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.
Categories: hep-th