Journal of Algebraic Geometry

Author(s): Jim Bryan; Dagan Karp
Journal: J. Algebraic Geom. 14 (2005), 529-542.
Posted: December 30, 2004
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Abstract: We compute the local Gromov-Witten invariants of the ``closed vertex'', that is, a configuration of three $\mathbb{P}^{1}$ 's meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of $\mathbb{P}^{3}$ and then to compute those invariants via the geometry of the Cremona transformation.

1.: Mina Aganagic, Albrecht Klemm, Marcos Marino, and Cumrun Vafa.
The Topological Vertex.
CALT-68-2439, HUTP-03/A032, HU-EP-03/24, CERN-TH/2003-111.
arXiv:hep-th/0305132.
2.: Jim Bryan, Sheldon Katz, and Naichung Conan Leung.
Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds.
J. Algebraic Geom., 10(3):549-568, 2001. MR 1832332 (2002j:14047)
3.: Jim Bryan and Naichung Conan Leung.
The enumerative geometry of surfaces and modular forms.
J. Amer. Math. Soc., 13(2):371-410, 2000. MR 1750955 (2001i:14071)
4.: Jim Bryan and Rahul Pandharipande.
Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory,
Duke Math. J. (to appear).
5.: Jim Bryan and Rahul Pandharipande.
BPS states of curves in Calabi-Yau 3-folds.
Geom. Topol., 5:287-318 (electronic), 2001. MR 1825668 (2002h:14092)
6.: Duiliu-Emanuel Diaconescu and Bogdan Florea.
Localization and Gluing of Topological Amplitudes.
arXiv:hep-th/0309143.
7.: C. Faber and R. Pandharipande.
Hodge integrals and Gromov-Witten theory.
Invent. Math., 139(1):173-199, 2000. MR 1728879 (2000m:14057)
8.: Andreas Gathmann.
Gromov-Witten invariants of blow-ups.
J. Algebraic Geom., 10(3):399-432, 2001. MR 1832328 (2002b:14069)
9.: Tom Graber and Eric Zaslow.
Open-String Gromov-Witten Invariants: Calculations and a Mirror ``Theorem''.
arXiv:hep-th/0109075.
10.: S. Hosono, M.-H. Saito, and A. Takahashi.
Holomorphic anomaly equation and BPS state counting of rational elliptic surface.
Adv. Theor. Math. Phys., 3(1):177-208, 1999. MR 1704198 (2001b:14085)
11.: J. Hu.
Gromov-Witten invariants of blow-ups along points and curves.
Math. Z., 233(4):709-739, 2000. MR 1759269 (2001c:53115)
12.: Sheldon Katz, Albrecht Klemm, and Cumrun Vafa.
M-theory, topological strings and spinning black holes.
Adv. Theor. Math. Phys., 3(5):1445-1537, 1999. MR 1796683 (2002b:81124)
13.: Jun Li, Chiu-Chu Liu, Kefeng Liu, and Jian Zhou.
A mathematical theory of the topological vertex.
In preparation.
14.: R. Pandharipande.
Hodge integrals and degenerate contributions.
Comm. Math. Phys., 208(2):489-506, 1999. MR 1729095 (2002k:14087)
15.: Jacob Shapiro.
On the Gopakumar-Vafa conjecture for local surfaces.
In preparation.

Jim Bryan
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
Email: jbryan@math.ubc.ca

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