A geometric introduction to the two-loop renormalization group flow

Abstract

The Ricci flow has been of fundamental importance in mathematics, most famously through its use as a tool for proving the Poincare conjecture and Thurston’s geometrization conjecture. It has a parallel life in physics, arising as the first-order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory. There recently has been interest in the second-order approximation of this flow, called the RG-2 flow, which mathematically appears as a natural nonlinear deformation of the Ricci flow. A curvature flow arising from quantum field theory seems to us to capture the spirit of Yvonne Choquet-Bruhat’s extensive work in mathematical physics, and so in this commemorative article we give a geometric introduction to the RG-2 flow.