Renormalisation Flow and Geodesics on the Moduli Space of Four Dimensional N=2 Supersymmetric Yang-Mills Theory

TL;DR

This paper demonstrates that the renormalization group flow lines in four-dimensional N=2 supersymmetric Yang-Mills theory correspond to geodesics on the moduli space, revealing geometric insights into the theory's coupling behavior.

Contribution

It establishes a geometric interpretation of beta function flows as geodesics on the moduli space in N=2 supersymmetric Yang-Mills theory.

Findings

Flow lines crossing from weak to strong coupling are geodesics.

Beta functions define integral curves that are geodesics.

Discussion of potential links to irreversibility in the theory.

Abstract

It is shown that the beta 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 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.