Two vague questions about TFT 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?
 A: 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.
