Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds

Lorenzo Foscolo, Mark Haskins, Johannes Nordstrom

Research output: Working paper

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Abstract

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics, and are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperk\"ahler geometry. We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known.
Original languageEnglish
Number of pages54
Publication statusPublished - 28 Dec 2020

Publication series

NameArxiv

Keywords

  • math.DG
  • hep-th
  • 53C10, 53C25, 53C29, 53C80

ASJC Scopus subject areas

  • Geometry and Topology
  • Mathematical Physics

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