The Spectral Curve of the Lens Space Matrix Model

TL;DR

The authors analyze the Chern-Simons matrix model corresponding to topological strings on $T^{*}(S^{3}/\mathbb{Z}_p)$, proving that the resolvent exhibits square-root cuts arranged into a $p$-cut structure and deriving a spectral curve that is a genus $(p-1)$ Riemann surface with four punctures. Using a regular-function approach, they show the resolvent is governed by a single algebraic curve whose deformation encodes the 't Hooft parameters and interplays with mirror symmetry through the Hori–Vafa construction. They perform explicit computations for the $p=2$ case, including a perturbative expansion of the spectral curve and a consistency check of the free energy against known results, while providing a general framework for arbitrary $p$ and relating it to large-$N transitions to $A_{p-1}$ fibrations. The results connect open-closed dualities, spectral geometry, and mirror symmetry in topological string theory, offering a path toward a matrix-model realization of the topological vertex beyond the $S^{3}$ case.

Abstract

Following hep-th/0211098 we study the matrix model which describes the topological A-model on T^{*}(S^{3}/\ZZ_p). We show that the resolvent has square root branch cuts and it follows that this is a p cut single matrix model. We solve for the resolvent and find the spectral curve. We comment on how this is related to large N transitions and mirror symmetry.

Figures (2)

• Figure 1: Large N dualities and mirror symmetry. a) $T^{*}(S^{3}/{\mathbb Z}_{p})$ represented by a deformation of a toric diagram Aganagic:2001ug b) The mirror to $T^{*}(S^{3}/{\mathbb Z}_{p})$ c) Schematic picture of the toric web of an $A_{p-1}$ fibration over ${\mathbb P}^{1}$ d) The Hori-Vafa mirror map gives a bundle over a genus $p-1$ Riemann surface, where the Riemann surface is simply a thickening of the toric web diagram
• Figure 2: Spectral curve for $S^3/{\mathbb Z}_2$ matrix model.