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

TL;DR

This paper derives exact topological string amplitudes for a broad class of toric Calabi-Yau threefolds related to Y(p,q) singularities, connecting string theory, gauge theories, and integrable systems.

Contribution

It provides explicit calculations of topological string amplitudes on Y(p,q) resolutions and explores their relation to gauge theories and integrable systems.

Findings

Exact topological string amplitudes for Y(p,q) geometries

Connection between mirror curves and relativistic Toda chains

Predictions for higher genus topological string amplitudes

Abstract

We obtain exact results in \alpha' 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.