Hyper-commutative algebras and cyclic cohomology

TL;DR

This paper develops a chain model for the Deligne-Mumford operad, establishing homotopy hyper-commutative algebra structures on Hochschild and cyclic cochains, and interprets gravity brackets as spectral sequence obstructions.

Contribution

It introduces a new chain model for the Deligne-Mumford operad and connects it to homotopy hyper-commutative structures on cyclic cohomology.

Findings

Constructed a chain model for the Deligne-Mumford operad.

Lifted the framed little disks action to the new chain model.

Interpreted gravity brackets as spectral sequence obstructions.

Abstract

This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our results are given in the language of deformation complexes of cyclic operads.