Pre-Calabi-Yau algebras and oriented gravity properad

TL;DR

This paper explores the structure of pre-Calabi-Yau algebras, establishing a connection between their cyclic Hochschild cohomology and a new oriented ribbon graph properad, with implications for moduli space cohomology and gravity operads.

Contribution

It introduces a novel dg properad of oriented ribbon graphs associated with pre-Calabi-Yau extensions and links it to moduli space cohomology, advancing understanding of algebraic and geometric structures.

Findings

The properad acts on cyclic cohomology of A-infinity algebras.

Cohomology of the properad relates to moduli space cohomology.

Gravity operad acts on higher Hochschild cohomology of pre-CY algebras.

Abstract

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