The Getzler-Gauss-Manin connection and Kontsevich-Soibelman operations on the periodic cyclic homology

Abstract

We study equivariant operations on the periodic cyclic homology of dg algebras that arise from the chain level action of the two-colored Kontsevich-Soibelman operad. The first main result is that these operations are covariantly constant with respect to the Getzler-Gauss-Manin connection on the periodic cyclic homology of a family of dg algebras. Then, using classical computations of Cohen \cite{Coh}, we explicitly compute a set of generators for these operations under composition, and show that these generators are closely related to the $p$-fold equivariant cap products previously studied by the author \cite{Che2} in relation to equivariant Gromov-Witten theory with mod $p$ coefficients. The main technical novelty is a re-formulation of the Kontsevich-Soibelman operad in terms of a two-colored version of the cacti operad, and a proof that it is \emph{equivariantly} quasi-equivalent to the two-colored operad of little disks on a disk/cylinder.

v3: Added clarification for Theorem 1.4. Fixed typos and minor mistakes