Relation between symplectic (co)homology and Hochschild (co)homology and deformations

Question

A very fluffy question in which I'm ignorant of homology/cohomology, grading etc:

The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the wrapped Fukaya category of some symplectic manifold, are proven to be (ring, not sure about BV-algebra?) isomorphisms in certain cases (see e.g. Ganatra, Ritter-Smith). It is also known (Abbondandolo-Schwarz, Abouzaid) that the symplectic (co)homology of a cotangent bundle is (BV-algebra) isomorphic to the homology of the free loop space of the base manifold, which is (BV-algebra) isomorphic to the Hochschild (co)homology of the singular cochains of the manifold (see e.g. Abbaspour).

My superficial understanding of Hochschild (co)homology is that it measures deformations, but I'm not very familiar with Hochschild (co)homology constructions mentioned above, nor with the constructions of Hochschild (co)homology for categories.

Question: Can I, and if so in what (geometric or algebraic) sense, interpret symplectic (co)homology as encoding deformations? I guess I can write down a Maurer-Cartan equation in symplectic (co)homology; what do solutions encode/measure?

Answer 1

First, the closed-open string map is expected to be an isomorphism between BV algebras. In the case when $X$ is a Weinstein manifold, it is known to be an isomorphism of BV algebras, just make sure that it is the wrapped Fukaya category $\mathcal{W}(X)$ that is under consideration.

Assume that $X$ is a smooth affine variety and admits a compactification $Y$ by a normal crossing divisor $D$. Then one can define the so-called relative Fukaya category $\mathcal{F}(Y,D)$, whose objects are objects in the compact Fukaya category $\mathcal{F}(X)$, and whose morphisms are defined by counting holomorphic polygons passing through $D$. In many known situations, $\mathcal{F}(Y,D)\subset\mathcal{F}(Y)$ is a generating full subcategory. It is expected that $\mathcal{F}(Y,D)$ is a deformation of $\mathcal{F}(X)$ with respect to the image of a cochain $s\in\mathit{SC}^\ast(X)$ under the closed open string map

$\mathit{CO}:\mathit{SC}^\ast(X)\rightarrow\mathit{CC}^\ast(\mathcal{F}(X))$

where $\mathit{SC}^\ast$ is the complex computing symplectic cohomology, and $\mathit{CC}^\ast$ is the Hochschild cochain complex. Note that $\mathit{CO}$ is in fact an $L_\infty$-homomorphism, and both $s$ and $\mathit{CO}(s)$ are Maurer-Cartan elements with respect to the corresponding $L_\infty$-structures, so one can use it to deform $\mathcal{F}(X)$.

The cochain $s$ is constructed by counting holomorphic thimbles in $Y$ passing through the divisor $D$ and asymptotic to Reeb orbits on the ideal contact boundary $\partial_\infty X$. In nice senarios, e.g. when $Y$ has Kodaira dimension $-\infty$ and $D$ is anticanonical, one can show that the cochain $s\in\mathit{SC}^\ast(X)$ is actually a cocycle, therefore defines a class $[s]\in\mathit{SH}^0(M)$ in the (in general even degree) symplectic cohomology. This class is called a Borman-Sheridan class. One can use it to deform the wrapped Fukaya category of $X$ as well, since the closed-open string map can be defined for $\mathcal{W}(X)$.

Elements in the odd degree symplectic cohomology (or the underlying cochain complex), on the other hand, can be regarded as the non-commutative analogues of algebraic vector fields, so one can deform objects in the Fukaya category along these vector fields. This is explored in detail by Seidel in his works on categorical dynamics, see for example https://arxiv.org/abs/1108.0394.