Gauge theory on T^*CP²: explicit Sp(2)-instantons, HYM connections, and Spin(7)-instantons

2025/08/23 by Alonso, Izar, Madnick, Jesse, Windes, Emily Autumn
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.17119

Abstract

We construct and classify SU(3)-invariant primitive Hermitian Yang-Mills connections and Sp(2)-instantons with gauge groups S = S¹ and S = SO(3) over the Calabi manifold X = T^*CP², the unique non-flat, complete, cohomogeneity-one hyperkahler 8-manifold. Moreover, in the case of S = S¹, we also classify 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 S¹-bundle \widetildeE_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 \widetildeE_k, which is non-flat when k ≠ 0.

Citations

• On Sp(n)-Instantons and the Fourier-Mukai Transform of Complex Lagrangians
• A singularity removal theorem for Yang-Mills fields in higher dimensions

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