Twisted Tensor Product of dg Categories and Kontsevich’s Swiss Cheese Conjecture

Abstract Let $A$ be a $1$-algebra. The Kontsevich Swiss Cheese conjecture [13] states that the homotopy category $\text{Ho}(\text{Act}(A))$ of actions of $2$-algebras on $A$ has a final object and that this object is weakly equivalent to the pair $(\text{Hoch}^{\bullet}(A), A),$ where $\text{Hoch}^{\bullet}(A)$ is the Hochschild complex of $A$. Here the category of actions is the category whose objects are pairs $(B,A)$ that are algebras of the chain Swiss Cheese operad such that the induced action of the little interval operad on the component $A$ coincides with the $1$-structure on $A$. We prove that there is a colored dg operad $\mathcal{O}$ with 2 colors, weakly equivalent to the chain Swiss Cheese operad for which the following “stricter” version of the Kontsevich Swiss Cheese conjecture holds. Denote the two colors of $\mathcal{O}$ by $a$ (for the 1-algebra argument) and $b$ (for the 2-algebra argument), denote by $E_{1}^{\mathcal{O}}$ the restriction of $\mathcal{O}$ to the color $a$, and by $E_{2}^{\mathcal{O}}$ the restriction of $\mathcal{O}$ to the color $b$. Let ${\text{Alg}}(\mathcal{O})$ be the category of dg algebras over $\mathcal{O}$. For a fixed $1$-algebra $A$ we also have the the category of action ${\text{Alg}}(\mathcal{O})_{A}$ (equal to $\text{Act}(A)$ in case of the Swiss Cheese operad). We prove that there is an equivalence of categories $$ \begin{align*} &{\text{Alg}}(\mathcal{O})_A \cong{\text{Alg}}(E_2^{\mathcal{O}})/\text{Hoch}^{\bullet}(A).\end{align*} $$ We stress that for this particular model of Swiss Cheese operad the statement holds on the chain level, without passage to the homotopy category.