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 ℙ and ℙ * associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors ∈ Z ² is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of ℙ and ℙ* 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 ℙ and ℙ*. The spectral parameters attached to Fock spaces play the role of the Kähler parameters.