General metrics of G2 holonomy and contraction limits

Abstract We obtain first-order equations for G 2 holonomy of a wide class of metrics with S 3 × S 3 principal orbits and SU (2)× 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 × 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 × 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 ×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.