Yang–Mills energy identity

Statement

Let $P\to X$ be a principal bundle with compact structure group over a closed oriented Riemannian four-manifold, and choose an Ad-invariant inner product on its Lie algebra. For any connection $A$ , write $F_A=F_A^++F_A^-$ for the Hodge decomposition of its curvature and set $\operatorname{YM}(A)=\tfrac12\|F_A\|_{L^2}^2$ . Then

\operatorname{YM}(A) =\frac12\int_X\langle F_A\wedge F_A\rangle+\|F_A^-\|_{L^2}^2 =-\frac12\int_X\langle F_A\wedge F_A\rangle+\|F_A^+\|_{L^2}^2.

The curvature integral depends only on the bundle and the chosen invariant polynomial, without requiring integral normalization; an integral normalization is needed only to identify it with an integer characteristic number. These identities hold for every smooth connection and immediately imply a sharp lower bound.

Here $F_A$ denotes the base-valued curvature: if $\Omega_A$ is the principal curvature on $P$ , then $F_A(x)(v,w)=[p,\Omega_A(\widetilde v,\widetilde w)]\in\operatorname{ad}(P)_x$ , with $p\in P_x$ and any lifts of $v,w$ . Horizontality and equivariance make this independent of the choices. The Hodge star acts on the differential-form factor.

Derivation

The Hodge star is an orthogonal involution on two-forms in dimension four. Consequently,

\|F_A\|_{L^2}^2=\|F_A^+\|_{L^2}^2+\|F_A^-\|_{L^2}^2, \qquad \int_X\langle F_A\wedge F_A\rangle =\|F_A^+\|_{L^2}^2-\|F_A^-\|_{L^2}^2.

Adding and subtracting these two equalities gives the identity. Chern–Weil theory makes the second integral independent of $A$ , so the identity compares connections in one topological bundle class.

Equality cases

The two remainder terms are nonnegative. Hence

\operatorname{YM}(A)\geq \frac12\left|\int_X\langle F_A\wedge F_A\rangle\right|.

Equality holds exactly when $F_A^-=0$ or $F_A^+=0$ , with the choice determined by the sign of the topological term. Thus self-dual and anti-self-dual connections are absolute energy minimizers within their topological class.

For an $SU(r)$ -bundle with inner product $\langle\xi,\eta\rangle=-\operatorname{tr}(\xi\eta)$ , the instanton number satisfies

k=\frac{1}{8\pi^2}\int_X\operatorname{tr}(F_A\wedge F_A),

so the convention $\operatorname{YM}(A)=\tfrac12\|F_A\|^2$ gives $\operatorname{YM}(A)\geq4\pi^2|k|$ .

Conventions and scope

References

Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §2.1, curvature decomposition and the topological energy bound.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 2, Chern–Weil normalization and Yang–Mills energy.