23 research outputs found
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 -algebras. We give an account of homology theory for manifolds
(and spaces), which give invariant of manifolds but also invariant of
-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