the form “You give me a perfect stack and I will give you a TFT”.

The symmetric monoidal structure comes from the tensor product of presentable stable categories with unit the $\infty$-category of $k$-modules. Why do we care? From here we can see what the TFT should assign to closed surfaces. In our example the answer will just be a $k$-module.

How does it work? Luckily, this is ‘monoidally easy’ so I can hopefully understand it. We note that the unit of $2Cob$ is the empty $1$-manifold, i.e., the boundary of a closed surface. So a TFT sends closed surfaces to endomorphisms of the unit of the target $\infty$-category $\mathcal{C}$. For our choice $\mathcal{C} = Cat_\infty^{ex}$, we obtain our previous answer of a $k$-module as the image of a closed surface, since the unit is $Mod_k$.

Now someone should give me a perfect stack called $X$ and we can state our theorem:

Theorem

For a perfect stack $X$, there is a $2$-dimensional TFT $\mathcal{Z}_X: 2Cob \rightarrow Cat_\infty^{ex}$ with $\mathcal{Z}_X(S^1) = QC(\mathcal{L}X)$ and $\mathcal{Z}_X(\Sigma) = \Gamma(X^\Sigma,\mathcal{O}_{X^\Sigma})$, for closed surfaces $\Sigma$.

Then there is a proof. I am not really going to read that right now, even though it seems like that is where the goodies are. For now let’s note some important points:

• Since all of the stacks involved are perfect, pullback and pushforward of quasicoherent sheaves along this correspondence defines a colimit-preserving functor.

Now I have not told you what the correspondence is, but we can still see why it is so nice to be perfect. A big theme has been the quest for function theories with nice pullback and pushforward conditions. In particular, that integral transforms should be the same as cocontinuous functors.

Second we note again the symmetric monoidal structure on $\mathcal{Z}_X$. In particular, our other favorite fact comes into play:

• $QC(X_1)\otimes_{QC(Y)}QC(X_2) \stackrel{\sim}\rightarrow QC(X_1\times_Y X_2)$,

where there are maps of perfect stacks $p_1: X_1 \rightarrow Y$ and $p_2: X_2 \rightarrow Y$.

It seems that we are all somewhat starving for examples of perfect/derived stacks. Maybe we are just looking too hard! BZFN give a nice example here of a TFT coming from the classifying space $BG$.

Example

Let $X = BG$ (in characteristic zero). Then the corresponding TFT assigns to the circle the $\infty$-category of sheaves on the adjoint quotient $G/G = Loc_G(S^1)$. That is, $G$-local systems on the circle. To a surface $\Sigma$, the TFT assigns the cohomology of the structure sheaf of the moduli stack $BG^\Sigma = Loc_G(\Sigma)$.

Again there is more to be said here but it will have to wait to get to the lab.

They note that the $2$-dimensional topological field theory defined above is a categorified analogue of the $2$-dimensional TFT’s defined by string topology on a compact oriented manifold, or the topological $B$-model defined by a Calabi–Yau variety.

I am not at all qualified to comment on this statement without doing a little digging, and since I am short on time I will just wait and see if someone else chimes in here.

BZFN note that the their TFT operations are not sensitive to smooth structure. They then go on to show how the proof of the existence of a TFT given a perfect stack can be extended to give proof of the existence of a TFT on spaces as well. This is the second proposition/theorem.

I will quit here and send this over to the $n$Lab for further work. Also, I will begin to say something there about the Deligne–Kontsevich conjectures for derived centers.