A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.q-alg/9507034

We derive a quantum deformation of the W N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.q-alg/9508011

We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory. 1

Abstract. We establish the equivalence between the refined topological vertex of IqbalKozcaz-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 ∈ 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.

We argue the connection of Nekrasov's partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N = 2 SU (2) YangMills theory the Nakrasov's partition function with equivariant parameters ǫ 1 , ǫ 2 of toric action on C 2 factorizes correctly as the character of SU (2) L × SU (2) R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F 0 . We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T 2 action allows us to obtain the generating functions of equivariant χ y and elliptic genera of the Hilbert scheme of n points on C 2 by the method of topological vertex.