Atiyah–Bott symplectic form

Let $\Sigma$ be a closed oriented surface, let $G$ be a compact Lie group whose Lie algebra has an $\operatorname{Ad}$ -invariant inner product $\langle-,-\rangle$ , and let $P\to\Sigma$ be a principal $G$ -bundle. The space $\mathcal A(P)$ of principal connections is affine, modeled on the space $\Omega^1(\Sigma;\operatorname{ad}P)$ of adjoint-bundle-valued one-forms. Thus each tangent space to $\mathcal A(P)$ is this vector space. Its Atiyah–Bott symplectic form is the constant two-form

\omega_A(a,b)=\int_\Sigma\langle a\wedge b\rangle, \qquad a,b\in T_A\mathcal A(P).

Here the coefficient pairing and wedge product produce an ordinary two-form. Orientation defines its integral, while compactness makes it finite without support conditions.

Why the form is symplectic

The formula does not depend on $A$ , so $\omega$ is closed. If $a\ne0$ , choose a Riemannian metric on $\Sigma$ and put $b=*a$ ; then

\omega_A(a,*a)=\int_\Sigma |a|^2\,d\operatorname{vol}>0.

Thus the pairing is nondegenerate on smooth tangent vectors. In the Fréchet setting this is commonly called a weak symplectic form: the induced map from the tangent space to its continuous dual need not be onto.

The gauge group preserves $\omega$ , because its action on $\operatorname{ad}P$ preserves the chosen inner product. Together these properties provide the symplectic setup for gauge-theoretic reduction.

Role in gauge theory

For a surface, curvature is an $\operatorname{ad}P$ -valued top-degree form. Using the same coefficient pairing, it represents a covector on the Lie algebra $\Omega^0(\Sigma;\operatorname{ad}P)$ of the gauge group. The resulting curvature moment map turns flatness into a moment-map equation, so the moduli space of flat connections is formally a symplectic quotient.

Conventions and scope

Multiplying the invariant inner product by a positive constant rescales $\omega$ . A sign may also be inserted, especially when authors choose the opposite convention for fundamental vector fields or moment maps. The construction extends to noncompact surfaces only after imposing support or decay conditions that make the integral and functional-analytic setting meaningful.

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, the symplectic structure on the space of connections and reduction by the gauge group.