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.

19 pages v2: typos are corrected