The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II
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.
scientific article; zbMATH DE number 912122
scientific article