Multi-valued potentials in topological field theory

In these lecture notes we review the original work of Riemann on multi-valued functions, Kelvin’s application to Green’s potential theory, and the result of Gauss on the interpretation of the magnetic potential in terms of solid angle to show the relevance of these earlier results in modern topological field theory. This is done by considering some particular case studies. First we re-derive the Biot-Savart induction law in terms of solid angle, and we discuss the interplay of topology and physics in the context of the celebrated Aharonov-Bohm experiment. By showing how the helicity is made gauge invariant in a multiply connected domain, we demonstrate how Riemann’s cuts are re-interpreted in terms of modern homological concepts. Finally, by considering Kleinert’s multi-valued gauge theory for defects we show how direct application of the theory of currents help to correct results in the hydrodynamic interpretation of vortex defects in condensed matter physics.