On dg properads of pre-CY algebras

We present new small models for the dg properads that govern pre-CY algebras: one for general pre-CY algebras and one for pre-CY algebras without curvature terms. These small models have only four and, respectively, three generators with valencies $\le 4$ (in contrast to the original properads which have infinitely many generators with arbitrarily large valencies). We prove that the valency upper bound $4$ is sharp. We study the cohomology group of the deformation complex of the dg properad governing general pre-CY algebras and prove that it contains the totality ${\prod }_{g\ge 1}{H}^{\text{bu}}({\text{M}}_{g,1})$ of cohomology groups of moduli spaces $M_{g,1}$ of genus $g$ algebraic curves with one marked point. As an application of this result we prove that the Tradler-Zeinalian properad of $V^{(d)}$-algebras is not Koszul.