Research

We construct and classify SU(3)-invariant Sp(2)-instantons with gauge groups S=S^1 and S=SO(3) over the Calabi manifold X = T^*CP^2, the unique non-flat, complete, irreducible hyperkahler 8-manifold admitting a cohomogeneity-one action by a compact simple Lie group. Moreover, in the case of S=S^1, we also classify SU(3)-invariant primitive Hermitian Yang-Mills connections, and the SU(3)-invariant Spin(7)-instantons over X in the following sense. Letting Φ_I, Φ_J, Φ_K denote the Spin(7)-structures on X induced from the complex structures I,J,K in the hyperkahler triple, we prove that on each invariant S^1-bundle Ẽ_k→X, k∈ℤ, the space of invariant Spin(7)-instantons with respect to Φ_L  forms a one-parameter family modulo gauge. Moreover, every pair of one-parameter families of Φ_I-, Φ_J-, and Φ_K-Spin(7)-instantons intersects only at the unique invariant Sp(2)-instanton on Ẽ_k, which is non-flat when k≠0. In the case of S=SO(3) and up to gauge transformations, we find exactly one invariant Sp(2)-instanton on each bundle Ẽ_k→X, k∈ℤ≥0, which is reducible, and a unique invariant Sp(2)-instanton on the bundle Ẽ_∧, which is irreducible. We also classify SU(3)-invariant primitive Hermitian Yang-Mills connections on Ẽ_0, which modulo gauge reduce to the flat connection. 

We study the existence of SU(2)^2-invariant G_2-instantons on R^4xS^3 with the coclosed G_2-structures found on [arXiv:2209.02761]. We find an explicit 1-parameter family of SU(2)^3-invariant G_2-instantons on the trivial bundle on R^4xS^3 and study its ''bubbling'' behaviour. We prove the existence a 1-parameter family on the identity bundle.  We also provide existence results for locally defined SU(2)^2-invariant G_2-instantons.

We consider two different SU(2)^2-invariant cohomogeneity one manifolds, one non-compact M=R^4xS^3 and one compact M=S^4xS^3, and study the existence of coclosed SU(2)^2-invariant G_2-structures constructed from half-flat SU(3)-structures. For each of these manifolds, we prove the existence of a family of coclosed G_2-structures (but not necessarily torsion-free) which is given by three smooth functions satisfying certain boundary conditions and a non-zero parameter. Moreover, any coclosed G_2-structure constructed from half-flat SU(3)-structures is in this family. 

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.