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

TL;DR

This paper explores the algebraic structures acting on periodic cyclic homology of dg algebras, connecting operadic actions with symplectic topology and Gromov-Witten theory, and introduces new operad reformulations.

Contribution

It reformulates the Kontsevich-Soibelman operad using a two-colored cacti operad and proves its equivariant quasi-equivalence to little disks, with applications to symplectic topology.

Findings

Explicit generators for equivariant operations computed

Operad reformulation and quasi-equivalence established

Applications to Fukaya categories and Lagrangian obstructions

Abstract

We study equivariant operations on the periodic cyclic homology of a dg algebra that arise from the chain level action of the two-colored Kontsevich-Soibelman operad. Using classical computations of Cohen [Coh], we explicitly compute a set of generators for these operations under composition, and show that they agree with the p-fold equivariant cap products previously studied by the author [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 equivariantly quasi-equivalent to the two-colored operad of little disks on a disk/cylinder. We give applications of the main results to symplectic topology, and more specifically, arithmetic aspects of Fukaya category and classical obstructions to realizing a middle cohomology class of a symplectic manifold by Lagrangian submanifold