Curvature as a moment map

Definition

Let $P\to\Sigma$ be a principal bundle over a closed oriented surface, with compact structure group $G$ and an $\operatorname{Ad}$ -invariant inner product on its Lie algebra. Give $\mathcal A(P)$ the Atiyah–Bott symplectic form. Identify the dual of the gauge Lie algebra $\Omega^0(\Sigma;\operatorname{ad}P)$ with $\Omega^2(\Sigma;\operatorname{ad}P)$ by integration. Then curvature is a moment map for the gauge-group action:

\mu\colon\mathcal A(P)\longrightarrow \Omega^2(\Sigma;\operatorname{ad}P),\qquad \mu(A)=F_A,

up to the common overall sign convention in the definition of a moment map.

Verification of the moment-map identity

For $\xi\in\Omega^0(\Sigma;\operatorname{ad}P)$ , define

\mu^\xi(A)=\int_\Sigma\langle F_A,\xi\rangle.

The curvature variation in the direction $a\in\Omega^1(\Sigma;\operatorname{ad}P)$ is $d_Aa$ . Integration by parts on the closed surface gives

d\mu^\xi_A(a) =\int_\Sigma\langle d_Aa,\xi\rangle =\int_\Sigma\langle a\wedge d_A\xi\rangle,

with the displayed sign corresponding to one standard convention. Since the infinitesimal gauge action is $d_A\xi$ or $-d_A\xi$ , depending on whether the action and fundamental vector fields are defined on the left or right, this is precisely the Hamiltonian identity. The calculation appears in Atiyah–Bott, §9.

Symplectic reduction and Yang–Mills

The zero level is the set of flat connections:

\mu^{-1}(0)=\{A:F_A=0\}.

Consequently, the flat moduli space is formally the symplectic quotient $\mathcal A(P)\mathbin{/\mkern-6mu/}\mathcal G(P)$ . At regular irreducible points this produces the familiar finite-dimensional symplectic structure; stabilizers and obstructions can make the quotient singular.

After a Riemannian metric is chosen on $\Sigma$ , the Yang–Mills functional is, up to normalization, the squared $L^2$ -norm $\|\mu(A)\|^2$ . This is why equivariant Morse theory for a norm-square of a moment map enters the Atiyah–Bott analysis.

Conventions and scope

Changing the sign of the symplectic form, the infinitesimal gauge action, or the moment-map identity changes $\mu$ to $-F_A$ . A central constant can also shift a moment map; fixed central-curvature equations are therefore nonzero moment levels. The direct identification above is special to a two-dimensional oriented base, where curvature has top degree.

References

Michael F. Atiyah and Raoul Bott, “The Yang–Mills Equations over Riemann Surfaces,” Philosophical Transactions of the Royal Society of London A 308 (1983), 523–615. DOI record. Relevant: §9, curvature as the moment map and the symplectic reduction of flat connections.

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