## 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.

### MathOverflow Questions:

Group cohomology version of Deligne-Beilinson cohomology

### MSC:

 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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### References:

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