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.