Twisted tensor product of dg categories and Kontsevich's Swiss Cheese conjecture

Abstract

Let $A$ be a $1$-algebra. The Kontsevich Swiss Cheese conjecture [K2] states that the homotopy category $\mathrm{Ho}(\mathrm{Act}(A))$ of actions of $2$-algebras on $A$ has a final object and that this object is weakly equivalent to the pair $(\mathrm{Hoch}(A), A),$ where $\mathrm{Hoch}(A)$ is the Hochschild complex of $A$. Here the category of actions is the category whose objects are pairs $(B,A)$ which 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 ${O}$ 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 ${O}$ by $a$ (for the 1-algebra argument) and $b$ (for the 2-algebra argument), denote by $E_1^{{O}}$ the restriction of ${O}$ to the color $a$, and by $E_2^{{O}}$ the restriction of ${O}$ to the color $b$. Let $\mathrm{Alg}({O})$ be the category of dg algebras over ${O}$. For a fixed $1$-algebra $A$ we also have the the category of action $\mathrm{Alg}({O})_A$ (equal to $\mathrm{Act}(A)$ in case of the Swiss Cheese operad). We prove that there is an equivalence of categories $$\mathrm{Alg}({O})_A \cong \mathrm{Alg}(E_2^{{O}})/\mathrm{Hoch}(A)$$ We stress that for this particular model of Swiss Cheese operad the statement holds on the chain level, without passage to the homotopy category.