Exact results for topological strings on resolved Y(p,q) singularities

TL;DR

This work develops exact $\alpha'$ results for open and closed A-model amplitudes on toric Calabi–Yau threefolds arising from minimal resolutions of $Y^{p,q}$ singularities by exploiting the M-theory/5D gauge theory correspondence. It introduces a method to obtain closed forms for the full set of mirror-period derivatives across the entire $B$-model moduli space using GKZ systems and Lauricella functions, enabling access to all phases, including orbifolds, and to higher-genus amplitudes. The authors apply the approach to local $\mathbb{F}_2$ and predict genus-zero, genus-one, and genus-two orbifold Gromov–Witten invariants of $\mathbb{C}^3/\mathbb{Z}_4$, while revealing a modular structure governed by $\Gamma(2)$ in elliptic cases and linking the mirror geometry to Seiberg–Witten curves of 5D gauge theories; for $p=q$ the curves coincide with the spectral curves of the Ruijsenaars relativistic Toda chain, suggesting a wider class of integrable systems associated to generic $Y^{p,q}$ geometries. The framework is general and extensible to higher $p$ and potentially to all toric Calabi–Yau threefolds, with implications for exact nonperturbative topological-string data across moduli space.

Abstract

We obtain exact results in α' for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yau's that we analyze are obtained as minimal resolution of cones over Y(p,q) manifolds and give rise via M-theory compactification to SU(p) gauge theories on R^4 x S^1. As an application we present a detailed study of the local F_2 case and compute open and closed genus zero Gromov-Witten invariants of the C^3/Z_4 orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes.The mirror curve in this case is the spectral curve of the relativistic A_1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y(p,q) geometries.

Figures (10)

• Figure 1: The fan of $C(Y^{p,q})$ for $p=5$, $q=2$.
• Figure 2: The fan of $\widetilde{C(Y^{p,q})}$ for $p=5$, $q=2$.
• Figure 3: The $pq$-web diagram of $\widetilde{C(Y^{p,q})}$ for $p=5$, $q=2$.
• Figure 4: The pq-web for $p=5$, $q=2$ with a lagrangian brane on an inner leg.
• Figure 5: Cuts and punctures of the $X$ plane in the genus $1$ case.