Skip to main content
Log in

Donaldson–Thomas theory and resolutions of toric A-singularities

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We prove the crepant resolution conjecture for Donaldson–Thomas invariants of toric Calabi–Yau 3-orbifolds with transverse A-singularities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Brini, A., Cavalieri, R., Ross, D.: Crepant resolutions and open strings. arXiv:1309.4438 (2013)

  2. Bryan, J., Cadman, C., Young, B.: The orbifold topological vertex. Adv. Math. 229(1), 531–595 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Behrend, K.: Donaldson–Thomas type invariants via microlocal geometry. Ann. Math. (2) 170(3), 1307–1338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bryan, J., Graber, T.: The crepant resolution conjecture. In: Algebraic Geometry—Seattle 2005. Part 1, Volume 80 of Proceedings of Symposium Pure Mathematics, pp. 23–42. Amer. Math. Soc., Providence (2009)

  5. Bryan, J., Steinberg, D.: Curve-counting invariants for crepant resolutions. arXiv:1208.0884 (2012)

  6. Calabrese, J.: On the crepant resolution conjecture for Donaldson–Thomas invariants. arXiv:1206.6524 (2012)

  7. Lam, T.: Loop symmetric functions and factorizing matrix polynomials. In: Fifth International Congress of Chinese Mathematicians. Part 1, 2, Volume 2 of AMS/IP Stud. Adv. Math., 51, pt. 1, pp. 609–627. Amer. Math. Soc., Providence (2012)

  8. Lam, T., Pylyavskyy, P.: Total positivity in loop groups, I: Whirls and curls. Adv. Math. 230(3), 1222–1271 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press Oxford University Press, New York. With contributions by A. Zelevinsky, Oxford Science Publications (1995)

  10. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142(5), 1263–1285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maulik, D., Oblomkov, A., Okounkov, A., Pandharipande, R.: Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds. Invent. Math. 186(2), 435–479 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals. In: The Unity of Mathematics, Volume 244 of Progr. Math., pp. 597–618. Birkhäuser, Boston (2006)

  13. Ross, D.: The loop Murnaghan–Nakayama rule. J. Algebraic Combin. Preprint arXiv:1208.4369 (2012)

  14. Ross, D.: On gw/dt and ruan’s conjecture in all genus for calabi-yau 3-orbifolds. Preprint arXiv:1409.7015 (2014)

  15. Ross, D., Zong, Z.: The gerby Gopakumar–Mariño–Vafa formula. Geom. Topol. 17(5), 2935–2976 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ross, D., Zong, Z.: Two-partition cyclic Hodge integrals and loop Schur functions. arXiv:1401.2217 (2014)

Download references

Acknowledgments

The author is greatly indebted to Jim Bryan and Ben Young for helpful conversation and encouragement. He is also grateful to Renzo Cavalieri for carefully listening to the main arguments appearing in this paper and providing helpful feedback. The author has been supported by NSF RTG Grants DMS-0943832 and DMS-1045119 and the NSF postdoctoral research fellowship DMS-1401873.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dustin Ross.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ross, D. Donaldson–Thomas theory and resolutions of toric A-singularities. Sel. Math. New Ser. 23, 15–37 (2017). https://doi.org/10.1007/s00029-016-0234-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-016-0234-1

Mathematics Subject Classification

Navigation