Non-\(c_i\)-self-dual quaternionic Yang-Mills connections and \(L_2\)-gap theory (Q1818024)

The author considers Yang-Mills theory on compact quaternion-Kähler manifolds. It is known that over these manifolds the bundle of 2-forms has a splitting in three parts (with respect to some irreducible decomposition as a representation of \(\text{Sp}(n)\cdot \text{Sp}(1))\). The curvature of a vector bundle is said to be \(c_j\)-self-dual if it has zero \(j\)th parts for \(j= 1,2\) in the above decomposition. Moreover, if the manifold is compact, every \(c_1\)- or \(c_2\)-self-dual connection minimizes the natural Yang-Mills functional. The main results are the existence of quaternionic Yang-Mills connections which are neither \(c_1\)- nor \(c_2\)-self-dual connections (this is given over compact quaternionic Kähler symmetric spaces and using invariant connections), and the study of the gap phenomena for quaternionic Yang-Mills connections by \(L_2\)-norms.