Gaëtan Borot (MPI): Rational 2d CFTs, CohFTs and topological recursion

vendredi 14 oct. 2016, 14:00 → 15:00 Europe/Paris

001 (batiment I)

In quantum field theories, sewing (and its reciprocal operation, factorization) axioms play a prominent role. For surfaces, factorization typically allows reducing the understanding of the theory on a surface of genus g with n boundaries/marked points, to simpler surfaces like disks, cylinders, pairs of pants. The complexity of surfaces is measured by the Euler characteristic (up to a sign) 2g - 2 + n. By "topological recursion", we mean an induction on 2g - 2 + n. I will describe 3 axiomatic structures where such factorization property are included in some sense, and what they have to do with one another, each time going one step up in geometry (but one step down in generality) First, the topological recursion (TR), where the quantities under study is a collection of numbers indexed by (g,n), and some labels for each marked point. The recursion itself allows to calculate the numbers for any (g,n) starting from initial data (g.n) = (0,1) and (0,2). This recursion is universal, and choosing various initial data leads to the solution of various problems in enumerative geometry, like 2d TQFT amplitudes (Verlinde formula) or volumes of Deligne-Mumford moduli spaces. Second, cohomological field theories (CohFT), which were invented by Kontsevich and Manin to capture the properties of Gromov-Witten invariants. The quantity under study here is a sequence of cohomology classes on \bar{M}_{g,n} with values in a 2d TQFT. A remarkable result following from Givental and Teleman classifies all semi-simple CohFT. As a consequence, the correlation functions of such CohFTs are computed by TR for (explicit) initial data. Third, 2d rational conformal field theories, whose mathematical definition here will be the data of a modular functor. The quantities under studies are the assignment for each surface (with some decorations) of a vector space, together with action of mapping class groups and compatible with factorization. I will show how, out of a 2d rational CFT, one can construct a bundle over \bar{M}_{g,n} (the bundle of conformal blocks), and that its Chern character provides a semi-simple CohFT. In particular, its intersection indices are computed by TR for an initial data I will describe. In the case of Wess-Zumino-Witten models, it hints at a family index interpretation of the topological recursion computation.