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 work by pioneers like Seidel, Abouzaid, and others, is:

The Hochschild cohomology should be isomorphic to the relative symplectic cohomology of the diagonal Lagrangian .

Let's break down what this means and why it's the expected answer.

1. The Mirror Symmetry Prediction

As you've beautifully laid out, the most powerful guide here is mirror symmetry. Let's formalize the correspondence you mentioned:

• B-side (Algebraic Geometry): Let be a smooth Fano variety. Its derived category of coherent sheaves is .
• A-side (Symplectic Geometry): The mirror to is a Landau-Ginzburg model , where is a Stein manifold (a non-compact Kähler manifold) and is a holomorphic function called the superpotential. The associated category is the Fukaya-Seidel category .

The core prediction of Homological Mirror Symmetry (HMS) in this context is the equivalence of triangulated categories:

Now we can compute invariants on both sides and expect them to match.

• Hochschild Cohomology on the B-side: As you stated, the Hochschild-Kostant-Rosenberg (HKR) theorem gives us: This is the cohomology of the complex of polyvector fields on .

• Hochschild Cohomology on the A-side: Therefore, by HMS, we must have:

The question then becomes: What is the geometric object on the A-side (the side) that computes this algebra of polyvector fields on the B-side?

2. The Symplectic Counterpart: Relative Symplectic Cohomology of the Diagonal

The modern understanding, primarily developed by Paul Seidel and Mohammed Abouzaid, identifies this A-side object.

Let's first consider the case without a potential, . The Hochschild cochain complex of an -category can often be identified with the endomorphism algebra of the diagonal bimodule. In the context of the Fukaya category of a manifold , this means looking at the Fukaya category of and studying the Floer cohomology of the diagonal Lagrangian with itself.

• corresponds to .
• corresponds to , where is the symplectic Serre functor.

For a compact Calabi-Yau manifold , this Serre functor is just a shift in grading, which is why and are so closely related. The object computes the quantum cohomology , as you noted.

Now, let's reintroduce the non-compactness and the superpotential . The appropriate setting is that of wrapped Fukaya categories. The morphisms are given by wrapped Floer cohomology. The Hochschild cohomology of the Fukaya-Seidel category is then given by the wrapped Floer cohomology of the diagonal, but now in a relative sense determined by the potential .

Consider the product manifold with the split symplectic form . The diagonal is a Lagrangian submanifold. We define a potential on by The Hochschild cohomology is then conjectured to be:

This object is the relative wrapped Floer cohomology of the diagonal.

• Chains: The underlying chain complex is generated by chords of a Hamiltonian flow that "wraps" at infinity, starting and ending on . These correspond to 1-periodic orbits of certain Hamiltonian vector fields on .
• Differential: The Floer differential counts pseudoholomorphic strips in with boundary on , but the count is weighted by . This potential vanishes on the diagonal , but its presence affects the differential for strips that move away from the diagonal.

This construction is the A-side mirror to the algebra of polyvector fields. The product structure on is given by counting pairs-of-pants, which mirrors the wedge product of polyvector fields.

3. Deformations and

Your motivation is exactly right. The space governs the infinitesimal deformations of the category.

• On the B-side (), the deformation space is .
  • corresponds to deformations of the complex structure of .
  • corresponds to "non-commutative" or gerby deformations.
• On the A-side (), we expect to parameterize deformations of the symplectic data.
  • Mirror to are deformations of the complex structure on . Because is Stein, these are often trivial, but the interaction with and the wrapping condition can make them non-trivial.
  • Mirror to the non-commutative deformations are often deformations of the B-field, which modify the symplectic form .

The rich structure of captures all these possible deformations of the A-side model.

Summary

The fact that is not Calabi-Yau is the A-side reflection of the fact that is Fano (i.e., is anti-ample, not trivial). This non-CY nature is precisely what makes and different and interesting, corresponding to differential forms vs. polyvector fields.

References for further reading:

• Paul Seidel, "A biased view of homological mirror symmetry": This is a foundational text explaining the role of the diagonal and how Hochschild (co)homology arises from Floer theory.
• Mohammed Abouzaid, "A geometric criterion for generating the Fukaya category": This and subsequent works by Abouzaid develop the machinery of wrapped Floer cohomology and its relation to Hochschild invariants.
• Denis Auroux, "A beginner's introduction to homological mirror symmetry": Provides a very clear overview of the subject.
• Sheel Ganatra, John Pardon, Vivek Shende, "Microlocal Morse theory of wrapped Fukaya categories": This recent work (and related preprints) provides a rigorous framework for many of these ideas, linking wrapped categories to sheaf theory on the manifold's skeleton.