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 W1+infinity introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors is an element of Z(2) is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi* give the refined topological vertex C lambda mu nu(t, q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C lambda mu(nu).(q, t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi*. The spectral parameters attached to Fock spaces play the role of the Kahler parameters.