| 研究概要 |
The main goal of my proposal was to find W-constraints for the total descendant potential of simple and simple elliptic singularities. I completed successfully the case of simple singularities, which in some sense was the preliminary case. The next case however is much more subtle and I still do not have an answer. Nevertheless, in FY 2012 I came up with a new idea, which I had managed to develop to some extend. Namely I proved that the total ancestor potential is characterized by the local Eynard-Orantin recursion. The total ancestor potential is a certain deformation of the total descendant potential depending on a set of deformation parameters. The local recursion applies when the deformation parameters are generic. On the other hand, the total descendant potential corresponds to the most degenerate value of the deformation parameters. In FY 2013 I managed to prove that the local Eynard-Orantin recursion extends for a generic degenerate value. Moreover, the extension seems to be governed by the W-algebra of type A2 . In particular, I obtained a proof of 10 years old conjecture of Givental about the analyticity of the total ancestor potential as a function on the deformation parameters. In the 2nd half of FY 2013, I understood that the vertex operators for simple elliptic singularities can be related via Iritani’s modification of the Chern character map with certain K-theory vector bundles. This simplifies to a great extend the analysis of the monodromy representation. I believe that all these ideas should help me to reach my original goal.
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