Kapustin's closed-string states and Hochschild cohomology conjecture

Let $\mathcal A$ be an associative algebra equipped with a differential

$$Q=\overline{\partial}+\partial_{\rm deform}.$$

For a theory with a single D-brane, the deformation data are encoded by $(\mathcal A,Q)$, and their equivalence classes are described by the Hochschild cohomology $HH^\bullet(\mathcal A,Q)$. Kapustin's conjecture. The space of physical closed-string states is isomorphic to the Hochschild cohomology of $(\mathcal A,Q)$.

The conjecture relates closed topological string states to infinitesimal deformations of open-string theory and suggests that closed-string correlators can be constructed from open-string data. It has been partially verified by computations of Hochschild cohomology in categories of D-branes; the general statement for perturbative deformations remains open.