In this paper, we describe a surprising link between the theory of the GoldmanTuraev Lie bialgebra on surfaces of genus zero and the Kashiwara-Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman-Turaev Lie bialgebra is defined by the Goldman bracket {−, −} and Turaev cobracket δ on the K-span of homotopy classes of free loops on Σ.Applying an expansion θ : Kπ → K x 1 , . . . , x n yields an algebraic description of the operations {−, −} and δ in terms of non-commutative variables x 1 , . . . , x n . If Σ is a surface of genus g = 0 the lowest degree parts {−, −} −1 and δ −1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that {−, −} = {−, −} −1 and δ = δ −1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral.In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).

Abstract. We define a family KV (g,n+1) of Kashiwara-Vergne problems associated with compact connected oriented 2-manifolds of genus g with n + 1 boundary components. The problem KV (0,3) is the classical Kashiwara-Vergne problem from Lie theory. We show the existence of solutions of KV (g,n+1) for arbitrary g and n. The key point is the solution of KV(1,1) based on the results by B. Enriquez on elliptic associators. Our construction is motivated by applications to the formality problem for the Goldman-Turaev Lie bialgebra g (g,n+1) . In more detail, we show that every solution of KV (g,n+1) induces a Lie bialgebra isomorphism between g (g,n+1) and its associated graded gr g (g,n+1) . For g = 0, a similar result was obtained by G. Massuyeau using the Kontsevich integral.This paper is a summary of our results. Details and proofs will appear elsewhere.

Given a closed manifold M. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincaré duality model of the manifold, and involves the configuration space of two points on M. We moreover, construct an I B L ∞ -structure on (a model of) cyclic chains on the cochain algebra of M, such that the natural comparison map to the S 1 -equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that M carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of M beyond its homotopy type. More precisely, the action of Diff(M) does not (in general) factor through aut(M). CONTENTS 1. Introduction 1.1. Statement of results 2.2. Operads, Com ∞ -and Com ∞ -algebras 2.3. Hochschild and cyclic complex 2.4. Pullback-pushout lemma 2.5. Fulton-MacPherson-Axelrod-Singer compactification of configuration spaces 2.6. The Lambrechts-Stanley model of configuration space 2.7. Graph complex models for configuration spaces 2.8. Graph complex (Lie algebra) GC H and GC M , following [9] 3. String topology operations 3.1. Preliminaries 3.2. String product (Cohomology coproduct) 3.3. String coproduct (cohomology product) 3.4. String coproduct (χ(M) = 0 case) 3.5. Definition of string bracket and cobracket 4. Cochain complex models 4.1. Iterated integrals and model for path spaces 4.2. Splitting map 4.3. Connes differential 4.4. Space of figure eights 4.5. S 1 -equivariant loops 5. Cochain zigzags for the string product and bracket 5.1. Product 5.2. Coproduct (relative to constant loops case) 5.3. Coproduct (χ(M) = 0 case) 6. Simply-connected case, and proofs of Theorems 1.2 and 1.3, and the first part of Theorem 1.5 7. Involutive homotopy Lie bialgebra structure on cyclic words 7.1. Lie bialgebra of cyclic words 7.2. Twisting by Maurer-Cartan elements 1

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In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a quadratic Lie algebra. We prove a uniqueness statement about these Poisson brackets with a given moment map. As an application we get formulae for the linearization of the quasi-Poisson structure of the moduli space of flat connections on a punctured sphere, and thereby identify their symplectic leaves with the reduction of coadjoint orbits. Equivalently, we get linearizations for the Goldman double Poisson bracket, our definition of Poisson brackets coincides with that of Van Den Bergh [2] in this case. This can furthermore be interpreted as giving a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces.