[Paper Review] $G_2$ Manifolds, Mirror Symmetry and Geometric Engineering

Mina Aganagic, Cumrun Vafa|ArXiv.org|Oct 18, 2001

Black Holes and Theoretical Physics45 references52 citations

TL;DR

This paper constructs $\mathcal{N}=1$ supersymmetric $A_r$ quiver theories via geometric engineering in type IIA string theory using Calabi-Yau geometries with wrapped D6 branes, and demonstrates a large $N$ duality between topological strings and Chern-Simons theory. It lifts the construction to M-theory, yielding smooth $G_2$ holonomy manifolds via quantum geometric transitions without branes or fluxes, and derives a mirror symmetry for $G_2$ manifolds using linear sigma models and T-duality on a compactified circle.

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.

Motivation & Objective

• To construct $\mathcal{N}=1$ supersymmetric $A_r$ quiver gauge theories using D6 branes wrapped on cycles in Calabi-Yau threefolds in type IIA string theory.
• To generalize the topological string/large $N$ Chern-Simons duality to more complex geometries via geometric transitions.
• To lift the type IIA setup to M-theory, obtaining smooth quantum geometric transitions in $G_2$ holonomy manifolds without branes or fluxes.
• To develop a linear sigma model realization for superstrings propagating on local $G_2$ holonomy manifolds.
• To derive a mirror geometry for this class of $G_2$ manifolds using T-duality and dualization of scalar fields and complex structure moduli.

Proposed method

• Uses the Higgs branch description of the conifold geometry as a $U(1)$ gauge theory with four chiral fields and a D-term condition parameterized by the complexified FI parameter $s$.
• Applies the geometric transition by deforming the D-term condition to relate the resolved conifold $\mathcal{M}_K$ to the deformed conifold $\mathcal{M}_C$, with the transition governed by $1 - e^{-s} = e^{-Y/N}$.
• Constructs a linear sigma model for $G_2$ holonomy manifolds by deforming an $\mathcal{N}=2$ $U(1)^N$ gauge theory with chiral fields $X_i$ and a real periodic scalar $\phi$ with logarithmic charge under the gauge group.
• Applies T-duality to the circle compactification of the M-theory background, dualizing the scalar $\phi$ to a gauge field and then to a dual scalar $\theta$, leading to a mirror geometry with shifted complex structure moduli.
• Derives the mirror geometry as $xz = F(u,v,t_i + iN_i\theta)$, where $t_i$ are complexified Kähler parameters and $\theta$ is the dual scalar on the $S^1$, yielding a $G_2$ mirror with fluxless, purely geometric description.
• Uses the adiabatic principle to treat the dual scalar $\theta$ as slowly varying, enabling the construction of a dual $G_2$ geometry that preserves the $\mathcal{N}=1$ supersymmetry of the original model.

Experimental results

Research questions

• RQ1 How can $\mathcal{N}=1$ $A_r$ quiver gauge theories be geometrically engineered in type IIA string theory using D6 branes and Calabi-Yau geometries?
• RQ2 What is the large $N$ duality between topological strings on resolved conifolds and Chern-Simons theory on $S^3$, and how can it be generalized to more complex geometries?
• RQ3 How do quantum geometric transitions in M-theory yield smooth $G_2$ holonomy manifolds without branes or fluxes?
• RQ4 What is the linear sigma model realization of superstrings propagating on local $G_2$ holonomy manifolds?
• RQ5 How can mirror symmetry be formulated for $G_2$ manifolds, and what is the resulting dual geometry in terms of complex structure and moduli?

Key findings

• The geometric transition between the resolved conifold $\mathcal{M}_K$ and the deformed conifold $\mathcal{M}_C$ is governed by the relation $1 - e^{-s} = e^{-Y/N}$, which interpolates between the D6-brane and fluxed geometry at finite $N$.
• In the large $N$ limit, the $\mathbb{P}^1$ in $\mathcal{M}_K$ grows to finite size, and the transition becomes a smooth quantum geometric transition in M-theory, yielding a smooth $G_2$ holonomy manifold.
• The linear sigma model for $G_2$ holonomy manifolds is constructed by deforming an $\mathcal{N}=2$ $U(1)^N$ gauge theory with chiral fields $X_i$ and a periodic scalar $\phi$ with logarithmic charge, breaking $\mathcal{N}=2$ to $\mathcal{N}=1$ without spoiling the conformal invariance of the IR fixed point.
• Mirror symmetry for $G_2$ manifolds is achieved by dualizing the circle compactification and the phases of the chiral fields, resulting in a mirror geometry defined by $xz = F(u,v,t_i + iN_i\theta)$, where $\theta$ is the dual scalar on $S^1$.
• The mirror construction is purely geometric and fluxless, with the mirror geometry being a fibration over $\mathbb{C}^2 \times \mathbb{C}^2 \times S^1$, and it preserves the type IIA/IIA and IIB/IIB nature of the original theory.
• The mirror symmetry exchanges the roles of the complex structure and Kähler moduli, and the resulting mirror is a $G_2$ manifold with a non-trivial complex structure that depends on the $U(1)$ fluxes $N_i$ via the dual scalar $\theta$.