Gravity formality

Abstract

We show that Willwacher's cyclic formality theorem can be extended to preserve natural Gravity operations on cyclic multivector fields and cyclic multidifferential operators. We express this in terms of a homotopy Gravity quasi-isomorphism with explicit local formulas. For this, we develop operadic tools related to mixed complexes and cyclic homology and prove that the operad M of natural operations on cyclic operators is formal and hence quasi-isomorphic to the Gravity operad.

Gravity algebra structure on the negative cyclic homology of Calabi–Yau algebras

“…In fact, in [37] Ward showed that the mixed cyclic cochain complex of symmetric Frobenius algebras is an algebra over the (chain) gravity operad, hence Theorem 1.1 is a direct corollary of his result in this case. In the subsequent paper [3] Campos and Ward proved a more general result saying that any mixed complex over the (chain) gravity operad induces the gravity algebra structure on its negative cyclic homology (see loc cit Corollary 1.33 and Example 1.34).…”

Six operations formalism for generalized operads

“…Both of these Feynman categories have symmetrizations given by proper maps, so we may apply Subsection 9.2 to conclude that the deformation complex of V coincides with the deformation complex of its double V ⊕ V * viewed as an algebra over a cyclic operad. It was shown in [War16] that any such deformation complex of planar cyclic operads carries an action of such a chain model for the gravity operad, which was shown to be formal in [CW18] hence the claim.…”

Intertwining for semidirect product operads

“…For evidence we recall that Conjecture 5.4 was based on an analogy with the algebraic setting, and we may update this analogy to include the recent results Theorem 1.1 of [KW17] and Theorem 2.8 of [CW18], which show formality can be made equivariant in the algebraic setting when n = 2.…”