Yang–Mills moduli space

Let $P\to M$ be a principal bundle with compact structure group over an oriented Riemannian manifold, equipped with an invariant inner product on the Lie algebra. The Yang–Mills moduli space of $P$ is the quotient

\mathcal M_{\mathrm{YM}}(P) = \{A\in\mathcal A(P)\mid d_A^*F_A=0\}/\mathcal G(P),

where $\mathcal A(P)$ is the space of connections and $\mathcal G(P)$ is the gauge group. Thus its points are gauge-equivalence classes of Yang–Mills connections on the fixed bundle $P$ . Equip the smooth connection space with its usual smooth topology and the displayed set with the quotient topology on its solution locus. On a closed base, analytic treatments use compatible Sobolev completions and elliptic regularity to recover smooth gauge classes. On noncompact bases or in the presence of boundary, one must specify regularity, decay, and boundary conditions.

Local structure

The slice theorem for the gauge action replaces the quotient locally by solutions in a Coulomb slice modulo the stabilizer. On a closed base, near an irreducible connection, after removing unavoidable central gauge transformations, elliptic deformation theory can give a finite-dimensional manifold when the obstruction space vanishes. Reducible connections have larger stabilizers and generally create singular strata.

The linearized Yang–Mills equation together with the Coulomb condition is elliptic at a solution, but the nonlinear quotient need not be smooth, connected, compact, or finite as a set. These properties depend on $M$ , $P$ , the metric, and the structure group.

Important subspaces

On an oriented four-manifold, self-dual and anti-self-dual connections define instanton moduli spaces inside $\mathcal M_{\mathrm{YM}}(P)$ . On closed four-manifolds they are absolute minima of the Yang–Mills functional in the appropriate topological sector by the Yang–Mills energy identity. General Yang–Mills critical points need not be self-dual or anti-self-dual, so the full Yang–Mills moduli space is usually larger.

Over a closed Riemann surface, the Yang–Mills functional and its gauge symmetry organize the space into Morse-theoretic strata; Atiyah and Bott use this structure to relate gauge theory to moduli of holomorphic bundles.

Compactification and conventions

Sequences of four-dimensional instantons can develop curvature concentration. The ordinary quotient is then enlarged by ideal connections carrying point-like bubbling data, as described by Uhlenbeck compactness.

References

Michael F. Atiyah and Raoul Bott, “The Yang–Mills Equations over Riemann Surfaces,” Philosophical Transactions of the Royal Society A 308 (1983), 523–615. DOI record. Relevant: §§3–9, the Yang–Mills functional, gauge quotient, critical sets, and moduli.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, spaces of connections, gauge quotients, and local moduli.