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D-branes on \(\mathbb C^3_6\). I: Prepotential and GW-invariants. (English) Zbl 1260.53139

Summary: This is the first of a set of papers having the aim to provide a detailed description of brane configurations on a family of non-compact three-dimensional Calabi-Yau manifolds. The starting point is the singular manifold defined by a given quotient \(\mathbb C^3/\mathbb Z_6\), which we called simply \(\mathbb C^3_6\) and which admits five distinct crepant resolutions. Here we apply local mirror symmetry to partially determine the prepotential encoding the \(GW\)-invariants of the resolved varieties. It results that such prepotential provides all numbers but the ones corresponding to curves having null intersection with the compact divisor. This is realized by means of a conjecture, due to S. Hosono, so that our results provide a check confirming at least in part the conjecture.

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33 Mirror symmetry (algebro-geometric aspects)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T45 Topological field theories in quantum mechanics
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