The local Gromov-Witten invariants of configurations of rational curves

Abstract

We compute the local Gromov-Witten invariants of certain configurations of rational curves in a Calabi-Yau threefold. These configurations are connected subcurves of the `minimal trivalent configuration', which is a particular tree of P^1's with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov-Witten invariants of a blowup of P^3 at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov--Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.

arXiv commentThis is the version published by Geometry & Topology on 7 March 2006

References in corpus (5)

• The Topological Vertex
• Topological Field Theory and Rational Curves
• Hodge integrals and degenerate contributions
• Hodge integrals and invariants of the unknot
• BPS states of curves in Calabi--Yau 3--folds

Cited by in corpus (5)

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• Refining the Shifted Topological Vertex
• Open string amplitudes of closed topological vertex