On Kontsevich's Hochschild cohomology conjecture

Generalizing a conjecture of Deligne, Kontsevich proposed that there should be a notion of Hochschild cohomology of algebras over the little cube operad (or its chain complex) which in a natural way generalizes Hochschild cohomology of associative algebras. He moreover conjectured that the Hochschild cohomology, in this new sense, of an algebra over the little k-cube operad is an algebra over the little (k + 1)-cube operad. In this paper, we precisely state and prove this conjecture.