Renormalisation flow and geodesics on the moduli space of four dimensional N=2 supersymmetric Yang–Mills theory

It is shown that the β-functions for four dimensional N=2 supersymmetric Yang–Mills theory without matter give integral curves on the moduli space some of which are geodesics of the natural metric on the moduli space. In particular the flow lines which cross-over from the weak coupling limit (asymptotically free theory) to the singular points, representing the strong coupling limit, are geodesics. A possible connection with irreversibility is discussed.