algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
For a scalar-valued function, a curve whose derivative is the negative of its gradient is called a gradient flow. It always points down the direction of steepest descent and hence is monotonically descreasing with respect to the scalar function. It is then possible to study its convergence to critical points, especially those that are local minima.
If the scalar function is the Yang-Mills action functional, then the gradient flow is called Yang-Mills flow. It is described by the Yang-Mills equation and can be used to find Yang-Mills connections, the critical points, as well as study stable Yang-Mills connections, the local minima.
The Yang-Mills flow is named after Chen Ning Yang and Robert Mills, who introduced Yang-Mills theory in 1954. But it was first studied by Michael Atiyah and Raoul Bott in 1982. It was also studied by Simon Donaldson in the context of the Kobayashi-Hitchin correspondence (or Donaldson-Uhlenbeck-Yau theorem).
Let be a compact Lie group with Lie algebra and be a principal -bundle with a compact orientable Riemannian manifold having a metric and a volume form . Let be its adjoint bundle. One has , which are either under the adjoint representation invariant Lie algbera valued or vector bundle valued differential forms. Since the Hodge star operator is defined on the base manifold as it requires the metric and the volume form , the second spaces are usually used.
All spaces are vector spaces, which from together with the choice of an -invariant pairing on (which for semisimple must be proportional to its Killing form) inherits a local pairing . It defined the Hodge star operator by for all . Through postcomposition with integration, there is furthermore a scalar product . Its induced norm is exactly the norm.
The Yang-Mills action functional is given by:
is called configuration space.
Hence the gradient of the Yang-Mills action functional gives exactly the Yang-Mills equations:
For an open interval , a map (hence continuously differentiable) fulfilling:
is a Yang-Mills flow.
For a Yang-Mills connection , the constant path on it is a Yang-Mills flow.
For a Yang-Mills flow , the composition with the Yang-Mills action functional is monotonically descreasing and in particular fulfills:
Using the flow equation yields:
For an infinitely long Yang-Mills flow , the limit exists and is a Yang-Mills connection.
According to the previous lemma, the limit exists, since the function is monotonically descreasing and bounded from below. The inequality in the previous lemma then yields , which implies that the limit exists. Using the flow equation then shows, that it is a Yang-Mills connection:
On ordinary Yang-Mills theory (YM):
Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)
Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation
Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem
On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):
Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow
Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory
3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM
4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM
topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory
See also:
Last revised on March 22, 2026 at 09:17:29. See the history of this page for a list of all contributions to it.