# Mirror symmetry for Gromov-Witten invariants of a quintic threefold

# Mirror symmetry for Gromov-Witten invariants of a quintic threefold

The mirror symmetry principle of string theory provides closed formulas for GW-invariants, with special attention devoted to a quintic threefold, $Q3$. The genus $0$ mirror prediction for $Q3$ was verified 12 years ago by using the Atiyah-Bott localization theorem. In this talk, I will outline how the analoguos genus 1 localization problem is solved by making use of a number of its relations with the genus $0$ localization problem. This approach confirms the 1993 BCOV mirror symmetry prediction for genus $1$ GW-invariants of $Q3$. It also produces mirror formulas for genus $1$ GW-invariants of a degree $n$ hypersurface in $P^{n-1} (Q3$ is $n=5)$, confirming a recent prediction of Klemm-Pandharipande for a sextic fourfold ($n=6$) and producing a puzzling combinatorial identity related to unbranched covers of tori ($n=3$).