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\(N=1\) mirror symmetry and open/closed string duality. (English) Zbl 1022.81046
Summary: We show that the exact \({\mathcal N}= 1\) superpotential of a class of four-dimensional string compactifications is computed by the closed topological string compactified to two dimensions. A relation to the open topological string is used to define a special geometry for \({\mathcal N}= 1\) mirror symmetry. Flat coordinates, an \({\mathcal N}= 1\) mirror map for chiral multiplets and the exact instanton corrected superpotential are obtained from the periods of a system of differential equations. The result points to a new class of open/closed string dualities which map individual string world-sheets with boundary to ones without. It predicts an mathematically unexpected coincidence of the closed string Gromov-Witten invariants of one Calabi-Yau geometry with the open string invariants of the dual Calabi-Yau.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30 String and superstring theories in gravitational theory
32Q25 Calabi-Yau theory (complex-analytic aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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