Gauge theory on $T^{\ast }C{P}^2$: explicit Sp(2)-instantons, HYM connections, and Spin(7)-instantons

Abstract

We construct and classify $SU(3)$-invariant $Sp(2)$-instantons with gauge groups $S=S^1$ and $S=SO(3)$ over the Calabi manifold $X=T^{\ast }C{P}^2$, 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 ${\Phi }_I$, ${\Phi }_J$, ${\Phi }_K$ 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 ${\tilde{E}}_k\to X$, $k\in \mathbb{Z}$, 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 ${\tilde{E}}_k$, which is non-flat when $k\ne 0$. In the case of $S=SO(3)$ and up to gauge transformations, we find exactly one invariant $Sp(2)$-instanton on each bundle ${\tilde{E}}_k$, $k\in {\mathbb{Z}}_{\ge 0}$, which is reducible, and a unique invariant $Sp(2)$-instanton on the bundle ${\tilde{E}}_{\wedge }$, which is irreducible. We also classify $SU(3)$-invariant primitive Hermitian Yang-Mills connections on ${\tilde{E}}_0$, which modulo gauge reduce to the flat connection.

37 pages. New case added, including the construction of an irreducible invariant Sp(2)-instanton with gauge group SO(3). Other minor changes