The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II
type: Item
instance of: scholarly article
title: The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II
zbMATH Open document ID: 0864.57026
DOI: 10.1215/S0012-7094-96-08305-2
author: Jean-Luc Brylinski
author: Dennis A. McLaughlin
published in: Duke Mathematical Journal
publication date: 1997-06-22T00:00:00Z
review text: This paper is the sequel of Part I [the authors, ibid. 75. No. 3, 603-638 (1994; Zbl 0844.57025)]. The authors continue the study begun there. The program is to find and explore some geometric objects (as the 2-gerbes) which represent classes \(\alpha\in H^4(BG; \mathbb{Z})\), where \(G\) is a compact Lie group. The authors construct the 2-gerbe associated to a class \(\alpha\) in the cases where \(G=S^1\) and where \(G\) is a finite group. They derive the Weil reciprocity law for Riemann surfaces as a special case of the Segal-Witten reciprocity law, which was proved in Part I. By considering a holomorphic line bundle with Hermitian structure over a complex manifold the authors study the compatibility between the Chern class \(\widehat C_2\) (depending only on the holomorphic structure) and the differential character of Chern-Cheeger-Simons \(\widetilde C_2^\nabla\) (associated to the connection \(\nabla\) compatible with both the holomorphic structure and the Hermitian structure). For a proper holomorphic fibration \(f:X\to Y\) whose fibers are connected Riemann surfaces of genus \(g\geq 2\) and for a Hermitian holomorphic vector bundle \(E\to X\), the compatibility between the classes \(\widehat C_2\) and \(\widetilde C^\nabla_2\) can be ``pushed forward'' along the fibers of \(f\) to produce a metrized line bundle on \(Y\). By an ingenious interpretation of the Narasimhan-Seshadri Theorem the authors apply the above ideas to construct the Quillen metric on a determinant line bundle over the moduli space \({\mathcal M} (r, {\mathcal L})\) of stable bundles of rank \(r\) and fixed determinant \({\mathcal L}\) on a Riemann surface, in the case where \((r,\deg {\mathcal L})=1\). The paper is rich in technical details and uses with competence the numerous references.
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 912122
zbMATH Keywords: compact Lie group
zbMATH Keywords: 2-gerbe
zbMATH Keywords: Weil reciprocity law
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: Segal-Witten reciprocity law
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: Chern class
zbMATH Keywords: holomorphic fibration
reviewed by: I. D. Albu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 912122