A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry

Details A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry

2010-06-19

arxiv.org/abs/1006.3827v3

Pacific J. Math. 254 (2011), no. 2, 275-293

Mathematics

Symplectic Geometry

v3: many minor changes, published in Pacific J. Math.; v2: 16 pages. Completely rewritten and improved

Scientific paper

We prove that open Gromov-Witten invariants for semi-Fano toric manifolds of the form $X=\mathbb{P}(K_Y\oplus\mathcal{O}_Y)$, where $Y$ is a toric Fano manifold, are equal to certain 1-pointed closed Gromov-Witten invariants of $X$. As applications, we compute the mirror superpotentials for these manifolds. In particular, this gives a simple proof for the formula of the mirror superpotential for the Hirzebruch surface $\mathbb{F}_2$.

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