Mathematical Aspects of String Duality

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AbstractAward: DMS-0405117Principal Investigator: Kefeng LiuBy using localization methods we plan to intensively studyvarious conjectures from string theory, in particular thosearised from string duality. These include the most general mirrorprinciple for higher genus from functorial localization, theGopakumar-Vafa integrality conjecture, the relationship betweenelliptic genus and Gromov-Witten invariants, both fromequivariant index theory. We have made significant progressestowards our goals, for which the functorial localization formulawe discovered during our proof of the mirror principle is one ofthe key techniques.The interactions of string theory and mathematics has been one ofthe very few most active and most exciting fields in mathematicsfor the past twenty years. String duality has created manyamazingly beautiful mathematical conjectures. Our localizationmethods have been very successful in proving them, inlcuding themirror conjecture, the Marino-Vafa conjecture, the Hori-Vafaconjecture and the mathematical construction of the topologicalvertex. Such interactions not only produce mathematical theoriesand results, but also give strong evidences to the physicaltheories.