Spectral Networks - A story of Wall-Crossing in Geometry and Physics

This thesis deals with the phenomenon of wall-crossing for BPS indices in \(d = 4\) \(\mathcal{N} = 2\) supersymmetric gauge theories with gauge group \(SU(K)\). Compactification over  yields a three-dimensional \(\sigma\)-model with target space \(M\) a fiber bundle over the Coulomb branch \(B\) of the four-dimensional theory. We demonstrate how the wall- crossing is captured by smoothness conditions on the Hyperkähler metric of \(M\). Three ways of determining the \(4d\) BPS spectrum are explained, drawing on the work of Gaiotto, Moore and Neitzke. Firstly, a twistor space construction reduces the problem to finding holomorphic Darboux coordinates which are obtained as solutions to a Riemann-Hilbert problem for large radii \(R\) of the circle. Secondly, for a subclass of theories obtained by compactifying a six-dimensional theory over a surface \(C\), the Darboux coordinates can be computed from Fock-Goncharov coordinates on certain triangulations of \(C\) for gauge group \(SU(2)\). Thirdly, a codimension one sublocus of \(C\) called a Spectral Network captures the BPS degeneracies in a more efficient way.