The geometry of degree-four characteristic classes and of line bundles on loop spaces. I. (English) Zbl 0844.57025

It is a very dense paper of a high level which supports the relationship between Chern-Simons theory and conformal field theory. This relationship is based on a geometric reciprocity law for loop groups and it is realized by constructing a geometric object (2-gerbe) corresponding to some characteristic class.
In the beginning the authors generalize the theory of line bundles with connection on a manifold \(M\) by introducing the concept of pseudo-line bundle. By assigning to each open set \(U \subset M\) all pseudo-line bundles on \(U\) they define a sheaf of categories which is an example of a gerbe (in the sense of Giraud). The authors describe the properties of the gerbes and the connective structures on the gerbe of pseudo-line bundles \({\mathcal C} (\nu)\) associated to a closed integral 3-form \(\nu\). As \({\mathcal C} (\nu)\) is equipped with a connection, the authors consider the holonomy of \(\mathcal C\) around any loop in \(M\) and give a geometric construction of an “anomaly line bundle” over the free loop space LM corresponding to \(\nu\). When the manifold is a Lie group \(G\) one obtains a description of the central extension of LG associated to \(\nu\). This leads to a proof of the reciprocity law of Segal-Witten.
Concerning the question if it is possible to find a gerbe on the total space \(P\) of a principal \(G\)-bundle \(P \to M\) whose restriction to each fiber coincides with \({\mathcal C} (\nu)\) on \(G\), the authors show that the obstruction to the existence of the gerbe is the cohomology class in \(H^4(M, \mathbb{Z})\) obtained by transgressing \(\nu\). A formula for an integer-valued Čech cocycle representing the first Pontryagin class \(p_1\) is deduced.
Finally the theory of the 2-gerbes and its classification by degree-3 sheaf cohomology are described. The authors give an interesting example of such a 2-gerbe and develop the differential geometry of this 2-gerbe in the holomorphic context.


57R20 Characteristic classes and numbers in differential topology
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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