General metrics of G2 holonomy and contraction limits Original Research Article

We obtain first-order equations for G2 holonomy of a wide class of metrics with S3×S3 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 S3 factors is subjected to an Abelian, Heisenberg or Euclidean-group contraction. In the Abelian contraction, we recover some recently-constructed G2 metrics with S3×T3 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 S3×T3 principal orbits that are complete and non-singular are either flat R4 times T3, or else the direct product of Eguchi–Hanson and T3, which is asymptotic to R4/Z2×T3. 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 G2 manifolds by associative 3-tori with either T4 or K3 as base.