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Higher gerbes, loop spaces and transgression

Banff International Research Station for Mathematical Innovation and Discovery

Complex line bundles are classified up to isomorphism by integer cohomology in degree two, and it is of interest to look for similarly geometric objects which are associated to higher degree cohomology. Gerbes (of which there are various versions, due respectively to Giraud, Brylinski, Hitchin and Chattergee and Murray) are such objects associated to H^3, and various notions of "higher gerbes" have been likewise defined. However, these often to run into technical difficulties and annoyances typically associated with higher categories. We propose a natural geometric notion of higher gerbes as "multi simplicial line bundles", which avoids many of the difficulties. Moreover, every cohomology class is represented by one of these objects in the guise of a line bundle on the iterated free loop space with a so-called "fusion product" for each loop factor. This is joint work with Richard Melrose.

Mathematics; Geometry; Partial differential equations; Differential geometry

video/mp4

Author affiliation: New College of Florida

2017-06-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61939

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/