Unifying Mirror Symmetry

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AbstractAward: DMS-0072504Principal Investigator: Eric ZaslowZaslow proposes research directed towards a unified understandingof mirror symmetry. Current ideas -- including ``classical''mirror symmetry; Kontsevich's conjecture; the work of Vafa; andthe conjecture of Strominger, Yau, and Zaslow -- are only looselyconnected and involve both perturbative and non-perturbativestring reasoning. An understanding of how non-perturbativestring theory relates to the classical mirror symmetry pictureand Gromov-Witten invariants will be an important step towardsunification. Five projects are proposed towards achieving thisgoal: 1) Developing a mathematical formulation of the newinvariants obtained by Gopakumar and Vafa from BPS statecounting. 2) Understanding the multiple-cover formulas whichyield integers from higher-genus Gromov-Witten invariants. 3)Resolving the holomorphic ambiguity by determining it from thestructure of singularities of Calabi-Yau moduli space. 4)Developing a physical understanding of Kontsevich's enlargementof Calabi-Yau moduli space to include A-infinity structures. 5)Defining a geometric Fourier-Mukai-like functor relatingspecial-Lagrangian cycles of one manifold to Hermitian-Yang-Millsconnections on bundles over the mirror. This functor could leadto a geometric proof of Kontsevich's formulation of mirrorsymmetry. All these projects aim to unify our still disparateapproaches to mirror symmetry.This project is directed towards unifying our mathematical andphysical understanding of the phenomenon of mirror symmetry,discovered by theoretical physicists working in string theory.String theory is a proposed physical theory with the promise ofincorporating Einstein's understanding of space and gravity intothe quantum theory. Mirror symmetry is a duality symmetry instring theory, whereby two very different physical theories areactually equivalent. When one of the theories is easilycomputable and the other hard, this leads to predictions ofanswers to difficult calculations. From the mathematical pointof view, this can lead to conjectures relating parallelstructures -- the structures involved in describing thedifferent, but equivalent, mathematical models of physicaltheories. The relationships unearthed by mirror symmetry aredeep and novel. A unified understanding of them may join notonly fields of research but different disciplines -- math andphysics -- as well. This award is partially funded by theprogram in Mathematical Physics of the Division of Physics.