The minimal model for the Batalin–Vilkovisky operad

Abstract

The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov-Kontsevich and Manin which states that the underlying homology groups of some Batalin-Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.

Homotopically trivializing the circle in the framed little disks

“…Different versions of this statement have been discussed by Costello as well as Kontsevich, and a rough outline of a partial argument is present in [15], although there is no indication of the homotopy invariance there. A cognate of this theorem was proved at the level of rational homology in [4,5], relying on earlier work of Getzler [7,8]. This theorem is a nice improvement; using adjunctions along the lines of those in [18], one can see that the topological story implies the same result at the level of integral, not just rational, homology.…”

Homotopy derivations

“…We recall the construction of [14] and [2] and briefly indicate changes needed for our purposes. We closely follow [2], where we wish to 1. specialize from many to single color, 2. generalize from 0 to arbitrary degree of the derivation, 3. generalize from non-Σ (a.k.a.…”

“…As an aside, recall that this ambiguity can be essentially removed (in many cases) by requiring the resolution to be minimal, since then it is unique up to an isomorphism by Proposition 3.7 of [3]. The Koszul resolutions are examples of minimal models.…”

Givental action and trivialisation of circle action

“…The operad HyperCom is well studied; for instance, it is proved to be Koszul in [Get95]. For our purposes, it is important that its internal structure can be described in terms of the intersection theory on the moduli spaces of curves of genus 0 and that it is a crucial ingredient in the homotopy theory of Batalin-Vilkovisky algebras [Man99,DV13,KMS13].…”