Moduli spaces of \(B_ 2\)-connections over quaternionic Kähler manifolds
Let M be a 4n-dimensional compact quaternion Kaehlerian manifold of positive scalar curvature, and \(p: Z\to M\) the corresponding twistor space. For a \(C^{\infty}\) complex vector bundle \((E,h)\) of rank r over M with a Hermitian metric h, we denote by D a particular connection on \((E,h)\) called by the author a Hermitian \(B_ 2\)-connection [\textit{T. Nitta}, Vector bundles over quaternion Kähler manifolds, Tohoku Math. J. (to appear)]. Then, there is an \({\mathfrak SU}\)-valued elliptic complex \(\Sigma_ p\) on M, and we denote by \(C'_ B\) the set of all such D for which \(H^ 0(M,\Sigma_ p)=\{0\}\). Since theframe bundle of \((E,h)\) can be reduced to a principal \(U(r)\)-bundle P, a \(C^{\infty}\)-section of \(U_ p\) may be called a gauge transformation of P, and the set of all such transformations forms a group \({\mathfrak G}_ p\) which is referred to as the gauge group. We can call the quotient space \({\mathfrak B}'=C'_ B/{\mathfrak G}_ p\) the moduli space. Then it is proven here that the moduli space has a natural real analytic structure. Further, if \(C''_ B\) is the set of all Hermitian \(B_ 2\)-connections D on \((E,h)\) for which \(H^ 0(M,\Sigma_ p)= H^ 2(M,\Sigma_ p)= \{0\}\), then \({\mathfrak B}''=C''_ B/{\mathfrak G}_ p\) is a smooth manifold with a natural Riemannian metric. A similar consideration is made for Einstein-Hermitian connections on \((p^*E,p^*h)\) over Z.
scientific article; zbMATH DE number 4046610