Renormalization group flow and parallel transport with non-metric compatible connections

TL;DR

This paper introduces a new family of non-metric compatible connections on the space of couplings in renormalizable field theories, analyzing their role in RG flows of specific models like N=2 SYM and O(N) models.

Contribution

It defines a novel class of connections derived from a generalized metric and RG equations, extending the geometric framework of renormalization group analysis.

Findings

RG flows analyzed via parallel transport with these connections

Connections are torsion free but generally not metric compatible

Application to N=2 supersymmetric Yang-Mills and O(N) models in large N limit

Abstract

A family of connections on the space of couplings for a renormalizable field theory is defined. The connections are obtained from a Levi-Civita connection, for a metric which is a generalisation of the Zamolodchikov metric in two dimensions, by adding a family of tensors which are solutions of the renormalization group equation for the operator product expansion co-efficients. The connections are torsion free, but not metric compatible in general. The renormalization group flows of N=2 supersymmetric Yang-Mills theory in four dimensions and the O(N)-model in three dimensions, in the large $N$ limit, are analysed in terms of parallel transport under these connections.