Math 2.0

Global analysis, analysis on manifolds Definition

Yang–Mills functional

Builds on Riemannian manifoldLie groupPrincipal bundleConnection on a principal bundleCurvature formYang–Mills equations

Definition

Let (M,g)(M, g) be a compact oriented Riemannian manifold without boundary, with volume form volg\mathrm{vol}_g. Let GG be a compact Lie group with Lie algebra 𝔤\mathfrak{g}. Fix an inner product on 𝔤\mathfrak{g} that is invariant under the adjoint action of GG. Let P→MP \to M be a principal GG-bundle, and let ad⁡P=P×G𝔤\operatorname{ad} P = P \times_G \mathfrak{g} be its adjoint bundle. For a connection AA on PP, let FAF_A be its curvature, a 22-form on MM with values in ad⁡P\operatorname{ad} P. The metric gg and the inner product on 𝔤\mathfrak{g} give a pointwise norm ∣FA∣\abs{F_A}. The Yang–Mills functional is the map from the space of connections on PP to [0,∞)[0, \infty) given by YM(A)=12∫M∣FA∣2 volg.\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{F_A}^2 \, \mathrm{vol}_g.

Remarks

Many authors omit the factor 12\frac{1}{2}; this does not change the critical points. The functional is invariant under gauge transformations, since Fu∗A=Ad⁡(u−1)FAF_{u^* A} = \operatorname{Ad}(u^{-1}) F_A and the inner product is invariant. Its Euler–Lagrange equation is dA∗FA=0d_A^* F_A = 0, the Yang–Mills equations.

Example: let G=U(1)G = U(1) with Lie algebra iℝi\R and inner product ⟨ia,ib⟩=ab\ip{ia}{ib} = ab, and let P=M×U(1)P = M \times U(1) be the trivial bundle. A connection is A=d+iαA = d + i\alpha for a real 11-form α\alpha, and its curvature is FA=i dαF_A = i \, d\alpha. So YM(A)=12∫M∣dα∣2 volg\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{d\alpha}^2 \, \mathrm{vol}_g, the energy of the electromagnetic field with potential α\alpha. It vanishes exactly when dα=0d\alpha = 0, that is, when AA is flat.

Non-example: the integral ∫Mtr⁡(FA∧FA)\int_M \tr(F_A \wedge F_A) for a matrix group GG on a 44-manifold is not the Yang–Mills functional. By Chern–Weil theory its value depends only on the bundle PP and not on AA, so it has no variational content.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Yang–Mills_equations
About this page
TagB6K478R9XE
URIm2://B6K478R9XE
Kinddefinition
WritingW0 stub
MSC58E15, 53C07, 81T13
DefinesYang–Mills functional
UsesRiemannian manifold, Lie group, Principal bundle, Connection on a principal bundle, Curvature form, Yang–Mills equations

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