Yang–Mills moduli space - Imported Knowlpedia Content

Definition

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 $P$ . Analytic treatments replace the smooth spaces by compatible Sobolev completions without changing the smooth gauge-equivalence classes.

Local structure

The slice theorem for the gauge action replaces the quotient locally by solutions in a Coulomb slice modulo the stabilizer. 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 $\mathcal M_{\mathrm{YM}}(P)$ . 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 Atiyah–Bott, §§3–9.

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.