The 2nd order renormalization group flow for non-linear sigma models in 2 dimensions

TL;DR

This paper studies the second order renormalization group flow for 2D non-linear sigma models, demonstrating invariant metric subsets, global solutions, and an eternal flow with UV and IR limits.

Contribution

It establishes the existence of invariant metric subsets, global solutions, and an eternal flow with UV and IR limits for the 2nd order RG flow in 2D.

Findings

Existence of invariant subsets of metrics under the flow

Global solutions on these invariant sets

Construction of an eternal solution with UV and IR limits

Abstract

We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the existence of an eternal solution that has both a UV and IR limit, and passes through regions where the flow is parabolic and backwards-parabolic.