Quantization of the Chem-Simons Gauge Theory on Four-manifolds

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The reporter of this note studied from the year 2004 to 2006 the quantization of the Chern-Simons gauge theory on four-manifolds, he gave the geometric quantization of the moduli space of connections on a four-manifolds generally with boundary. He gave a pre- symplectic structure on the moduli-space of connections. He constructed an hermitian line bundle with connection on the moduli space whose curvature is given by the pre-symplectic form. The transition function of the line bundle is described by the 5- dimensional Chern-Simons functional. On the space of connections there is a Hamiltonian action of the group of gauge transformations that are identity on the boundary, whose symplectic reduction becomes the space of flat connections, this is the geometric quantization of the space of flat connections. The mapping group from the boundary three-manifold to the structure Lie group acts infinitesimally symplectic way on this moduli space of flat connections, and the author showed that the abelian extension of the mapping group lifts the action to the quantium line bundle. The last extension was constructed by Mickelsson and extended by the author's previous research on four-dimensional Wess-Zumino-Witten theory. The results were submitted to a journal of differential geometry. After this research the reporter investigated also the quasi-symplectic structure on the space of flat connections on three- manifolds and the condition for a connection to be extended to the four-manifold that cobord the first one and decided the class of such connections. Other than these research the author investigated the vortex representation of the Hamilton-Yang-Mills equation and the relation of it to the helicity of Hamiltonian flows.