Speaker: Bertrand Eynard (IPHT CEA Saclay)
Title: Topological recursion for Gromov-Witten invariants, a proof of the BKMP “remodeling” conjecture
Abstract: BKMP (Bouchard Klemm Marino Pasquetti) in 2008, proposed that open and closed Gromov-Witten invariants of toric Calabi-Yau 3-folds satisfy the same recursion relation as the large size expansion of matrix integrals, called “topological recursion”. This conjecture is an example of Mirror symmetry to all genus. It is useful because it allows an explicit computation of Gromov-Witten invariants of given genus. They checked this conjecture in many examples for low genus. The conjecture was proved to all genus by Chen and Zhou in 2010, only for the simplest Calabi-Yau 3-fold C^3. In 2012, we (Eynard and Orantin) have proved the general case, using localization, and using properties of the recursion. In this talk, we shall introduce the notion of Gromov-Witten invariants on a simple example (the local P^2 3-fold), and we shall show how the proof works for that case. We shall use that the localization formula expresses Gromov-Witten invariants as sums of weighted graphs (A-model side). On the other hand, the recursion (B-model side) is naturally encoded in terms of weighted graphs. On both sides, the graphs contain vertices labeled by a genus and a valence (g,n), however, the graphs and weights on the 2 sides are different. In particular, on the B-side, there is no graph with vertices of type (0,1) or (0,2). A standard method in graphs combinatorics allows to rewrite sums of weighted graphs with 1-valent and/or 2-valent vertices in terms of other graphs without those vertices. After doing this combinatorial resummation, it remains to compare the weights on both sides. Using the “special geometry” property of the recursion (similar to Seiberg-Witten equations), we show that the weights on both side satisfy the same differential equation with respect to the Kahler radius. And in the large radius limit, the weights on the two sides coincide because the toric graph of the Calabi-Yau 3-fold, is the tropical limit of its mirror. This completes the proof.