M-theory on G_2 manifolds and the method of (p,q) brane webs

TL;DR

This paper explores the construction of G_2 holonomy manifolds via toric hyper-K"ahler quotients and connects these geometries to four-dimensional N=1 quiver gauge theories using a reformulated (p,q) web method.

Contribution

It introduces a new approach to relate G_2 manifolds constructed from toric hyper-K"ahler spaces with quiver gauge theories through a reformulated (p,q) web technique.

Findings

G_2 manifolds are constructed as cones over S^2 bundles on toric varieties.

The gauge group structure is derived as a product of unitary groups constrained by anomaly cancellation.

A reformulation of (p,q) webs links geometry with gauge theory content.

Abstract

Using a reformulation of the method of (p,q) webs, we study the four-dimensional N=1 quiver theories from M-theory on seven-dimensional manifolds with G_2 holonomy. We first construct such manifolds as U(1) quotients of eight-dimensional toric hyper-K\"ahler manifolds, using N=4 supersymmetric sigma models. We show that these geometries, in general, are given by real cones on \bf S^2 bundles over complex two-dimensional toric varieties, \cal \bf V^2= {{\bf C}^{r+2}/ {{\bf C}^*}^r}. Then we discuss the connection between the physics content of M-theory on such G_2 manifolds and the method of (p,q) webs. Motivated by a result of Acharya and Witten [hep-th/0109152], we reformulate the method of $(p,q)$ webs and reconsider the derivation of the gauge theories using toric geometry Mori vectors of \cal \bf V^2 and brane charge constraints. For {\bf WP^2}_{w_1,w_2, w_3}, we find that the gauge group is given by G=U(w_1n)\times U(w_2n)\times U(w_3n). This is required by the anomaly cancellation condition.