Refined Topological Vertex and Instanton Counting

TL;DR

This paper verifies that the refined topological vertex method accurately computes A-model string amplitudes for SU(N) geometries, matching Nekrasov's partition functions for N=2 gauge theories.

Contribution

It provides the first explicit verification that the refined topological vertex reproduces Nekrasov's partition functions for SU(N) geometries.

Findings

Confirmed the equivalence between refined A-model amplitudes and K-theoretic Nekrasov partition functions.

Validated the refined topological vertex as a tool for computing gauge theory partition functions.

Extended the computation to SU(N) geometries, generalizing previous results.

Abstract

It has been proposed recently that topological A-model string amplitudes for toric Calabi-Yau 3-folds in non self-dual graviphoton background can be caluculated by a diagrammatic method that is called the ``refined topological vertex''. We compute the extended A-model amplitudes for SU(N)-geometries using the proposed vertex. If the refined topological vertex is valid, these computations should give rise to the Nekrasov's partition functions of N=2 SU(N) gauge theories via the geometric engineering. In this article, we verify the proposal by confirming the equivalence between the refined A-model amplitude and the K-theoretic version of the Nekrasov's partition function by explicit computation.