Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane

TL;DR

This paper investigates the thermodynamic limit of the Nekrasov partition function for 5d $Sp(N)$ gauge theories with an $O5$-plane, establishing a connection with Seiberg-Witten curves and mirror symmetry.

Contribution

It derives the Seiberg-Witten curve from the Nekrasov partition function with an $O5$-plane using topological vertex formalism and saddle point analysis, confirming mirror symmetry in orientifold settings.

Findings

Seiberg-Witten curve matches dual graph analysis.

Nekrasov partition function relates to the prepotential.

Supports mirror symmetry with orientifold planes.

Abstract

In this paper, we study 5d $\mathcal{N}=1$ $Sp(N)$ gauge theory with $N_f ( \leq 2N + 3 )$ flavors based on 5-brane web diagram with $O5$-plane. On the one hand, we discuss Seiberg-Witten curve based on the dual graph of the 5-brane web with $O5$-plane. On the other hand, we compute the Nekrasov partition function based on the topological vertex formalism with $O5$-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten curve and its boundary conditions as well as its relation to the prepotential in terms of the cycle integrals. They coincide with those directly obtained from the dual graph of the 5-brane web with $O5$-plane. This agreement gives further evidence for mirror symmetry which relates Nekrasov partition function with Seiberg-Witten curve in the case with orientifold plane and shed light on the non-toric Calabi-Yau 3-folds including D-type singularities.