$G_2$ Manifolds, Mirror Symmetry and Geometric Engineering

TL;DR

This paper explores the construction of Calabi-Yau and $G_2$ holonomy manifolds, revealing new dualities and geometric transitions relevant for string theory and M-theory, and introduces mirror symmetry for $G_2$ manifolds.

Contribution

It presents new Calabi-Yau geometries with D6 branes, studies their geometric transitions, and constructs mirror geometries for $G_2$ holonomy manifolds, advancing understanding of string dualities.

Findings

Realization of ${ m N}=1$ supersymmetric $A_r$ quiver theories via Calabi-Yau geometries with wrapped D6 branes.

Discovery of new large $N$ dualities for topological strings extending Chern-Simons duality.

Construction of smooth quantum geometric transitions in $G_2$ holonomy manifolds without branes or fluxes.

Abstract

We construct Calabi-Yau geometries with wrapped D6 branes which realize ${\cal N}=1$ supersymmetric $A_r$ quiver theories, and study the corresponding geometric transitions. This also yields new large $N$ dualities for topological strings generalizing topological strings/large $N$ Chern-Simons duality. Lifting up to M-theory yields smooth quantum geometric transitions without branes or fluxes, in the context of $G_2$ holonomy manifolds. In addition we construct a linear sigma model realization which is relevant for the worldsheet theory of superstrings propagating in local manifolds with $G_2$ holonomy, and obtain mirror geometries for this class of supersymmetric sigma models.