Pre-Calabi-Yau algebras and oriented gravity properad
Sergei Merkulov
TL;DR
The paper builds a bridge between pre-Calabi–Yau algebras and geometric moduli spaces by introducing the oriented gravity properad $O\mathcal{RG}raphs_{d+1}$ and proving its action on the dual cyclic Hochschild complex $Cyc^\bullet(A,\mathbb{K})$ for a degree $d$ pre-CY extension. It then shows that the cohomology $H^\bullet(O\mathcal{RG}raphs_{d+1}(m,n))$ matches the compactly supported moduli-space cohomology $\prod_g H_c^{\bullet-m+d(2g-2+m+n)}(\mathcal{M}_{g,m+n})$, thereby giving a combinatorial model for gravity-type cohomology. The work also provides a morphism from the minimal resolution $\mathcal{H}\!olie_d$ into $O\mathcal{RG}raphs_{d+1}$, yielding an $\mathcal{H}\!olie_d$-algebra structure on higher Hochschild cohomology, and demonstrates a natural gravity operad action on higher Hochschild cohomology of pre-CY algebras. An interpolation framework between undirected and oriented ribbon-graph formalisms clarifies the functorial and geometric content, linking gravity-type algebraic structures to the topology of moduli spaces. Overall, the results elucidate how pre-CY data encode rich operadic and properadic actions that reflect the geometry of moduli spaces and gravity-type operads.
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 $(A, π)$.