Mathematische Annalen

, Volume 360, Issue 1–2, pp 67–78 | Cite as

Counting curves on surfaces in Calabi–Yau 3-folds

  • Amin GholampourEmail author
  • Artan Sheshmani
  • Richard Thomas


Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs \(Z\subset H\) in a Calabi–Yau threefold \(X\). Here \(H\) is a member of a sufficiently positive linear system and \(Z\) is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of \(Z\subset H\), pushed forward to \(X\) and considered as a certain Joyce–Song pair in the derived category of \(X\). We express these invariants in terms of the MNOP invariants of \(X\).


Modulus Space Line Bundle Modular Form Chern Character Ideal Sheaf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Kai Behrend, Vincent Bouchard, Jim Bryan, Tudor Dimofte, Dagan Karp, Jan Manschot, Davesh Maulik and Yukinobu Toda for useful discussions. The second author would like to thank the University of British Columbia, the Max Planck Institute, Bonn and the Isaac Newton Institute for Mathematical Sciences, Cambridge for their hospitality. The third author is supported by the EPSRC programme Grant EP/G06170X/1.


  1. 1.
    Behrend, K.: Donaldson–Thomas invariants via microlocal geometry. Ann. Math. 170, 1307–1338 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cheng, M., DeBoer, J., Dijkgraaf, R., Manschot, J., Verlinde, E.: A farey tail for attractor black holes. J. High Energy Phys. 2006(11), 1–28 (2006)Google Scholar
  3. 3.
    Denef, F., Moore, G.: Split states, entropy enigmas, holes and halos. J. High Energy Phys. 2011(11), 1–152 (2011)Google Scholar
  4. 4.
    Gaiotto, D., Strominger, A., Yin, X.: The M5-Brane elliptic genus: modularity and BPS states. J. High Energy Phys. 2007(8), 1–18 (2007)Google Scholar
  5. 5.
    Gaiotto, D., Yin, X.: Examples of M5-Brane elliptic genera. J. High Energy Phys. 2007(11), 1–12 (2007)Google Scholar
  6. 6.
    Huybrechts, D., Thomas, R.P.: Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes. Math. Ann. 346(3), 545–569 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. Memoirs AMS 217(1020), iv+199 (2011). ISBN 978-0-8218-5279-8Google Scholar
  8. 8.
    Kollár, J.: Projectivity of complete moduli. J. Differ. Geom. 32, 235–268 (1990)zbMATHGoogle Scholar
  9. 9.
    Maldacena, J., Strominger, A., Witten, E.: Black hole entropy in M-theory. J. High Energy Phys. 1997(12), 1–16 (1997)Google Scholar
  10. 10.
    Manschot, J.: Wall-crossing of D4-branes using flow trees. Adv. Theor. Math. Phys. 15(1), 1–42 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 11, 1263–1285 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70(10), 106–119 (2001)MathSciNetGoogle Scholar
  13. 13.
    Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Thomas, R.P.: A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations. J. Differ. Geom. 54, 367–438 (2000)zbMATHGoogle Scholar
  15. 15.
    Toda, Y.: Bogomolov–Gieseker type inequality and counting invariants. J. Topol. 6(1), 217–250 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Amin Gholampour
    • 1
    Email author
  • Artan Sheshmani
    • 2
  • Richard Thomas
    • 3
  1. 1.University of MarylandCollege ParkUSA
  2. 2.Department of mathematicsOhio State UniversityColumbusUSA
  3. 3.Imperial College LondonLondon UK

Personalised recommendations