Chern–Simons pre-quantizations over four-manifolds

Abstract

We endow the space of connections on an SU ( n ) -principal bundle over a four-manifold with a pre-symplectic structure and define a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. Then we consider the trivial SU ( n ) -principal bundle for n ⩾ 3 over the four-manifold that is a submanifold of a null-cobordant four-manifold, and we construct on the moduli space of flat connections a hermitian line bundle with connection whose curvature is given by the pre-symplectic form. This is the Chern–Simons pre-quantization of moduli spaces. The group of gauge transformations on the boundary of the four-manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. When the four-manifold is a 4-dimensional disc we show that this action is lifted to the pre-quantization by its Lie group extension. The geometric description of the latter is related to the 4-dimensional Wess–Zumino–Witten model.