Konrad Waldorf. Classical Gauge Theory and Matter Fields. References: G. Naber: Topology, Geometry, and Gauge Fields. D. Bleecker: Gauge Theory and Variational Principles. Notation: M is a spacetime (oriented Lorentz manifold). Definition. A Yang-Mills theory over M consists of a Lie group G (gauge group), Ad-invariant bilinear

form κ: g⊗g → R on the Lie algebra g of G, principal G-bundle P over M, fields are connections ω ∈ Ω^{1}(P, g), action functional is S_{YM}(ω) = 2^{−1} ∫_{M} ∥F_{ω}∥_{κ}^{2}.

Yang-Mills equation: D^{ω} ∗ F_{ω} = 0 and D^{ω}F_{ω} = 0. Definition: A gauge transformation is a bundle automorphism g: P → P. Remark: We can identify gauge

transformations with G-equivariant map g̃: P → G. We have rg̃ = g. Equivariance means g̃(pγ) = γ^{−1}g̃(p)γ. Remark: If P is a principal G-bundle, ρ: G → GL(V), then we have a bijection between Ω_{ρ}^{k}(P, V) and Ω^{k}(M, P ×_{ρ} V). We have ψ ∈ Ω_{ρ}^{k}(P, V) if r_{γ}^{∗}ψ = ρ(γ^{−1})(ψ).

Theorem: S_{YM} is gauge-invariant: S_{YM}(ω) = S_{YM}(g^{∗}ω). Definition: A classical electromagnetic field theory is a Yang-Mills theory with G = U(1). If we remove

a line from 3-dimensional space, then we observe Aharonov-Bohm effect. If we remove a point, then we observe magnetic monopoles.

Definition: If G is a gauge group, then a matter type for G is a quadruple T = (V, h, ρ, f) such that V is a finite-dimensional real vector space (internal state space), h: V ⊗ V → R is a real bilinear form on V, ρ: G → GL(V) is an isometric representation of G: h(ρ(g)v, ρ(g)w) = h(v, w) (transformation behavior), f: V → R is a smooth ρ-invariant function: f(ρ(g)v) = f(v) (self-interaction potential).

Definition: Let (P, G) be a Yang-Mills theory over M, T a matter type for G. A field for P of type T is a section ϕ: M → P ×_{ρ} V. The action functional is S_{T} (ω, ϕ) = 2^{−1} ∫_{M} ∥D^{ω}ϕ∥_{h}^{2} + ∗(f(ϕ)).

Remark: We have two options: (a) Keep ω fixed, consider S_{ω}(ϕ) = S_{T} (ω, ϕ). (b) Consider S(ω, ϕ) = S_{YM}(ω) + S_{T} (ω, ϕ). Notice that (ω, ϕ) extremizes S iff (1) ϕ extremizes S_{ω} (Euler-Lagrange); (2) δ^{ω}F_{ω} = J^{ω}(ϕ) (inhomegeneous field equation). Here δ^{ω} = ∗D^{ω}∗.

Example 1: G = {e}, P = M. A scalar field is a field of matter type (R, h, id, f) with f(x) = −m^{2}x^{2}. S(ϕ) = ∫_{M} ∥dϕ∥ + m^{2}ϕ^{2}. Euler-Lagrange equation in this case is Klein-Gordon equation (Δ^{2} + m^{2})ϕ = 0.

Example 2: π^{−}-meson: charge n ∈ Z, no spin. Yang-Mills theory (P, U(1)). Matter type (C, h, ρ_{n}, f). Here ρ_{n} is the nth power represetation of U(1) on C, f(z) = −m^{2}|z|^{2} and h(z_{1}, z_{2}) is left as an exercise. Euler-Lagrange equation is (Δ^{ω} + m^{2})ϕ = 0. Here Δ^{ω} = δ^{ω}D^{ω}.