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Article

Quantum Algebraic Approach to Refined Topological Vertex

Journal of High Energy Physics. 2012. No. 3. P. 41-68.
Feigin B. L., Awata H., Shiraishi J.

We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W1+∞ 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 ∈ Z2 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 Cλμν (t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex Cλμν (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 Kahler parameters.