Pre-Symplectic Structure on the Space of Connections

Abstract

Let $X$ be a four-manifold with boundary three-manifold $M$. We shall describe (i) a pre-symplectic structure on the sapce $\mathcal{A}(X)$ of connections on the bundle $X\times \mathrm{SU}(n)$ that comes from the canonical symplectic structure on the cotangent space $T^{\ast}\mathcal{A}(X)$. By the boundary restriction of this pre-symplectic structure we obtain a pre-symplectic structure on the space $\mathcal{A}^{\flat}_0(M)$ of flat connections on $M\times \mathrm{SU}(n)$ that have null charge.