The minimal model for the Batalin–Vilkovisky operad
نویسندگان
چکیده
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.
منابع مشابه
Operadic Formulation of Topological Vertex Algebras and Gerstenhaber or Batalin-vilkovisky Algebras
We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak...
متن کاملFramed Discs Operads and Batalin–vilkovisky Algebras
The framed n-discs operad f Dn is studied as semidirect product of SO(n) and the little n-discs operad. Our equivariant recognition principle says that a grouplike space acted on by f Dn is equivalent to the n-fold loop space on an SO(n)-space. Examples of f D2-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f Dn , which produces higher Batalin–Vilko...
متن کاملHigher String Topology on General Spaces
In [2], Chas and Sullivan considered the free loop space LM = Maps(S,M) for a smooth orientable compact manifold M . They used geometric methods to show that H∗LM has, among other things, the structure of a Batalin-Vilkovisky algebra. More precisely, C∗LM has an action by the operad of chain complexes C∗D2, where D2 is the framed 2-dimensional little disk operad. Each element of D2(n) is an ord...
متن کاملBatalin-vilkovisky Coalgebra of String Topology
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gr...
متن کاملOn Operad Structures of Moduli Spaces and String Theory
We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012