Volume 10, issue 1 (2006)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The local Gromov–Witten invariants of configurations of rational curves

Dagan Karp, Chiu-Chu Melissa Liu and Marcos Mariño

Geometry & Topology 10 (2006) 115–168

arXiv: math.AG/0506488

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 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 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.

Keywords
Gromov–Witten invariants, Calabi–Yau threefolds, topological vertex
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 53D45
References
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Publication
Received: 23 August 2005
Accepted: 25 November 2005
Published: 7 March 2006
Proposed: Jim Bryan
Seconded: Paul Goerss, Eleny Ionel
Authors
Dagan Karp
Department of Mathematics
University of California at Berkeley
California 94720-3840
USA
Chiu-Chu Melissa Liu
Department of Mathematics
Northwestern University
Evanston
Illinois 60208-2370
USA
Marcos Mariño
Department of Physics
CERN
Geneva 23
CH-1211
Switzerland