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Question 1. Take a smooth projective Calabi-Yau $X$. Then $D^b(X)$ is a fully-dualizable category and there's an associated 2d TFT. This the usual 2d B-model with target $X$.

But $D^b(X)$ is actually a monoidal category, so there's a 2-category $D^b(X)$-mod of categories with an action of $D^b(X)$. Apparently this 2-category is also fully-dualizable, and it defines a 3d TFT. This is the Rozansky-Witten theory with target $T^*X$. (Here I'm just repeating words I've been told.)

But $D^b(X)$ is actually symmetric monoidal, so $D^b(X)$-mod is monoidal, so there's a 3-category ($D^b(X)$-mod)-mod. Does this give us a 4d TFT? And if so does this ever stop? $\mathcal{O}_X$ is genuinely commutative (or $E_\infty$) so for every $n$ there should be an $n$-category of modules over modules over $\;$ ...$\;$ over modules over $\mathcal{O}_X$. So potentially an $n$-d TFT.

Does this (a) work, and (b) produce anything interesting?

And a related:

Question 2. Evaluating an $n$-d TFT on the $n-1$ sphere apparently gives us an $E_n$-algebra. If the $n$-d theory came from modules over ... an $E_{n-1}$ algebra then this is some version of taking the centre. For example the 2d B-model on $X$, evaluated on $S^1$, produces the $E_2$ algebra $HH^*(X)$.

What happens if I take modules over modules over ... this $E_n$ algebra? If I'm lucky I might get a dualizable $n$-category and then an $(n+1)$-TFT.

Does this work in general? (Presumably not). Does it ever work?

For example is there a 3d TFT arising from modules over modules over the $E_2$-algebra $HH^*(X)$, and if so does it have a name?

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I will only attempt to answer the first question.

Given a symplectic manifold $M$, there is a TQFT in any odd dimension. In dimension 1 this is the topological quantum mechanics and in dimension 3 this is the Rozansky--Witten theory. In higher dimensions these have no name and do not arise from topological twists of supersymmetric sigma-models. These are all examples of AKSZ theories.

If you squint your eyes, you can think of $M=B G$ as a symplectic manifold. The corresponding TQFTs are the Chern--Simons theories: in 1d it is the theory considered in https://arxiv.org/abs/1005.2111, in 3d is is the usual Chern--Simons theory, in 5d it is the theory considered in https://arxiv.org/abs/1710.02841 (a topological twist of 5-dimensional superysmmetric Yang--Mills). In higher dimensions it again does not arise from topological twisting.

Given any manifold $X$ you can consider $M=T^* X$. So, you get TQFTs associated to any manifold $X$. These will not be fully defined in the top dimension as $M$ is noncompact. For $M = T^* BG$ these are called the BF theories.

Similarly, if you have an odd symplectic manifold $M$, you get the AKSZ TQFT in any even dimension. E.g. for $M = \Pi T^* X$ for a manifold $X$ you get the $B$-model.

To conclude: the TQFTs you mention (obtained from $D^b(X)$ as a symmetric monoidal category) exist in any dimension. The partition functions will not be well-defined in dimensions $\dim\geq 3$ (the corresponding object is not fully dualizable), but they have interesting spaces of states, categories etc.

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    $\begingroup$ How do you show that the higher-dimensional Chern-Simons theories don't arise from topological twisting? (Or, more generally, why do the odd-dimensional TQFTs associated to symplectic manifolds not arise as topological twists?) $\endgroup$
    – skd
    Commented Mar 28, 2022 at 13:42
  • $\begingroup$ Fantastic thanks Pavel! I wasn't sure if anyone was still looking at MO... $\endgroup$
    – Ed Segal
    Commented Mar 28, 2022 at 13:52
  • $\begingroup$ That is a pretty comprehensive answer to part 1. For part 2, I wouldn't be surprised if E_2 modules (properly understood) over HH(X) are equivalent to D(X), and maybe that's all that happens. But I'm kind of guessing here. $\endgroup$
    – Ed Segal
    Commented Mar 28, 2022 at 13:53
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    $\begingroup$ @skd: the only explanation I have is that people have computed twists of nonlinear gauged sigma-models and the higher-dimensional Chern--Simons/RW theories do not arise. It might be possible that they arise from twisting some higher spin theories, but I know much less about them. $\endgroup$
    – Pavel Safronov
    Commented Mar 28, 2022 at 15:42
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    $\begingroup$ @EdSegal: the 2-category $\mathrm{HH}^\bullet(X)\mathrm{-mod}\mathrm{-mod}$ is extensively studied by Dario Beraldo in relation to singular support of coherent sheaves. If $X$ is smooth, then $\mathrm{HH}^\bullet(X)\mathrm{-mod}\mathrm{-mod}$ is equivalent to $\mathrm{ShvCat}(X_{dR})$, the 2-category of crystals of categories over $X$. I don't know a good name for the corresponding 3d TQFT. $\endgroup$
    – Pavel Safronov
    Commented Mar 28, 2022 at 16:15

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