Pre-Calabi-Yau algebras and oriented gravity properad

Abstract

We study the dual cyclic Hochschild complex $Cy{c}^{\bullet }(A,\mathbb{K})$ of a (possibly, infinite-dimensional) $A_{\infty }$-algebra $(A,\hat{I}¼)$ and prove that any pre-Calabi-Yau extension $\ddot{I}€$ of the given $A_{\infty }$ structure $\hat{I}¼$ in $A$ induces on the cyclic cohomology of $(A,\hat{I}¼)$ 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 $(A,\ddot{I}€)$.

Small corrections