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

TL;DR

This paper constructs a dg operad model for the Swiss Cheese operad, proving a chain-level equivalence of action categories that confirms a version of Kontsevich's Swiss Cheese conjecture.

Contribution

It introduces a colored dg operad model for the Swiss Cheese operad, establishing a chain-level equivalence that verifies the conjecture in a stricter form.

Findings

Constructed a colored dg operad ${O}$ weakly equivalent to the Swiss Cheese operad.

Proved an equivalence of categories between algebra actions and Hochschild complexes.

Validated the conjecture at the chain level without passing to homotopy categories.

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.