Hermitian and G2-structures with large symmetry groups

Thesis

In the context of Hermitian geometry, the Hull--Strominger system is a system of non-linear PDEs on heterotic string theory, over a six-dimensional manifold endowed with an SU(3)-structure. Its seven-dimensional analogue, the heterotic G₂ system, is a system for both geometric fields and gauge fields over a manifold with a G₂-structure. In this thesis, we study manifolds with geometric structures compatible with the Hull--Strominger system and the heterotic G₂ system in the cohomogeneity one setting. In the former case, we develop a case-by-case analysis to provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. In the latter case, we study two different SU(2)²-invariant cohomogeneity one manifolds, one non-compact M = R⁴ x S³, and one compact M = S⁴ x S³. For R⁴ x S³, we prove the existence of a family of coclosed (but not necessarily torsion-free) G₂-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed G₂-structure constructed from a half-flat SU(3)-structure is in this family. For S⁴ x S³, we prove that there are no SU(2)²-invariant coclosed G₂-structures constructed from half-flat SU(3)-structures. Then, we study the existence of SU(2)²-invariant G₂-instantons on R⁴ x S³ manifold with the coclosed G₂-structures found. We find two 1-parameter families of smooth SU(2)³-invariant G₂-instantons with gauge group SU(2) on R⁴ x S³ and study its ``bubbling'' behaviour. We also provide existence results for locally defined SU(2)²-invariant G₂-instantons.

Authors: Alonso Lorenzo, I

• DOI: 10.5287/ora-agey9zzaj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: balanced metrics; G2-structures; SU(3)-structures; cohomogeneity one actions; G2-instantons
• Subjects: Differential geometry; Mathematics