  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. A relevant concept in the
vertex operator algebra and the BRST cohomology is that of the elliptic genera
(the one-loop string partition function). We show that the elliptic genera can
be written in terms of spectral functions of the hyperbolic three-geometry
(which inherits the cohomology structure of BRST-like operator). We show that
equivalence classes of deformations are described 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 discuss the
identification of the harmonic structure $(HT^\bullet(X); H\Omega_\bullet(X))$
of affine space $X$ and the group ${\rm Ext}_{X}^n({\cal O}_{\triangle}, {\cal
O}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.
