In differential geometry, the Yang–Mills–Higgs flow is a gradient flow described by the Yang–Mills–Higgs equations, hence a method to describe a gradient descent of the Yang–Mills–Higgs action functional. Simply put, the Yang–Mills–Higgs 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–Higgs pairs, which solve the Yang–Mills–Higgs 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–Higgs flow is named after Yang Chen-Ning, Robert Mills and Peter Higgs with the former two having formulated the underlying Yang–Mills theory in 1954 and the latter having proposed the coupling to the Higgs field in 1964.
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 (not canonically) isomorphic to , is the space of connections. These are under the adjoint representation invariant -valued (Lie algebra–valued) differential forms on and through pullback along smooth sections differ by -valued (vector bundle–valued) differential forms on .
The Yang–Mills–Higgs action functional is given by:Zhang 2020, Eq. (1.1)Changpeng, Zhenghan & Zhang 2023, Eq. (1.2)
Its first term is also called Yang–Mills action.
Hence the gradient of the Yang–Mills–Higgs action functional gives exactly the Yang–Mills–Higgs equations:
For an open interval , two maps and (hence continuously differentiable) fulfilling:Zhang 2020, Eq. (1.3)Changpeng, Zhenghan & Zhang 2023, Eq. (1.4)
are a Yang–Mills–Higgs flow.
Properties For a Yang–Mills–Higgs pair , the constant path on it is a Yang–Mills–Higgs flow. For a Yang–Mills–Higgs flow one has:
Hence is a monotonically decreasing function. Since the Yang–Mills–Higgs action functional is always positive, a Yang–Mills–Higgs flow which is continued towards infinity must inevitably converge to vanishing derivatives and hence a Yang–Mills–Higgs pair according to the above equations. For any pair , there is a unique Yang–Mills–Higgs flow with . Then is a Yang–Mills–Higgs pair. For a stable Yang–Mills–Higgs pair , there exists a neighborhood so that every unique Yang–Mills–Higgs flow with initial condition in it fulfills:
Ginzburg–Landau flow A generalization of the Yang–Mills–Higgs flow is the Ginzburg–Landau flow, named after Vitaly Ginzburg and Lev Landau, with an additional potential term for the Higgs field.
Literature
See also Yang–Mills flow Seiberg–Witten flow
External links Yang-Mills-Higgs flow at the nLab
References