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This is a blog I am starting to keep track of the notes I take during talks. Usually, I won't include a long discussion, but will instead mention an interesting idea or two.

The latex rendering is done by mathjax. I followed the instructions I found at irreducible representations.

Masahiro Futaki (Universiteit Antwerpen) On the Sebastiani-Thom theorem for directed Fukaya categories

The abstract:

The directed Fukaya category is an A-model counterpart for the derived category of toric Fano B-model under the Fano/Landau-Ginzburg mirror correspondence. Auroux-Katzarkov-Orlov conjectured that the directed Fukaya category of the direct sum of two potentials splits as tensor product (up to derived equivalence) and pointed out that this is the case for the mirror of $\mathbb{P}^1 \times \mathbb{P}^1$. We show that the conjecture holds if one of the potentials is of complex dimension 1, partially generalizing Seidel's suspension theorem for directed Fukaya categories.


If I understood the talk correctly, part of the challenge is to make sense of the tensor product for $A_\infty$ categories. Alas, most of the aspects of this I understand superficially at best.

This conjecture of AKO is related to some things I do understand, however. Oancea, in his thesis, proved a Künneth formula for the symplectic homology of the product of two Liouville manifolds. By the appendix to Bourgeois-Ekholm-Eliashberg's surgery exact sequence written by Ganatra and Maydanski (and conjectured by Seidel), symplectic homology of the total space of an exact Lefschetz fibration is the Hochschild [Co?]homology of the derived Fukaya category. In this sense, the AKO conjecture is saying that Oancea's theorem actually lives at the level of the Fukaya category.

I wonder if it would be possible to prove this theorem by carefully using the BEE surgery construction.