Selecta Mathematica

, Volume 24, Issue 2, pp 1527–1548 | Cite as

Trace identities for the topological vertex



The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson–Thomas theory of toric Calabi–Yau threefolds, or the open string partition function of \({\mathbb {C}}^{3}\). We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson–Thomas invariants of elliptically fibered Calabi–Yau threefolds at the end of the paper.

Mathematics Subject Classification

Primary: 05A15 Secondary: 14N35 11F50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254(2), 425–478 (2005). arXiv:hep-th/0305132 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Behrend, K.: Donaldson–Thomas type invariants via microlocal geometry. Ann. Math. (2) 170(3), 1307–1338 (2009). arXiv:math/0507523 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bloch, S., Okounkov, A.: The character of the infinite wedge representation. Adv. Math. 149(1), 1–60 (2000). arXiv:alg-geom/9712009 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bouttier, J., Chapuy, G., Sylvie C.: From Aztec diamonds to pyramids: steep tilings. arXiv:1407.0665
  5. 5.
    Bryan, J.: The Donaldson–Thomas theory of \(K3\times E\) via the topological vertex. arXiv:1504.02920
  6. 6.
    Bryan, J., Kool, M.: Donaldson–Thomas invariants of local elliptic surfaces via the topological vertex. arXiv:1608.07369
  7. 7.
    Bryan, J., Oberdieck, G., Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds. arXiv:1506.00841
  8. 8.
    Huang, M., Katz, S., Klemm, A.: Topological string on elliptic CY 3-folds and the ring of Jacobi forms. arXiv:1501.04891
  9. 9.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2nd edn, With contributions by A. Zelevinsky, Oxford Science Publications (1995)Google Scholar
  10. 10.
    MacMahon, P.A.: Combinatory Analysis. Two Volumes (Bound as One). Chelsea Publishing Co., New York (1960)Google Scholar
  11. 11.
    Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math 142(5), 1263–1285 (2006). arXiv:math.AG/0312059 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Okounkov, A.: Infinite wedge and random partitions. Selecta Math. (N.S.) 7(1), 57–81 (2001). arXiv:math/9907127
  13. 13.
    Okounkov, A., Pandharipande, R.: Gromov–Witten theory, Hurwitz theory, and completed cycles. Ann. Math. (2) 163(2), 517–560 (2006). arXiv:math.AG/0204305 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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). arXiv:hep-th/0309208

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands
  3. 3.Department of Mathematics, Fenton HallUniversity of OregonEugeneUSA

Personalised recommendations