Quaternionic geometry of a nilpotent variety
Let \({\mathcal N}\) be the set of all \(A \in {\mathfrak sl} (3,C)\) such that \(A^ 3 = 0\). The group \(SL (3,C)\) acts on \({\mathcal N}\) via the adjoint representation and has two nontrivial orbits one of which is the regular orbit \({\mathcal O}_ r\), which is 12-dimensional and is open and dense in \({\mathcal N}\). The authors investigate the quaternionic geometry of \({\mathcal O}_ r\) as explicitly as possible. Now, the Grassmannian manifold \(\widetilde {Gr} ({\mathfrak su} (3))\) of oriented three-planes in \({\mathfrak su} (3) = \mathbb{R}^ 8\) carries an \(SU (3)\)-invariant functional \(\varphi\) defined by \(\varphi (V) = \varphi (e_ 1, e_ 2, e_ 3) = - \langle e_ 1, [e_ 2, e_ 3] \rangle\), where \(\langle, \rangle\) denotes the metric determined by the Killing form of \({\mathfrak su} (3)\) and \((e_ 1, e_ 2, e_ 3)\) is the oriented orthonormal basis of \(V\). Consider the gradient flow in the direction of increasing \(\varphi\). Let \(C_ r\) be the critical set corresponding to the least critical values of \(\varphi\), and let \(M_ r\) be the associated unstable manifold. The authors prove that \(M_ r\) is eight-dimensional and admits a quaternionic Kähler structure whose twister space \(Z_ r\) is the projectivisation \(P({\mathcal O}_ r)\) of the regular nilpotent orbit \({\mathcal O}_ r\) of \({\mathfrak sl} (3,C)\). The critical set \(C_ r\) is the symmetric space \(PSU (3)/SO (3)\) and \(M_ r\) is isomorphic to the rank-three vector bundle associated to the Lie algebra of \(SO (3)\). This result is applied to describe a class of harmonic maps of a compact Riemann surface into \(G_ 2/SO (4)\), which is locally isometric to \(M_ r\).
scientific article; zbMATH DE number 537342
scientific article