Gromov-Witten invariants and mirror symmetry

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AbstractAward: DMS-1105871Principal Investigator: Mark GrossThe principal investigator plans to study the geometry of mirrorsymmetry for Calabi-Yau manifolds. This will be done from theperspective of a program developed jointly with B. Siebert, which"tropicalizes" Calabi-Yau manifolds, using an algebro-geometricversion of the Strominger-Yau-Zaslow conjecture. In particular,the principal investigator proposes to resolve a number of theremaining key steps of this program; these steps should be thecomponent pieces of a complete conceptual proof of mirrorsymmetry at genus zero. These steps include a general gluingformula for the theory of logarithmic Gromov-Witten invariantsand a generalization of the tropical vertex of the PI,Pandharipande and Siebert to all dimensions. In addition, theprogram with Siebert has led to the construction of canonicaltheta functions for lines bundles on (degenerating families) ofCalabi-Yau manifolds. The existence of such theta functions isexpected to have various applications, such as geometriccompactifications of the moduli space of K3 surfaces (beingpursued with P. Hacking and S. Keel) and a tropical approach tohomological mirror symmetry (being pursued with Siebert andM. Abouzaid).The work described in this proposal lies at the intersection ofstring theory and geometry. String theory replaces thetraditional notion of the point particle with a small loop ofstring, moving through space-time. To make string theorycompatible with quantum mechanics, space-time must beten-dimensional. Since space-time appears four-dimensional, oneexpects six of these dimensions to be a very small "curled up"geometric object. These geometric objects are called Calabi-Yaumanifolds. In the early 1990s, string theorists proposed aremarkable association between completely different Calabi-Yaumanifolds: certain calculations extremely difficult to perform onone Calabi-Yau manifold could be completed by performingcompletely different, and much easier, calculations on adifferent Calabi-Yau manifold. This discovery was known as mirrorsymmetry. Since this time, many geometers have been trying tounderstand the mathematics behind this miraculousobservation. New insights and explanations for this phenomenonare developing rapidly, and these insights are leading to newapproaches to old problems in algebraic geometry. The proposedwork aims to both further understanding of mirror symmetry andapply this understanding to produce new developments in algebraicgeometry.