Yang–Mills flow


In differential geometry, the Yang–Mills flow is a gradient flow described by the Yang–Mills equations, hence a method to describe a gradient descent of the Yang–Mills action functional. Simply put, the Yang–Mills flow is a path always going in the direction of steepest descent, similar to the path of a ball rolling down a hill. This helps to find critical points, called Yang–Mills connections or instantons, which solve the Yang–Mills equations, as well as to study their stability. Illustratively, they are the points on the hill on which the ball can rest.
The Yang–Mills flow is named after Yang Chen-Ning and Robert Mills, who formulated the underlying Yang–Mills theory in 1954, although 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.

Definition

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., an affine vector space isomorphic to, is the space of connections. These are under the adjoint representation invariant -valued differential forms on and through pullback along smooth sections differ by -valued differential forms on.
All spaces are vector spaces, which from together with the choice of an invariant pairing on inherit a local pairing. It defines the Hodge star operator by for all. Through postcomposition with integration there is furthermore a scalar product. Its induced norm is exactly the norm.
A connection induces a differential, which has an adjoint codifferential. Unlike the Cartan differential with, the differential fulfills with the curvature form:
The Yang–Mills action functional is given by:
Hence the gradient of the Yang–Mills action functional gives exactly the Yang–Mills equations:
For an open interval, a map fulfilling:
is a Yang–Mills flow''.''

Properties

Literature

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