Journal of Fixed Point Theory and Applications

, Volume 14, Issue 1, pp 3–20 | Cite as

A geometric introduction to the two-loop renormalization group flow

  • Karsten Gimre
  • Christine Guenther
  • James Isenberg


The Ricci flow has been of fundamental importance in mathematics, most famously through 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 Kij 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 analogue 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.

Mathematics Subject Classification

35K55 53C44 53C80 58Z05 81T17 


Renormalization group flow geometric flows nonlinear parabolic equations Ricci flow 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Karsten Gimre
    • 1
  • Christine Guenther
    • 2
  • James Isenberg
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of Mathematics and Computer SciencePacific UniversityForest GroveUSA
  3. 3.Department of MathematicsUniversity of OregonEugeneUSA

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