Visualization of gradient descent with one flow line
Gradient flow of the Yang–Mills action functional
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.
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) 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:[1][2][3]
Hence the gradient of the Yang–Mills action functional gives exactly the Yang–Mills equations:
For a Yang–Mills connection , the constant path on it is a Yang–Mills flow.
For a Yang–Mills flow one has:
Hence is a monotonically decreasing function. Alternatively with the above equation, the derivative can be connected to the Bi-Yang–Mills action functional:
Since the Yang–Mills action functional is always positive, a Yang–Mills flow which is continued towards infinity must inevitably converge to a vanishing derivative and hence a Yang–Mills connection according to the above equation.
For any connection , there is a unique Yang–Mills flow with . Then is a Yang–Mills connection.