Let $X$ be a compact, oriented Riemann manifold. Let $\pi_{P}: P \rightarrow X$ be a principal $G$-bundle over $X$, for a compact Lie group $G$. Let $(M, \omega)$ be a symplectic manifold endowed with a symplectic action of $G$. Denote by $\mathcal{N}:=C^{\infty}(P,M)^{U(1)}$ the space of smooth $G$-equivariant maps $u:P \rightarrow M$. Then $C^{\infty}(P,M)^{G}$ is a smooth Frechet manifold. The total space of the tangent bundle $T\mathcal{N} = C^{\infty}(P,TM)^{G}$. At a point $u \in \mathcal{N}$, $T_{u}\mathcal{N} = \Gamma(P, u^{\ast}TM)^{G}$.

For $\xi_{1}, \xi_{2} \in T_{u}\mathcal{N}$, define $\Omega(\xi_{1}, \xi_{2}) = \displaystyle \int_{X} \omega_{u}(\xi_{1}, \xi_{2}) ~ dvol_{\scriptscriptstyle X}$, where $\omega_{u}(\cdot, \cdot)$ denotes the restric tion of $\omega$ along $u$. Will $\Omega(\cdot, \cdot)$ be a symplectic form on $\mathcal{N}$? More precisely, is $\Omega(\cdot, \cdot)$ closed?