XXIX Workshop on Geometric Methods in Physics 27.06-03.07.2010
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# Andrey Bytsenko

## Deformation of complex structures in topological field theory

We study a Lie algebra of formal vector fields $W_n$ with it 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 DG-algebra $\cA = (A, Q)$, $Q =\overline{\partial}+\partial_{\rm deform}$, which is defind to be the cohomology of $(-1)^n Q +d_{\rm Hoch}$. Here $\overline{\partial}$ 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 ${\rm Ext}_{X?}^n(\cO_{\triangle}, \cO_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.