Geometric quantization of Chern-Simons gauge theory
type: Item
instance of: scholarly article
title: Geometric quantization of Chern-Simons gauge theory
zbMATH Open document ID: 0697.53061
author: Scott Axelrod
author: Steve Della Pietra
author: Edward Witten
published in: Journal of Differential Geometry
publication date: 1991-01-01T00:00:00Z
review text: We present a new construction of the quantum Hilbert space of Chern- Simons gauge theory using methods which are natural from the three dimensional point of view. To show that the quantum Hilbert space associated to a Riemann surface \(\Sigma\) is independent of the choice of complex structure on \(\Sigma\), we construct a natural projectively flat connection on the quantum Hilbert bundle over Teichmüller space. This connection has been previously constructed in the context of two dimensional conformal field theory where it is interpreted as the stress energy tensor. Our construction thus gives a \(2+1\) dimensional derivation of the basic properties of \(1+1\) dimensional current algebra. To construct the connection we show generally that for affine symplectic quotients the natural projectively flat connection on the quantum Hilbert bundle may be expressed purely in terms of the intrinsic Kähler geometry of the quotient and the Quillen connection on a certain determinant line bundle. The proof of most of the properties of the connection we construct follows surprisingly simply from the index theorem identities for the curvature of the Quillen connection. As an example, we treat the case when \(\Sigma\) has genus one explicitly. We also make some preliminary comments concerning the Hilbert space structure.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 81S10
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 53C80
zbMATH DE Number: 4142976
zbMATH Keywords: quantum Hilbert space
zbMATH Keywords: Chern-Simons gauge theory
zbMATH Keywords: projectively flat connection
zbMATH Keywords: symplectic quotients
zbMATH Keywords: Kaehler geometry
zbMATH Keywords: Quillen connection
zbMATH Keywords: index theorem
Wikidata QID: Q115181595
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214446565
OpenAlex ID: W1897085511
DOI: 10.4310/JDG/1214446565
description: scientific article; zbMATH DE number 4142976