Pre-Calabi-Yau algebras and oriented gravity properad

Abstract

We study the dual cyclic Hochschild complex $Cyc^\bullet(A,\mathbb{K})$ of a (possibly, infinite-dimensional) $A_\infty$-algebra $(A,μ)$ and prove that any pre-Calabi-Yau extension $π$ of the given $A_\infty$ structure $μ$ in $A$ induces on the cyclic cohomology of $(A,μ)$ a representation of a new dg properad of oriented ribbon graphs. We compute the cohomology of that properad in terms of the compactly supported cohomology groups of moduli spaces $\mathcal{M}_{g,m+n}$ of algebraic curves of genus $g$ with $m+n$ marked points. We also show that the gravity operad acts naturally on the higher Hochschild cohomology of any pre-CY algebra $(A, π)$.