General Metrics of G_2 Holonomy and Contraction Limits

TL;DR

This paper derives first-order equations for G_2 holonomy metrics with specific symmetries, explores group contractions leading to new solutions, and classifies complete non-singular solutions, advancing understanding of G_2 geometries.

Contribution

It extends previous G_2 metric constructions using Hitchin's method, introduces contraction limits, and classifies certain complete solutions with specific orbit structures.

Findings

Derived new first-order equations for G_2 holonomy metrics.

Found explicit solutions in contraction limits with additional U(1) symmetry.

Classified all complete, non-singular solutions with S^3×T^3 principal orbits.

Abstract

We obtain first-order equations for G_2 holonomy of a wide class of metrics with S^3\times S^3 principal orbits and SU(2)\times SU(2) isometry, using a method recently introduced by Hitchin. The new construction extends previous results, and encompasses all previously-obtained first-order systems for such metrics. We also study various group contractions of the principal orbits, focusing on cases where one of the S^3 factors is subjected to an Abelian, Heisenberg or Euclidean-group contraction. In the Abelian contraction, we recover some recently-constructed G_2 metrics with S^3\times T^3 principal orbits. We obtain explicit solutions of these contracted equations in cases where there is an additional U(1) isometry. We also demonstrate that the only solutions of the full system with S^3\times T^3 principal orbits that are complete and non-singular are either flat R^4 times T^3, or else the direct product of Eguchi-Hanson and T^3, which is asymptotic to R^4/Z_2\times T^3. These examples are in accord with a general discussion of isometric fibrations by tori which, as we show, in general split off as direct products. We also give some (incomplete) examples of fibrations of G_2 manifolds by associative 3-tori with either T^4 or K3 as base.