Joint IAS/Princeton University Symplectic Geometry Seminar

More than twenty years ago, Witten proposed a remarkable conjecture connecting the geometry of moduli space of curve or Gromov-Witten theory of point to KdV integrable hierarchies (solved by Kontsevich). Since then, Witten's conjecture opened up a rather mysterious area of geometry and physics which can be described as "geometry/integrable hierarchies mirror symmetry". So far we only understand a few examples of this type of mirror symmetry such as \(P^1\) via 2-Toda (solved by Okounkov-Pandharipande). Those examples provided some of deepest theorems in Gromov-Witten theory. Given a geometry, there is no receipt to associate an integrable hierarchy, A more interesting question is to locate the geometry associated with some of classical integrable hierarchies. A prominent family of examples are Drinfeld-Sokolov-hiearchies which are indexed by affine Dykin diagram. At the same time for his conjecture of KdV hierarchy, Witten generalized his conjecture to ADE Drinfel-Sokolov hierachies. His generalization includes a proposal for the corresponding geometry and a conjecture that the new geometries is governed by ADE Drinfeld-Sokolov hierarchies. During last twenty years, a great deal of effort were spent to construct these geometries rigorously and now Witten's ADE conjecture is a theorem. In this talk, we extend Witten ADE conjecture to \(B_n, C_n, F_4,G_2\) series. This is a joint work with Si-Qi Liu and Youjin Zhang of Tsinghua University.