Surveys in Differential Geometry

Volume 15 (2010)

A loop of SU(2) gauge fields stable under the Yang-Mills flow

Pages: 163 – 236



Daniel Friedan (Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854-8019, U.S.A.)


The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial SU(2) gauge transformation. Local stability is demonstrated by calculating the flow equations to leading order in perturbations of the loop. The stable loop might play a role in physics as a classical winding mode of the lambda model, a 2-d quantum field theory that was proposed as a mechanism for generating spacetime quantum field theory. We also present evidence for 2-manifolds of SU(3) and SU(2) gauge fields that are stable under the Yang-Mills flow. These might provide 2-d instanton corrections in the lambda model.

Published 1 January 2010