Half density quantization of the moduli space of flat connections and Witten's semiclassical manifold invariants
Let \(\Sigma\) be a closed surface and \(N\) a 3-manifold having \(\Sigma\) as boundary. A topological field theory would assign to \(\Sigma\) a vector space \(V_ \Sigma\) and to \(N\) a vector \(v_ N\) in \(V_ \Sigma\) in such a way that certain symmetries are preserved. The paper under review is meant to be a step towards constructing a topological field theory from the geometry of the SU(2)-moduli space \({\mathcal S}\) via geometric quantization in a real polarization. The requisite line bundle \({\mathcal L}\) is nowadays a well studied object in relationship with determinant bundles etc. A gauge theory incarnation thereof may be found in a paper of the second named author with \textit{T. R. Ramadas} and \textit{I. M. Singer} [Commun. Math. Phys. 126, No. 2, 409-420 (1989; Zbl 0686.53066)]. A handlebody \(H\) having \(\Sigma\) as boundary gives rise to a Lagrangian subspace \(L_ H\) of \({\mathcal S}\) which is in fact a leaf of a certain Lagrangian foliation constructed in earlier work of the second named author. On taking this foliation as real polarization, it is shown that there exists a global covariant section of \({\mathcal L}\) restricted to \(L_ H\) and in this way an element of the quantization to the bounding handlebody \(H\) is assigned. However, the action of the group of diffeomorphisms of \(\Sigma\) does not preserve this quantization whence the assignment to \(H\) of \(L_ H\) does not lead directly to three- manifold invariants. This difficulty is overcome by means of a form of the Blattner-Kostant-Sternberg pairing appropriate for non-transverse intersections. This procedure then leads to certain 3-manifold invariants which are shown to be combinations of Reidemeister torsion and Chern- Simons invariants. They are related to 3-manifold invariants of Witten and Reshetikhin-Turaev.
scientific article; zbMATH DE number 488109