Algebraic string bracket as a Poisson bracket

14 PagesInternational audienceIn this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold

The Hodge Chern character of holomorphic connections as a map of simplicial presheaves

We define a map of simplicial presheaves, the Chern character, that assignsto every sequence of composable non connection preserving isomorphisms ofvector bundles with holomorphic connections an appropriate sequence ofholomorphic forms. We apply this Chern character map to the Cech nerve of agood cover of a complex manifold and assemble the data by passing to thetotalization to obtain a map of simplicial sets. In simplicial degree 0, thismap gives a formula for the Chern character of a bundle in terms of theclutching functions. In simplicial degree 1, this map gives a formula for theChern character of bundle maps. In each simplicial degree beyond 1, theseinvariants, defined in terms of the transition functions, govern thecompatibilities between the invariants assigned in previous simplicial degrees.In addition to this, we also apply this Chern character to complex Liegroupoids to obtain invariants of bundles on them in terms of the simplicialdata. For group actions, these invariants land in suitable complexescalculating various Hodge equivariant cohomologies. In contrast, the de RhamChern character formula involves additional terms and will appear in a sequelpaper. In a sense, these constructions build on a point of view of"characteristic classes in terms of transition functions" advocated by RaoulBott, which has been addressed over the years in various forms and degrees,concerning the existence of formulae for the Hodge and de Rham characteristicclasses of bundles solely in terms of their clutching functions.<br

Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry

We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.Comment: v2: slightly expanded expositio

Formality of the chain operad of framed little disks

We extend Tamarkin's formality of the little disk operad to the framed little disk operad.Comment: 5 page

Notes on factorization algebras, factorization homology and applications

These notes are an expanded version of two series of lectures given at the winter school in mathematical physics at les Houches and at the Vietnamese Institute for Mathematical Sciences. They are an introduction to factorization algebras, factorization homology and some of their applications, notably for studying $E_n$-algebras. We give an account of homology theory for manifolds (and spaces), which give invariant of manifolds but also invariant of $E_n$-algebras. We particularly emphasize the point of view of factorization algebras (a structure originating from quantum field theory) which plays, with respect to homology theory for manifolds, the role of sheaves with respect to singular cohomology. We mention some applications to the study of mapping spaces and study several examples, including some over stratified spaces.Comment: 122 pages. A few examples adde