Hypercommutative algebras and cyclic cohomology (Q2408341)

The \textit{hypercommutative operad} is defined by using the homology of genus \(0\) moduli spaces of surfaces with boundary; hypercommutative algebras are known to be equivalent to Batalin-Vilkovisky algebras enhanced with a homotopy trivialization of the \(\Delta\)-operator (see for example [\textit{A. Khoroshkin} et al., Commun. Math. Phys. 322, No. 3, 697--729 (2013; Zbl 1281.55011)] about this topic). For the notion of \textit{cyclic operad} which appears in the statement below, the reader can look at [\textit{E. Getzler} and \textit{M. M. Kapranov}, in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. 167--201 (1995; Zbl 0883.18013)]. The associated complexes are constructed in [\textit{B. C. Ward}, J. Noncommut. Geom. 10, No. 4, 1403--1464 (2016; Zbl 1375.18056)]; one can also see the reminders of Section 3 of the article under review. From the author's introduction, the main result of the paper is the following: Theorem. Let \(\mu : \mathcal{A}_\infty\to\mathcal{O}\) be a map of cyclic operads. Let \(\mathrm{CH}^*(\mathcal{O},\mu)\) and \(C^*_\lambda(\mathcal{O},\mu)\) be the associated deformation and cyclic deformation complexes of \(\mu\). If the morphism \(\mu\) is cyclically degenerate (Definition 3.4), then the homotopy BV algebra structure on \(\mathrm{CH}^*(\mathcal{O},\mu)\) lifts to a compatible hypercommutative algebra. Moreover, \(C^*_\lambda(\mathcal{O},\mu)\) carries the structure of a \(\mathcal{H}y\mathcal{C}om_\infty\)-algebra for which the inclusion \(C^*_\lambda(\mathcal{O},\mu)\to \mathrm{CH}^*(\mathcal{O},\mu)\) extends to an \(\infty\)-morphism of \(\mathcal{H}y\mathcal{C}om_\infty\)-algebras.