The closed topological vertex via the Cremona transform
Authors:
Jim Bryan and Dagan Karp
Journal:
J. Algebraic Geom. 14 (2005), 529-542
DOI:
https://doi.org/10.1090/S1056-3911-04-00394-7
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 $\mathbb {P}^{1}$’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.
AKMV 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.
- 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
- 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. MR 1750955, DOI https://doi.org/10.1090/S0894-0347-00-00326-X
BrPaTQFT Jim Bryan and Rahul Pandharipande. Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory, Duke Math. J. (to appear).
- Jim Bryan and Rahul Pandharipande, BPS states of curves in Calabi-Yau 3-folds, Geom. Topol. 5 (2001), 287–318. MR 1825668, DOI https://doi.org/10.2140/gt.2001.5.287
DiaconescuFlorea Duiliu-Emanuel Diaconescu and Bogdan Florea. Localization and Gluing of Topological Amplitudes. arXiv:hep-th/0309143.
- C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI https://doi.org/10.1007/s002229900028
- Andreas Gathmann, Gromov-Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001), no. 3, 399–432. MR 1832328
GraberZaslow Tom Graber and Eric Zaslow. Open-String Gromov-Witten Invariants: Calculations and a Mirror “Theorem”. arXiv:hep-th/0109075.
- 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, DOI https://doi.org/10.4310/ATMP.1999.v3.n1.a7
- J. Hu, Gromov-Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000), no. 4, 709–739. MR 1759269, DOI https://doi.org/10.1007/s002090050495
- 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, DOI https://doi.org/10.4310/ATMP.1999.v3.n5.a6
LiLiuLiuZhou Jun Li, Chiu-Chu Liu, Kefeng Liu, and Jian Zhou. A mathematical theory of the topological vertex. In preparation.
- R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), no. 2, 489–506. MR 1729095, DOI https://doi.org/10.1007/s002200050766
Shapiro Jacob Shapiro. On the Gopakumar-Vafa conjecture for local $K3$ surfaces. In preparation.
AKMV 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.
BKL 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.
BrLe1 Jim Bryan and Naichung Conan Leung. The enumerative geometry of ${K}3$ surfaces and modular forms. J. Amer. Math. Soc., 13(2):371–410, 2000.
BrPaTQFT Jim Bryan and Rahul Pandharipande. Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory, Duke Math. J. (to appear).
Br:Pa Jim Bryan and Rahul Pandharipande. BPS states of curves in Calabi-Yau 3-folds. Geom. Topol., 5:287–318 (electronic), 2001.
DiaconescuFlorea Duiliu-Emanuel Diaconescu and Bogdan Florea. Localization and Gluing of Topological Amplitudes. arXiv:hep-th/0309143.
FaPa C. Faber and R. Pandharipande. Hodge integrals and Gromov-Witten theory. Invent. Math., 139(1):173–199, 2000.
Gathmann Andreas Gathmann. Gromov-Witten invariants of blow-ups. J. Algebraic Geom., 10(3):399–432, 2001.
GraberZaslow Tom Graber and Eric Zaslow. Open-String Gromov-Witten Invariants: Calculations and a Mirror “Theorem”. arXiv:hep-th/0109075.
HST 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.
Hu J. Hu. Gromov-Witten invariants of blow-ups along points and curves. Math. Z., 233(4):709–739, 2000.
KatzKlemmVafa Sheldon Katz, Albrecht Klemm, and Cumrun Vafa. M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys., 3(5):1445–1537, 1999.
LiLiuLiuZhou Jun Li, Chiu-Chu Liu, Kefeng Liu, and Jian Zhou. A mathematical theory of the topological vertex. In preparation.
Pand R. Pandharipande. Hodge integrals and degenerate contributions. Comm. Math. Phys., 208(2):489–506, 1999.
Shapiro Jacob Shapiro. On the Gopakumar-Vafa conjecture for local $K3$ surfaces. In preparation.
Additional Information
Jim Bryan
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
ORCID:
0000-0003-2541-5678
Email:
jbryan@math.ubc.ca
Dagan Karp
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
Email:
dkarp@math.ubc.ca
Received by editor(s):
January 1, 2004
Received by editor(s) in revised form:
April 1, 2004
Published electronically:
December 30, 2004
