the Yang-Mills functional
/ yang-mills /
The Yang-Mills functional asks: among all the ways to put a connection on a bundle, which ones are 'as flat as possible'? It assigns to each connection a single number — the total squared size of its curvature — and the connections that minimize or are critical for this number are the geometrically and physically preferred ones. In physics this is the action of gauge theory, and electromagnetism, the weak and strong nuclear forces are all governed by minimizing (extremizing) it.
Precisely, fix a principal G-bundle with G compact (so its Lie algebra has an invariant inner product) over a Riemannian manifold M. For a connection A with curvature 2-form F, the Yang-Mills functional is YM(A) = integral over M of |F|^2, the L^2-norm-squared of the curvature, where |F|^2 combines the Hodge star of the metric with the invariant inner product on the Lie algebra. Critical points are connections satisfying the Yang-Mills equation D_A * F = 0 (the covariant divergence of the curvature vanishes), which together with the always-true Bianchi identity D_A F = 0 forms a coupled system — the nonabelian generalization of Maxwell's equations. Gauge transformations (bundle automorphisms) leave YM invariant, so one really studies the functional on the space of connections modulo gauge.
The Yang-Mills functional is a meeting point of geometry, topology, and physics. In dimension 4 it has special structure: the curvature splits into self-dual and anti-self-dual parts, YM is bounded below by a topological number (a Pontryagin/instanton number), and the bound is achieved exactly by (anti-)self-dual connections — the instantons. Donaldson built revolutionary smooth-4-manifold invariants from the moduli space of these solutions. One honest caveat: minimizers do not always exist — in dimension 4 the conformal invariance of YM allows curvature to concentrate and 'bubble off', so the calculus of variations is delicate; and the quantum Yang-Mills mass-gap problem (existence and a spectral gap of the quantized theory) remains an open Millennium Prize problem, quite separate from the classical variational story described here.
On a 4-manifold, write the curvature as F = F^+ + F^- (self-dual plus anti-self-dual parts). Then YM(A) = integral of (|F^+|^2 + |F^-|^2), while the second Chern (instanton) number is proportional to integral of (|F^+|^2 - |F^-|^2). Adding and subtracting shows YM >= 8 pi^2 |k| with equality iff F^+ = 0 (an anti-self-dual instanton) or F^- = 0. So the absolute minimizers in each topological class are exactly the (anti-)self-dual connections — solutions of a first-order equation, far easier than the full second-order Yang-Mills equation.
In dimension 4, instantons minimize Yang-Mills within a topological class by solving a first-order (anti-)self-duality equation.
Yang-Mills minimizers need not exist: in dimension 4 the functional is conformally invariant, so a minimizing sequence can concentrate curvature into a shrinking ball and 'bubble', losing energy to a point — the limit may be a connection plus point singularities, not a smooth minimizer. This non-compactness is a real analytic difficulty, separate from the open quantum mass-gap problem.