Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 2 (2009), 345-360.

Local Gromov–Witten invariants of cubic surfaces via nef toric degeneration

Yukiko Konishi and Satoshi Minabe

Full-text: Open access


We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.

Article information

Ark. Mat., Volume 47, Number 2 (2009), 345-360.

Received: 25 April 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2008 © Institut Mittag-Leffler


Konishi, Yukiko; Minabe, Satoshi. Local Gromov–Witten invariants of cubic surfaces via nef toric degeneration. Ark. Mat. 47 (2009), no. 2, 345--360. doi:10.1007/s11512-007-0064-7.

Export citation


  • Aganagic, M., Mariño, M. and Vafa, C., All loop topological string amplitudes from Chern–Simons theory, Comm. Math. Phys. 247 (2004), 467–512.
  • Barth, W., Peters, C. and Van de Ven, A., Compact Complex Surfaces, Ergeb. Math. Grenzgeb. 4, Springer, Berlin–Heidelberg, 1984.
  • Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493–535 (Preprint version: Univ.-GH-Essen, 1992).
  • Chiang, T. M., Klemm, A., Yau, S.-T. and Zaslow, E., Local mirror symmetry: calculations and interpretations, Adv. Theor. Math. Phys. 3 (1999), 495–565.
  • Diaconescu, D.-E. and Florea, B., The ruled vertex and nontoric del Pezzo surfaces, J. High Energy Phys. 12 (2006), 19 pp.
  • Diaconescu, D.-E., Florea, B. and Grassi, A., Geometric transitions and open string instantons, Adv. Theor. Math. Phys. 6 (2002), 619–642.
  • Diaconescu, D.-E., Florea, B. and Grassi, A., Geometric transitions, del Pezzo surfaces and open string instantons, Adv. Theor. Math. Phys. 6 (2002), 643–702.
  • Diaconescu, D.-E., Florea, B. and Saulina, N., A vertex formalism for local ruled surfaces, Comm. Math. Phys. 265 (2006), 201–226.
  • Dolgachev, I. V., Weyl groups and Cremona transformations, in Singularities, Part 1 (Arcata, CA, 1981), Proc. Sympos. Pure Math. 40, pp. 283–294, Amer. Math. Soc., Providence, RI, 1983.
  • Gopakumar, R. and Vafa, C., M-theory and topological strings–II, Preprint, 1998. arXiv:hep-th/9812127.
  • Graber, T. and Pandharipande, R., Localization of virtual classes, Invent. Math. 135 (1999), 487–518.
  • Hartshorne, R., Algebraic Geometry, Grad. Texts Math. 52, Springer, New York, 1977.
  • Hosono, S., Counting BPS states via holomorphic anomaly equations, in Calabi–Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), Fields Inst. Commun. 38, pp. 57–86, Amer. Math. Soc., Providence, RI, 2003.
  • Iqbal, A., All genus topological string amplitudes and 5-brane webs as Feynman diagrams, Preprint, 2002. arXiv:hep-th/0207114.
  • Katz, S., Klemm, A. and Vafa, C., M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999), 1445–1537.
  • Klemm, A. and Zaslow, E., Local mirror symmetry at higher genus, in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math. 23, pp. 183–207, Amer. Math. Soc., Providence, RI, 2001.
  • Kodaira, K., On stability of compact submanifolds of complex manifolds, Amer. J. Math. 85 (1963), 79–94.
  • Kodaira, K., Nirenberg, L. and Spencer, D. C., On the existence of deformations of complex analytic structures, Ann. of Math. 68 (1958), 450–459.
  • Konishi, Y., Pole structure of topological string free energy, Publ. Res. Inst. Math. Sci. 42 (2006), 173–219.
  • Konishi, Y. and Minabe, S., Flop invariance of the topological vertex, Internat. J. Math. 19 (2008), 27–45.
  • Kontsevich, M., Enumeration of rational curves via torus actions, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math. 129, pp. 335–368, Birkhäuser, Boston, MA, 1995.
  • Lerche, W., Mayr, P. and Warner, N. P., Non-critical strings, del Pezzo singularities and Seiberg–Witten curves, Nuclear Phys. B 499 (1997), 125–148.
  • Li, J. and Tian, G., Comparison of algebraic and symplectic Gromov–Witten invariants, Asian J. Math. 3 (1999), 689–728.
  • Liu, C. C. M., Liu, K. and Zhou, J., A formula of two-partition Hodge integrals, J. Amer. Math. Soc. 20 (2007), 149–184.
  • Minahan, J. A., Nemeschansky, D. and Warner, N. P., Investigating the BPS spectrum of non-critical En strings, Nuclear Phys. B 508 (1997), 64–106.
  • Mori, S. and Mukai, S., On Fano 3-folds with B2≥2, in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, pp. 101–129, North-Holland, Amsterdam, 1983.
  • Oda, T., Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb. 15, Springer, Berlin–Heidelberg, 1988.
  • Peng, P., A simple proof of Gopakumar–Vafa conjecture for local toric Calabi–Yau manifolds, Comm. Math. Phys. 276 (2007), 551–569.
  • Siebert, B., Algebraic and symplectic Gromov–Witten invariants coincide, Ann. Inst. Fourier (Grenoble) 49 (1999), 1743–1795.
  • Ueda, K., Stokes matrix for the quantum cohomology of cubic surfaces, Preprint, 2005. arXiv:math.AG/0505350.
  • Zhou, J., Localizations on moduli spaces and free field realizations of Feynman rules, Preprint, 2003. arXiv:math.AG/0310283.