Geometry/topology Seminar
Tuesday, January 20, 2009, 4:30pm, 119 Physics
Aleksey Zinger (Stony Brook)
The Geometry of Genus-One Gromov-Witten Invariants and Mirror Symmetry

Abstract:

In 1991, Candelas, de la Ossa, Green, and Parkes used physical principles to give an amazing formula for counts of genus 0 curves in a quintic hypersurface in P4, even before these counts were defined mathematically. Two years later, Bershadsky, Cecotti, Ooguri, and Vafa followed up with a theory determining such counts in all genera, modulo a few low-degree terms; for the quintic, these were determined in genus 1 by BCOV then and up to genus 51 by Huang, Klemm, and Quackenbush recently. Mathematically, these formulas became known as mirror symmetry predictions for Gromov-Witten invariants. The genus 0 case was confirmed mathematically in several different ways in the mid-late 90s, with all but one of the approaches using a standard relation between the genus 0 GW-invariants of the quintic and of P4. In this talk, I will describe geometric properties of genus 1 GW-invariants that mimic the properties of genus 0 invariants, including a long elusive relation between genus 1 GW-invariants of a hypersurface and of the ambient projective space. These properties have led to the confirmation of the genus 1 BCOV prediction for the quintic and should extend to higher genera.

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