Implosions and hypertoric geometry (Q2870995)

In this paper, a hyper-Kähler analogue of \textit{V. Guillemin} et al.'s construction of symplectic implosion [Transform. Groups 7, No. 2, 155--184 (2002; Zbl 1015.53054)] is studied. This is a continuation of a previous paper by the authors [Compos. Math. 149, No. 9, 1592--1630 (2013; Zbl 1286.53056)], developing the ideas of toric geometry. In particular, in the case \(K = \mathrm{SU}(n)\), the existence of a hypertoric variety inside the implosion, which has a natural description in terms of quivers is proved. This is a hyper-Kähler analogue of the result of [Zbl 1015.53054] that the universal symplectic implosion \((T^*K)_{\mathrm{impl}}\) naturally contains the toric variety associated to a positive Weyl chamber for \(K\). NEWLINENEWLINENEWLINEIn Section 1, the authors recall the construction of the symplectic implosion: Given a symplectic manifold \(M\) with a Hamiltonian action of a compact group \(K\), the implosion \(M_{\mathrm{impl}}\) is a stratified symplectic space with an action of the maximal torus \(T\) of \(K\), such that the symplectic reductions of \(M_{\mathrm{impl}}\) by \(T\) equals to the reductions of \(M\) by \(K\). Implosions of Hamiltonian \(K\)-manifolds can be defined using the symplectic implosion of the cotangent bundle \((T^*K)_{\mathrm{impl}}\), which acts as a universal object. In Sections 2, 3 and 4, following [Zbl 1286.53056], a hyper-Kähler analogue of the universal implosion for \(K=\mathrm{SU}(n)\) actions is presented. Further, a hypertoric variety which maps naturally to the universal hyper-Kähler implosion \(Q\) for \(K =\mathrm{SU}(n)\), and the stratification of \(Q\) into strata which are hyper-Kähler manifolds, are described. In Section 5, this stratification is refined to obtain strata \(Q_{[\sim,\mathcal O]}\) which are not hyper-Kähler but which reflect the group structure of \(K =\mathrm{SU}(n)\). In Sections 6, 7 and 8 open subsets of the refined strata are described using the Jordan canonical form. In Section 9, the authors discuss the relationship between the universal hyper-Kähler implosion \(Q\) for \(K = \mathrm{SU}(n)\) and the Nahm equations.