Large N Duality, Lens Spaces and the Chern-Simons Matrix Model

TL;DR

The paper establishes a concrete bridge between open Chern-Simons theory on the lens space $S^{3}/\mathbb{Z}_p$ and a closed topological-string geometry by showing the CS matrix-model spectral curve exactly matches the mirror Riemann surface of the blown-up, orbifolded conifold. It derives the spectral curve as a deformation of $F_p=0$ from a multi-matrix model and uses toric geometry with the Hori-Vafa mirror to identify the dual geometry as a blowup of the ${\mathbb Z}_p$ orbifold of the conifold, a nontrivial $A_p$ fibration over ${\mathbb P}^{1}$. The results demonstrate agreement at leading order in the string coupling and concretely realize the large-$N$ duality for $T^{*}(S^{3}/\mathbb{Z}_p)$ with $p>2$, extending prior checks that were restricted to $p=2$. This deepens the understanding of open/closed string duality in orbifold settings and clarifies the geometric nature of the duality beyond the $p=2$ case.

Abstract

We demonsrate that the spectral curve of the matrix model for Chern-Simons theory on the Lens space S^{3}/\ZZ_p is precisely the Riemann surface which appears in the mirror to the blownup, orbifolded conifold. This provides the first check of the $A$-model large $N$ duality for T^{*}(S^{3}/\ZZ_p), p>2.

Figures (2)

• Figure 1: The fan for the resolution of the ${\mathbb Z}_{2}$ orbifold of ${\cal O}_{-1} + {\cal O}_{-1} \rightarrow {\mathbb P}^{1}$.
• Figure 2: The toric web diagrams of the large $N$ dual of $T^{*}(S^{3}/{\mathbb Z}_{p})$ for some values of $p$.