Pre-Calabi-Yau algebras and oriented gravity properad

We study the dual cyclic Hochschild complex $Cy{c}^{\bullet }(A,\mathbb{K})$ of a (possibly, infinite-dimensional) $A_{\mathrm{\infty }}$-algebra $(A,\mu )$ and prove that any pre-Calabi-Yau extension $\pi$ of the given $A_{\mathrm{\infty }}$ structure $\mu$ in $A$ induces on the cyclic cohomology of $(A,\mu )$ 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 $(A,\pi )$.