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 connections, as well as on that 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. MSC: 53D30, 53D50, 81R10, 81S10, 81T50. Subj. Class: Global analysis, Quantum field theory.

Abstract.The present paper is concerned with extensions of complex analysis on the complex plane C to conformally flat 4-manifolds. We shall give in an explicit form a fundamental system of spinors that will serve as the basis vectors for the Laurent expansion. Restricted to a sphere around the center of the expansion these spinors form a complete orthonormal system of eigenspinors of the tangential Dirac operator on the sphere, and give a basis of the representations of Spin(4). We shall also give the definition of meromorphic spinors and residues, and prove under some hypothesis that, on a compact conformally flat 4-manifold, the sum of the residues of a meromorphic spinor is zero.

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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.