Half density quantization of the moduli space of flat connections and witten's semiclassical manifold invariants

Abstract

LET PpB denote the moduli space of flat connections on (the trivial SU(2) bundle on) a compact, oriented two-manifold P. The space s”, is a stratified space containing an open dense set Yg which is a symplectic manifold of (real) dimension 6g - 6. There exists a natural line bundle 9 -+ ,Fb, with hermitian metric ( , ), whose restriction to Y9 has a connection V with curvature given by the symplectic form o on Y9. If we equip the two-manifold Xg with a metric, the resulting Hodge star operator turns ,Yg into a Klhler manifold, with w as the Kahler form. Setting aside for the moment concern with the singularities of Pg,, we arrive at the usual setting for geometric quantization: we are given a symplectic manifold (qg, o), a line bundle 3’ + 9, with connection of curvature CO, and a polarization of the space of sections of 3. The quantization CYY~(C~) = H”(Pg, Yk) of this system has been of importance in attempts to construct a topological field theory. A topological field theory would assign to every two-manifold Cg the quantization XDk(Cg) of the system (Pg, CO, 9, V), which must be proved independent of the choice of Kahler structure on -Fg, coming from a choice of metric on X9. The space Xk(Xg) can be given the structure of a Hilbert space with the metric given by ((s1,s~)) = j(si>s~)~~~-~. A topological field theory would also assign an element sN of the quantization Xk(Xg) to every three-manifold N3 whose boundary is given by 8N3 = Cg. If this assignment satisfied the axioms of topological field theory, the space Zk(Eg) would be equipped with a representation of the group Diff(Xg) of diffeomorphisms of Eg. Furthermore, we would obtain invariants of closed three-manifolds as follows. Let N be a closed three-manifold, and let N = H be a