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

Abstract

In this paper, we study 5d $\mathcal{N}=1$ $Sp(N)$ gauge theory with $N_f(\le 2N+3)$ 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.

arXiv commentv1:60 pages, 22 figures, v2: 67 pages, 26 figures, a new subsection and an appendix added

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• Quantum toroidal algebras and solvable structures in gauge/string theory
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• Seiberg-Witten curves with O7${}^{\pm }$-planes
• Five-dimensional gauge theories and the local B-model
• O-vertex, O7${}^{+}$-plane, and Topological Vertex
• D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants