Quantum Algebraic Approach to Refined Topological Vertex

TL;DR

This paper connects the refined topological vertex, a key concept in string theory, with quantum algebra representations, providing a new algebraic framework and explicit constructions for the vertex functions.

Contribution

It establishes an equivalence between the refined topological vertex and a representation theory of quantum W_{1+infty} algebra, introducing intertwining operators and explicit matrix element formulas.

Findings

Matrix elements of intertwining operators reproduce refined topological vertex functions.

The algebraic construction recovers known refined topological vertices by different basis choices.

Gluing factors match the composition of intertwining operators, linking algebraic and geometric perspectives.

Abstract

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+infty} 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 in 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 K"ahler parameters.