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

TL;DR

This paper constructs and classifies invariant Hermitian Yang-Mills connections, Sp(2)-instantons, and Spin(7)-instantons on the Calabi manifold $T^*CP^2$, revealing a rich structure of solutions and their intersections.

Contribution

It provides explicit classifications of invariant instantons and HYM connections on $T^*CP^2$, including the unique non-flat Sp(2)-instanton and Spin(7)-instantons, expanding understanding of gauge theory on hyperkahler manifolds.

Findings

Classification of $SU(3)$-invariant primitive HYM connections.

Explicit construction of $Sp(2)$-instantons with gauge groups $S^1$ and $SO(3)$.

One-parameter families of invariant Spin(7)-instantons intersect only at the unique non-flat $Sp(2)$-instanton.

Abstract

We construct and classify $SU(3)$-invariant primitive Hermitian Yang-Mills connections and $Sp(2)$-instantons with gauge groups $S = S^1$ and $S = SO(3)$ over the Calabi manifold $X = T^*CP^2$, the unique non-flat, complete, cohomogeneity-one hyperkahler 8-manifold. Moreover, in the case of $S = S^1$, we also classify the $SU(3)$-invariant $Spin(7)$-instantons over $X$ in the following sense. Letting $\Phi_I$, $\Phi_J$, $\Phi_K$ 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 $\widetilde{E}_k \to X$, $k \in \mathbb{Z}$, the space of invariant $Spin(7)$-instantons with respect to $\Phi_L$ forms a one-parameter family modulo gauge. Moreover, every pair of one-parameter families of $\Phi_I$-, $\Phi_J$-, and $\Phi_K$-$Spin(7)$-instantons intersects only at the unique invariant $Sp(2)$-instanton on $\widetilde{E}_k$, which is non-flat when $k \neq 0$.