  We study a Lie algebra of formal vector fields $W_n$ with its application to
the perturbative deformed holomorphic symplectic structure in the A-model, and
a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent
classes of deformations are describing by a Hochschild cohomology theory of the
DG-algebra ${\mathfrak A} = (A, Q)$, $Q =\bar{\partial}+\partial_{\rm deform}$,
which is defined to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here
$\bar{\partial}$ is the initial non-deformed BRST operator while $\partial_{\rm
deform}$ is the deformed part whose algebra is a Lie algebra of linear vector
fields ${\rm gl}_n$. We show that equivalent classes of deformations are
described by a Hochschild cohomology of ${\mathfrak A}$, an important geometric
invariant of the (anti)holomorphic structure on $X$. We discuss the
identification of the harmonic structure $(HT^\bullet(X); H\Omega_\bullet(X))$
of affine space $X$ and the group ${\rm Ext}_{X^{2}}^n({\cO}_{\triangle},
{\cO}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation
pairing.
