Pre-symplectic structures 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 A(X) of connections on the bundle X × SU (n) that comes from the canonical symplectic structure on the cotangent space T * A(X), and (ii) a presymplectic structure on the space A ♭ 0 (M ) of flat connections on M × SU (n) that have null charge. These two structures are related by the boundary restriction map. We discuss also the Hamiltonian feature of the space of connections A(X) with the action of the group of gauge transformations.

Dirac structures on the space of connections

“…The symplectic structure on the space of connections over a Riemann surface was introduced in 1983 by M. Atiyah and L. Bott in their study of the geometry and topology of moduli spaces of gauge fields (see [1]). In [6], we introduced a pre-symplectic structure on the space A X of irreducible connections over a four-manifold X. That was given by the 2-form:…”

“…The case for t = 1 was discussed in [6]. Every principal G-bundle over a three-manifold M is extended to a principal G-bundle over a four-manifold X that cobords M, and for a connection A ∈ A M there is a connection A ∈ A X that extends A.…”

“…In [6], it is proved that the space of flat connections A ♭ X is mapped by r onto the space of flat connections over M that are of degree 0:…”

“…In [6] we saw that the action of G M on the space of flat connections A ♭ M is infinitesimally pre-symplectic, that is, the Lie derivative of the 2-form ω in (3.3) by the fundamental vector field d A ξ vanishes:…”

“…In [6] one of the author proved that there is a pre-symplectic structure on A X that is induced from the canonical symplectic structure on the cotangent bundle T * A X by a generating function CS : A X −→ T * A X which is given by the Chern-Simons form. Let ω be the boundary reduction of this pre-symplectic form to A M .…”