Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry

Abstract

We carry out the SYZ program for the local Calabi--Yau manifolds of type $\tilde{A}$ by developing an equivariant SYZ theory for the toric Calabi--Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov--Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood--Iqbal--Vafa.

v3: 43 pages, 12 figures, improved exposition

References in corpus (5)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
• Topological Strings and (Almost) Modular Forms
• LG/CY Correspondence for Elliptic Orbifold Curves via Modularity
• Local mirror symmetry in the tropics

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