BRST-Invariant Deformations of Geometric Structures in Sigma Models

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.