Unifying Mirror Symmetry

Northwestern University, Evanston IL

Investigators

Abstract

Abstract Award: DMS-0072504 Principal Investigator: Eric Zaslow Zaslow proposes research directed towards a unified understanding of mirror symmetry. Current ideas -- including ``classical'' mirror symmetry; Kontsevich's conjecture; the work of Vafa; and the conjecture of Strominger, Yau, and Zaslow -- are only loosely connected and involve both perturbative and non-perturbative string reasoning. An understanding of how non-perturbative string theory relates to the classical mirror symmetry picture and Gromov-Witten invariants will be an important step towards unification. Five projects are proposed towards achieving this goal: 1) Developing a mathematical formulation of the new invariants obtained by Gopakumar and Vafa from BPS state counting. 2) Understanding the multiple-cover formulas which yield integers from higher-genus Gromov-Witten invariants. 3) Resolving the holomorphic ambiguity by determining it from the structure of singularities of Calabi-Yau moduli space. 4) Developing a physical understanding of Kontsevich's enlargement of Calabi-Yau moduli space to include A-infinity structures. 5) Defining a geometric Fourier-Mukai-like functor relating special-Lagrangian cycles of one manifold to Hermitian-Yang-Mills connections on bundles over the mirror. This functor could lead to a geometric proof of Kontsevich's formulation of mirror symmetry. All these projects aim to unify our still disparate approaches to mirror symmetry. This project is directed towards unifying our mathematical and physical 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 of incorporating Einstein's understanding of space and gravity into the quantum theory. Mirror symmetry is a duality symmetry in string theory, whereby two very different physical theories are actually equivalent. When one of the theories is easily computable and the other hard, this leads to predictions of answers to difficult calculations. From the mathematical point of view, this can lead to conjectures relating parallel structures -- the structures involved in describing the different, but equivalent, mathematical models of physical theories. The relationships unearthed by mirror symmetry are deep and novel. A unified understanding of them may join not only fields of research but different disciplines -- math and physics -- as well. This award is partially funded by the program in Mathematical Physics of the Division of Physics.