Exceptional holonomy on vector bundles with two-dimensional fibers

TL;DR

This paper demonstrates that certain 6-dimensional manifolds with special geometric structures can be extended to higher-dimensional manifolds with parallel Spin(7) or Spin_0(3,4) structures, highlighting the extension process and non-uniqueness.

Contribution

It proves the extension of analytic SU(3) or SU(1,2)-structures to parallel Spin(7) or Spin_0(3,4)-structures on trivial bundles, and discusses non-uniqueness and potential generalizations.

Findings

Extension of structures is possible on trivial disc bundles.

The extended structure is not uniquely determined by the initial data.

Discussion on potential generalizations to non-trivial bundles.

Abstract

An SU(3)- or SU(1,2)-structure on a 6-dimensional manifold N^6 can be defined as a pair of a 2-form omega and a 3-form rho. We prove that any analytic SU(3)- or SU(1,2)-structure on N^6 with d omega^2 =0 can be extended to a parallel Spin(7)- or Spin_0(3,4)-structure Phi that is defined on the trivial disc bundle N^6\times B_epsilon(0) for a sufficiently small epsilon>0. Furthermore, we show by an example that Phi is not uniquely determined by (omega,rho) and discuss if our result can be generalized to non-trivial bundles.