A flat SL(2,C) bundle on a punctured torus isn't much to look at: it's basically just a pair of matrices. If you give the torus a flat metric, though, you can do a lot of interesting things by following flat sections of the bundle along straight paths. If you're a complex symplectic geometer, you can find the bundle's spectral coordinates. If you're a hyperbolic geometer, you can often build a pleated hyperbolic structure on the torus. If you're a condensed matter physicist, you can sometimes make a model of electrons hopping around on a chain of atoms. I'll present evidence for a surprising connection between these three things. This will be an informal discussion of work in progress, rather than a polished talk.