Selecta Mathematica

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

Trace identities for the topological vertex

  • Jim Bryan
  • Martijn Kool
  • Benjamin Young


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 


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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

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