Abstract
We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W 1+∞ introduced by Miki. Our construction involves trivalent intertwining operators Φ and Φ* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors ∈ \( {\mathbb{Z}^2} \) is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Φ and Φ* give the refined topological vertex C λμν (t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C λμ ν (q, t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Φ and Φ*. The spectral parameters attached to Fock spaces play the role of the Kähler parameters.
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Awata, H., Feigin, B. & Shiraishi, J. Quantum algebraic approach to refined topological vertex. J. High Energ. Phys. 2012, 41 (2012). https://doi.org/10.1007/JHEP03(2012)041
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DOI: https://doi.org/10.1007/JHEP03(2012)041