On the deformation complex of homotopy affine actions

An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A\to V⋊\mathrm{E}\mathrm{n}\mathrm{d}(V)$, where $V$ is a vector space and $V⋊\mathrm{E}\mathrm{n}\mathrm{d}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathbf{s}\mathbf{c}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathbf{s}\mathbf{c}}_1{)}_{\mathrm{\infty }}$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_{\mathrm{\infty }}$-morphism $A \to V \rtimes {\rm End}(V)$,where$A$isan$A_\infty$-algebra. In this paper we prove a relative version of Deligne’s conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an ${\mathrm{S}\mathrm{C}}_2$ operad. That structure is naturally compatible with the ${\mathrm{E}}_2$ structure on the deformation complex of the $A_{\mathrm{\infty }}$-algebra.