The closed topological vertex via the Cremona transform

Authors:
Jim Bryan and Dagan Karp

Journal:
J. Algebraic Geom. **14** (2005), 529-542

Published electronically:
December 30, 2004

MathSciNet review:
2129009

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Abstract | References | Additional Information

Abstract: We compute the local Gromov-Witten invariants of the ``closed vertex'', that is, a configuration of three '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 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**(2001), no. 3, 549–568. MR**1832332****3.**Jim Bryan and Naichung Conan Leung,*The enumerative geometry of 𝐾3 surfaces and modular forms*, J. Amer. Math. Soc.**13**(2000), no. 2, 371–410 (electronic). MR**1750955**, 10.1090/S0894-0347-00-00326-X**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**(2001), 287–318 (electronic). MR**1825668**, 10.2140/gt.2001.5.287**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**(2000), no. 1, 173–199. MR**1728879**, 10.1007/s002229900028**8.**Andreas Gathmann,*Gromov-Witten invariants of blow-ups*, J. Algebraic Geom.**10**(2001), no. 3, 399–432. MR**1832328****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**(1999), no. 1, 177–208. MR**1704198**, 10.4310/ATMP.1999.v3.n1.a7**11.**J. Hu,*Gromov-Witten invariants of blow-ups along points and curves*, Math. Z.**233**(2000), no. 4, 709–739. MR**1759269**, 10.1007/s002090050495**12.**Sheldon Katz, Albrecht Klemm, and Cumrun Vafa,*M-theory, topological strings and spinning black holes*, Adv. Theor. Math. Phys.**3**(1999), no. 5, 1445–1537. MR**1796683**, 10.4310/ATMP.1999.v3.n5.a6**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**(1999), no. 2, 489–506. MR**1729095**, 10.1007/s002200050766**15.**Jacob Shapiro.

On the Gopakumar-Vafa conjecture for local surfaces.

In preparation.

Additional Information

**Jim Bryan**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Email:
jbryan@math.ubc.ca

**Dagan Karp**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Email:
dkarp@math.ubc.ca

DOI:
https://doi.org/10.1090/S1056-3911-04-00394-7

Received by editor(s):
January 1, 2004

Received by editor(s) in revised form:
April 1, 2004

Published electronically:
December 30, 2004