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The closed string correlators can be constructed from the open ones using topological string theories as a model. The space of physical closed string states is isomorphic to the Hochschild cohomology of (A,Q) (operator Q of ghost number one), - this statement has been verified by means of computation of the Hochschild cohomology of the category of D -branes. 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, [Formula: see text], [Formula: see text], which is defined to be the cohomology of (-1) n Q+d  Hoch  . Here [Formula: see text] is the initial non-deformed BRST operator while ∂  deform   is the deformed part whose algebra is a Lie algebra of linear vector fields gl  n . We assume that if in the theory exists a single D -brane then all the information associated with deformations is encoded in an associative algebra A equipped with a differential [Formula: see text]. In addition equivalence classes of deformations of these data are described by a Hochschild cohomology of (A,Q), an important geometric invariant of the (anti)holomorphic structure on X. We also discuss the identification of the harmonic structure (HT • (X); HΩ • (X)) of affine space X and the group [Formula: see text] (the HKR isomorphism), and bulk-boundary deformation pairing.

BRST-INVARIANT DEFORMATIONS OF GEOMETRIC STRUCTURES IN SIGMA MODELS