Quantum algebraic approach to refined topological vertex

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 _{λ μν} ( t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C _{λ μ} ^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.