$D$-branes on $C^3_6$ Part I: prepotential and $GW$-invariants

Abstract

This is the first of a set of papers having the aim to provide a detailed description of brane configurations on a family of noncompact threedimensional Calabi–Yau manifolds. The starting point is the singular manifold defined by a given quotient $C3/Z6$, which we called simply $C^3_6$ 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.