Towards Joyce Structures on Moduli Spaces of Meromorphic Quadratic Differentials

In this thesis, we construct a class of geometric structures on moduli spaces of
meromorphic quadratic differentials on wild algebraic curves, generalising the work
of Bridgeland from the holomorphic to the irregular meromorphic setting. The
type of geometric structure we are after is a family of non-linear, flat, meromorphic
Ehresmann connections defined on a moduli space X that parametrises irregular, wild
quasi-parabolic SL2(C)-Higgs bundles and flat connections on wild curves. The space
X fibres over the space of quadratic differentials and resembles a complexification of
the Hitchin system for irregular Higgs bundles. The family of Ehresmann connections
is the essential object underlying a Joyce structure on spaces of stability conditions in
the sense of Bridgeland and arises naturally from the geometry of isomonodromic
deformations of irregular connections, prescribing a complex Hyperkähler structure
on the underlying integrable system. The main corpus of the thesis is dedicated to
building the requisite theory to support the construction of Joyce structures on spaces
of meromorphic quadratic differentials with arbitrary poles. Namely, we study deco-
rated wild quasi-parabolic bundles on wild algebraic curves and prove a generalised
spectral correspondence between X and the space of anti-invariant, wild branched
connections on spectral curves, a result deeply rooted in the ever so popular wild
version of the geometric Langlands program. Then we construct an ε-family of flat
Ehresmann connections on the resulting integrable system, in the case of wild curves
of genus 0. Lastly, we demonstrate the effectiveness of the general theory developed
in this thesis by recovering the ad-hoc results of Bridgeland-Masoero as a special
case of our hereby established framework.