Stratified hyperkähler spaces and Nahm's equations - Mayrand, M (2019)
When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden-Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space is usually fairly singular, but Sjamaar and Lerman showed that it can still be stratified into smooth symplectic manifolds which “fit together nicely”, in a precise sense. In this thesis, we prove analogues of Sjamaar-Lerman's results in hyperkähler geometry, yielding to the notion of stratified hyperkähler spaces. We also study examples of stratified hyperkähler spaces coming from hyperkähler quotients of certain moduli spaces of solutions to the so-called Nahm equations. In particular, we prove a Kempf-Ness type theorem which realises these spaces as quasi-projective algebraic varieties. We then focus on an interesting family of examples whose stratification structure can be described explicitly by combinatorial data associated with the root system of a complex semisimple Lie algebra. Leaving stratified spaces aside, we investigate how Nahm's equations, which are non-linear systems of ODEs, can generate groupoid structures by concatenation of paths, in a manner analogous to the fundamental groupoid of a topological space. Finally, we study another moduli space of solutions to Nahm's equations, which is a smooth hyperkähler manifold diffeomorphic to a variety studied in geometric representation theory called the universal centraliser. We draw analogies between this space and the moduli space of Higgs bundles, and explain how mirror symmetry naturally enters into the picture.