12 noon, Seminar Hall
Toward a new way of computing Gromov-Witten invariants
University of Michigan, Ann Arbor, USA.
This will be an introductory talk about the "A-model Landau-Ginzburg/Calabi-Yau correspondence." This is a (largely conjectural) correspondence between the Gromov-Witten theory of a Calabi-Yau manifold (or orbifold) and a new theory of Fan, Jarvis, and Ruan based on ideas by Witten. Roughly, the Gromov-Witten theory of a complex manifold X counts the number of holomorphic curves of a given degree and genus in X, satisfying some specified conditions. They are expressed as integrals over moduli spaces of stable maps. For example, the number 609250 of degree 2 curves on a quintic threefold, calculated by Kontsevich in the early 90s (and previously predicted by physicists), is a Gromov-Witten invariant. Fan-Jarvis-Ruan-Witten invariants are more subtle, and are expressed as integrals over covers of moduli spaces of stable curves. Since moduli spaces of curves are more well-understood than spaces of stable maps, these integrals should in theory be easier to compute directly than Gromov-Witten invariants. There is a wide conjecture with origins in physics that the two theories should be recoverable from each other. This was proven by Chiodo and Ruan for the special case of the quintic threefold in 4-dimensional projective space, and some other examples have since been worked out. If I have time I will discuss basic computational evidence toward a generalization to a nonabelian orbifold.