The ASD equations in split signature and hypersymplectic geometry - Roeser, MK (2012)
This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on 𝐑2,2. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on S2 × S2, which can be given an interpretation as a hypersymplectic quotient.