A geometric introduction to the 2-loop renormalization group flow

Abstract

The Ricci flow has been of fundamental importance in mathematics, most famously though its use as a tool for proving the Poincaré 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. A number of new results are presented as part of this narrative: short-time existence and uniqueness results in all dimensions if the sectional curvatures $K_{ij}$ satisfy certain inequalities; the calculation of fixed points for $n = 3$ dimensions; a reformulation of constant curvature solutions in terms of the Lambert W function; a classification of the solutions that evolve only by homothety; an analog for RG-flow of the 2-dimensional Ricci flow solution known to mathematicians as the cigar soliton, and discussed in the physics literature as Witten's black hole. We conclude with a list of Open Problems whose resolutions would substantially increase our understanding of the RG-2 flow both physically and mathematically.