Yang–Mills equations

In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics.

This article discusses the Yang–Mills equations from a mathematical perspective. For the physics perspective, see Yang–Mills theory
The dx1⊗σ3 coefficient of a BPST instanton on the (x1,x2)-slice of R4 where σ3 is the third Pauli matrix (top left). The dx2⊗σ3 coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2, ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S4 of R4 (bottom right). The BPST instanton is a solution to the anti-self duality equations, and therefore of the Yang–Mills equations, on R4. This solution can be extended by Uhlenbeck's removable singularity theorem to a topologically non-trivial ASD connection on S4.

Solutions of the equations are called Yang–Mills connections or instantons. The moduli space of instantons was used by Simon Donaldson to prove Donaldson's theorem.

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