Tensor product of cyclic A$_\infty$-algebras and their Kontsevich classes

TL;DR

This paper establishes that the tensor product of two cyclic A$_ $-algebras admits a unique cyclic A$_ $-structure up to quasi-isomorphism and that its Kontsevich class equals the cup product of the individual classes.

Contribution

It proves the existence and uniqueness of a cyclic A$_ $-structure on tensor products and relates the Kontsevich class to the cup product on moduli space.

Findings

Existence of a cyclic A$_ $-structure on tensor products.

Uniqueness up to cyclic A$_ $-quasi-isomorphism.

Kontsevich class of the tensor product equals the cup product.

Abstract

Given two cyclic A$_\infty$-algebras $A$ and $B$, we prove that there exists a cyclic A$_\infty$-algebra structure on their tensor product $A\otimes B$ which is unique up to a cyclic A$_\infty$-quasi-isomorphism. Furthermore, the Kontsevich class of $A\otimes B$ is equal to the cup product of the Kontsevich classes of $A$ and $B$ on the moduli space of curves.