Supersymmetric quantum mechanics on vortex moduli spaces

I will report on an ongoing project aiming at uncovering fundamental
features of N=(2,2) supersymmetric quantum mechanics on moduli spaces
of vortices on compact Riemann surfaces, in analogy with the spectrum
of quantum dyon-monopole bound states that emerged in connection with
Sen's S-duality conjectures in the 1990s. My focus in this talk will
be on the geometry underlying the coupling of waveforms to local
systems in effective theories for supersymmetric two-dimensional sigma-models
with toric gauge symmetry. I will consider models with
both linear and nonlinear targets; the corresponding ground states
can be investigated by means of the theory of L^{^2}-invariants. I shall
explain why the quanta of such abelian gauge theories can nontrivially
realize nonabelian statistics, and motivate a conjecture regarding a
nonlinear superposition principle for the ground states.