Some comments on Chern-Simons gauge theory (Q1262562)

Let M denote a compact 2-manifold without boundary and with genus \(g\geq 3\). With the space \({\mathcal A}\) of connections on the trivial SU(2) bundle on M, there is associated the subspace \({\mathcal A}_ F\) of flat connections and the submanifold \({\mathcal A}^ s_ F\) of irreducible flat connections. The group of gauge transformations is denoted by \({\mathcal G}\); it is well-known that \({\mathcal A}^ s_ F/{\mathcal G}\) is in a natural way a symplectic manifold. The authors show that there exists a natural hermitian line bundle on \({\mathcal A}_ F/{\mathcal G}\). Restricted to \({\mathcal A}^ s_ F/{\mathcal G}\), this line bundle carries a natural connection whose curvature is (up to a factor i) the standard symplectic form. This theorem does not require a choice of conformal structure on M. If M is endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. It is demonstrated how path-integral quantization of the Chern-Simons action yields holomorphic sections of this bundle.