The Hochschild cohomology of the Fukaya-Seidel category ( FS((Y, \omega), W) ), where ( (Y, \omega) ) is a non-compact Kähler manifold and ( W: Y \to \mathbb{C} ) is a holomorphic function with isolated non-degenerate critical points, is deeply connected to mirror symmetry.

Under the mirror symmetry conjecture, if ( (Y, W) ) is the mirror of a smooth projective Fano variety ( X ), then...

Short answer.

• For a Lefschetz Landau–Ginzburg model (Y, ω, W) with only Morse critical points (the setting in which the Fukaya–Seidel category FS(Y, W) is defined), the “closed string” A-model state space is the relative cohomology H*(Y, Y−∞; C), and there are natural open–closed and closed–open maps OC: HH∗(FS(Y, W)) → H∗+n(Y, Y−∞), CO: H∗(Y, Y−∞) → HH∗(FS(Y, W)), where n = dimC Y.
• The ...

Below is what is known at present about the “closed string” algebra which controls deformations of the Fukaya–Seidel category of a Landau–Ginzburg model.

1. Set–up

   Let
   (Y, ω, λ) exact symplectic (= Liouville) manifold,
   W : Y → ℂ proper, J–holomorphic, with only Morse (or Morse–Bott) critical points,

and assume that outside a compact set Re W is a rad...

This is an excellent and insightful question that gets to the heart of modern homological mirror symmetry. You've correctly identified the key players and the main difficulty: the Fukaya-Seidel category is generally not Calabi-Yau, so its Hochschild homology and cohomology are distinct and interesting objects.

The short answer, based on the principles of homological mirror symmetry and ...

Let
 W : (Y,ω) ⟶ ℂ

be a (tame, exact) Lefschetz fibration; write
 Y_{-\infty}=W^{-1}(z),  Re z≪0.
Put a Liouville structure on Y which is conical outside a compact set and coincides with –d(Re W)◦J near Y_{-\infty}.
With this choice the Fukaya–Seidel A∞–category (constructed e.g. by Seidel) may be viewed, in the sense of Ganatra–Pardon–Shende, as a wrapped Fukaya category of the **stop...

Here is what one “expects’’ (and in many cases can now prove) for the Hochschild cohomology of the directed Fukaya–Seidel category. We will denote by FS = FS((Y,ω),W) the A∞–category of vanishing thimbles of the Lefschetz fibration W : Y→ℂ, and by W(Y) its wrapped Fukaya category. The key input is the (now proved, in the Weinstein case) isomorphism between symplectic cohomology and Hochschild co...

Hochschild Cohomology of the Fukaya-Seidel Category

The Hochschild cohomology of the Fukaya-Seidel category is an important but subtle object in mirror symmetry. I'll attempt to provide a comprehensive answer based on current understanding in the field.

Expected Formula

For a Landau-Ginzburg model with isolated critical points, the Hochschild cohomology of the Fukaya-Seidel ...

This is an excellent question about the Hochschild cohomology of Fukaya-Seidel categories. Let me provide what is known and expected.

The General Picture

For a Fukaya-Seidel category associated to a Landau-Ginzburg model, the Hochschild cohomology should be computed by a version of the open-closed map that relates it to certain cohomology groups of the ambient...