Refined Topological Vertex and Instanton Counting

TL;DR

This work demonstrates that the refined topological vertex, a two-parameter extension of the A-model vertex, precisely reproduces the K-theoretic Nekrasov partition functions for ${\cal N}=2$ SU(N) gauge theories arising from toric Calabi–Yau geometries. By computing refined A-model amplitudes and implementing a specific framing modification, the authors establish $Z_{inst}^{A\text{-model}}(Q_B,Q_F) = Z_{inst}^{Nek,SU(N)}(\hat{Q},Q_{ab})$ with $\hat{Q}$ given by a particular reparameterization, thereby validating the refinement proposal. They extend the construction to include matter via strip geometries, showing the approach yields Nekrasov functions for theories with $N_f=2N$ and can be used to generate quiver gauge theories. The results solidify the link between refined topological strings and refined instanton counting, supporting the refined vertex as a robust tool for two-parameter geometric engineering.

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.

Figures (6)

• Figure 1: The toric diagram obtained by gluing the vertices $C_{\lambda \mu \nu}$ and $C_{\lambda '\mu '\nu^t }$
• Figure 2: The local Hirzebruch surface which is a line bundle over $\mathop {\mathbb P}^{1} \times {\mathbb P}^{1}$
• Figure 3: The refined topological vertex $\mathop C\nolimits_{\lambda \mu \nu } (t,q)$
• Figure 4: (a)The toric diagram of $SU(N)$ geometry (b)The building block of $SU(N)$ geometry, and refined vertex on this geometry implies $\mathop K\nolimits_{\mathop \mu \nolimits_1 \cdots \mathop \mu \nolimits_N } \left( {\mathop Q\nolimits_{F,1} , \cdots ,\mathop Q\nolimits_{F,N - 1} } \right)$
• Figure 5: A toric diagram of a strip geometry which is obtained from triangulation of a strip toric data