Properties of Gromov-Witten Invariants

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AbstractAward: DMS-0306299Principal Investigator: Eleny-Nicoleta IonelThis proposal aims to understand the structure of theGromov-Witten invariants of symplectic manifolds by combiningtogether ideas coming from different fields of research. Thefirst project is motivated by a conjecture made by twophysicists, Gopakumar and Vafa. The conjecture implies severalsurprising restrictions on the Gromov invariants of a Calabi-Yau3-fold. The goal is to prove the conjectured formula by adaptingsome analytical techiques developed by Taubes to relate theSeiberg-Witten and Gromov invariants in 4-dimensions. The secondproject's goal is to obtain new relations in the cohomology ringof the moduli space of complex structures on a marked Riemannsurface. Using the techniques introduced in a previous paper, thePI found several families of interesting relations, one of whichproves a ten year old conjecture of Faber. The last project seeksto extend the sum formula for Gromov-Witten invariants todeformations more general then those appearing from a symplecticsum. There already seems to be several interesting new phenomenaappearing in the general case.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for a unifying theory of the universe, which extendsEistein relativity theory. It is based on the idea thatelementary particles (like electrons, photons) should be thoughtnot as points, but rather small vibrating loops. Working out thedetails of this theory turned out to be quite delicate, and hasin turn inspired many remarkable results in mathematics. But alsofundamental results in mathematics have inspired many newdiscoveries in physics. It is hoped that this project willcontribute to the increased interaction between mathematics andhigh energy physics. In the same time, one of the projects willinvolve graduate students.