Half density quantization of the moduli space of flat connections and Witten's semiclassical manifold invariants
type: Item
instance of: scholarly article
title: Half density quantization of the moduli space of flat connections and Witten's semiclassical manifold invariants
zbMATH Open document ID: 0811.57017
DOI: 10.1016/0040-9383(93)90003-E
author: Lisa C. Jeffrey
author: Jonathan Weitsman
published in: Topology
publication date: 1995-05-01T00:00:00Z
review text: 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.
reviewed by: J. Huebschmann
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 32G13
zbMATH DE Number: 488109
zbMATH Keywords: topological field theory
zbMATH Keywords: SU(2)-moduli space
zbMATH Keywords: geometric quantization
zbMATH Keywords: real polarization
zbMATH Keywords: Lagrangian foliation
zbMATH Keywords: handlebody
zbMATH Keywords: three-manifold invariants
zbMATH Keywords: Blattner-Kostant-Sternberg pairing
zbMATH Keywords: Reidemeister torsion
zbMATH Keywords: Chern-Simons invariants
MaRDI profile type: Publication
full work available at URL: 0040-9383(93)90003-e
OpenAlex ID: W2021739867
description: scientific article; zbMATH DE number 488109