Mirror maps equal SYZ maps for toric Calabi-Yau surfaces

Abstract

We prove that the mirror map is the SYZ map for every toric Calabi-Yau surface. As a consequence one obtains an enumerative meaning of the mirror map. This involves computing genus-zero open Gromov-Witten invariants, which is done by relating them with closed Gromov-Witten invariants via compactification and using an earlier computation by Bryan-Leung.

v2: final version, typos and citations corrected, published in Bull. London Math. Soc. (2011) doi: 10.1112/blms/bdr090

References in corpus (3)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• A relation for Gromov-Witten invariants of local Calabi-Yau threefolds

Cited by in corpus (11)

• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• Open Gromov-Witten invariants and SYZ under local conifold transitions
• Homological mirror symmetry for $A_n$-resolutions as a $T$-duality
• Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry
• Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces
• Geometric transitions and SYZ mirror symmetry
• Twin Lagrangian fibrations in mirror symmetry
• The Strominger-Yau-Zaslow conjecture and its impact
• Affine elliptic surfaces with type-A singularities and orbi-conifolds