Principal Investigator: Kefeng Liu
By using localization methods we plan to intensively study various conjectures from string theory, in particular those arised from string duality. These include the most general mirror principle for higher genus from functorial localization, the Gopakumar-Vafa integrality conjecture, the relationship between elliptic genus and Gromov-Witten invariants, both from equivariant index theory. We have made significant progresses towards our goals, for which the functorial localization formula we discovered during our proof of the mirror principle is one of the key techniques.
The interactions of string theory and mathematics has been one of the very few most active and most exciting fields in mathematics for the past twenty years. String duality has created many amazingly beautiful mathematical conjectures. Our localization methods have been very successful in proving them, inlcuding the mirror conjecture, the Marino-Vafa conjecture, the Hori-Vafa conjecture and the mathematical construction of the topological vertex. Such interactions not only produce mathematical theories and results, but also give strong evidences to the physical theories.
